diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzpmoq" "b/data_all_eng_slimpj/shuffled/split2/finalzzpmoq" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzpmoq" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction and main results}\nLiouville conformal field theory (LCFT) is a family of conformal field theories which arises in a wide variety of contexts ranging form random planar maps to 4d gauge theory. It was introduced by Polyakov in 1981 \\cite{Pol} in his attempt to construct string theory; in this paper, LCFT appears under the form of a 2d version of a Feynman path integral. Recently, in a series of works, a rigorous probabilistic construction of the path integral was provided using the theory of the Gaussian Free Field and the theory of Gaussian multiplicative chaos: see \\cite{DKRV} for the sphere, \\cite{DRV} for the torus and \\cite{GRV} for higher genus surfaces. In this theory, the main objects are the correlation functions of fields $V_\\alpha$, denoted $\\langle V_{\\alpha_1}(z_1)\\dots V_{\\alpha_m}(z_m)\\rangle_{(\\Sigma,g)}$, associated to $m$ marked points $z_1,\\dots,z_m$ on a closed Riemannian surface $(\\Sigma,g)$. \nA general formalism, called \\emph{conformal bootstrap} in physics by Belavin-Polyakov-Zamolodchikov \\cite{BPZ}, was developed in order to find explicit formulas for these correlation functions in terms of the $3$-point function on the sphere $\\mathbb{S}^2$ and the so-called \\emph{conformal blocks}, which are holomorphic functions of the points $z_j$ and the moduli of the Riemann surface $(\\Sigma,g)$. The conformal bootstrap method relies on the representation theory of an infinite dimensional Lie algebra of operators, called the \\emph{Virasoro algebra}, encoding the conformal symmetries of the system. It is formally generated by elements $({\\bf L}_n)_{n\\in \\mathbb{Z}}$ together with a central element denoted $c_L$, called \\emph{the central charge}, with commutation relations \n\\[ [\\mathbf{L}_n,\\mathbf{L}_m]=(n-m)\\mathbf{L}_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}, \\quad [\\mathbf{L}_n, c_L]=0. \\]\n\nIn the context of the probabilistic construction of LCFT, the central charge $c_L$ is just a scalar and the conformal bootstrap has recently been established by the last 4 authors in the papers \\cite{dozz,GKRV,GKRV1}. To implement this programme, \na Hilbert space $\\mathcal{H}=L^2(H^{-s}(\\mathbb{T}),\\mu_0)$ was introduced where $H^{-s}(\\mathbb{T})$ is the Sobolev space of order $-s<0$ on the unit circle $\\mathbb{T}=\\{z\\in \\mathbb{C}\\,|\\, |z|=1\\}$ \nand $\\mu_0$ is the distribution of the Gaussian Free Field on $\\hat{\\mathbb{C}}$ restricted to $\\mathbb{T}$ times the Lebesgue measure for the zero Fourier mode on $\\mathbb{T}$; the Hilbert space $\\mathcal{H}$ is therefore a space of fields $\\varphi$ on the circle. \n\nThe dilation $s_{e^{-t}}: z\\mapsto e^{-t}z$ in $\\mathbb{C}$ for $t\\geq 0$ is a conformal transformation that can be obtained as the flow of the holomorphic vector field $-z\\partial_z$. By composing the Gaussian Free Field and Gaussian multiplicative chaos on the unit disk $\\mathbb{D}\\subset \\mathbb{C}$ by $s_{e^{-t}}$, \na Markovian dynamic was constructed in \\cite{GKRV} which produces a contraction semi-group $e^{-t{\\bf H}}:\\mathcal{H}\\to \\mathcal{H}$, for some operator $\\mathbf{H}$ called the Hamiltonian of LCFT. The diagonalisation in $\\mathcal{H}$ of this Hamiltonian was then performed in \\cite{GKRV} and \nused to decompose the correlation functions into pieces, leading to the explicit formulas involving the conformal blocks in \\cite{GKRV,GKRV1}. The operator product expansion used in the physics literature can then be interpreted in terms of the \n eigenbasis of $\\mathbf{H}$. The family of eigenfunctions is denoted \n $(\\Psi_{Q+iP,\\nu,\\tilde{\\nu}})_{P,\\nu,\\tilde{\\nu}}$\n $P\\in \\mathbb{R}_+$ and $\\nu,\\tilde{\\nu}$ are Young diagrams: the spectrum of $\\mathbf{H}$ is absolutely continuous and equal to $[Q^2\/2,\\infty)$ (with $Q>2$).\nThis family carries an important algebraic structure, namely that the Young diagrams encode the action of the Virasoro generators on the highest weight vector $\\Psi_{Q+iP}:=\\Psi_{Q+iP,0,0}$:\n\\begin{equation}\\label{descendantsLn}\n\\Psi_{Q+iP,\\nu,\\tilde{\\nu}} = {\\bf L}_{-\\nu_k}\\dots {\\bf L}_{-\\nu_1}\\tilde{\\bf L}_{-\\tilde{\\nu}_{k'}}\\dots\\tilde{\\bf L}_{-\\tilde{\\nu}_1}\\Psi_{Q+iP}\n\\end{equation}\nwhere $\\nu=(\\nu_1,\\dots,\\nu_k)\\in \\mathbb{N}^k$ with $\\nu_j\\geq \\nu_{j+1}$ for all $j$, $\\tilde{\\nu}=(\\tilde{\\nu}_1,\\dots,\\tilde{\\nu}_{k'})\\in \\mathbb{N}^{k'}$ with $\\tilde{\\nu}_j\\geq \\tilde{\\nu}_{j+1}$ for all $j$, with ${\\bf L}_n,\\tilde{\\bf L}_m$ being two representations in $\\mathcal{H}$ of the Virasoro algebra with central charge $c_L=1+6Q^2$, \nsuch that $[{\\bf L}_n,\\tilde{\\bf L}_m]=0$. This can be compared to the way the harmonic oscillator in $\\mathbb{R}^n$ is diagonalised by the action of the Heisenberg algebra on the constant function (the ground state), except that here one has a further continuous parameter $P>0$ in addition.\n\nYet the construction in \\cite{GKRV} of these descendant states $\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}$ was not performed that way, due to the fact that some terms involved (those coming from the potential, i.e. Gaussian multiplicative chaos) in the construction of the operators ${\\bf L}_n$ are very singular and thus technically difficult to handle. Instead, we used the diagonalisation of the free field Hamiltonian and used scattering theory to construct the descendant states \n$\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}$, leading to certain restrictions on the analytic extension of $ \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ from the line \n$\\alpha \\in Q+i\\mathbb{R}$ to the complex plane. \n\nThe physics literature links the Virasoro generators $({\\bf L}_n)_{n\\in \\mathbb{Z}}$ (or $(\\tilde{\\bf L}_n)_{n\\in \\mathbb{Z}}$) to the dynamics obtained as flows of holomorphic vector fields of the form $v(z)\\partial_z$, generalizing this way the picture drawn in the case of dilations (see for instance \\cite[Ch. 9]{polch} or \\cite{Gaw}). Yet, as stressed in \\cite{Gaw}, a rigorous interpretation of these operators from this angle is far from being straightforward due to the intricate structure of their domains. In this work and inspired by these ideas, we give a probabilistic construction of the operators ${\\bf L}_n$ using a family of Markovian dynamics generated by some flows of holomorphic vector fields, as suggested in the first author's doctoral dissertation \\cite[Section 1.5.2]{these}. The crux of the matter is that the Markovian nature leads to perfectly well defined operators when seen as generators of contraction semigroups: this property is valid under some conditions of the holomorphic vector field, in which case the vector field will be called Markovian, and \nthe case of non Markovian vector field is then treated via polarization type arguments. This leads to an explicit construction of the whole Virasoro algebra as unbounded operators ${\\bf L}_n$ acting on $\\mathcal{H}$, \nand we show that the descendant states \n$\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}$ constructed by scattering theory in \\cite{GKRV} are related to the highest weight state $\\Psi_{Q+iP}$ by the relation \n\\eqref{descendantsLn}. This point of view is new and offers a different perspective from the algebraic approach usually developed in physics. Let us now explain this construction and its applications. \n\n\n\n\\subsection{Hilbert space of Liouville CFT}\n\nThe Hilbert space $\\mathcal{H}$ can be constructed explicitly as follows. Let $\\mathbb{T}$ denote the standard unit circle in the complex plane. We consider the space $\\mathbb{R} \\times \\Omega_\\mathbb{T}$ where $\\Omega_\\mathbb{T}$ is the set of real sequences $(x_n)_{n\\geq 1}$ and $(y_n)_{n\\geq 1}$. We introduce $(\\varphi_{n})_{n \\in \\mathbb{Z}^{\\ast}}$ (with $\\varphi_{-n}=\\overline\\varphi_{n}$) such that $\\varphi_n:=\\frac{1}{2\\sqrt{n}}(x_n+iy_n)$. The space $\\mathcal{H}$ is then constructed as the $L^2$-space $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$ equipped with the measure $\\d c \\otimes \\P_\\mathbb{T}$ (and the standard Borel sigma algebra) where $\\d c$ denotes the Lebesgue measure and $\\P_\\mathbb{T}$ is the Gaussian measure\n \\begin{align}\\label{Pdefin}\n \\P_\\mathbb{T}:=\\bigotimes_{n\\geq 1}\\frac{1}{2\\pi}e^{-\\frac{1}{2}(x_n^2+y_n^2)}\\text{\\rm d} x_n\\text{\\rm d} y_n.\n\\end{align}\nThe scalar product on $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$ is denoted $\\langle \\cdot , \\cdot\\rangle_2$ and the norm $ \\| \\cdot \\|_2$. Expectation with respect to $\\P_\\mathbb{T}$ will be denoted $\\mathbb{E}[\\cdot]$. Under $ \\P_\\mathbb{T}$ the random variable \n\\begin{equation}\\label{GFFcircle}\n\\varphi(\\theta)=\\sum_{n\\not=0}\\varphi_ne^{in\\theta} \n\\end{equation}\nis the Gaussian Free Field on the circle with covariance \n\\begin{equation}\n\\mathbb{E}[\\varphi({\\theta})\\varphi({\\theta'})]=-\\log |e^{i\\theta}-e^{i\\theta'}|.\n\\end{equation}\nIn the sequel, we will often identify the sequence $(c,(\\varphi_{n})_{n \\in \\mathbb{Z}^{\\star}})$ with the corresponding Fourier series $c+ \\sum_{n\\not=0}\\varphi_ne^{in\\theta} $, which is almost surely an element of $H^{-s}(\\mathbb{T}):=\\{\\sum a_ne^{in\\theta}\\,|\\,\\sum_n (1+|n|)^{-2s}|a_n|^2<\\infty\\}$\n for $s>0$. We denote by $\\norm{\\cdot}_{H^{-s}(\\mathbb{T})}$ and $\\langle\\cdot,\\cdot\\rangle_{H^{-s}(\\mathbb{T})}$ the norm and inner-product defined by the sum in the definition of $H^{-s}(\\mathbb{T})$. The space $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$ is then equivalent to $L^2(H^{-s}(\\mathbb{T}),\\mu_0)$ where $\\mu_0=\\varphi_*(\\text{\\rm d} c\\otimes \\P_\\mathbb{T})$ ($\\varphi_*()$ denotes pushforward). \n\n\\subsection{Semi-groups of Liouville CFT and representations of Virasoro algebra}\nThe Virasoro algebra ${\\rm Vir}(c_L)$ is by definition a central extension of the Witt algebra, whose elements are represented by vector fields \n$-z^{n+1}\\partial_z$ for $z\\in \\mathbb{C}\\setminus \\{0\\}$ if $n\\in \\mathbb{Z}$. The central element, being denoted $c_L$ in our setting, will simply be a constant $c_L:=1+6Q^2$ where $Q\\in(2,\\infty)$. To represent ${\\rm Vir}(c_L)$ into $\\mathcal{H}$, we first consider a certain family of holomorphic vector fields \n$\\mathbf{v}$, called \\emph{Markovian} and defined in the unit disk $\\mathbb{D}$, of the form $\\mathbf{v}=v(z)\\partial_z$ where $v(z)=-\\sum_{n=-1}^\\infty v_nz^{n+1}$, satisfying the property \n${\\rm Re}(\\bar{z}v(z))<0$ for $ z \\in \\mathbb{T}$. Each such Markovian vector field generates a flow of holomorphic transformations $f_t: \\mathbb{D}\\to \\mathbb{D}$ solving \n$\\partial_t f_t(z)=v(f_t(z))$ with initial condition $f_0(z)=z$, with a unique fixed point in $\\mathbb{D}$ and such that $f_{t'}(\\mathbb{D})\\subset f_t(\\mathbb{D})$ if $t'\\geq t$: as $t\\to +\\infty$, $f_t$ contracts $\\mathbb{D}$ to the unique zero of $v$ in $\\mathbb{D}$ which is the attractor of the flow $f_t$. Up to composing $v$ with a M\\\"obius transformation, we shall choose $v$ so that $v_{-1}=v(0)=0$ and $v_0=-v'(0)=\\omega>0$. \n\n We consider $X= P\\varphi+X_\\mathbb{D}$ where $P\\varphi$ is the harmonic extension of $\\varphi$ on $\\mathbb{D}$ and $X_\\mathbb{D}$ an independent Dirichlet Gaussian Free Field. One can then define a semigroup $P_t$ on $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$ as follows: for $F=F(c,\\varphi)$ \n depending only on finitely many variables $(\\varphi_n)_{|n|\\leq N}$ and decaying faster than any exponentials when $|c|\\to \\infty$ (this set of functions will be denoted $\\mathcal{C}_{\\rm exp}$ and is defined rigorously in Section \\ref{freeham}), let \n\\begin{equation} \\label{FeynmanKacintro}\n P_t F(c,\\varphi) := |f'_t(0)|^{\\frac{Q^2}{2}} \\mathbb{E}_\\varphi\\Big[ F\\Big ( c+ \\big(X \\circ f_t + Q \\log \\frac {|f_t'|} {|f_t|}\\big)\\Big|_{\\mathbb{T}} \\Big ) e^{- \\mu e^{\\gamma c} \\int_{\\mathbb{D}\\setminus f_t(\\mathbb{D})} \\frac{e^{\\gamma X(x)}} {|x| ^{\\gamma Q} } \\text{\\rm d} x } \\Big] \n \\end{equation}\nwhere for a function $u$, $u|_{\\mathbb{T}} $ denotes restriction of the function to the unit circle $\\mathbb{T}$ , $\\mathbb{E}_\\varphi[ \\cdot ]$ denotes the conditional expectation with respect to $\\varphi$, $\\mu>0, \\gamma\\in (0,2)$ are some parameters and $Q=2\/\\gamma+\\gamma\/2$. Since the Gaussian field $X$ is not defined pointwise, the term $ \\int_{\\mathbb{D}\\setminus f_t(\\mathbb{D})} \\frac{ e^{\\gamma X(x)}} {|x| ^{\\gamma Q} } \\text{\\rm d} x $ is defined via a renormalisation procedure and yields a non trivial quantity, i.e. non zero, for $\\gamma \\in (0,2)$ (see \\eqref{GMCsphere})\\footnote{The condition $\\gamma \\in (0,2)$ in this paper comes from this non triviality result on Gaussian multiplicative chaos. It is a restriction of the probabilistic approach; indeed LCFT in the physics literature is studied for all $\\gamma \\in \\mathbb{C}$. }; \nit is called the Gaussian multiplicative chaos measure of $\\mathbb{D}\\setminus f_t(\\mathbb{D})$. One can check that $P_t$ indeed defines a Markovian semigroup.\n\nIn the case where $v(z)=-z$, one has $f_t(z)=e^{-t}z$ and the semigroup $P_t=e^{-t\\mathbf{H}}$ where ${\\bf H}$ is the self-adjoint Hamiltonian defined and studied \nin \\cite{GKRV}, given by the expression\n\\[\\mathbf{H}=-\\frac{1}{2}\\partial_c^2+\\frac{Q^2}{2}+{\\bf P}+\\mu e^{\\gamma c}V(\\varphi)\\]\nwhere ${\\bf P}$ and $V$ are unbounded non-negative operators on $L^2(\\Omega_\\mathbb{T})$ defined by\n\\[{\\bf P}:=\\sum_{n\\geq 1} {\\bf A}_n^*{\\bf A}_n+\\tilde{\\bf A}_n^*\\tilde{\\bf A}_n, \\quad V(\\varphi) =\\int_0^{2\\pi}e^{\\gamma \\varphi(\\theta)}\\text{\\rm d} \\theta \\]\nand ${\\bf A}_n:=\\sqrt{n}(\\partial_{x_n}-i\\partial_{y_n})$, $\\tilde{\\bf A}_n:=\\sqrt{n}(\\partial_{x_n}+i\\partial_{y_n})$. \nHere, ${\\bf P}$ has discrete spectrum equal to $\\mathbb{N}$ and $V$ has to be defined using Gaussian multiplicative chaos theory with a renormalisation procedure (see \\eqref{GMCcircle}): it is a non-negative unbounded operator which becomes a multiplication by an $L^{p}(\\Omega_\\mathbb{T})$ function for $p<2\/\\gamma^2$ when $\\gamma<\\sqrt{2}$. The operator ${\\bf H}$ defines a quadratic form $\\mathcal{Q}(F,F):=\\langle \\mathbf{H} F,F\\rangle_2$ and we denote $\\mathcal{D}(\\mathcal{Q})$ its domain, and $\\mathcal{D}'(\\mathcal{Q})$ its dual.\nWe will also use the notation ${\\bf A}_0=\\tilde{\\bf A}_0:=\\frac{i}{2}(\\partial_c+Q)$, ${\\bf A}_{-n}:={\\bf A}_n^*$ and $\\tilde{\\bf A}_{-n}:=\\tilde{\\bf A}_n^*$ if $n>0$, where the adjoint is taken with respect to the scalar product on $\\mathcal{H}$.\n\nOur first main theorem is: \n\\begin{theorem}\\label{theoremfreefieldintro}\n Let $\\mathbf{v}=v(z)\\partial_z$ be a Markovian vector field with $v(z)=-\\sum_{n=0}^\\infty v_{n}z^{n+1}$ and $v'(0)=-\\omega$ for $\\omega>0$ such that $v$ admits a holomorphic extension in a neighborhood of $\\mathbb{D}$. If $\\omega> 0$ is large enough\\footnote{One could in fact work with a general complex $\\omega$ with a large real part and still get the same results but we will stick to real $\\omega$ for simplicity.}, then the operator $P_t$ is a contraction semi-group on $L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})$, it admits an invariant measure $\\mu_h$ that is absolutely continuous with respect to $\\mu_0$ and its generator $\\mathbf{H}_\\mathbf{v}$ has the form \n\\[\\mathbf{H}_\\mathbf{v}= \\omega {\\bf H}+ \\sum_{n\\geq 1}v_n\\, \\mathbf{L}_n + \\sum_{n\\geq 1}\\overline{v_n} \\, \\widetilde{\\mathbf{L}}_n\\]\nwhere $\\mathbf{H}_\\mathbf{v}, {\\bf L}_n, \\tilde{\\bf L}_n$ are bounded as linear maps $\\mathcal{D}(\\mathcal{Q})\\to \\mathcal{D}'(\\mathcal{Q})$ but unbounded on $L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})$. Moreover, they are given by the formula \n\\begin{equation}\\label{virasoro}\n\\begin{gathered}\n\\mathbf{L}_n:=-i(n+1)Q\\mathbf{A}_n+\\sum_{m\\in\\mathbb{Z}}:\\mathbf{A}_{n-m}\\mathbf{A}_m: + \\frac{\\mu}{2} e^{\\gamma c} \\int_0^{2 \\pi} e^{in \\theta} e^{\\gamma \\varphi(\\theta)} \\text{\\rm d} \\theta,\\\\\n\\widetilde{\\mathbf{L}}_n:=-i(n+1)Q\\widetilde{\\mathbf{A}}_n+\\sum_{m\\in\\mathbb{Z}}:\\widetilde{\\mathbf{A}}_{n-m}\\widetilde{\\mathbf{A}}_m: \n+ \\frac{\\mu}{2} e^{\\gamma c} \\int_0^{2 \\pi} e^{-i n \\theta} e^{\\gamma \\varphi(\\theta)} \\text{\\rm d} \\theta \n\\end{gathered}\n\\end{equation}\nwhere the normal order product $:{\\bf A}_n{\\bf A}_m:$ is defined as ${\\bf A}_n{\\bf A}_m$ if $m>0$ or ${\\bf A}_m{\\bf A}_n$ if $n>0$.\n \\end{theorem}\n\nWe recover this way the formulas announced by J. Teschner in \\cite[Section 10]{Tesc1}. Also, we notice that if $\\mathbf{v}_n=-z^{n+1}\\partial_z$ and $\\mathbf{v}_0=-z\\partial_z$, then we can recover ${\\bf L}_n$ by the expression \n\\[{\\bf L}_n=\\frac{1}{2}({\\bf H}_{\\omega \\mathbf{v}_0+\\mathbf{v}_n}-i{\\bf H}_{\\omega\\mathbf{v}_0+i\\mathbf{v}_n})-\\frac{1}{2}\\omega(1-i){\\bf H}\\]\nwhere $\\omega>0$ is chosen large enough so that $\\omega \\mathbf{v}_0+\\mathbf{v}_n$ is Markovian. We define for $n>0$ \n\\[{\\bf L}_{-n}:= {\\bf L}_n^* , \\quad \\tilde{\\bf L}_{-n}:=\\tilde{\\bf L}_n^* \\]\nwhere the adjoints are taken with respect the Hermitian product on $\\mathcal{H}$ while acting on a dense set of regular functions: \n$\\langle{\\bf L}_{-n}F,F'\\rangle_2=\\langle F,{\\bf L}_{n}F'\\rangle_2$ for all $F,F'\\in \\mathcal{D}(\\mathcal{Q})$.\nWe will explain below that $({\\bf L}_n)_{n\\in \\mathbb{Z}}$ and $(\\tilde{\\bf L}_n)_{n\\in \\mathbb{Z}}$ are two commuting representations of ${\\rm Vir}(c_L)$ into $\\mathcal{H}$. A first look at these operators prevents us to compose them, even when acting on very regular functions $F$, which makes it difficult to define \nthe commutators $[{\\bf L}_n,{\\bf L}_m]$ on a dense set. The technical difficulty in dealing with these operators was already stressed in \\cite{Gaw}. We will however construct infinite dimensional vector spaces, called Verma modules, \ncontained in $e^{N|c|}L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})$ for $N>0$ which are preserved by all ${\\bf L}_n$.\n\n\n\\subsection{Highest weight states, descendant states and scattering coefficients} \n In \\cite{GKRV}, we constructed a family of eigenfunctions of $\\mathbf{H}$ for $\\alpha|{\\rm Re}(\\alpha)-Q|$ and $c_-=\\min(c,0)$, satisfying\n\\[ \\big(\\mathbf{H} -\\alpha(Q-\\frac{\\alpha}{2})\\big)\\Psi_{\\alpha}=0, \\quad \\Psi_\\alpha(c,\\varphi)=e^{(\\alpha-Q)c}+\\mathcal{O}(e^{(\\alpha-Q+\\epsilon)c}) \\textrm{ as }c\\to -\\infty.\\]\nMoreover, we proved in \\cite{GKRV} that \n\\[ \\alpha \\mapsto \\Psi_{\\alpha}\\]\nadmits an analytic continuation to the region $\\{{\\rm Re}(\\alpha)\\leq Q\\}\\setminus \\cup_{j>1}\\{Q\\pm i\\sqrt{2j}\\}$, it is continuous at the points \n$Q\\pm i\\sqrt{2j}$ with possible square root singularities. We also constructed using scattring theory a whole family \n\\[\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}\\] \nof eigenfunctions of $\\mathbf{H}$ with eigenvalues $\\alpha(Q-\\frac{\\alpha}{2})+|\\nu|+|\\tilde{\\nu}|$\nfor each Young diagram $\\nu=(\\nu_1,\\dots,\\nu_k)$, $\\tilde{\\nu}=(\\tilde{\\nu}_1,\\dots,\\tilde{\\nu}_{k'})$, with $|\\nu|=\\sum_j\\nu_j$ and $|\\tilde{\\nu}|=\\sum_j\\tilde{\\nu}_j$, we showed that they are analytic in an open set $W_{\\ell}\\subset \\{{\\rm Re}(\\alpha)\\leq Q\\}\\setminus \\cup_{j\\geq 0}\\{Q\\pm i\\sqrt{2j}\\}$ \ncontaining $Q+i\\mathbb{R}$, where $\\ell=|\\nu|+|\\tilde{\\nu}|$. Moreover the following Plancherel type formula holds: for $u,u'\\in \\mathcal{H}$\n\\begin{equation}\\label{diagonalisation}\n\\langle u,u'\\rangle_{2}= \\frac{1}{2\\pi}\\sum_{\\substack{\\nu,\\nu',\\tilde{\\nu},\\tilde{\\nu}'\\in \\mathcal{T}\\\\\n|\\nu|=|\\nu'|, |\\tilde{\\nu}|=|\\tilde{\\nu}'|}}\\int_0^\\infty \\langle u,\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}\\rangle_2 \\langle \\Psi_{Q+iP,\\nu',\\tilde{\\nu}'},u'\\rangle_2 F_{Q+iP}^{-1}(\\nu,\\nu')F_{Q+iP}^{-1}(\\tilde{\\nu},\\tilde{\\nu}')\\text{\\rm d} P \\end{equation}\n where $\\mathcal{T}$ denotes the set of Young diagrams, and $(F_{Q+iP}^{-1}(\\nu,\\nu'))_{\\nu,\\nu'\\in \\mathcal{T}_n}$ \n are positive definite matrices for each $n>0$ if $\\mathcal{T}_n=\\{\\nu \\in \\mathcal{T}\\,|\\, |\\nu|=n\\}$ is the set of Young diagrams of size $n$. In our convention $\\mathcal{T}$ contains $\\{0\\}$ and $\\Psi_{Q+iP,0,0}=\\Psi_{Q+iP}$.\n \nIn this paper, we prove that the $\\Psi_{\\alpha,\\nu,\\tilde\\nu}$ can be obtained from $\\Psi_\\alpha$ by \napplying the direct sum of two copies of the Virasoro algebra ${\\rm Vir}(c_L)$ to it. \n\\begin{theorem}\\label{descendantsandLn} The following properties hold:\\\\\n 1) The function $\\Psi_{\\alpha}$ defined by \\eqref{Psialphadef} admits an analytic extension to $\\alpha \\in \\mathbb{C}$, with values in $e^{-\\beta c_-}\\mathcal{D}(\\mathcal{Q})$ for \n$\\beta>|{\\rm Re}(\\alpha)-Q|$, and for each $n\\in \\mathbb{Z}$, ${\\bf L}_{n}\\Psi_\\alpha$ is well-defined as an element in $e^{-\\beta c_-}\\mathcal{D}(\\mathcal{Q})$ for\n$\\beta>|{\\rm Re}(\\alpha)-Q|$, it is analytic in $\\alpha$ and equal to $0$ when $n>0$.\\\\\n2) The functions $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ appearing in \\eqref{diagonalisation} admit an analytic extension to \n$\\alpha\\in \\mathbb{C}$ as elements of $e^{-\\beta c_-}\\mathcal{D}(\\mathcal{Q})$ for $\\beta>|{\\rm Re}(\\alpha)-Q|$, \n and for each $n\\in \\mathbb{Z}$, ${\\bf L}_{n}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ is well-defined as an element in $e^{-\\beta c_-}\\mathcal{D}(\\mathcal{Q})$ for\n$\\beta>|{\\rm Re}(\\alpha)-Q|$, analytic in $\\alpha$, and is equal to $0$ when $n>|\\nu|+|\\tilde{\\nu}|$.\\\\\n3) These functions are related by the formula\n\\[ \\Psi_{\\alpha,\\nu,\\tilde{\\nu}} = {\\bf L}_{-\\nu_k}\\dots {\\bf L}_{-\\nu_1}\\tilde{\\bf L}_{-\\tilde{\\nu}_{k'}}\\dots\\tilde{\\bf L}_{-\\tilde{\\nu}_1}\\Psi_{\\alpha}\\]\nand for each $n\\in \\mathbb{Z}$, and $\\alpha\\notin Q\\pm (\\frac{2}{\\gamma}\\mathbb{N}+\\frac{\\gamma}{2}\\mathbb{N})$\n\\[{\\bf L}_{n} \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}\\in {\\rm span}\\{ \\Psi_{\\alpha,\\nu',\\tilde{\\nu}'}\\,|\\, \\nu' \\in \\mathcal{T},\\, |\\nu|-n=|\\nu'|\n\\}.\\]\n4) They satisfy the functional equation for all $\\alpha\\in \\mathbb{C} \\setminus (Q\\pm (\\frac{2}{\\gamma}\\mathbb{N}+\\frac{\\gamma}{2}\\mathbb{N}))$ and all $\\nu,\\tilde{\\nu}\\in \\mathcal{T}$\n\\[ \\Psi_{2Q-\\alpha,\\nu,\\tilde{\\nu}}=R(2Q-\\alpha)\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}\\]\nwhere $R(\\alpha)$ is the scattering coefficient given by \n\\[R(\\alpha)=-\\Big(\\pi \\mu \\frac{\\Gamma(\\frac{\\gamma^2}{4})}{\\Gamma(1-\\frac{\\gamma^2}{4})}\\Big)^{2\\frac{(Q-\\alpha)}{\\gamma}}\\frac{\\Gamma(-\\frac{\\gamma(Q-\\alpha)}{2})\\Gamma(-\\frac{2(Q-\\alpha)}{\\gamma})}{\\Gamma(\\frac{\\gamma(Q-\\alpha)}{2})\\Gamma(\\frac{2(Q-\\alpha)}{\\gamma})}\\]\n5) For $P>0$, $\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}$ satisfy the asymptotic expansion as $c\\to -\\infty$\n\\[ \\Psi_{Q+iP,\\nu,\\tilde{\\nu}}=e^{iPc}\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}(\\varphi)+R(Q+iP)e^{-iPc}\\mathcal{Q}_{Q-iP,\\nu,\\tilde{\\nu}}(\\varphi) +G_{Q+iP}(c,\\varphi)\\]\nwhere $G_{Q+iP}(c,\\varphi)\\in \\mathcal{D}(\\mathcal{Q})\\subset L^2$ and $\\mathcal{Q}_{Q\\pm iP,\\nu,\\tilde{\\nu}}\\in L^2(\\Omega_\\mathbb{T})$ are particular eigenfunctions of ${\\bf P}$ with eigenvalues $|\\nu|+|\\tilde{\\nu}|$ (see Section \\ref{Vermafreefield} for their definition).\n\\end{theorem}\n\nWe notice that Statement 5) means that the scattering matrix is essentially (up to identifying $\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}$ with \n$\\mathcal{Q}_{Q-iP,\\nu,\\tilde{\\nu}}$) a constant $R(Q+iP)$ times the Identity map, a feature that is important in physics and that is related to the integrability of LCFT.\n\nThis result allows us in particular to define the vector space $\\mathcal{W}_\\alpha:={\\rm span}\\{\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}\\,|\\, \\nu,\\tilde{\\nu}\\in \\mathcal{T}\\}$ and to show that the operators ${\\bf L}_n,\\tilde{{\\bf L}}_m$ preserve $\\mathcal{W}_\\alpha$, with the commutation relations \n\\[ [\\mathbf{L}_n,\\mathbf{L}_m]=(n-m)\\mathbf{L}_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}, \\quad [\\tilde{\\bf L}_n,\\tilde{\\bf L}_m]=(n-m)\\tilde{\\bf L}_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}\\]\nand $[{\\bf L}_n,\\tilde{\\bf L}_m]=0$.\nThe space $\\mathcal{W}_\\alpha$ is a Verma module for a highest weight representation \nof the direct sum ${\\rm Vir}(c_L)\\oplus {\\rm Vir}(c_L)$ of two Virasoro algebras with central charge $c_L=1+6Q^2$, it can be splitted as a tensor product of two Verma modules $\\mathcal{W}_{\\alpha}=\\mathcal{V}_\\alpha\\otimes \\overline{\\mathcal{V}}_\\alpha$, each one associated to the representations ${\\bf L}_n$ and $\\tilde{\\bf L}_n$ of ${\\rm Vir}(c_L)$. We refer to Section \\ref{VermaLiouville} for more details on these aspects related to representation theory.\n\n\n\n\n\n\n\n\n\n\\subsection{Future applications}\nThis approach will be instrumental in several subsequent works in the context of the conformal bootstrap for LCFT. First it will serve as a key tool in a programme aiming at establishing the conformal bootstrap for open surfaces: the analyticity of the eigenstates over the full complex plane as well as their asymptotics is needed to analyze the contribution of the one point bulk correlator, the FZZ structure constant introduced in \\cite{ARS}. This issue already appears in \\cite{wu} in the case of the annulus and will be further developed in a forthcoming work treating the case of general open surfaces. \n\nAlso, recall that the conformal blocks appearing in the conformal bootstrap formulae depend on a choice of pant decomposition of the underlying Riemann surface. As stated in \\cite{GKRV1}, the conformal blocks depend on the splitting curves used to obtain the pant decomposition. Therefore they are not yet fully understood as analytic functions on Teichm\\\"uller space as they depend on the choice of local coordinates (the splitting curves). In a forthcoming work, we will use the Markovian dynamics introduced in this manuscript to show that the conformal blocks do not depend on local deformations of the splitting curves. This will show that conformal blocks only depend on the homotopy classes of the splitting curves, and therefore are well-defined on Teichm\\\"uller space. As an intermediate step, we will identify the annulus amplitudes with the kernels of the semigroups introduced in the present work. This is in agreement with Segal's axioms \\cite{Gaw} and generalizes \\cite[Section 6]{GKRV1}.\n\n On the other hand, conformal blocks are not expected to be single-valued on moduli space (contrarily to correlation functions), but it is conjectured that there exists a representation of the mapping class group describing the variation of the blocks under a change of pants decomposition. This is related to a conjecture of Verlinde identifying the space of conformal blocks of Liouville theory with the Hilbert space of quantum Teichm\\\"uller theory as isomorphic representations of the mapping class group (see \\cite{Teschner04} for an overview). In geometric terms, the Virasoro generators introduced in this article should define a (projectively) flat connection on the bundle of conformal blocks, whose monodromy is described by the above-mentioned representation. Our interpretation of the Virasoro operators as generators of diffusion processes is reminiscent to a type of connection on the moduli space introduced by Hitchin \\cite{Hitchin}, for which parallel transport is given by the heat kernel. Finally, the full analycity of the descendants $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ will be used to bridge the gap between the probabilistic construction of LCFT and the Vertex Operator Algebra approach.\n\n \n\\subsubsection{Organization of the paper}\nThe paper is organized as follows. In section 2, we introduce the main notations and the framework of the paper; in this section, we also construct the semigroup associated to the free field case $\\mu=0$. In section 3, we construct the semigroup associated to the general case, i.e. with the Liouville potential, via a Feynman-Kac type formula, and we study its stationary measure $\\mu_h$.\n In this section, we use the dynamics to show that the basis $\\Psi_{\\alpha, \\nu,\\tilde \\nu}$ can be constructed via the action of the Virasoro algebra: this is the content of Proposition \\ref{intertwining_for_descendants}. In section 4, we use these results to study the scattering coefficients and show the analytic extension and functional equations for the eigenstates $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$. \n \n \n{\\bf Acknowledgements:} Colin Guillarmou acknowledges the support of European Research Council (ERC) Consolidator grant 725967 and Antti Kupiainen the support of the ERC Advanced Grant 741487. R\\'emi Rhodes is\npartially supported by the Institut Universitaire de France (IUF). R\\'emi Rhodes and Vincent Vargas acknowledge\nthe support of the French National Research Agency (ANR) ANR-21-CE40-003.\n\n \n\\section{Segal-Sugawara representation and the Gaussian Free Field}\\label{sec:free_field}\n\n\\subsection{Free Hamiltonian and Virasoro generators for the free field}\\label{freeham}\n\nLet us introduce some material taken from \\cite[Section 4]{GKRV}. \nAs mentionned in the Introduction, we consider the space $\\Omega_\\mathbb{T}=(\\mathbb{R}^2)^\\mathbb{N}$ with the Gaussian probability measure \n \\[\n \\P_\\mathbb{T}:=\\bigotimes_{n\\geq 1}\\frac{1}{2\\pi}e^{-\\frac{1}{2}(x_n^2+y_n^2)}\\text{\\rm d} x_n\\text{\\rm d} y_n.\n\\]\nThe space $\\mathcal{H}$ is then defined to be the space $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$ equipped with the measure $\\mu_0:=\\text{\\rm d} c \\times \\P_\\mathbb{T}$ (and the standard Borel sigma algebra) where $\\text{\\rm d} c$ denotes the Lebesgue measure. Consider the real valued random variable on $\\mathbb{T}$\n\\[\n\\varphi(\\theta)=\\sum_{n\\not=0}\\varphi_ne^{in\\theta} , \\quad \\varphi_n=\\frac{x_n+iy_n}{2\\sqrt{n}} \\textrm{ for }n>0 \n\\]\nNote that $\\varphi\\in H^{-s}(\\mathbb{T})$ for all $s>0$ almost surely. The space $H^{-s}(\\mathbb{T})$ can be equipped with the pushforward measure $\\mu_0$ (still denoted $\\mu_0$ for simplicity) by the map \n$(c,(\\varphi_n)_n)\\in \\mathbb{R}\\times \\Omega_\\mathbb{T}\\mapsto c+\\varphi\\in H^{-s}(\\mathbb{T})$ so that $\\mathcal{H}\\simeq L^2(H^{-s}(\\mathbb{T}),\\mu_0)$. \n\nWe set $\\partial_n:= \\partial_{\\varphi_n}= \\sqrt{n} (\\partial_{x_n}-i \\partial_{y_n})$ (and $\\partial_{-n}:= \\partial_{\\varphi_{-n}}= \\sqrt{n} (\\partial_{x_n}+i \\partial_{y_n})$) and introduce $\\mathcal{S}$ the set of smooth functions of the form $F(x_1,y_1, \\cdots, x_N,y_N)$ for some $N \\geq 1$ where $F\\in C^\\infty((\\mathbb{R}^2)^N)$ with at most polynomial growth at infinity for $F$ and its derivatives. We consider the vector space $\\mathcal{C}_{\\rm exp}$ \nof smooth functions $F:\\mathbb{R}\\times \\Omega_\\mathbb{T}\\to \\mathbb{C}$ so that there is $N>0$ such that $F(c,\\varphi)=F(c,x_1,y_1, \\cdots, x_N,y_N)$ for all $c\\in\\mathbb{R}$, and (with $\\mathbb{N}_0:=\\mathbb{N}\\cup\\{0\\}$)\n\\[ \\forall k\\in \\mathbb{N}_0, \\forall \\alpha,\\beta \\in \\mathbb{N}_0^{N}, \\exists L\\geq 0, \\forall M\\geq 0, \\exists C>0, \n\\quad |\\partial_c^k \\partial_{x}^\\alpha\\partial_y^\\beta F(c,\\varphi)|\\leq Ce^{-M|c|}\\langle \\varphi\\rangle_N^{L}\n\\]\nif $\\partial_{x}^\\alpha=\\partial_{x_1}^{\\alpha_1}\\dots \\partial_{x_N}^{\\alpha_N}$ and $\\langle \\varphi\\rangle_N=(1+\\sum_{|n|\\leq N}|\\varphi_n|^2)^{1\/2}$ is the Japanese bracket.\n\nThe space $\\mathcal{C}_{\\rm exp}$ is a dense vector subspace of $\\mathcal{H}$ and \nthe free Hamiltonian is the Friedrichs extension of the operator defined on $\\mathcal{C}_{\\rm exp}$ by the expression\n\\begin{equation}\\label{defH0}\n{\\bf H}^0=\\frac{1}{2}(-\\partial_c^2+Q^2+2{\\bf P}) ,\\quad \\textrm{ with }{\\bf P}=\\sum_{n=1}^\\infty n(\\partial_{x_n}^*\\partial_{x_n}+\\partial_{y_n}^*\\partial_{y_n}),\n\\end{equation}\nwhere $\\partial_{x_n}^*$ denotes the adjoint of $\\partial_{x_n}$ with respect to $\\mu_0$. Recall that the construction of the Friedrichs extension relies on the quadratic form denoted $\\mathcal{Q}_0(F,F):=\\langle{\\bf H}^0F,F\\rangle_2$ for $F\\in \\mathcal{C}_{\\rm exp}$. By \\cite[Proposition 4.3]{GKRV}, \nthe quadratic form is closable with closure still denoted $\\mathcal{Q}_0$ and domain denoted $\\mathcal{D}(\\mathcal{Q}_0)$, which becomes a Hilbert space when equipped with the norm $\\|F\\|_{\\mathcal{Q}_0}:=\\sqrt{\\mathcal{Q}_0(F,F)}$. \nThis quadratic form produces a self-adjoint extension (Friedrichs extension) for ${\\bf H}^0$ on a domain $\\mathcal{D}(\\mathcal{H}^0)\\subset \\mathcal{D}(\\mathcal{Q}_0)$. \nMoreover ${\\bf H}^0$ extends as a bounded operator ${\\bf H}^0:\\mathcal{D}(\\mathcal{Q}_0)\\to \\mathcal{D}'(\\mathcal{Q}_0)$ where $\\mathcal{D}'(\\mathcal{Q}_0)$ \nis the dual to $\\mathcal{D}(\\mathcal{Q}_0)$. \n \nNext we recall the construction of the representation of two copies of the Virasoro algebra as operators on the space $\\mathcal{C}_{\\rm exp}$. Consider first\n the operators for $n\\geq 1$\n\\begin{equation}\\label{defAn}\n\\mathbf{A}_n= \\tfrac{i\\sqrt{n}}{2}(\\partial_{x_n}-i\\partial_{y_n}),\\, \\, \\mathbf{A}_{-n}={\\bf A}_n^*, \\qquad \\widetilde{\\mathbf{A}}_n= \\tfrac{i\\sqrt{n}}{2}(\\partial_{x_n}+i\\partial_{y_n}),\\,\\,\n \\widetilde{\\mathbf{A}}_{-n}=\\widetilde{\\bf A}_n^*,\n \\end{equation} \nwhere the adjoint here is formal and has to be understood in the sense of pairing on $\\mathcal{C}_{\\rm exp}$ with respect to the measure $\\mu_0$, and let \n\\[\\mathbf{A}_0=\\widetilde{\\mathbf{A}}_0=\\tfrac{i}{2}(\\partial_c+Q).\\] \nThen for $n\\in \\mathbb{Z}$ define the operators acting on $\\mathcal{C}_{\\rm exp}$ \n\\[\\mathbf{L}_n^0=-i(n+1)Q\\mathbf{A}_n+\\sum_{m\\in\\mathbb{Z}}:\\!\\mathbf{A}_{n-m}\\mathbf{A}_m\\!:, \\quad \\widetilde{\\mathbf{L}}_n^0:=-i(n+1)Q\\widetilde{\\mathbf{A}}_n+\\sum_{m\\in\\mathbb{Z}}:\\!\\widetilde{\\mathbf{A}}_{n-m}\\widetilde{\\mathbf{A}}_m\\!: \n\\]\nand where the normal ordering is defined by $:\\!\\mathbf{A}_n\\mathbf{A}_m\\!\\!:\\,=\\mathbf{A}_n\\mathbf{A}_m$ if $m>0$ and $\\mathbf{A}_m\\mathbf{A}_n$ if $n>0$ (i.e. annihilation operators are on the right). In particular, for $n<0$ the only $c$ derivative in ${\\bf L}_n^0$ is the term involving ${\\bf A}_0$, which is given by \n${\\bf A}_n{\\bf A}_0+{\\bf A}_0{\\bf A}_n=2{\\bf A}_n{\\bf A}_0$.\nNotice that $({\\bf L}_n^0)^*= {\\bf L}_{-n}^0$ in the weak sense, i.e. for each $F_1,F_2\\in \\mathcal{C}_{\\rm exp}$ the following relation holds \n\\[ \\langle {\\bf L}_n^0F_1,F_2\\rangle_2 = \\langle F_1, {\\bf L}_{-n}^0 F_2\\rangle_2.\\]\nA direct computation allows us to express the Hamiltonian in terms of the $n=0$ Virasoro elements\n\\[ {\\bf H}^0={\\bf L}_0^0+ \\widetilde{\\mathbf{L}}_0^0=\\frac{1}{2}(-\\partial_c^2+Q^2)+2\\sum_{n=1}^\\infty {\\bf A}_n^*{\\bf A}_n+\\widetilde{\\bf A}_n^*\\widetilde{\\bf A}_n\\]\nand the commutation relations (for $c_L=1+6Q^2)$\n\\[ [\\mathbf{L}^0_n,\\mathbf{L}^0_m]=(n-m)\\mathbf{L}^0_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}, \\quad [\\widetilde{\\mathbf{L}}^0_n,\\widetilde{\\mathbf{L}}^0_m]=(n-m)\\widetilde{\\mathbf{L}}^0_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}.\\] \nNotice that $\\mathcal{Q}_0$ can be written in terms of the ${\\bf A}_n$'s as \n\\[\\begin{split} \n\\mathcal{Q}_0(F,F)= \\frac{1}{2}(\\|\\partial_cF\\|_2^2+Q^2\\|F\\|^2_2)+2\\sum_{n=1}^\\infty (\\|{\\bf A}_nF\\|^2_{2}+\\|\\widetilde{\\bf A}_nF\\|^2_{2})= 2\\|{\\bf A}_0F\\|_2^2+2\\sum_{n=1}^\\infty (\\|{\\bf A}_nF\\|^2_{2}+\\|\\widetilde{\\bf A}_nF\\|^2_{2}).\n\\end{split}\\]\nWe first show the following: \n\\begin{lemma}\\label{boundonLn}\nThere exist some constant $C>0$ such that for all $n \\geq 1$ and all $F,G\\in\\mathcal{C}_{\\rm exp}$ we have \n\\begin{equation}\\label{bilinearboundLn}\n | \\langle {\\bf L}^0_n F,G \\rangle_2|\\leq C(1+n)^{3\/2}\\|F\\|_{\\mathcal{Q}_0}\\|G\\|_{\\mathcal{Q}_0} \n \\end{equation}\nand the operator ${\\bf L}_n^0$ extends as a bounded operator ${\\bf L}_n^0: \\mathcal{D}(\\mathcal{Q}_0)\\to \\mathcal{D}'(\\mathcal{Q}_0)$.\n\\end{lemma}\n\\begin{proof}\nFirst we note that the extension of ${\\bf L}_n^0$ follows directly from the estimate \\eqref{bilinearboundLn}.\nUsing Cauchy-Schwarz, we have for $F,G\\in\\mathcal{C}_{\\rm exp}$\n\\begin{align*}\n | \\langle {\\bf L}_n^0 F,G \\rangle_2 | & = \\Big| -i (n+1) Q \\langle {\\bf A}_n F , G \\rangle_2 + 2 \\langle {\\bf A}_0 F, {\\bf A}_n G \\rangle_2 + \\sum_{k=1}^{n-1} \\langle {\\bf A}_{n-k}F , {\\bf A}_{-k}G \\rangle_2 + 2 \\sum_{m \\geq 1} \\langle {\\bf A}_{n+m} F, {\\bf A}_m G \\rangle_2 \\Big| \\\\\n& \\leq \\Big ( (n+1) Q \\|{\\bf A}_nF\\|^2_2+ 2 \\|{\\bf A}_0F\\|^2_2 + \\sum_{k=1}^{n-1} \\| {\\bf A}_kF \\|^2_2+ 2 \\sum_{m \\geq 1} \\|{\\bf A}_{n+m} F \\|^2_2 \\Big)^{\\frac{1}{2}} \\\\\n& \\quad \\times \\Big ( (n+1) Q \\| G \\|^2_2 + 2 \\| {\\bf A}_n G \\|^2_2 + \\sum_{k=1}^{n-1} \\| {\\bf A}_{-k}G \\|^2_2 + 2 \\sum_{m \\geq 1} \\| {\\bf A}_m G \\|^2_2\\Big)^{\\frac{1}{2}} \n\\end{align*}\nand using the commutation relation $[{\\bf A}_k,{\\bf A}_{-k}]= \\frac{k}{2}$ which implies $\\| {\\bf A}_{-k} G \\|^2= \\|{\\bf A}_{k} G \\|^2+ \\frac{k}{2} \\| G \\|^2 $ we obtain \n\\[ \\begin{split} \n| \\langle {\\bf L}_n^0 F,G \\rangle_2 | & \\leq \\Big ( (n+1) Q \\|{\\bf A}_nF\\|^2+ 2 \\|{\\bf A}_0F\\|^2_2 + \\sum_{k=1}^{n-1} \\| {\\bf A}_{k}F \\|^2_2+ 2 \\sum_{m \\geq 1} \\| {\\bf A}_{n+m} F \\|^2_2 \\Big)^{\\frac{1}{2}} \\\\\n& \\quad \\times \\Big ( ( (n+1) Q+ \\frac{n(n-1)}{4}) \\| G \\|^2_2 + 2 \\| {\\bf A}_n G \\|^2_2 + \\sum_{k=1}^{n-1} \\| {\\bf A}_{k}G \\|^2_2 + 2 \\sum_{m \\geq 1} \\| {\\bf A}_m G \\|^2_2\\Big)^{\\frac{1}{2}} \\end{split}\\]\nThe first term is bounded by $C(1+n)^{1\/2}\\|F\\|_{\\mathcal{Q}_0}$ and the second by $C(1+n)\\|G\\|_{\\mathcal{Q}_0}$ for some uniform $C>0$, which concludes the proof.\n\\end{proof}\n\n\\subsection{Gaussian Free field and Gaussian multiplicative chaos}\n\nOn the Riemann sphere $\\hat{\\mathbb{C}}$, we put the Riemannian metric $g_0=|dz|^2\/|z|_+^4$ (with $|z|_+:=\\max(1,|z|)$)\n which is invariant by the inversion $z\\mapsto 1\/z$. We shall use freely the complex variable $z$ in $\\mathbb{C}$ or $x\\in \\mathbb{R}^2$ when we think of it as a real variable.\nThe Gaussian Free Field $X$ (GFF in short) in the metric $g_0$ is a random variable in the Sobolev space $H^{s}(\\mathbb{C})$ for $s<0$ \nwith mean $0$ on $\\mathbb{T}$ and covariance kernel \n \\begin{equation*}\n \\mathbb{E}[X(z)X(z')]= \\ln \\frac{|z|_+|z'|_+}{|z-z'|}.\n \\end{equation*}\nIf $\\varphi$ is the random variable \\eqref{GFFcircle} on the unit circle $\\mathbb{T}:=\\{z\\in \\mathbb{C}\\,|\\, z=1\\}$, it is easily checked (see \\cite{GKRV}) that \n\\begin{equation}\\label{decomposGFF}\n X= P\\varphi+ X_\\mathbb{D}+X_{\\mathbb{D}^c}\n \\end{equation} \n where $P\\varphi$ is the harmonic extension of $\\varphi$ and $X_\\mathbb{D},X_{\\mathbb{D}^c}$ are two independent GFFs on $\\mathbb{D}:=\\{z\\in \\mathbb{C}\\,|\\, |z|<1\\}$ \n and $\\mathbb{D}^c:=\\{z\\in \\hat{\\mathbb{C}}\\,|\\, |z|>1\\}$ with Dirichlet boundary conditions defined respectively on the probability spaces $ (\\Omega_\\mathbb{D}, \\Sigma_\\mathbb{D},\\P_\\mathbb{D})$ and $ (\\Omega_{\\mathbb{D}^c}, \\Sigma_{\\mathbb{D}^c}, \\P_{\\mathbb{D}^c})$\\footnote{With a slight abuse of notations, we will assume that these spaces are canonically embedded in the product space $(\\Omega,\\Sigma)$ and we will identify them with the respective images of the respective embeddings.}. \n The covariance of $X_{\\mathbb{D}}$ is the Dirichlet Green's function on $\\mathbb{D}$, given by \n \\[ \\mathbb{E}[X_{\\mathbb{D}}(z)X_{\\mathbb{D}}(z')]=G_\\mathbb{D}(z,z')= \\ln \\frac{|1-z\\bar{z}'|}{|z-z'|}.\\]\n The random variable $X$ is defined on a probability space $(\\Omega, \\Sigma, \\P)$ (with expectation $\\mathbb{E}[.]$) where $\\Omega= \\Omega_\\mathbb{T} \\times \\Omega_\\mathbb{D} \\times \\Omega_{\\mathbb{D}^c} $, $\\Sigma= \\Sigma_\\mathbb{T} \\otimes \\Sigma_\\mathbb{D} \\otimes \\Sigma_{\\mathbb{D}^c}$ and $\\P$ is a product measure $\\P=\\P_\\mathbb{T}\\otimes \\P_{\\mathbb{D}}\\otimes \\P_{\\mathbb{D}^c}$. We will write $\\mathbb{E}_\\varphi[\\cdot ]$ for conditional expectation with respect to the GFF on the circle $\\varphi$.\n \nWe introduce the Gaussian multiplicative chaos measure (GMC in short), introduced by Kahane \\cite{cf:Kah},\n\\begin{equation}\\label{GMCsphere}\ne^{\\gamma X(x)}\\text{\\rm d} x:= \\underset{\\epsilon \\to 0} {\\lim} \\; \\; e^{\\gamma X_\\epsilon(x)-\\frac{\\gamma^2}{2} \\mathbb{E}[X_\\epsilon(x)^2]}\\text{\\rm d} x\n\\end{equation}\nwhere $X_\\epsilon= X \\ast \\theta_\\epsilon$ is the mollification of $X$ with an approximation $(\\theta_\\epsilon)_{\\epsilon>0}$ of the Dirac mass $\\delta_0$; indeed, one can show that the limit \\eqref{GMCsphere} exists in probability in the space of Radon measures on $\\hat\\mathbb{C}$ and that the limit does not depend on the mollifier $\\theta_\\epsilon$: see \\cite{RoV, review, Ber}. The condition $\\gamma \\in (0,2)$ stems from the fact that the random measure $M_\\gamma$ is different from zero if and only if $\\gamma \\in (0,2)$. We will also consider the GMC measure on the circle $\\mathbb{T}$ associated to $\\varphi$ defined via \n\\begin{equation}\\label{GMCcircle}\ne^{\\gamma \\varphi(\\theta)} \\text{\\rm d} \\theta:= \\underset{\\epsilon \\to 0} {\\lim} \\; \\; e^{\\gamma \\varphi_\\epsilon(\\theta)-\\frac{\\gamma^2}{2} \\mathbb{E}[\\varphi_\\epsilon(\\theta)^2]} \\text{\\rm d} \\theta\n\\end{equation}\nwhere $\\varphi_\\epsilon$ is a mollification at scale $\\epsilon$ of $\\varphi$. The measure $e^{\\gamma \\varphi(\\theta)} \\text{\\rm d} \\theta$ is different from zero if and only if $\\gamma \\in (0,\\sqrt{2})$ and therefore for $\\gamma \\in [\\sqrt{2},2)$, the Fourier coefficients in \\eqref{virasoro} will not act as multiplication by a variable and will rather be defined via the Girsanov transform in the context of Dirichlet forms. \n\n\\subsection{Liouville Hamiltonian and its quadratic form} \nIn \\cite[Section 5]{GKRV}, an Hamiltonian ${\\bf H}$ associated to Liouville CFT is defined as an unbounded\n operator acting on the Hilbert space $\\mathcal{H}$, it is formally given by the expression on $\\mathcal{C}_{\\rm exp}$ \n\\begin{equation}\\label{LiouvilleH}\n{\\bf H}={\\bf H}^0+\\mu e^{\\gamma c}V \n\\end{equation}\nwhere $V$ is a non-negative unbounded operator on $L^2(\\Omega_\\mathbb{T},\\mathbb{P}_\\mathbb{T})$ associated to a symmetric \nquadratic form $\\mathcal{Q}_V$; when $\\gamma<\\sqrt{2}$, $V$ is a multiplication operator by the potential \n\\[V(\\varphi)=\\int_0^{2\\pi} e^{\\gamma \\varphi(\\theta)}d\\theta \\in L^{\\frac{2}{\\gamma^2}-\\epsilon}(\\Omega_\\mathbb{T}), \\quad \\forall \\epsilon>0\\]\nwith $e^{\\gamma \\varphi(\\theta)}d\\theta$ being the GMC measure on the circle defined in \\eqref{GMCcircle}.\nThe rigorous construction of ${\\bf H}$ uses again the notion of Friedrichs extension. Consider the symmetric quadratic form \n\\begin{equation}\\label{quadformQ}\n\\mathcal{Q}(F,F):=\\mathcal{Q}_0(F,F)+\\mu \\int_{\\mathbb{R}} e^{\\gamma c}\\mathcal{Q}_V(F,F)\\text{\\rm d} c \\geq \\mathcal{Q}_0(F,F).\n\\end{equation}\nThis quadratic form admits a closure on a domain $\\mathcal{D}(\\mathcal{Q})\\subset \\mathcal{D}(\\mathcal{Q}_0)$ and one can view ${\\bf H}$ as a self-adjoint operator using the Friedrichs extension \\cite[Proposition 5.5]{GKRV}. The operator ${\\bf H}$ has a domain $\\mathcal{D}({\\bf H})\\subset \\mathcal{D}(\\mathcal{Q})$ but it is also bounded as a map \n\\[ {\\bf H} : \\mathcal{D}(\\mathcal{Q})\\to \\mathcal{D}'(\\mathcal{Q})\\]\nwhere $ \\mathcal{D}'(\\mathcal{Q})$ is the dual space to $\\mathcal{D}(\\mathcal{Q})$. The propagator $e^{-t{\\bf H}}$ for $t\\geq 0$ is a contraction \nsemi-group that has the Markov property \\cite[Proposition 5.2]{GKRV}, it will appear also below as a special case of \nMarkovian vector field.\n\\subsection{Markovian holomorphic vector fields}\nLet $\\mathcal{M}$ be the space of univalent maps on $\\mathbb{D}$ extending smoothly \nto the boundary. By Carath\\'eodory's conformal mapping theorem, $f(\\mathbb{T})$ is a Jordan curve for each $f\\in\\mathcal{M}$, and this Jordan curve is smooth. Infinitesimal variations around the identity in $\\mathcal{M}$ give rise to \nthe space $\\mathrm{Vect}(\\mathbb{D})$ of holomorphic vector fields on $\\mathbb{D}$ extending smoothly to the boundary. \nThese vector fields can be written as $\\mathbf{v}=v\\partial_z$ for some holomorphic function $v$, with power series expansion at $z=0$\n\\begin{equation}\\label{eq:coordinates_vector_fields}\nv(z)=-\\sum_{n=-1}^\\infty v_nz^{n+1}.\n\\end{equation}\nThese vector fields encode the infinitesimal motion $f_t(z)=z+tv(z)+o(t)$ on $\\mathcal{M}$.\n\nLet us introduce the following subspace of $\\mathrm{Vect}(\\mathbb{D})$:\n\\begin{equation}\\label{eq:def_vect_plus}\n\\begin{aligned}\n&\\mathrm{Vect}_+(\\mathbb{D}):=\\left\\lbrace\\mathbf{v}\\in\\mathrm{Vect}(\\mathbb{D})|\\, \\forall z\\in\\mathbb{T}, \\Re\\left(\\bar{z}v(z)\\right)<0\\,\\right\\rbrace.\\\\\n\\end{aligned}\n\\end{equation}\nWe will refer to such vector fields as \\emph{Markovian}. Notice that for each $\\mathbf{v}\\in\\mathrm{Vect}(\\mathbb{D})$ with $v(0)=0$, one can find $\\omega>0$ large enough such that $\\omega \\mathbf{v}_0+\\mathbf{v}$ is Markovian, if $\\mathbf{v}_0=-z\\partial_z$.\n\nThe basis elements \n\\[\\mathbf{v}_n=-z^{n+1}\\partial_z \\in\\mathrm{Vect}(\\mathbb{D}) ,\\quad n\\geq-1\\] \nspan a Lie subalgebra of the complex Witt algebra. Recall that the complex Witt algebra is the complex Lie algebra of polynomial vector fields on $\\mathbb{T}$, i.e.vector fields of the form $\\sum_{|k|\\leq N}a_k e^{ik\\theta}\\partial_{\\theta}$ for $a_k\\in \\mathbb{C}$ and some $N<\\infty$. This is a subalgebra of the Lie algebra $\\mathbb{C}(z)\\partial_z$ of meromorphic vector fields in $\\mathbb{D}$. \nThere is an antilinear involution on $\\mathbb{C}(z)\\partial_z$ given by $\\mathbf{v}\\mapsto \\mathbf{v}^*:=z^2\\bar{v}(1\/z)\\partial_z$. On the canonical generators, we have $\\mathbf{v}_n^*=\\mathbf{v}_{-n}$ and $(i\\mathbf{v}_n)^*=-i\\mathbf{v}_{-n}$. The vector fields with negative powers encode the deformations near the identity of the space of conformal transformations of the outer disc. The involution $^*$ relates the two types of deformations.\n\n\\begin{lemma}\nFor $\\mathbf{v}=v(z)\\partial_z \\in \\mathrm{Vect}_+(\\mathbb{D})$, let $f_t(z)$ be the solution of the complex ordinary differential equation $\\partial_tf_t(z)=v(f_t(z))$ for $t\\geq 0$ with $f_0(z)=z$. \nThe family $(f_t)_{t\\geq 0}$ is a family of holomorphic maps in $\\mathcal{M}$ satisfying $f_{t}(\\mathbb{D})\\subset f_{s}(\\mathbb{D})$ for all \n$t\\geq s$ and $f_t(\\mathbb{D})$ converges exponentially fast to $a\\in \\mathbb{D}$, the unique zero of $v(z)$ in $\\mathbb{D}$.\n\\end{lemma}\n\\begin{proof}\nThe functions $(f_t)_{t\\geq 0}$ are holomorphic in $\\mathbb{D}$ as solution of an ODE with holomorphic coefficients. \nWe first show that there exists an $a \\in \\mathbb{D}$ such that $v(a)=0$. If this is not the case then $1\/v(z)$ is holomorphic in $\\mathbb{D}$ and hence by Cauchy's theorem\n\\begin{equation*}\n\\int_{0}^{2\\pi} \\frac{e^{i \\theta} \\overline{v (e^{i \\theta})}}{|v (e^{i \\theta})|^2} d \\theta = \\int_{0}^{2\\pi} \\frac{e^{i \\theta} }{v (e^{i \\theta})} d \\theta =0\n\\end{equation*} \nwhich is in contradiction with the fact that $\\Re (e^{i \\theta} \\overline{v (e^{i \\theta})} ) <0$ for all $\\theta \\in [0,2\\pi]$.\n\nWe first consider a vector field $\\mathbf{v} \\in \\mathrm{Vect}_+(\\mathbb{D})$ which satisfies $v(0)=0$. By the maximum principle for harmonic functions applied to $\\Re\\left( \\frac{v(z)}{z}\\right)$, there exists $c>0$ such that\n\\begin{equation}\\label{inequa}\n\\sup_{|z| \\leq 1} \\Re\\left( \\frac{v(z)}{z}\\right) \\leq -c.\n\\end{equation} \nThe flow satisfies $\\partial_t f_t(z)= v( f_t(z) )$ and $f_{0}(z)=z$. \nThanks to \\eqref{inequa}, one has the following \n\\begin{equation*}\n\\partial_t |f_t(z)|^2 =2{\\rm Re}\\Big (\\frac{\\partial_tf_t(z)}{f_t(z)}\\Big)|f_t(z)|^2=2 {\\rm Re}\\Big(\\frac{v(f_t(z))}{f_t(z)}\\Big)|f_t(z)|^2 \\leq -2c |f_t(z)|^2\n\\end{equation*} \nand therefore the flow is defined for all $t \\geq 0$ and one has $|f_t(z)| \\leq e^{-ct}|z|$.\nNext we consider the case where there exists some $a \\in \\mathbb{D}$ such that $v(a)=0$. The curve $(x(t),y(t))=({\\rm Re}(f_t(z)),{\\rm Im}(f_t(z))$ is the integral curve of the vector field $V={\\rm Re}(v)\\partial_x+{\\rm Im}(v)\\partial_y$ and $V$ is pointing inside $\\mathbb{D}$ at $\\mathbb{T}=\\partial\\mathbb{D}$, thus $f_t(z)$ is defined for all $t\\geq 0$.\nWe then introduce the M\\\"obius map $\\psi(z)= \\frac{z-a}{1-\\bar a z}$ preserving $\\mathbb{D}$ and such that $\\psi(a)=0$ and consider the vector field $v_\\psi= (\\psi' \\circ \\psi^{-1}) v \\circ \\psi^{-1} $ which satisfies $v_\\psi(0)=0$ and \n\\begin{equation*}\n\\sup_{|z| = 1} \\Re\\left( \\bar{z} v_\\psi(z) \\right) = \\sup_{|z| = 1} \\Re\\left( \\overline{\\psi (z)} \\psi'(z) v(z) \\right)= \\sup_{|z| = 1} \n\\Re\\left( \\bar{ z} v(z) \\right) \\frac{1-|a|^2}{|1-a\\bar{z}|^2}< 0\n\\end{equation*} \nby noticing that $ \\overline{\\psi (z)} \\psi'(z)= \\bar{z}\\frac{1-|a|^2}{|1-\\bar{z}a|^2}$ when $|z| = 1$. The vector field $v_\\psi$ has $ \\psi \\circ f_t \\circ \\psi^{-1}$ as flow. This concludes the proof.\n\\end{proof}\n\n\n\n\n\\begin{lemma}\\label{limith}\nAssume that $v\\in {\\rm Vect}_+(\\mathbb{D})$, that $v(0)=0$, $v'(0)=-\\omega$ for $\\omega>0$ and that $v$ admits a holomorphic extension in a neighborhood of $\\mathbb{D}$. \nFor each $k$, the function $e^{\\omega t}f_t$ converges in $C^k(\\mathbb{D})$ norm, as $t\\to \\infty$, towards a function $h$ which is holomorphic near $\\mathbb{D}$ and univalent on $\\mathbb{D}$.\nFinally, $f'_t(0)=e^{-\\omega t}$ and $h'(0)=1$.\n \\end{lemma}\n\\begin{proof}\nLet $U$ be the neighborhood of $\\mathbb{D}$ so that $v$ is holomorphic. We can assume that $v$ has no other zeros than $z=0$, by choosing $U$ appropriately.\nSince $v(z)=-\\omega z+\\mathcal{O}(|z|^2)$ at $z=0$, the linearisation of the ODE satisfied by $f_t$ at the fixed point $z=0$ is given by \n$\\partial_t f_t(z)=-\\omega f_t(z)$. Therefore, by standard ODE arguments we obtain a bound $|f_t(z)|\\leq Ce^{-\\omega t}|z|$ \nfor some $C>0$ uniform and that $g_t(z)=e^{\\omega t}f_t(z)$ satisfies, uniformly in $z\\in \\mathbb{D}$,\n\\[ \\partial_t g_t(z)=e^{\\omega t}(f_t(z)+\\omega v(f_t(z)))=\\mathcal{O}(e^{\\omega t}|f_t(z)|^2)=\\mathcal{O}(e^{-\\omega t}).\\]\nThus, $t\\to +\\infty$, there is some function $h: U \\to \\mathbb{C}$ so that \n\\[ e^{\\omega t}f_t(z)=g_t(z)= z+\\int_{0}^{t} \\partial_s g_s(z)ds \\to_{t\\to \\infty} h(z). \\]\nThe convergence is uniform on compact sets of $U $and the function $h$ is then holomorphic in $U$ \nas uniform limit of holomorphic functions, moreover $g_t\\to h$ in all $C^k(\\mathbb{D})$ norms \n(using Cauchy formula, the derivatives are dealt with using the $C^0$ convergence). The fact that $h$ is univalent on $\\mathbb{D}$ follows from Carath\\'eodory's kernel theorem, since $g_t$ is univalent on $\\mathbb{D}$ and converges uniformly on compact sets of $\\mathbb{D}$.\nExpanding $f_t(z)=\\sum_{n\\geq 1}a_n(t)z^n$ at $z=0$, we see that $a_n(t)$ must solve the ODE $\\partial_t a_1(t)=-\\omega a_1(t)$ with $a_1(0)=1$, \nthus $a_1(t)=e^{-\\omega t}$ and this implies that $h'(0)=1$. \n\\end{proof}\n\n\n\nWe end with the following lemma which will be useful in the sequel:\n\n\\begin{lemma}\\label{limitandh}\nAssume that $v\\in {\\rm Vect}_+(\\mathbb{D})$, that $v(0)=0$, $v'(0)=-\\omega$ and that $v$ admits a holomorphic extension in a neighborhood of $\\mathbb{D}$. Then for all $t\\geq 0$, $v(f_t(z))=f_t'(z)v(z)$ and the function $h$ of Lemma \\ref{limith} solves the differential equation $\\omega h(z)= -h'(z) v(z)$.\n\\end{lemma}\n\\begin{proof}\nSince $f_t(z)$ is one-to-one, $f_t'(z)$ never vanishes in $\\mathbb{D}$. We then have \n\\begin{equation*}\n\\partial_t \\frac{v(f_t(z))}{f_t'(z)}= \\frac{ \\partial_t f_t(z) v'( f_t(z) )}{f_t'(z)}- \\frac{ ( \\partial_t f_t )'(z) v( f_t(z) )}{(f_t'(z))^2} = \\frac{ v( f_t(z) ) v'( f_t(z) )}{f_t'(z)}-\\frac{ f_t'(z) v'( f_t(z)) v( f_t(z) )}{(f_t'(z))^2}=0.\n\\end{equation*}\nThis shows that $v(f_t(z))=f_t'(z)v(z)$ and thus as $t\\to +\\infty$ \n\\[\ne^{\\omega t}f'_t(z)v(z)= e^{\\omega t}v(f_t(z))=-e^{\\omega t}f_t(z)+\\mathcal{O}(e^{-\\omega t})\n\\]\nand by letting $t\\to +\\infty$, we get $h'(z)v(z)=-\\omega h(z)$.\n\\end{proof}\n\n\n\n\\subsection{The probabilistic formula for the vector field dynamics}\n\n\n\nLet $\\mathbf{v}=v(z)\\partial_z$ be a Markovian vector field, and we assume that $v(0)=0$ and $v'(0)=-\\omega$ for $\\omega>0$. This vector field induces a family of small conformal transformations $f_t(z)=z+tv(z)+o(t)$ on $\\mathbb{D}$ such that $f_t(\\mathbb{T})\\subset\\mathbb{D}$ for all $t$ and $f'_t(0)=e^{-\\omega t}$. Given an initial condition $\\varphi \\in H^{-s}(\\mathbb{T})$ with $s>0$, we define a stochastic process $(B_t,\\varphi_t)\\in H^{-s}(\\mathbb{T})$ via the following formula\n\\begin{equation}\\label{eq:process}\n(B_t,\\varphi_t):=\\left((P\\varphi+X_\\mathbb{D})\\circ f_t+Q\\log \\frac{|f_t'|}{|f_t|}\\right)\\Big|_{\\mathbb{T}}.\n\\end{equation}\nwhere $|_{\\mathbb{T}}$ denotes restriction to the unit circle and $(B_t,\\varphi_t)$ is the standard decomposition into the average and the average zero part. That is, we look at the trace of the field $P\\varphi+X_\\mathbb{D}$ on the deformed circle $f_t(\\mathbb{T})$ and pull this field back to $\\mathbb{T}$ using $f_t$. The extra term $Q\\log \\frac{|f_t'|}{|f_t|}$ is here to conform with the transformation properties of the Liouville field. \nNotice that $X_{\\mathbb{D}}\\circ f_t=\\sum_{n}X_n(t)e^{in\\theta}$ where \n\\begin{equation}\\label{X_n(t)}\nX_n(t)=\\frac{1}{2\\pi}\\int_0^{2\\pi} X_{\\mathbb{D}}(f_t(e^{i\\theta}))e^{-in\\theta}d\\theta\n\\end{equation} \n\\begin{equation}\\label{covarianceXn} \n\\mathbb{E}[ X_n(t)X_{m}(t')]= \\frac{1}{4\\pi^2}\\int_0^{2\\pi}\\int_0^{2\\pi} e^{-in\\theta-im\\theta'}\\log \\frac{| 1-f_t(e^{i\\theta})\\overline{f_{t'}(e^{i\\theta'})}|}{|f_t(e^{i\\theta})-f_{t'}(e^{i\\theta'})|}d\\theta d\\theta'.\n\\end{equation}\nWe can now register a few formulas: for all $x \\in \\mathbb{D} \\setminus \\mathbb{D}_t$\n\\begin{equation*}\n\\frac{1}{2 \\pi} \\int_0^{2 \\pi} G_{\\mathbb{D}} (x,f_t(e^{i \\theta'}) ) d \\theta'= \\log \\frac{| 1- f_t(0) \\overline{x})|}{|f_t(0)-x|}= \\log \\frac{1}{|x|}. \n\\end{equation*}\nThis formula is just the average principle applied to the harmonic function $z \\mapsto G_{\\mathbb{D}} (x,f_t(z) )$ for $z \\in \\mathbb{D}$. Taking $x \\to f_t(e^{i \\theta})$ yields by continuity for all $\\theta$\n\\begin{equation*}\n\\frac{1}{2 \\pi} \\int_0^{2 \\pi} G_{\\mathbb{D}} (f_t(e^{i \\theta}),f_t(e^{i \\theta'}) ) d \\theta'= \\log \\frac{ 1 }{|f_t(e^{i \\theta}) |}. \n\\end{equation*}\nNow using the average principle applied to the harmonic function $z \\mapsto \\log \\frac {|f_t(z) |}{|z|} $ for $z \\in \\mathbb{D}$, we get\n\\begin{equation*}\n\\frac{1}{2 \\pi} \\frac{1}{2 \\pi} \\int_0^{2 \\pi} \\int_0^{2 \\pi} G_{\\mathbb{D}} (f_t(e^{i \\theta}),f_t(e^{i \\nu}) ) d \\theta d \\nu= - \\frac{1}{2 \\pi} \\int_0^{2 \\pi} \\log \\frac {|f_t(e^{i \\theta}) |} { |e^{i \\theta}| } d \\theta = -\\log |f_t'(0)|\n\\end{equation*}\nSimilar considerations yield for $s \\leq t$\n\\begin{equation*}\n\\frac{1}{2 \\pi} \\frac{1}{2 \\pi} \\int_0^{2 \\pi} \\int_0^{2 \\pi} G_{\\mathbb{D}} (f_t(e^{i \\theta}),f_s(e^{i \\nu}) ) d \\theta d \\nu= - \\frac{1}{2 \\pi} \\int_0^{2 \\pi} \\log \\frac {|f_s(e^{i \\theta}) |} { |e^{i \\theta}| } d \\theta = -\\log |f_s'(0)|\n\\end{equation*}\nIn particular this shows that $\\mathbb{E}[ X_0(s) X_0(t) ]= \\omega \\min( s,t) $ and hence $X_0(t)=B_t$ is a Brownian motion with diffusivity $\\omega$. \n\n\\begin{proposition}\\label{continuousprocess}\nLet $\\mathbf{v}=v(z)\\partial_z$ with $v(0)=0$ and $v'(0)=-\\omega$, and let $f_t$ be the associated flow. Then the field $(B_t,\\varphi_t)$ is a continuous process with values in $H^{-s}(\\mathbb{T})$ for $s>0$ and the operator $P_t^0$ associated to this process, defined for all bounded and continuous $F$ by\n\\begin{equation}\\label{definitionPt0}\n P_t^0F(c,\\varphi):= |f_t'(0)|^{\\frac{Q^2}{2}} \\mathbb{E}_\\varphi[ F( c+B_t+\\varphi_t )] \n\\end{equation} \nis a Markov semi-group.\n\\end{proposition}\n\\begin{proof}\nFirst, the field $\\left(P\\varphi \\circ f_t+Q\\log \\frac{|f_t'|}{|f_t|}\\right)|_{\\mathbb{T}}$ is clearly continuous in $H^{-s}(\\mathbb{T})$. The property of the \nsecond part of the process is a consequence of the following lemma\n\\begin{lemma}\nLet $s>0$. There exist $\\alpha>0$ and $C>0$ such that for all $t,t'\\geq 0$\n\\[\n\\mathbb{E}[ \\| (X_\\mathbb{D} \\circ f_t)|_{\\mathbb{T}} - (X_\\mathbb{D} \\circ f_{t'})|_{\\mathbb{T}} \\|_{H^{-s}(\\mathbb{T})}^2 ] \\leq C |t-t'|^\\alpha\n\\]\n\\end{lemma}\n\\proof \nWe will show the result for the stochastic part. We have writing $\\langle n\\rangle=(1+|n|)$\n\\begin{align*}\n& \\mathbb{E}\\Big[ \\sum_{n \\in \\mathbb{Z}} \\langle n\\rangle^{-2s} \\Big| \\int_0^{2 \\pi } (X_\\mathbb{D} \\circ f_t) (e^{i \\theta}) e^{-i n \\theta} d \\theta - \\int_0^{2 \\pi } (X_\\mathbb{D} \\circ f_{t'}) (e^{i \\theta}) e^{-i n \\theta} d \\theta \\Big| ^2 \\Big] \\\\\n& = \\sum_{n \\in \\mathbb{Z}} \\frac{\\langle n\\rangle^{-2s}}{2 \\pi} \\int_{[0,2\\pi]^2} \\left ( G_\\mathbb{D} (f_t (e^{i \\theta}), f_t (e^{i \\theta'}))+G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) -2 G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) \\right ) e^{in (\\theta-\\theta')} d\\theta d \\theta' \\\\\n&= \\int_{[0,2\\pi]^2} \\left ( G_\\mathbb{D} (f_t (e^{i \\theta}), f_t (e^{i \\theta'}))+G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) -2 G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) \\right ) \\left ( \\sum_{n \\in \\mathbb{Z}} \\frac{\\langle n\\rangle^{-2s}}{2\\pi} e^{in (\\theta-\\theta')} \\right ) d\\theta d \\theta' \\\\\n& \\leq C \\int_0^{2 \\pi } \\int_0^{2 \\pi } | G_\\mathbb{D} (f_t (e^{i \\theta}), f_t (e^{i \\theta'}))+G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) -2 G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) | \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\\n& \\leq 2C \\int_0^{2 \\pi } \\int_0^{2 \\pi } | G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) - G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) | \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\\n& \\quad + 2C \\int_0^{2 \\pi } \\int_0^{2 \\pi } | G_\\mathbb{D} (f_t (e^{i \\theta}), f_t (e^{i \\theta'}))- G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) | \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta', \n\\end{align*}\nwhere we used the bound $ \\sum_{n \\in \\mathbb{Z}} \\langle n\\rangle^{-2s} e^{in (\\theta-\\theta')} \\leq C|\\theta-\\theta'|^{-1+2s}$. We only deal with the first term since and the second is similar. Now we have\n\\begin{align*}\n & \\int_0^{2 \\pi } \\int_0^{2 \\pi } \\Big| G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) - G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'}(e^{i \\theta'})) \\Big| \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\\n&= \\int_{|t-t'| \\leq |\\theta-\\theta'|^2} \\Big| G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) - G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) \\Big | \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\\n& \\quad +\\int_{|t-t'| > |\\theta-\\theta'|^2} \\Big | G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) - G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) \\Big| \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta'.\n\\end{align*}\nBut, using the explicit formula $G_{\\mathbb{D}}(z,z')=\\log \\frac{|z-z'|}{|1-\\bar{z}z'|}$, the bounds \n\\[|f_t (e^{i \\theta})-f_{t'} (e^{i \\theta'})|\\geq C^{-1}(|t-t'|+|\\theta-\\theta'|),\\quad |1-\\overline{f_t (e^{i \\theta})}f_{t'} (e^{i \\theta'})|\\geq c(|t-t'|+|\\theta-\\theta'|)\n\\] \nfor some $c>0$ locally uniform in $t',t$, there is $C>0$ locally uniform in $t,t'$ so that\n\\begin{align*}\n& \\int_{|\\theta-\\theta'|^2 < |t-t'|} \\Big| G_\\mathbb{D} (f_{t'} (e^{i \\theta}), f_{t'} (e^{i \\theta'})) - G_\\mathbb{D} (f_t (e^{i \\theta}), f_{t'} (e^{i \\theta'})) \\Big| \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\\n& \\leq C \\int_{|\\theta-\\theta'|^2 < |t-t'|} |\\ln (|\\theta-\\theta'|)| \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\leq C |t-t'|^{s} \n\\end{align*}\nSimilarly, one gets\n\\begin{align*}\n& \\int_{|t-t'| \\leq |\\theta-\\theta'|^2} \\Big| \\ln \\Big|\\frac{1-f_{t'}(e^{i \\theta}) \\overline{ f_t(e^{i \\theta'}) } }{ 1- f_t(e^{i \\theta}) \\overline{ f_t(e^{i \\theta'}) } }\\Big| \\Big| \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\ \n& = \\int_{|t-t'| \\leq |\\theta-\\theta'|^2} \\Big | \\ln\\Big |1+ \\frac{ ( f_t(e^{i \\theta})-f_{t'}(e^{i \\theta}) ) \\overline{ f_t(e^{i \\theta'}) } }{ 1- f_t(e^{i \\theta}) \\overline{ f_t(e^{i \\theta'}) } }\\Big| \\Big | \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\\n& \\leq \\int_{|t-t'| \\leq |\\theta-\\theta'|^2} \\Big| \\frac { ( f_t(e^{i \\theta})-f_{t'}(e^{i \\theta}) ) \\overline{ f_t(e^{i \\theta'}) } }{ 1- f_t(e^{i \\theta}) \\overline{ f_t(e^{i \\theta'}) } } \\Big | \\frac{1}{|\\theta-\\theta'|^{1-2s}} d\\theta d \\theta' \\\\\n& \\leq C \\int_{|t-t'| \\leq |\\theta-\\theta'|^2} |t-t'| \\frac{1}{|\\theta-\\theta'|^{2-2s}} d\\theta d \\theta' \\leq C |t-t'|^{s}\n\\end{align*}\nThe term in $\\ln | \\frac{f_t(e^{i \\theta}) -f_{t'}(e^{i \\theta'}) }{f_t(e^{i \\theta}) -f_t(e^{i \\theta'})} |$ can be dealt with similarly.\\qed\n\nOn Gaussian spaces, the $L^p$ norms are all equivalent on polynomials of fixed bounded degree and therefore for all $p \\geq 1$ there exists some $C>0$ such that for $t,t'\\geq 0$ \n\\begin{equation*}\n\\mathbb{E}[ \\| (X_\\mathbb{D} \\circ f_t)|_{\\mathbb{T}} - (X_\\mathbb{D} \\circ f_{t'})|_{\\mathbb{T}} \\|_{H^{-s}(\\mathbb{T})}^p ] \\leq C |t-t'|^{p\\frac{\\alpha}{2}}\n\\end{equation*}\nhence by choosing $p \\alpha >2$ the Kolmogorov continuity theorem ensures that the process $(\\varphi_t)_{t \\geq 0}$ admits a continuous \nversion in $H^{-s}(\\mathbb{T})$ for $s>0$. The operator thus defines a Markovian semi-group.\n\\end{proof}\n\n\nFirst, we identify the kernel $-\\log |h(e^{i \\theta})-h(e^{i \\theta'})|$ with the resolvant of a jump operator across the curve $h(\\mathbb{T})$ and deduce the existence of a Gaussian field with this covariance.\n\\begin{lemma}\\label{firstlemma1}\nThere exists a random variable $X_h\\in H^{-s}(\\mathbb{T})$, with vanishing mean and covariance\n\\[\n\\mathbb{E}[ X_h(e^{i \\theta}) X_h(e^{i \\theta'}) ] = \\log \\frac{1}{|h(e^{i \\theta})-h(e^{i \\theta'}) |}.\n\\]\n\\end{lemma}\n\\begin{proof}\nIt suffices to show that for all real non zero $f\\in L^2(\\mathbb{T})$ with $\\int_{0}^{2 \\pi}f(e^{i \\theta})\\text{\\rm d} \\theta=0$,\n\\begin{equation}\\label{positivecovariance}\n\\int_0^{2 \\pi} \\int_0^{2 \\pi} \\log \\frac{1}{|h(e^{i \\theta})-h(e^{i \\theta'}) |} f(e^{i\\theta}) f(e^{i\\theta'}) \\text{\\rm d} \\theta \\text{\\rm d} \\theta' >0\n\\end{equation}\nLet $K$ be the single layer operator $K:C^\\infty(\\mathbb{T})\\to C^0(\\mathbb{R}^2)$\n\\[ Kf(x)=\\int_{\\mathbb{T}} G(x,h(e^{i\\theta}))f(e^{i\\theta}) \\text{\\rm d} \\theta \\]\nwith $G(x,x'):=(2\\pi)^{-1} \\log \\frac{1}{|x-x'|}$ the Green's function on $\\mathbb{R}^2$. We have $\\Delta Kf=0$ in $\\mathbb{R}^2\\setminus h(\\mathbb{T})$ and \n$((\\partial_{\\nu}^+-\\partial_{\\nu}^-)Kf)(h(e^{i\\theta}))=\\frac{f(e^{i\\theta})}{ |h'(e^{i\\theta})| }$ where $\\partial_{\\nu}^+$ is the outward unit normal derivative in $h(\\mathbb{D})$ and $\\partial_{\\nu}^-$ the outward unit normal derivative in $\\mathbb{R}^2\\setminus h(\\mathbb{D})$. Thus $Kf$ solves a Neumann problem \n\\begin{equation}\\label{Neumannprb}\n \\Delta Kf=0 \\textrm{ in } \\mathbb{R}^2 \\setminus h(\\mathbb{T}) ,\\quad (\\partial_{\\nu}^+-\\partial_{\\nu}^-)Kf |_{h(\\mathbb{T})} =\\frac{f\\circ h^{-1}}{|h'|\\circ h^{-1}},\n \\end{equation}\nmoreover we see that \n\\[ Kf(x)=-(2\\pi)^{-1}\\log(|x|) \\int_{0}^{2 \\pi} f(e^{i\\theta}) \\text{\\rm d} \\theta + \\mathcal{O}(1\/|x|) \\textrm{ as }|x|\\to \\infty\\]\nso $Kf(x)\\to 0$ as $|x|\\to \\infty$ if $\\int_{0}^{2 \\pi} f(e^{i\\theta}) \\text{\\rm d} \\theta=0$. It is also direct to check that $|\\nabla Kf(z)|=\\mathcal{O}(1\/|z|^2)$ for large $|z|$ so that we can write using Green's formula and \\eqref{Neumannprb}\n\\[ \\int_{\\mathbb{R}^2} |\\nabla Kf(x)|^2 \\text{\\rm d} x=\\int_{\\mathbb{R}^2\\setminus h(\\mathbb{D})}|\\nabla Kf(x)|^2 \\text{\\rm d} x+\\int_{h(\\mathbb{D})}|\\nabla Kf(x)|^2\\text{\\rm d} x= \\int_{0}^{2\\pi}\\int_{0}^{2\\pi} f(e^{i\\theta})(Kf)(h(e^{i\\theta}))\\text{\\rm d} \\theta \\]\nwhich is nothing more than the left hand side of \\eqref{positivecovariance}. \n\\end{proof}\n\nLet $\\P_h$ be the probability measure induced by $X_h-Q \\log |v||_{\\mathbb{T}}$ on $H^{-s}(\\mathbb{T})$ for $s>0$\n and denote \n \\begin{equation}\\label{defmuh}\n \\mu_h:= \\text{\\rm d} c \\otimes \\P_h\n \\end{equation} \n the product probability measure on $H^{-s}(\\mathbb{T})$.\n\nIf ${\\bf v}=v(z)\\partial_z$ with $v(z)=-\\omega z-\\sum_{n=1}^\\infty v_nz^{n+1}$, we define the operator $\\mathbf{H}_\\mathbf{v}^0$ via the following formula: \n\\begin{equation}\\label{eq:def_L}\n \\forall F\\in\\mathcal{C}_\\mathrm{exp},\\quad \\mathbf{H}_\\mathbf{v}^0 F=\\omega \\mathbf{H}^0 F + \\sum_{n=1}^{\\infty} \\Re (v_n) (\\mathbf{L}_n^0+\\widetilde{\\mathbf{L}}_n^0 ) F + i \\sum_{n=1}^{\\infty} \\Im (v_n) (\\mathbf{L}_n^0-\\widetilde{\\mathbf{L}}_n^0)F.\n\\end{equation}\nOne should notice that in the two sums $\\sum_{n = 1}^{\\infty}$ only a finite number of terms are not equal to $0$ because $F \\in \\mathcal{C}_\\mathrm{exp}$ and therefore the above quantity is well defined. In particular, $\\mathbf{H}^0:=\\mathbf{H}_{\\mathbf{v}_0}^0$ (recall $\\mathbf{v}_0=-z\\partial_z$) is the free field Hamiltonian whose expression is \\eqref{defH0}.\n\n\nThe first main result of this section is the following theorem which establishes the link between $\\mathbf{H}_\\mathbf{v}^0$ and $P_t^0$:\n\n\n\\begin{theorem}\\label{theoremfreefield}\n Let $\\mathbf{v}=v(z)\\partial_z$ be a Markovian vector field such that $v(0)=0$, $v'(0)=-\\omega$ and $v$ admits a holomorphic extension in a neighborhood of $\\mathbb{D}$. There exists an absolute constant $K>0$ such that if $\\omega> K \\sum_{n \\geq 1}|v_n|n^2 $ then the operator $\\mathbf{H}_\\mathbf{v}^0$ defined in \\eqref{eq:def_L} admits a closed extension such that $e^{-t \\mathbf{H}_\\mathbf{v}^0 }$ is a continuous contraction semigroup on $\\mathcal{H}=L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})$. The semigroup $P_t^0$ admits $\\mu_h$ of \\eqref{defmuh} as invariant measure, this measure is absolutely continuous with respect to $\\mu_0$ and $P_t^0$ coincides with $e^{-t \\mathbf{H}_\\mathbf{v}^0 }$ on $L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})$. \n \\end{theorem}\n\n\n\nWe will first prove the following intermediate lemma:\n\n\\begin{lemma}\nThe semigroup $P_t^0$ defined by \\eqref{definitionPt0} admits $\\mu_h$ as an invariant measure and the measure $\\mu_h$ is absolutely continuous with respect to $\\mu_0$.\n\\end{lemma}\n\n\n\\proof\n\nWe first want to apply Example 3.8.15 in \\cite{boga} to show convergence in law of $\\varphi_t$ towards $X_h-Q \\log |v||_{\\mathbb{T}}$.\nLet $s>0$. For all $g(\\theta)= \\sum_{k \\in \\mathbb{Z}^\\ast} z_k e^{ik \\theta} \\in H^{-s}(\\mathbb{T})$ and $g'(\\theta)= \\sum_{k \\in \\mathbb{Z}^\\ast} z_k' e^{ik \\theta}\\in H^{-s}(\\mathbb{T})$, we have\n\\[\n\\mathbb{E}\\Big[ \\langle g , (X_\\mathbb{D} \\circ f_t)|_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})} \\langle g' , (X_\\mathbb{D} \\circ f_t)|_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})} \\Big] = \\frac{1}{(2 \\pi)^2}\\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\tilde{g}(\\theta) \\tilde{g}'(\\theta') \\log \\frac{| 1-f_t(e^{i\\theta})\\overline{f_t(e^{i\\theta'})}|}{|f_t(e^{i \\theta})-f_t(e^{i \\theta'}) |} d \\theta d\\theta'\n\\]\nwhere $\\tilde{g}(\\theta)= \\sum_{k \\in \\mathbb{Z}^\\ast} (1+|k|)^{-2s} z_k e^{ik \\theta} $ and $\\tilde{g}'(\\theta')= \\sum_{k \\in \\mathbb{Z}^\\ast} (1+|k|)^{-2s} z_k' e^{ik \\theta'}$. Since $g,g'$ have average $0$,\n\\begin{equation*}\n\\mathbb{E}\\Big[ \\langle g , (X_\\mathbb{D} \\circ f_t)|_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})} \\langle g' , (X_\\mathbb{D} \\circ f_t)|_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})}\\Big ] = \\frac{1}{(2 \\pi)^2}\\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\tilde{g}(\\theta) \\tilde{g}'(\\theta') \\log \\frac{| 1-f_t(e^{i\\theta})\\overline{f_t(e^{i\\theta'})}|}{|e^{t} f_t(e^{i \\theta})-e^{t}f_t(e^{i \\theta'}) |} d \\theta d\\theta'.\n\\end{equation*}\nWe have the following convergence (recall $f_t(z)\\to 0$ as $t\\to \\infty$ uniformly in $z$)\n\\begin{equation}\\label{HSconvergence}\n\\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\Big| \\log \\frac{| 1-f_t(e^{i\\theta})\\overline{f_t(e^{i\\theta'})}|}{|e^{t} f_t(e^{i \\theta})-e^{t}f_t(e^{i \\theta'}) |} - \\log \\frac{1}{|h(e^{i \\theta})-h(e^{i \\theta'}) |} \\Big|^2 \\text{\\rm d} \\theta \\text{\\rm d} \\theta' \\underset{t \\to \\infty}{ \\rightarrow} 0 .\n\\end{equation}\nNow, since $ \\| \\tilde{g} \\|^2_{L^2(\\mathbb{T})} \\leq \\| g \\|_{H^{-2s}(\\mathbb{T})}^2\\leq \\| g \\|_{H^{-s}(\\mathbb{T})}^2$, we get from the convergence \\eqref{HSconvergence} that \n\\begin{align*}\n& \\sup_{ \\| g \\|_{H^{s}(\\mathbb{T})} \\leq 1 } \\big | \\mathbb{E}[ \\langle g , (X_\\mathbb{D} \\circ f_t)|_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})}^2 ] - \\mathbb{E}[ \\langle g , X_h \\rangle_{H^{-s}(\\mathbb{T})}^2 ] \\big| \\\\ \n& \\leq \\sup_{ \\| \\tilde{g} \\|_{L^2(\\mathbb{T})}^2 \\leq 1 } \\Big| \\frac{1}{(2 \\pi)^2}\\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\tilde{g}(\\theta) \\tilde{g}(\\theta') \\Big ( \\log \\frac{| 1-f_t(e^{i\\theta})\\overline{f_{t}(e^{i\\theta'})}|}{|e^{t} f_t(e^{i \\theta})-e^{t}f_t(e^{i \\theta'}) |}- \\log \\frac{1}{|h(e^{i \\theta})-h(e^{i \\theta'}) |} \\Big ) d \\theta d\\theta' \\Big| \\underset{t \\to \\infty}{ \\rightarrow} 0 .\n\\end{align*}\nThis establishes (ii) of Example 3.8.15 in \\cite{boga}. \nWe also have that\n\\begin{align*}\n \\mathbb{E}[ \\| (X_\\mathbb{D} \\circ f_t) |_{\\mathbb{T}} \\|_{H^{-s}(\\mathbb{T})}^2 ] & = \\frac{1}{(2 \\pi)^2} \\int_{0}^{2 \\pi} \\int_0^{2 \\pi} G_{\\mathbb{D}} ( f_t(e^{i \\theta}), f_t(e^{i \\theta'}) ) \\Big ( \\sum_{n \\in \\mathbb{Z}} \\frac{\\langle n\\rangle^{-2s}}{2\\pi} e^{in (\\theta-\\theta')} \\Big ) \\text{\\rm d} \\theta \\text{\\rm d} \\theta' \\\\\n& =\\frac{1}{(2 \\pi)^2} \\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\log |1- f_t(e^{i \\theta}) \\overline{f_t(e^{i \\theta'})} | \\Big ( \\sum_{n \\in \\mathbb{Z}} \\frac{\\langle n\\rangle^{-2s}}{2\\pi} e^{in (\\theta-\\theta')} \\Big ) \\text{\\rm d} \\theta \\text{\\rm d} \\theta' \\\\\n& + \\frac{1}{(2 \\pi)^2} \\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\log \\frac{1}{|e^{t} f_t(e^{i \\theta})-e^{t}f_t(e^{i \\theta'}) |} \\Big ( \\sum_{n \\in \\mathbb{Z}} \\frac{\\langle n\\rangle^{-2s}}{2\\pi} e^{in (\\theta-\\theta')} \\Big ) \\text{\\rm d} \\theta \\text{\\rm d} \\theta'.\n\\end{align*}\nThe first term is bounded by $C e^{-2 \\alpha t} \\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\left ( \\sum_{n \\in \\mathbb{Z}} \\frac{\\langle n\\rangle^{-2s}}{2\\pi} e^{in (\\theta-\\theta')} \\right ) \\text{\\rm d} \\theta \\text{\\rm d} \\theta' $ hence converges to $0$ as $t$ goes to infinity and the second term converges to $ \\frac{1}{(2 \\pi)^2} \\int_{0}^{2 \\pi} \\int_0^{2 \\pi} \\log \\frac{1}{|h(e^{i \\theta})-h(e^{i \\theta'}) |} \\left ( \\sum_{n \\in \\mathbb{Z}} \\frac{\\langle n\\rangle^{-2s}}{2\\pi} e^{in (\\theta-\\theta')} \\right ) \\text{\\rm d} \\theta \\text{\\rm d} \\theta' $. This establishes (iii) of Example 3.8.15 in \\cite{boga}.\n\nFinally, by the proof of Lemma \\ref{limitandh} and the fact that $P\\varphi(0)=0$, the sequence $(P\\varphi \\circ f_t+Q \\log \\frac{|f'_t|}{|f_t|}) |_{\\mathbb{T}}$ converges in $H^{-s}(\\mathbb{T})$ towards $-Q \\log |v||_{\\mathbb{T}}$ as $t$ goes to infinity; this establishes (i) of Example 3.8.15 in \\cite{boga}. Hence we have established convergence of $\\varphi_t$ towards $X_h-Q \\log |v||_{\\mathbb{T}}$. \n\nNow we turn to the convergence of the couple $(B_t=X_0(t), \\varphi_t)$. We denote by $\\varphi_{n,t}$ the Fourier modes of $\\varphi_t$ and $X_n(t)$ the random part defined in \\eqref{X_n(t)}.\n\nConsider a continuous function $(u,\\varphi) \\to F(u,\\varphi)$ such that there exists a function $G(u)=\\mathcal{O}(\\langle u\\rangle^{-\\infty})$ such that $|F(u,\\varphi)| \\leq G(u)$. Let us decompose \n\\begin{align*}\n& \\varphi_{n,t}= \\tilde{\\varphi}_{n,t}+ c_{n,t} B_t, \\quad \\textrm{ with }\\\\ \n& c_{n,t}:= \\frac{\\mathbb{E}[ B_t X_{n}(t) ]}{\\omega t}=\\frac{1}{\\omega t (2 \\pi)^2} \\int_{0}^{2 \\pi} \\int_{0}^{2\\pi} \\log \\frac{|1- f_t(e^{i \\theta}) \\overline{f_t(e^{i \\theta'})} | }{| e^{t}f_t(e^{i \\theta}) -e^{t}f_t(e^{i \\theta'}) |} e^{-in \\theta} \\text{\\rm d} \\theta \\text{\\rm d} \\theta'=\\mathcal{O}(t^{-1})\n\\end{align*}\nand $\\tilde{\\varphi}_{n,t}$ is independent of $B_t$. Let $c_t= \\sum_{n\\in \\mathbb{Z}^\\ast}c_{n,t}e^{in\\theta}$ \nand $\\tilde{\\varphi}_t =\\sum_{n\\in \\mathbb{Z}^\\ast }\\tilde{\\varphi}_{n,t}e^{in\\theta}$.\nWe have by conditioning on $B_t$ \n\\begin{align*}\n| \\mathbb{E}[ \\sqrt{2 \\pi \\omega t} F(c+B_t, \\varphi_t) - \\int_{\\mathbb{R}} F(u, c_t u+ \\tilde{\\varphi}_t) \\text{\\rm d} u ] | & \\leq \\mathbb{E}\\Big[ \\int_{\\mathbb{R}} | e^{-\\frac{(u-c)^2}{2 \\omega t}} -1| |F(u, c_t (u-c)+ \\tilde{\\varphi}_t)| \\text{\\rm d} u \\Big] \\\\\n& \\leq \\int_{\\mathbb{R}} | e^{-\\frac{(u-c)^2}{2 \\omega t}} -1| G(u) \\text{\\rm d} u \\underset{t \\to \\infty}{\\rightarrow} 0 .\n\\end{align*}\nNow, one can show that $\\tilde{\\varphi}_t$ converges in law towards $ X_h-Q \\log |v||_{\\mathbb{T}}$ (since $\\varphi_t-\\tilde{\\varphi}_t$ converges in probability towards $0$) and $c_t$ converges as $t$ goes to infinity towards $0$ in $H^s(\\mathbb{T})$.\nWe have\n\\begin{equation*}\n\\mathbb{E}[ \\int_{\\mathbb{R}} F(u, c_t(u-c)+ \\tilde{\\varphi}_t) \\text{\\rm d} u ]= \\int_{\\mathbb{R}} \\mathbb{E}[ F(u, c_t(u-c)+ \\tilde{\\varphi}_t) ] \\text{\\rm d} u \\underset{t \\to \\infty}{\\rightarrow} \\int_{\\mathbb{R}} \\mathbb{E}[ F(u, X_h-Q \\log |v||_{\\mathbb{T}}) ] \\text{\\rm d} u\n\\end{equation*}\nwhere we have used the weak convergence at $u$ fixed and the dominated convergence theorem.\n\nIf we fix $t_0>0$ the continuous function $(u, \\varphi) \\to P_{t_0}F(u, \\varphi)$ satisfies also that its norm is dominated by a positive \nfunction $\\tilde{G}(u)=\\mathcal{O}(\\langle u\\rangle^{-\\infty})$ hence we get that\n\\begin{equation*}\n \\sqrt{2 \\pi \\omega t} P_t(P_{t_0}F)(c,\\varphi) \\underset{t \\to \\infty}{\\rightarrow} \\int_{\\mathbb{R}} \\mathbb{E}[ (P_{t_0}F)(u, X_h-Q \\log |v||_{\\mathbb{T}}) ] \\text{\\rm d} u\n\\end{equation*} \nBy using the semigroup property, we deduce the following identity\n\\begin{equation}\\label{invariance}\n\\int_{\\mathbb{R}} \\mathbb{E}[ (P_{t_0}F)(u, X_h) ] \\text{\\rm d} u= \\int_{\\mathbb{R}} \\mathbb{E}[ F(u, X_h) ] \\text{\\rm d} u.\n\\end{equation}\nTo summarize, we have proved that \\eqref{invariance} holds for all continuous function $(u,\\varphi) \\to F(u,\\varphi)$ such that there exists a function \n$G(u)=\\mathcal{O}(\\langle u\\rangle^{-\\infty})$ for which $|F(u,\\varphi)| \\leq G(u)$. We can extend the above identity to all bounded Borelian functions by a density argument.\n\nFinally, we have to show that the measure $\\mu_h$ is absolutely continuous with respect to $\\mu_0$. In order to do so, we have to show that $\\P_h$ is absolutely continuous with respect to $\\P_\\mathbb{T}$. We have the following lemma:\n\\begin{lemma}\\label{secondlemma1}\nThere exists a constant $C>0$ and $\\rho \\in (0,1)$ such that for all $n, p \\in \\mathbb{Z}^*$\n\\begin{equation}\\label{secondlemma1eq}\n \\Big|\\int_0^{2 \\pi} \\int_0^{2 \\pi} \\log \\frac{|e^{i \\theta}- e^{i \\theta'}|}{|h(e^{i \\theta})-h(e^{i \\theta'}) |} e^{- in \\theta} e^{-ip \\theta'} \\text{\\rm d} \\theta \\text{\\rm d} \\theta' \\Big| \\leq C \\rho^{|n|+|p|}\n\\end{equation}\n\\end{lemma}\n\\begin{proof} \nThe function $h$ can be extended to a univalent function on $(1+\\delta) \\mathbb{D}$ for some $\\delta >0$. Therefore, the holomorphic function $(z,w) \\mapsto \\frac{h(z)- h(w)}{z-w}$ is equal to $\\exp( \\sum_{j,k \\geq 0} a_{j,k} z^j w^k)$ where \n $|a_{j,k}| \\leq C \\frac{1}{(1+\\delta\/2)^{j+k}}$. Relation \\eqref{secondlemma1eq} can then be seen by writing $ \\log \\frac{|h(e^{i \\theta})-h(e^{i \\theta'}) |} {|e^{i \\theta}- e^{i \\theta'}|}= {\\rm Re} \\left ( \\sum_{j,k \\geq 0} a_{j,k} e^{ij \\theta} e^{i k \\theta'} \\right )$.\n\\end{proof}\nLet $G_{\\rm Id}$ be the operator on $L^2(\\mathbb{T})$ whose integral kernel is given by the covariance $-\\log|e^{i\\theta}-e^{i\\theta'}|$ of $X_{{\\rm Id}}$: a direct computation shows that it is equal to the Fourier multiplier $G_{\\rm Id}=\\pi |D|^{-1}$ where we set $|D|^{-s}e^{in\\theta}:= |n|^{-s}e^{in\\theta}$ and $|D|^{-s}1:=0$ for $s\\in \\mathbb{R}$. Let $G_h$ be the operator on $L^2(\\mathbb{T})$ whose integral kernel is given by the covariance of $X_h$. By \nLemma \\eqref{secondlemma1}, we have \n\\[ G_{h}=G_{\\rm Id}+W\\]\nwhere $W$ is a smoothing operator, bounded as operators $H^{-N}(\\mathbb{T})\\to H^N(\\mathbb{T})$ for all $N>0$. We can then write \n$G_h=G^{1\/2}_{\\rm Id}({\\rm Id}+\\tilde{W})G^{1\/2}_{\\rm Id}$ for some $\\tilde{W}$ satisfying the same properties as $W$, and ${\\rm Id}+\\tilde{W}$ is a positive self-adjoint Fredholm operator on $H_0^{-1-2s}(\\mathbb{T})=\\{f\\in H^{-1-2s}(\\mathbb{T})\\,|\\, \\langle f,1\\rangle=0\\}$ for $s>0$ thus there is $C>0$ such that for all $f\\in H_0^{-2s}(\\mathbb{T})$\n\\[ C^{-1} \\langle G_{\\rm Id}f,f\\rangle_{L^2} \\leq \\langle G_hf,f\\rangle_{L^2} \\leq C \\langle G_{\\rm Id}f,f\\rangle_{L^2}.\\]\nSince $\\mathbb{E}[ \\langle g , X_{h} |_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})}^2 ] = \\langle G_h |D|^{-2s}g,|D|^{-2s}g\\rangle_{L^2}= \\langle G_hf,f\\rangle_{L^2}$ with $f:=|D|^{-2s}g\\in H_0^{-2s}(\\mathbb{T})$, we get\n\\begin{equation*}\nC^{-1}\\mathbb{E}[ \\langle g , X_{\\rm Id} |_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})}^2 ] \\leq \\mathbb{E}[ \\langle g , X_{h} |_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})}^2 ] \\leq C \\mathbb{E}[ \\langle g, X_{\\rm Id} |_{\\mathbb{T}} \\rangle_{H^{-s}(\\mathbb{T})}^2 ]\n\\end{equation*}\nand so $X_h$ and $X_{\\rm Id}$ have the same Cameron-Martin space. Therefore we can conclude that both fields yield equivalent probability measures by using the discussion at the bottom of p.294 in \\cite{boga}. \\qed\n\n\n\n\n\n\n\n\\noindent\n\\emph{Proof of Theorem \\ref{theoremfreefield}}. \nConsider the Hilbert space $(\\mathcal{D}(\\mathcal{Q}_0),\\mathcal{Q}_0)$ and the quadratic form \n$\\mathcal{Q}_{0}^{\\mathbf{v}}(F,F):= \\langle \\mathbf{H}_\\mathbf{v}^0 F | F \\rangle_2$ on $\\mathcal{C}_{\\rm exp}$. \n\nUsing Lemma \\ref{boundonLn}, there is $C>0$ such that for for all $F \\in \\mathcal{C}$ \n\\begin{align*}\n {\\rm Re}(\\mathcal{Q}_{0}^{\\mathbf{v}}(F,F)) & = \\omega \\langle \\mathbf{H}^0 F | F \\rangle_2 + {\\rm Re}(\\sum_{n \\geq 1} \\Re (v_n) \\langle (\\mathbf{L}_n^0+\\widetilde{\\mathbf{L}}_n^0 ) F | F \\rangle_2+ i \\sum_{n \\geq 1} \\Im (v_n) \\langle (\\mathbf{L}_n^0-\\widetilde{\\mathbf{L}}_n^0 ) F | F \\rangle_2) \\\\\n& \\geq \\omega \\mathcal{Q}_0 (F, F) - \\sum_{n \\geq 1} | \\Re (v_n) | | \\langle (\\mathbf{L}_n^0+\\widetilde{\\mathbf{L}}_n^0 ) F | F \\rangle_2 | - \\sum_{n \\geq 1} | \\Im (v_n) | | \\langle (\\mathbf{L}_n^0-\\widetilde{\\mathbf{L}}_n^0 ) F | F \\rangle_2 \\\\\n& \\geq \\omega \\mathcal{Q}_0 (F, F) - 2C\\Big( \\sum_{n \\geq 1} n^2 ( | \\Re (v_n) |+ |\\Im v_n| )\\Big ) \\mathcal{Q}_0 (F, F) \\\\\n& \\geq \\Big( \\omega - 2C \\sum_{n \\geq 1} n^2 |v_n|\\Big) \\mathcal{Q}_0 (F, F).\n\\end{align*}\nOn the other hand one also has by the same argument \n\\[ |\\mathcal{Q}_{0}^{\\mathbf{v}}(F,F)| \\leq \\Big(\\omega +2C\\sum_{n \\geq 1} n^2 |v_n|\\Big)\\mathcal{Q}_0(F,F).\\]\nChoosing $\\omega >2C \\left ( \\sum_{n \\geq 1} n^2 |v_n| \\right )+1$, we see that there is $C_0>1$ such that \n\\[ C_0^{-1}\\mathcal{Q}_0(F,F) \\leq |\\mathcal{Q}_{0}^{\\mathbf{v}}(F,F)| \\leq C_0\\mathcal{Q}_0(F,F)\\]\nwhich implies that $\\mathcal{Q}_{0}^{\\mathbf{v}}$ is a closed quadratic form, that extends to $\\mathcal{D}(\\mathcal{Q}_0)$, \nand also that there is $\\theta\\in (0,\\pi\/2)$ so that \n $|{\\rm arg}(\\mathcal{Q}_{0}^{\\mathbf{v}}(F,F)|\\leq \\theta$ for all $F\\in \\mathcal{D}(\\mathcal{Q}_0)$. This means that $\\mathcal{Q}_0^{\\mathbf{v}}$ is strictly m-accretive \\cite[Chapter VIII.6]{rs1}. By Theorem VIII.16 and the following Lemma in \\cite{rs1}, there is a unique closed operator extending ${\\bf H}^0_{\\mathbf{v}}$ defined in a dense domain $\\mathcal{D}({\\bf H}^0_{\\mathbf{v}})\\subset\\mathcal{D}(\\mathcal{Q}_0)$, with $({\\bf H}^0_{\\mathbf{v}}-\\lambda)^{-1}$ invertible if ${\\rm Re}(\\lambda)<0$, and resolvent \n bound $\\|({\\bf H}^0_{\\mathbf{v}}-\\lambda)^{-1}\\|_{\\mathcal{H}\\to \\mathcal{H}}\\leq (-{\\rm Re}(\\lambda))^{-1}$. By the Hille-Yosida theorem, ${\\bf H}^0_{\\mathbf{v}}$ is the generator of a contraction semi-group denoted $e^{-t{\\bf H}^0_{\\mathbf{v}}}$. \nNext, we notice that if $\\mathcal{H}_{\\mathbb{R}}$ is the real Hilbert space consisting of the real valued elements $F\\in \\mathcal{H}$, we can \nrestrict $\\mathcal{Q}_0^{\\mathbf{v}}$ to $\\mathcal{D}_{\\mathbb{R}}(\\mathcal{Q}_0):= \\mathcal{D}(\\mathcal{Q}_0)\\cap \\mathcal{H}_{\\mathbb{R}}$, then it is easily seen that \n $\\mathcal{Q}_{0}^{\\mathbf{v}}(F,F)>0$. In view of the discussion above, it is then a coercive closed form in the sense of \\cite[Definition 2.4]{MaRockner}.\n \nNow we show that $e^{-t{\\bf H}_{\\mathbf{v}}^0}$ and $P_t^0$ coincide on $\\mathcal{C}_{\\rm exp}$.\nWe first consider a function $F$ in $\\mathcal{C}_{\\rm exp}$ of the form $f(c, (\\varphi_n)_{n \\in [-N,N]})$. In this case, we get that for all $t$, with $B_t$ the Brownian motion,\n\\begin{equation*}\nP_t^0F(c,\\varphi)= \\mathbb{E}_{\\varphi} [ f(c+B_t, \\varphi_{-N,t},\\dots, \\varphi_{N,t}) ]\n\\end{equation*}\nwhere $\\varphi_{n,t}$ denotes the Fourier modes of $\\varphi_t$.\nFrom the previous discussions on $f_t$, we have the series representation\n\\begin{equation*}\nf_t(e^{i \\theta})= e^{-\\omega t} e^{i \\theta}+ \\sum_{j \\geq 2} \\alpha_j(t) e^{ij \\theta}\n\\end{equation*}\nfor some $\\alpha_j(t)=\\mathcal{O}(e^{- \\omega t})$ and therefore for $n \\geq 1$\n\\begin{equation}\\label{varphi_tn}\n\\begin{split}\n\\varphi_{n,t}= & \\frac{1}{2 \\pi} \\int_0^{2 \\pi} P\\varphi( f_t(e^{i \\theta}) ) e^{-i n \\theta} d \\theta + \\frac{1}{2 \\pi} \\int_0^{2 \\pi} X_{\\mathbb{D}}( f_t(e^{i \\theta}) ) e^{- i n \\theta} d \\theta \n= e^{-\\omega n t} \\varphi_n+ \\sum_{k=1}^{n-1} \\tilde{\\alpha}_k(t) \\varphi_k + X_n(t)\n\\end{split}\n\\end{equation}\nwhere \n\\begin{equation*}\n\\tilde{\\alpha}_k(t)= \\sum_{\\substack{j_1, \\dots, j_k \\geq 1\\\\ j_1+ \\cdots +j_k=n}} \\alpha_{j_1}(t) \\cdots \\alpha_{j_k}(t). \n\\end{equation*}\nA similar relation holds for negative $n$ (with $\\varphi_{-n}$ in place of $\\varphi_n$). We deduce that \nfor fixed $t$ the function $P_t^0F$ depends only on $(c,(\\varphi_n)_{|n|\\leq N})$. From \\eqref{varphi_tn} and using that $B_t$ is a Brownian motion, for all \n$k,\\beta,M$, there is $C>0$ (depending locally uniformly on $t$ and whose value changes from line to line) so that\n\\[\\begin{split} \n|\\partial_c^k \\partial_{(x,y)}^\\beta P_t^0F(c,\\varphi)| \\leq & C\\langle (x,y)\\rangle^L\\mathbb{E}_{\\varphi}[ e^{-M|c+B_t|}\\langle (X_{-N}(t),\\dots,X_N(t))\\rangle^L]\\\\\n\\leq & Ce^{-M|c|}\\langle (x,y)\\rangle^L\n\\mathbb{E}_{\\varphi}[ e^{M|B_t|}\\langle (X_{-N}(t),\\dots,X_N(t))\\rangle^L]\\\\\n\\leq & Ce^{-M|c|}\\langle (x,y)\\rangle^L\n\\end{split}\\]\nthus $P_t^0F\\in \\mathcal{C}_{\\rm exp}$, i.e. $P_t^0:\\mathcal{C}_{\\rm exp}\\to \\mathcal{C}_{\\rm exp}$. \n\n\nIn this case, we can apply Propositions \\ref{prop:first_order} and \\ref{prop:second_order} to show that the following holds in $L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T},\\mu_0)$\n\\begin{equation}\\label{itsevolutionbaby}\n\\partial_t P_t^0 F= \\partial_{s}|_{s=0} P_{s}^0 ( P_t^0F ) = - \\mathbf{H}_\\mathbf{v}^0 P_t^0F, \\quad P_{t}^0 F|_{t=0}= F\n\\end{equation}\nThis is because for fixed $t$ the function $P_t^0F$ belongs to $\\mathcal{C}_{\\rm exp}$. By using uniqueness of the solution in equation \\eqref{itsevolutionbaby}, we see that for all $t \\geq 0$ the identity $e^{-t{\\bf H}_{\\mathbf{v}}^0}F=P_t^0F$ holds everywhere. This identity can be extended to $L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})$ by a density argument.\n\n\n\n\n\n\n\n\n\n\nNow, we prove the key relation \\eqref{itsevolutionbaby}. This is the purpose of the following two Propositions:\n\n\n\n\\begin{proposition}\\label{prop:first_order}\nFor all $\\mathbf{v}\\in\\mathrm{Vect}_+(\\mathbb{D})$, let us define the operator $\\nabla_\\mathbf{v}$ on $\\mathcal{C}$ by\n\\[\\nabla_\\mathbf{v} F(\\varphi):=-\\frac{\\d}{\\d t}_{|t=0} F\\Big((P\\varphi\\circ f_t+Q\\log\\frac{|f_t'|}{|f_t|})|_{\\mathbb{T}}\\Big).\\]\nThe above quantity exists in the classical sense for all $\\varphi \\in H^{s}(\\mathbb{T})$ and the limit which defines the derivative converges in $L^2$ if $F \\in \\mathcal{C}_\\mathrm{exp}$. We have $\\nabla_\\mathbf{v}= \\sum_{n \\geq 0} v_n \\nabla_n $ where\n\n\\[\\nabla_n=\\Re\\left(Qn\\partial_n+2\\sum_{m=1}^\\infty m\\varphi_m\\partial_{n+m}\\right).\\]\n The $\\mathbb{C}$-linear and $\\mathbb{C}$-antilinear parts of $\\nabla_n$ are\n\\begin{align*}\n&\\nabla_n^{1,0}=\\frac{1}{2}Qn\\partial_n+\\sum_{m=1}^\\infty m\\varphi_m\\partial_{n+m},\\quad \\nabla_n^{0,1}=\\frac{1}{2}Qn\\partial_{-n}+\\sum_{m=1}^\\infty m\\varphi_{-m}\\partial_{-n-m}.\n\\end{align*}\n\\end{proposition}\n\n\n\n\\begin{proof}\nLet $\\varphi\\in\\mathcal{C}^\\infty(\\mathbb{T})$ (we can suppose that $\\varphi\\in\\mathcal{C}^\\infty(\\mathbb{T})$ because $F$ depends on a finite number of variables), which we write in Fourier expansion\n\\[\\varphi(\\theta)=\\sum_{n\\in\\mathbb{Z}}\\varphi_n e^{in\\theta},\\]\nwhere $\\varphi_{-n}=\\bar{\\varphi}_n$ for all $n\\in\\mathbb{Z}$.\n\nLet $\\mathbf{v}=v\\partial_z\\in\\mathrm{Vect}_+(\\mathbb{D})$ generating the infinitesimal deformation $f_t(z)=z+tv(z)+o(t)$. We have\n\\begin{align*}\nP\\varphi\\circ f_t(z)+Q\\log \\frac{|f_t'(z)|}{|f_t(z)|}\n&=P\\varphi(z+tv(z))+Q\\Re\\log(1+tv'(z))-Q \\log |z| - Q\\Re\\log\\Big(1+t \\frac{v(z)}{z}\\Big)+o(t) \\\\\n&=P\\varphi(z)+2t{\\rm Re}(v(z)\\partial_z P\\varphi(z))+tQ\\Re(v'(z))-Q \\log |z| -tQ \\Re(\\frac{v(z)}{z}) +o(t)\\\\\n&=P\\varphi(z)-Q\\log |z|+t\\Re\\Big(2v(z)\\partial_z\\varphi(z)+Qv'(z) -Q \\frac{v(z)}{z}) \\Big)+o(t).\n\\end{align*}\n\nWe can consider the case $\\mathbf{v}_n=-z^{n+1}\\partial_z$ since the general case is just a linear combination of this case. Specialising to $\\mathbf{v}_n=-z^{n+1}\\partial_z$, we obtain for all $k \\in \\mathbb{Z}$\n\\begin{equation*}\n\\begin{split}\n\\frac{1}{2 \\pi}\\int_{0}^{2 \\pi} \\left ( P\\varphi\\circ f_t(e^{i \\theta})+Q\\log \\frac{|f_t'(e^{i \\theta})|}{|f_t(e^{i \\theta})|} \\right ) e^{-ik \\theta} d \\theta = & \n\\varphi_k -t(k-n) \\varphi_{k-n}{\\bf 1}_{k \\geq n+1}- tn\\frac{Q}{2}\\delta_{|k|-n}\\\\\n& +t(k+n) \\varphi_{k+n}{\\bf 1}_{k\\leq -n-1}+o(t) \n\\end{split}\n\\end{equation*}\nFor $F\\in\\mathcal{C}_\\mathrm{exp}$, we then deduce the following limit in $L^2$\n\\[ -\\frac{\\d}{\\d t}_{|t=0} F\\Big(\\big(P\\varphi\\circ f_t+Q\\log\\frac{|f_t'|}{|f_t|}\\big)\\Big|_{\\mathbb{T}}\\Big)=\\frac{nQ}{2}(\\partial_nF+\\partial_{-n}F)+\\sum_{m\\geq 1}\n(m\\varphi_m\\partial_{n+m}+m\\varphi_{-m}\\partial_{-m-n})F.\\qedhere\\]\n\\end{proof}\n\n\\begin{proposition}\\label{prop:second_order}\nFor all $\\mathbf{v}\\in\\mathrm{Vect}_+(\\mathbb{D})$, let us define the operator $\\Delta_\\mathbf{v}$ on $\\mathcal{C}_\\mathrm{exp}$ by\n\\[\\Delta_\\mathbf{v} F(\\varphi)=-\\frac{\\d}{\\d t}_{|t=0}\\mathbb{E}_\\varphi\\left[F\\left(\\varphi+(X_\\mathbb{D}\\circ f_t)|_{\\mathbb{T}}\\right)\\right].\\]\nThe above quantity exists in the classical sense for all $\\varphi \\in H^{s}(\\mathbb{T})$ and the limit which defines the derivative converges in $L^2$ if $F \\in \\mathcal{C}$. This definition extends uniquely to all $\\mathbf{v}\\in\\mathrm{Vect}(\\mathbb{D})$, where $\\Delta_n:=\\Delta_{\\mathbf{v}_n}$ is given for all $n\\geq0$ by\n\\[\\Delta_n=-\\frac{1}{2}\\Re\\left(\\sum_{m\\in\\mathbb{Z}}\\partial_{n-m}\\partial_m\\right).\\]\nThe $\\mathbb{C}$-linear and $\\mathbb{C}$-antilinear parts of $\\Delta_n$ are\n\\begin{align*}\n&\\Delta_n^+=\\frac{1}{4}\\sum_{m\\in\\mathbb{Z}}\\partial_{n-m}\\partial_m;\\qquad\\Delta_n^-=\\frac{1}{4}\\sum_{m\\in\\mathbb{Z}}\\partial_{-n-m}\\partial_m.\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof}\nWe write\n\\[(X_\\mathbb{D}\\circ f_t)(e^{i \\theta})=\\sum_{n\\in\\mathbb{Z}}X_n(t)e^{i n \\theta},\\]\nwhere $X_n(t)$ is given by \\eqref{X_n(t)} with covariance \\eqref{covarianceXn}. \nNext, we show that the covariance is differentiable with respect to $t$ and that the differential exists at $t=0$. First, we have for $z\\neq z'\\in\\overline{\\mathbb{D}}$,\n\\begin{align*}\n\\partial_t\\log|f_t(z)-f_t(z')|\n=\\partial_t\\Re\\log(f_t(z)-f_t(z'))\n=\\Re\\frac{\\partial_tf_t(z)-\\partial_tf_t(z')}{f_t(z)-f_t(z')}\n=\\Re\\frac{v(f_t(z))-v(f_t(z'))}{f_t(z)-f_t(z')}.\n\\end{align*}\nThis term has the finite limit ${\\rm Re}(f_t'(z)v'(f_t(z)))$ as $z'\\to z$, and it follows that $\\partial_t\\log|f_t(z)-f_t(z')|$ is uniformly bounded with respect to $t$ (in compact sets) on $\\overline{\\mathbb{D}}^2$. Applying the dominated convergence theorem, we have\n\\begin{align*}\n\\partial_t\\int_0^{2\\pi}\\int_0^{2\\pi}\\log|f_t(e^{i\\theta})-f_t(e^{i\\theta'})|e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}\n&=\\int_0^{2\\pi}\\int_0^{2\\pi}\\Re\\left(\\frac{v(f_t(e^{i\\theta}))-v(f_t(e^{i\\theta'}))}{f_t(e^{i\\theta})-f_t(e^{i\\theta'})}\\right)e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}\\\\\n&\\underset{t\\to0}{\\to}\\int_0^{2\\pi}\\int_0^{2\\pi}\\Re\\left(\\frac{v(e^{i\\theta})-v(e^{i\\theta'})}{e^{i\\theta}-e^{i\\theta'}}\\right)e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}.\n\\end{align*}\n\nNow we treat the term $\\log|1-f_t(z)\\overline{f_t(z')}|$, which is the real part of a holomorphic (resp. antiholomorphic) function in $z$ (resp. $z'$) converging in the disc. The values $\\int\\int\\log(1-f_t(e^{i\\theta})\\overline{f_t(e^{i\\theta'})})e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}$ are the coefficients of the power series expansion of this function at $0$. The derivatives with respect to $t$ of these coefficients are the coefficients of the function $\\partial_t\\log(1-f_t(z)\\overline{f_t(z')})=-\\frac{v(f_t(z))\\overline{f_t(z')}+f_t(z)\\overline{v(f_t(z'))}}{1-f_t(z)\\overline{f_t(z')}}$. As $t\\to0$, this function converges uniformly on compact sets of $\\mathbb{D}^2$ to $-\\frac{v(z)\\bar{z}'+z\\bar{v}(z')}{1-z\\bar{z}'}$, which is holomorphic (resp. antiholomorphic) in $z$ (resp. $z'$). Therefore, the coefficients of the expansion converge individually. Moreover, the limiting function has a continuation to $\\overline{\\mathbb{D}}^2$ with simple poles at $z=z'\\in\\mathbb{T}$. Thus,\n\\[\\partial_t|_{t=0}\\int_{[0,2\\pi]^2}\\log|1-f_t(e^{i\\theta})\\overline{f_t(e^{i\\theta'})}|e^{-ip\\theta+iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}=-\\int_{[0,2\\pi]^2}\\Re\\left(\\frac{v(e^{i\\theta})e^{-i\\theta'}+e^{i\\theta}\\overline{v(e^{i\\theta'})}}{1-e^{i(\\theta-\\theta')}}\\right)e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2},\\]\nwhere the integral is understood in the principal value sense (the formula computes the coefficients of the power series expansion of the integrand).\n\nPutting the two contributions together, we get\n\\begin{equation}\\label{eq:correl_asymp}\n\\begin{aligned}\nh_{p,q}\n&:=\\partial_t|_{t=0}\\mathbb{E}\\left[X_p(t)X_{-q}(t)\\right]\\\\\n&=-\\int_0^{2\\pi}\\int_0^{2\\pi}\\Re\\left(\\frac{v(e^{i\\theta})-v(e^{i\\theta'})}{e^{i\\theta}-e^{i\\theta'}}+\\frac{e^{i\\theta}\\overline{v(e^{i\\theta'})}}{1-e^{i(\\theta-\\theta')}}+\\frac{e^{-i\\theta'}v(e^{i\\theta})}{1-e^{i(\\theta-\\theta')}}\\right)e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}.\n\\end{aligned}\n\\end{equation}\n\nTo get explicit formulas for these operators, we need to compute $h_{p,q}$ for $v(z)=-z-z^{n+1}$, $n\\geq0$. Since the derivative of the \n covariance this is linear in $v$, it suffices to compute each term in \\eqref{eq:correl_asymp} separately for $v(z)=-z$ and $v(z)=-z^{n+1}$. We thus do it for $v(z)=-z^{n+1}$ for $n\\geq 0$. The first term is given by\n\\begin{equation}\\label{eq:first_term}\n\\begin{aligned}\n\\int_0^{2\\pi}\\int_0^{2\\pi}\\frac{e^{i(n+1)\\theta}-e^{i(n+1)\\theta'}}{e^{i\\theta}-e^{i\\theta'}}e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}\n&=\\sum_{k=0}^n\\int_0^{2\\pi}\\int_0^{2\\pi}e^{i(k-p)\\theta}e^{i(n-k+q)\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}\\\\\n&=\\left\\lbrace\\begin{aligned}\n&1\\text{ if }0\\leq p\\leq n\\text{ and }q=p-n\\\\\n&0\\text{ otherwise}.\n\\end{aligned}\\right.\n\\end{aligned}\n\\end{equation}\nand the conjugate term by\n\\begin{equation}\\label{eq:conjugate}\n\\begin{aligned}\n\\int_0^{2\\pi}\\int_0^{2\\pi}\\frac{e^{-i(n+1)\\theta}-e^{-i(n+1)\\theta'}}{e^{-i\\theta}-e^{-i\\theta'}}e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}\n&=\\left\\lbrace\\begin{aligned}\n&1\\text{ if }-n\\leq p\\leq0\\text{ and }q=p+n\\\\\n&0\\text{ otherwise}.\n\\end{aligned}\\right.\n\\end{aligned}\n\\end{equation}\nTo compute the other terms, we look at the power series expansion of the integrand and extract the relevant coefficient. For instance, we have\n\\begin{align*}\n\\int_0^{2\\pi}\\int_0^{2\\pi}\\frac{e^{i\\theta}e^{-i(n+1)\\theta'}}{1-e^{i(\\theta-\\theta')}}e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}\n&=\\int_0^{2\\pi}\\int_0^{2\\pi}\\sum_{k=0}^{\\infty}e^{-i(p-1-k)\\theta}e^{i(q-n-1-k)\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}\\\\\n&=\\left\\lbrace\\begin{aligned}\n&1\\text{ if }p>0\\text{ and }q=p+n\\\\\n&0\\text{ otherwise}.\n\\end{aligned}\\right.\n\\end{align*}\nWe proceed similarly for the other terms and we collect all the results below:\n\\begin{equation}\\label{eq:residues}\n\\begin{aligned}\n&\\int_0^{2\\pi}\\int_0^{2\\pi}\\frac{e^{-i\\theta}e^{i(n+1)\\theta'}}{1-e^{-i(\\theta-\\theta')}}e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}=\\left\\lbrace\\begin{aligned}\n&1\\text{ if }p<0\\text{ and }q=p-n\\\\\n&0\\text{ otherwise}.\n\\end{aligned}\\right.\\\\\n&\\int_0^{2\\pi}\\int_0^{2\\pi}\\frac{e^{-i\\theta'}e^{i(n+1)\\theta}}{1-e^{i(\\theta-\\theta')}}e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}=\\left\\lbrace\\begin{aligned}\n&1\\text{ if }p>n\\text{ and }q=p-n\\\\\n&0\\text{ otherwise}.\n\\end{aligned}\\right.\\\\\n&\\int_0^{2\\pi}\\int_0^{2\\pi}\\frac{e^{i\\theta'}e^{-i(n+1)\\theta}}{1-e^{-i(\\theta-\\theta')}}e^{-ip\\theta}e^{iq\\theta'}\\frac{\\d\\theta\\d\\theta'}{4\\pi^2}=\\left\\lbrace\\begin{aligned}\n&1\\text{ if }p<-n\\text{ and }q=p+n\\\\\n&0\\text{ otherwise}.\n\\end{aligned}\\right.\n\\end{aligned}\n\\end{equation}\n\n For Markovian $\\mathbf{v}$, the operator $\\Delta_\\mathbf{v}$ is the generator of a (complex) Brownian motion with covariance matrix $(h_{p,q})_{p,q\\in\\mathbb{Z}}$, i.e.\n\\[\\Delta_\\mathbf{v}=\\frac{1}{2}\\sum_{p,q\\in\\mathbb{Z}}h_{p,q}\\partial_p\\partial_{-q}.\\]\nFor $F\\in\\mathcal{C}_\\mathrm{exp}$, this reduces to a finite sum, and we deduce the claim about the convergence to $\\Delta_\\mathbf{v} F$ in $L^2$.\n\nNow, summing up all six terms appearing in \\eqref{eq:first_term}, \\eqref{eq:conjugate} and \\eqref{eq:residues}, we obtain for $\\mathbf{v}=\\mathbf{v}_n$:\n\\begin{equation}\\label{eq:covariance_wn}\nh_{p,q}=\\frac{1}{2}(\\delta_{q,p-n}+\\delta_{q,p+n})\n\\end{equation}\nso that\n\\begin{align*}\n\\Delta_n=\\frac{1}{4}\\sum_{p,q\\in\\mathbb{Z}}(\\delta_{q,p+n}+\\delta_{q,p-n})\\partial_p\\partial_{-q}\n&=\\frac{1}{4}\\sum_{m\\in\\mathbb{Z}}\\partial_{n-m}\\partial_m+\\partial_{-n-m}\\partial_m\\\\\n&=\\frac{1}{2}\\Re\\left(\\sum_{m\\in\\mathbb{Z}}\\partial_{n-m}\\partial_m\\right).\n\\end{align*}\n\nNotice that \\eqref{eq:correl_asymp} is the real part of a complex linear map in $\\mathbf{v}$; this gives the decomposition $\\Delta_\\mathbf{v}=\\Delta_\\mathbf{v}^++\\Delta_\\mathbf{v}^-$. Retaining only the complex linear (resp. antilinear) terms in $\\mathbf{v}$, we have\n\\[h_{p,q}^+=\\frac{1}{2}\\delta_{q,p-n};\\qquad h_{p,q}^-=\\frac{1}{2}\\delta_{q,p+n},\\]\nso that\n\\[\\Delta_n^+=\\frac{1}{4}\\sum_{m\\in\\mathbb{Z}}\\partial_{n-m}\\partial_m;\\qquad\\Delta_n^-=\\frac{1}{4}\\sum_{m\\in\\mathbb{Z}}\\partial_{-n-m}\\partial_m.\\qedhere\\]\n\\end{proof}\n\n\n\n\n\n\\section{Liouville operators}\\label{subsec:operators}\n\n\t\\subsection{Definition and Feynman-Kac formula}\n\t\nIn this section we define the Liouville operators. These will be densely defined, unbounded operators on the Liouville Hilbert space $L^2(\\d c\\otimes\\P)$. The operators associated with Markovian vector fields are closable operators. They form the natural generalisation of the Liouville Hamiltonian of \\cite[Section 5]{GKRV}, and they are the basis for the definition of operators associated with non-Markovian vector fields.\n\nLet $\\mathbf{v}=v(z)\\partial_z\\in\\mathrm{Vect}_+(\\mathbb{D})$ with $v(z)=-\\sum_{n=-1}^\\infty v_nz^{n+1}$ such that $v(0)=0$, $v'(0)=-\\omega$ and satisfying the conditions of Theorem \\ref{theoremfreefield}. Recall that $\\omega> K \\sum_{n \\geq 1}|v_n|n^2 $ where $K>0$ is the constant which appears in Theorem \\ref{theoremfreefield}. We consider the positive measure $\\d\\varrho_\\mathbf{v}(\\theta)=-\\Re(e^{-i\\theta}v(e^{i\\theta}))\\d\\theta$. One can then define the Dirichlet form on $\\mathcal{C}_{\\rm exp}$ via the formula\n\\begin{equation}\\label{defform}\n\\mathcal{Q}_{\\mathbf{v}}(F,G): = \\mathcal{Q}_{\\mathbf{v} }^0(F,G)+ \\mu e^{\\gamma c} \\langle V_\\mathbf{v} F , G \\rangle ,\\quad \\textrm{with } V_\\mathbf{v}:=\\frac{1}{2\\pi}\\int_{0}^{2 \\pi} e^{\\gamma\\varphi(\\theta)}\\d\\varrho_\\mathbf{v}(\\theta).\n \\end{equation} \n When $\\gamma<\\sqrt{2}$, this potential is an $L^{p}(\\Omega_\\mathbb{T})$ function if $1\\leq p<2\/\\gamma^2$, while when $\\gamma\\in [\\sqrt{2},2)$, we define the potential term in \\eqref{defform} by using the Girsanov formula to shift the variables\nsimilarly to \\cite[Section 5]{GKRV}. Since $\\d\\varrho_\\mathbf{v}$ has a positive smooth density with respect to $\\d\\theta$, there is a constant $C>1$ such that $C^{-1}V_\\mathbf{v}\\leq V\\leq CV_\\mathbf{v}$ as bounded operators $\\mathcal{D}(\\mathcal{Q})\\to\\mathcal{D}'(\\mathcal{Q})$, namely\n \\begin{equation}\\label{fundinequality}\n C^{-1} \\mathcal{Q}(F,F) \\leq \\mathcal{Q}_{\\mathbf{v}}(F,F) \\leq C \\mathcal{Q}(F,F).\n \\end{equation} \nThe form $\\mathcal{Q}_{\\mathbf{v}}$ is a positive definite bilinear form satisfying the weak sector condition. By inequality \\eqref{fundinequality} and using the fact that $\\mathcal{Q}$ is closable, we know that $ \\mathcal{Q}_{\\mathbf{v}}$ is closable and the domain of $\\mathcal{Q}_{\\mathbf{v}}$ coincides with the domain of $\\mathcal{Q}$. Hence $\\mathcal{Q}_{\\mathbf{v}}$ is a coercive closed form: there exists a unique continuous contraction semigroup associated to $\\mathcal{Q}_{\\mathbf{v}}$, \ndenoted $e^{-t \\mathbf{H}_\\mathbf{v} }$ and its generator ${\\bf H}_{\\mathbf{v}}: \\mathcal{D}(\\mathcal{Q})\\to \\mathcal{D}'(\\mathcal{Q})$ is the operator \n\\begin{equation}\\label{defHv} \n\\begin{split}\n\\mathbf{H}_\\mathbf{v}=& \\mathbf{H}_\\mathbf{v}^0+ \\mu V_{\\bf v}= \\omega \\mathbf{H}^0 + \\sum_{n \\geq 1} \\Re (v_n) (\\mathbf{L}_n^0+\\widetilde{\\mathbf{L}}_n^0 ) + i \\sum_{n \\geq 1} \\Im (v_n) (\\mathbf{L}_n^0-\\widetilde{\\mathbf{L}}_n^0)+ \\mu e^{\\gamma c} V_{\\bf v}\\\\\n=& \\omega {\\bf H}+ \\sum_{n\\geq 1}v_n\\, \\mathbf{L}_n + \\sum_{n\\geq 1}\\overline{v_n} \\, \\widetilde{\\mathbf{L}}_n\n\\end{split}\n\\end{equation} \nwhere \n\\begin{equation}\\label{formulaLn} \n\\mathbf{L}_n = \\mathbf{L}_n^0+ \\frac{\\mu}{4\\pi}\\int_0^{2\\pi} e^{\\gamma \\varphi(\\theta)}e^{in\\theta}d\\theta,\\quad \\widetilde{\\mathbf{L}}_n = \n\\widetilde{\\mathbf{L}}_n^0+ \\frac{\\mu}{4\\pi}\\int_0^{2\\pi} e^{\\gamma \\varphi(\\theta)}e^{-in\\theta}d\\theta.\n\\end{equation}\nNotice that ${\\bf L}_n$ can be recovered as a sum of operators ${\\bf H}_{\\mathbf{v}}$ with $\\mathbf{v}$ Markovian by the formula \n\\begin{equation}\\label{Ln_en_terme_de_Hv_n} \n{\\bf L}_n=\\frac{1}{2}({\\bf H}_{\\omega \\mathbf{v}_0+\\mathbf{v}_n}-i{\\bf H}_{\\omega\\mathbf{v}_0+i\\mathbf{v}_n})-\\frac{1}{2}\\omega(1-i){\\bf H}.\n\\end{equation}\nNow, we consider the following semigroup $P_t$ defined for $F$ bounded on $H^{-s}(\\mathbb{T})$ (with $s>0$) by the formula:\n\\begin{equation} \\label{FeynmanKac}\n P_t F := |f'_t(0)|^{\\frac{Q^2}{2}} \\mathbb{E}_\\varphi \\Big [ F\\Big( (X \\circ f_t + Q \\ln | \\frac { f_t'} {f_t} | )|_{\\mathbb{T}} \\Big) e^{- \\mu e^{\\gamma c} \\int_{\\mathbb{D}_t^c} e^{\\gamma X(x)} \\frac{1} {|x| ^{\\gamma Q} } dx } \\Big] \n \\end{equation}\n \n\\begin{theorem}\\label{FeynmanKacPt}\nThe semigroup $P_t$ coincides with $e^{-t \\mathbf{H}_\\mathbf{v}}$ on $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$.\n\\end{theorem} \n\\begin{proof} \n The map $(s,\\theta)\\mapsto f_s(e^{i\\theta})$ is a diffeomorphism from $(0,t)\\times \\mathbb{T}$ to $\\mathbb{D}_t^c$, and its Jacobian is given by \n $-\\Re (e^{- i \\theta} v(f_s (e^{i \\theta}))\\overline{f'_s(e^{i \\theta})})$. Therefore, for $\\epsilon>0$ we get that\n \\begin{equation*}\n \\int_{\\mathbb{D}_t^c} \\epsilon^{\\frac{\\gamma^2}{2}} e^{\\gamma X_\\epsilon(x)} \\frac{1} {|x| ^{\\gamma Q} } dx= -\\int_{0}^t \\int_{0}^{2 \\pi} \\epsilon^{\\frac{\\gamma^2}{2}} e^{\\gamma X_{\\epsilon} (f_s(e^{i \\theta})) } \\Re \\Big(e^{- i \\theta} v(f_s (e^{i \\theta})) \\overline{f'_s(e^{i \\theta})}\\Big) \\text{\\rm d} \\theta \\text{\\rm d} s\n \\end{equation*}\nwhere the cutoff $X_\\epsilon$ is defined via circle averages $X_\\epsilon(x)= \\frac{1}{2 \\pi} \\int_0^{2 \\pi} X(x+ \\epsilon e^{i \\theta}) \\text{\\rm d} \\theta$. Now we have up to a neglectable term $X_{\\epsilon} (f_s(e^{i \\theta}))= (X \\circ f_s) _{\\epsilon_{s, \\theta}} (e^{i \\theta})$ where $\\epsilon_{s, \\theta}= \\frac{\\epsilon}{|f_s'(e^{i \\theta})|}$. This leads to (up to a neglectable term) \n \\begin{equation*}\n \\int_{\\mathbb{D}_t^c} \\epsilon^{\\frac{\\gamma^2}{2}} e^{\\gamma X_\\epsilon(x)} \\frac{1} {|x| ^{\\gamma Q} } dx= -\\int_{0}^t \\int_{0}^{2 \\pi} \\epsilon_{s, \\theta}^{\\frac{\\gamma^2}{2}} e^{\\gamma ( (X \\circ f_s) _{\\epsilon_{s, \\theta}} (e^{i \\theta}) + Q \\log \\frac{|f_s'(e^{i \\theta}|}{|f_s(e^{i \\theta})|} )} \\Re \\Big(e^{- i \\theta} \\frac{v(f_s (e^{i \\theta}))}{f'_s(e^{i \\theta})} \\Big) \\text{\\rm d} \\theta \\text{\\rm d} s\n \\end{equation*}\nSince $\\log \\frac{|f_s'(e^{i \\theta}|}{|f_s(e^{i \\theta})|} $ is regular, we get by a change a variable on the cutoff that\n\\begin{equation}\\label{polardecomposition}\n \\int_{\\mathbb{D}_t^c} e^{\\gamma X(x)} \\frac{1} {|x| ^{\\gamma Q} } dx= -\\underset{\\epsilon \\to 0}{\\lim} \\int_{0}^t \\int_{0}^{2 \\pi} \\epsilon^{\\frac{\\gamma^2}{2}} e^{\\gamma \\varphi_{s,\\epsilon}(\\theta)} \\Re \\Big(e^{- i \\theta} \\frac{v(f_s (e^{i \\theta}))}{f'_s(e^{i \\theta})}\\Big) \\text{\\rm d} \\theta \\text{\\rm d} s\n\\end{equation}\nwhere we recall that $\\varphi_t(\\theta)= (X \\circ f_t)(e^{i \\theta}) + Q \\ln | \\frac { f_t'(e^{i \\theta})} {f_t(e^{i \\theta})} |$ and $\\varphi_{t,\\epsilon}(\\theta)$ is its regularization at scale $\\epsilon$. \nRecall from Lemma \\ref{limitandh} that $\\frac{v(f_s (e^{i \\theta}))}{f'_s(e^{i \\theta})}$ does not depend on $s$ and therefore\n\\begin{equation}\\label{polardecompositionctd}\n \\int_{^c \\mathbb{D}_t} e^{\\gamma X(v)} \\frac{1} {|v| ^{\\gamma Q} } dv= -\\underset{\\epsilon \\to 0}{\\lim} \\int_{0}^t \\int_{0}^{2 \\pi} \\epsilon^{\\gamma^2\/2} e^{\\gamma \\varphi^s_\\epsilon(\\theta)} \\Re (e^{- i \\theta} v(e^{i \\theta})) \\text{\\rm d} \\theta \\text{\\rm d} s\n\\end{equation}\n\n Thanks to the free case, we know that for $F \\in \\mathcal{C}_{\\rm exp}$, $\\|P_tF\\|_{2}\\leq \\|P_t^0F\\|_{2}\\leq \\|F\\|_2$ and hence $P_t$ can be extended on $L^2$ into a contraction semigroup $\\tilde{T}_t$. By adapting the arguments of \\cite[Section 5]{GKRV} one can show that for all $F,G \\in \\mathcal{C}_{\\rm exp}$ one has\n \\begin{equation*}\n \\langle \\frac{(I-\\tilde{T}_t)F}{t} | G \\rangle_2= \\langle \\frac{(I-P_t)F}{t} | G \\rangle_2 \\underset{t \\to 0}{\\rightarrow} \\langle \\mathbf{H}_\\mathbf{v} F | G \\rangle_2 \n \\end{equation*} \nThe above convergence is based on the the weak convergence of measures $\\frac{\\mathds{1}_{\\mathbb{D}_t^c}}{t}\\d x\\to\\varrho_\\mathbf{v}$ on $\\bar{\\mathbb{D}}$, and the rest of the proof follows \\cite[Section 5]{GKRV} since we know that $\\frac{(I-P_t^0)F}{t}$ converges in $L^2$ to $\\mathbf{H}_\\mathbf{v}^0 F$ as $t$ goes to $0$.\n\nLet $P_t^N$ be the semigroup of the form \\eqref{FeynmanKac} where one works with first $N$ harmonics of $\\varphi$ and similarly for $\\mathbf{H}_\\mathbf{v}^N, \\mathcal{Q}_{\\mathbf{v}}^N$. One can adapt the arguments of \\cite[Section 5]{GKRV} to show that $P_t^N=e^{-t\\mathbf{H}_\\mathbf{v}^N}$. In the sequel, we will consider the semigroups and the quadratic forms on the space of real functions (this no restriction by a linearity argument). Let $\\lambda>0$ and $F \\in \\mathcal C$ be real valued. We have for all real valued $G \\in \\mathcal C_\\mathrm{exp}$\n\\begin{equation*}\n\\lambda \\langle R_\\lambda^N F | G \\rangle_2+ \\mathcal{Q}_{\\mathbf{v}}^N(R_\\lambda^N F, G)= \\langle F | G \\rangle_2\n\\end{equation*} \nwhere $R_\\lambda^N$ is the resolvant associated to $P_t^N$. Since $R_\\lambda^N F$ and $G$ depends on the first $N$ harmonics for $N$ large, the following holds by definition of $\\mathcal{Q}_{\\mathbf{v}}$ if $G$ depends on the first $N$ harmonics\n\\begin{equation}\\label{quadrelation}\n\\lambda \\langle R_\\lambda^N F | G \\rangle_2+ \\mathcal{Q}_{\\mathbf{v}}(R_\\lambda^N F, G)= \\langle F | G \\rangle_2\n\\end{equation} \nThe above identity for $G= R_\\lambda^N F$ shows that $\\sup_{N \\geq 1} \\mathcal{Q}_{\\mathbf{v}}(R_\\lambda^N F,R_\\lambda^N F ) < \\infty$. Moreover, by adapting \\cite[Section 5]{GKRV}, we know that $R_\\lambda^N F$ converges in $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$ to $R_\\lambda F$ where $R_\\lambda$ is the resolvent associated to $P_t$. Therefore, we can apply Lemma 2.12 in \\cite{MaRockner} which yields that $R_\\lambda F \\in \\mathcal{D}(\\mathcal{Q}_{\\mathbf{v}} )$ and one can find a subsequence $(R_\\lambda^{N_k}F)_{k \\geq 1}$ which converges in Cesaro mean to $R_\\lambda F$ with respect to $\\mathcal{Q}_{\\mathbf{v}}$. Hence, we can take the limit in \\eqref{quadrelation} which yields for all $G \\in \\mathcal C_\\mathrm{exp}$\n\\begin{equation}\\label{quadrelationlimit}\n\\lambda \\langle R_\\lambda F | G \\rangle_2+ \\mathcal{Q}_{\\mathbf{v}}(R_\\lambda F, G)= \\langle F | G \\rangle_2\n\\end{equation} \nSince $\\mathcal C_\\mathrm{exp}$ is dense in the domain of $\\mathcal{Q}_{\\mathbf{v}}$, this shows that $R_\\lambda$ is equal to the resolvant associated to $\\mathcal{Q}_{\\mathbf{v}}$ on $\\mathcal{C}_\\mathrm{exp}$. Since $\\mathcal{C}_\\mathrm{exp}$ is dense in $L^2(\\mathbb{R} \\times \\Omega_\\mathbb{T})$, this implies that $R_\\lambda$ is the resolvant associated to $\\mathcal{Q}_{\\mathbf{v}}$. By unicity of the resolvent, we have proved the desired result.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Liouville descendant states}\\label{subsec:poisson}\n\nIn this section we recall the definition and properties of the Poisson operator introduced in \\cite[Section 6]{GKRV}. This will be used to describe the diagonalization of the Liouville Hamiltonian and obtain a representation of the Virasoro algebra.\n \n\\subsubsection{The spectral Riemann surface}\n\nLet $\\mathcal{D}_0:=\\{\\alpha_{\\pm j}:=Q\\pm i \\sqrt{2j} \\, |\\, j\\in \\mathbb{N}\\}$ and consider the non-compact Riemann surface $\\Sigma$ so that $r_j(\\alpha):=\\sqrt{-(\\alpha-Q)^2+2j}$ are \nholomorphic functions for all $2j>0$ and so that for each $\\alpha\\in \\Sigma$, ${\\rm Im}(\\sqrt{-(\\alpha-Q)^2+2j})>0$ for $j\\in\\mathbb{N}$ large enough. Here we use the convention that for $j=0$, $r_0(\\alpha)=-i(\\alpha-Q)$.\nThe surface $\\Sigma$ is as a ramified covering $\\pi:\\Sigma\\to \\mathbb{C}$ with ramifications of order $2$ at $\\mathcal{D}_0$. We identify \nthe half-space $\\{\\alpha \\in\\mathbb{C} \\,|\\, {\\rm Re}(\\alpha)0$ for all $j\\in\\mathbb{N}_0$ if ${\\rm Re}(\\alpha)0$.\n\n\\begin{proposition}[Proposition 6.19 in \\cite{GKRV}]\n\\label{poissonprop}\nLet $0<\\beta<\\gamma\/2$ and $\\ell\\in\\mathbb{N}$.\nThen there is an analytic family of operators $\\mathcal{P}_\\ell(\\alpha)$ \n\\[\\mathcal{P}_\\ell(\\alpha): E_\\ell\\to e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q})\\]\nin the region \n\\begin{equation}\\label{regionvalide}\n\\Big\\{\\alpha=Q+ip\\in \\mathbb{C}\\,\\Big|\\, {\\rm Re}(\\alpha)< Q, {\\rm Im}\\sqrt{p^2-2\\ell}<\\beta\\Big\\}\\cup \\Big\\{Q+ip\\in Q+i\\mathbb{R}\\, \\Big| \\, |p|\\in \\bigcup_{j\\geq \\ell} (\\sqrt{2j},\\sqrt{2(j+1)})\\Big\\},\n\\end{equation}\ncontinous at each $Q\\pm i\\sqrt{2j}$ for $j\\geq \\ell$, satisfying $(\\mathbf{H}-\\tfrac{Q^2+p^2}{2})\\mathcal{P}_\\ell(\\alpha)F=0$ and in the region $\\{c\\leq 0\\}$\n\\begin{equation}\\label{expansionP} \n\\mathcal{P}_\\ell(\\alpha)F=\\sum_{j\\leq \\ell}\\Big(F^-_je^{ic\\sqrt{p^2-2j}}+F_j^+(\\alpha)\ne^{-ic\\sqrt{p^2-2j}}\\Big)+G_\\ell(\\alpha,F)\n\\end{equation}\nwith $F^-_j=\\Pi_{\\ker(\\mathbf{P}-j)}F$, $F_j^+(\\alpha)\\in \\ker (\\mathbf{P}-j)$, and\n$G_\\ell(\\alpha,F)\\in \\mathcal{D}(\\mathcal{Q})$. \nSuch a solution $u\\in e^{-\\beta\\rho}L^2(\\mathbb{R}\\times\\Omega_\\mathbb{T})$ to the equation $(\\mathbf{H}-2\\Delta_{Q+ip})u=0$ with the asymptotic expansion \\eqref{expansionP} is unique.\nFinally, $F_j^+(\\alpha)$ depends analytically on $\\alpha$ in the region \\eqref{regionvalide} and the operator $\\mathcal{P}_\\ell(\\alpha)$ admits a meromorphic extension to the region \n\\begin{equation}\\label{regiondextension}\n\\Big\\{\\alpha=Q+ip \\in \\Sigma\\, |\\, \\forall j\\leq \\ell, {\\rm Im}\\sqrt{p^2-2j}\\in (\\beta\/2-\\gamma,\\beta\/2)\\Big\\}\\end{equation}\nand $\\mathcal{P}_\\ell(\\alpha)F$ satisfies \\eqref{expansionP} in that region. \n\\end{proposition}\nWe notice that, by uniqueness of the expansion \\eqref{expansionP}, for $j<\\ell$ and $|p|>\\sqrt{2\\ell}$, we have \n\\begin{equation}\\label{egalitepoissonell} \n\\mathcal{P}_\\ell(Q+ip)|_{\\ker ({\\bf P}-j)}= \\mathcal{P}_{j}(Q+ip)|_{\\ker ({\\bf P}-j)}.\n\\end{equation}\n\n\\subsubsection{Verma modules and highest weight vectors for the free field}\\label{Vermafreefield}\n\nWe shall use a particular eigenbasis of ${\\bf P}$ that is parametrized by the Young diagrams and that is adapted to the Virasoro algebra. \nRecall that a Young diagram $\\nu$ is a non-increasing finite sequence of integers $\\nu_1\\geq \\dots\\geq \\nu_k>0$. We denote by $\\mathcal{T}$ the union of $\\{0\\}$ and the set of Young diagrams and we denote by $|\\nu|=\\sum_{j}\\nu_j$ its length. We shall write $\\mathcal{T}_n:=\\{\\nu \\in \\mathcal{T}\\,|\\, |\\nu|=n\\}$. \nNotice that $\\mathcal{T}$ can be viewed as a subset of $\\mathcal{N}$, where $\\mathcal{N}$ is the set \nof sequences $(k_n)_{n\\geq 0}\\in \\mathbb{N}_0^{\\mathbb{N}_0}$ so that there is $n_0$ with $k_n=0$ for all $n\\geq n_0$.\nGiven two elements $\\nu= (\\nu_i)_{i \\in [1,k]}\\in\\mathcal{N}$ and $\\tilde{\\nu}= (\\tilde{\\nu}_i)_{i \\in [1,j]}\\in\\mathcal{N}$ \nwe define the operators on $\\mathcal{C}_{\\rm exp}$ \n\\begin{equation}\\label{LnuL-nu}\n\\begin{gathered}\n \\mathbf{L}_{-\\nu}^0:=\\mathbf{L}_{-\\nu_k}^0 \\cdots \\, \\mathbf{L}_{-\\nu_1}^0, \\qquad \\tilde{\\mathbf{L}}_{-\\tilde \\nu}^0:=\\tilde{\\mathbf{L}}_{-\\tilde\\nu_j}^0 \\cdots\\, \\tilde{\\mathbf{L}}_{-\\tilde\\nu_1}^0\\\\\n \\mathbf{L}_{\\nu}^0:=\\mathbf{L}_{\\nu_k}^0 \\cdots \\, \\mathbf{L}_{\\nu_1}^0, \\qquad \\tilde{\\mathbf{L}}_{\\tilde \\nu}^0:=\\tilde{\\mathbf{L}}_{\\tilde\\nu_j}^0 \\cdots\\, \\tilde{\\mathbf{L}}_{\\tilde\\nu_1}^0\n\\end{gathered}\\end{equation}\nand define the functions\n\\begin{align}\\label{psibasis}\n\\Psi^0_{\\alpha,\\nu, \\tilde\\nu}:=\\mathbf{L}_{-\\nu}^0\\tilde{\\mathbf{L}}_{-\\tilde\\nu}^0 \\: \\Psi^0_\\alpha,\n\\end{align}\nwith the convention that $\\Psi^0_{\\alpha,0,0}:=\\Psi^0_{\\alpha}=e^{(\\alpha-Q)c}$. These functions satisfy \n\\[ {\\bf H}^0\\Psi^0_{\\alpha,\\nu, \\tilde\\nu}=(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)\\Psi^0_{\\alpha,\\nu, \\tilde\\nu}\\quad \\textrm{ with }\\Delta_\\alpha:=\\frac{\\alpha}{2}(Q-\\frac{\\alpha}{2}).\\]\nBy \\cite[Proposition 4.9]{GKRV}, for each $\\nu,\\tilde{\\nu}\\in \\mathcal{T}$, there is a polynomial $\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}$ in the variables $(x_n,y_n)_n$ so that \n\\begin{equation}\\label{descendantfree}\n\\Psi^0_{\\alpha,\\nu, \\tilde\\nu}=\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}\\Psi_{\\alpha}^0, \\qquad {\\bf P} \\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}=(|\\nu|+|\\tilde{\\nu}|) \\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}.\n\\end{equation}\nIt is also shown that ${\\rm span}\\{ \\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}\\, |\\, \\nu,\\tilde{\\nu}\\in \\mathcal{T},\\, |\\nu|+|\\tilde{\\nu}|=\\ell\\}=\\ker({\\bf P}-\\ell)$ if $\\alpha\\notin Q-\\frac{\\gamma}{2}\\mathbb{N}-\\frac{2}{\\gamma}\\mathbb{N}$. More generally, a relation of the form \\eqref{descendantfree} holds if $\\nu,\\tilde{\\nu}\\in \\mathcal{N}$ with \n$\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}\\in \\ker ({\\bf P}-|\\nu|-|\\tilde{\\nu}|)$ where $|\\nu|=\\sum_j \\nu_j$ and $|\\tilde{\\nu}|=\\sum_j \\tilde{\\nu}_j$. In fact if\n$n_1,\\dots,n_k\\in \\mathbb{N}$ and $\\tilde{n}_1,\\dots,\\tilde{n}_{k'}\\in \\mathbb{N}$, then\n\\[\n{\\bf L}_{-n_k}^0\\dots {\\bf L}_{-n_1}^0\\tilde{{\\bf L}}_{-\\tilde{n}_{k'}}^0\\dots \\tilde{{\\bf L}}_{-\\tilde{n}_1}^0\\Psi^0_\\alpha\\in {\\rm span}\\Big\\{ \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}\\,|\\, \\nu,\\tilde{\\nu}\\in \\mathcal{T}, |\\nu|=\\sum_{j=0}^kn_j, \\; |\\tilde{\\nu}|=\\sum_{j=0}^{k'}\\tilde{n}_j\\Big\\}\n\\]\nif $\\alpha\\notin Q-\\frac{\\gamma}{2}\\mathbb{N}-\\frac{2}{\\gamma}\\mathbb{N}$. In particular, for $n\\in \\mathbb{Z}$, if $\\nu,\\tilde{\\nu}\\in\\mathcal{T}$, there are \ncoefficients $\\ell_{n}^\\alpha (\\nu,\\nu')\\in \\mathbb{C}$, that are $0$ when $|\\nu|-n\\not=|\\nu'|$, so that \n\\begin{equation}\\label{descendant_general} \n{\\bf L}_{n}^0 \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0=\\sum_{\\nu'\\in \\mathcal{T}}\\ell_{n}^\\alpha (\\nu,\\nu')\\Psi_{\\alpha,\\nu',\\tilde{\\nu}}^0 , \n\\quad \\tilde{{\\bf L}}_{n}^0 \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0=\\sum_{\\tilde{\\nu}'\\in \\mathcal{T}}\\ell_{n}^\\alpha (\\tilde{\\nu},\\tilde{\\nu}')\\Psi_{\\alpha,\\nu,\\tilde{\\nu}'}^0.\n\\end{equation}\nMoreover, when $0|\\nu|$, \n$\\ell_{n}^{\\alpha}(\\nu,\\cdot)=0$. \nThe functions $\\ell_{n}^{\\alpha}$ can be computed by \ncommuting recursively ${\\bf L}^0_{n}$ with the ${\\bf L}_{-\\nu_j}^0$ for those $\\nu_j\\leq n$ when writing $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0$ under the form \\eqref{psibasis}. In the proof of \\cite[Proposition 4.9]{GKRV}, it is shown that \n\\[ {\\bf L}_{n}^0 \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0= e^{(\\alpha-Q)c} {\\bf L}_{n}^{0,\\alpha}\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}},\\qquad \\tilde{{\\bf L}}_{n}^0 \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0= e^{(\\alpha-Q)c} \\tilde{{\\bf L}}_{n}^{0,\\alpha}\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}\\]\nwhere ${\\bf L}_n^{0,\\alpha}= e^{(Q-\\alpha)c}[{\\bf L}_{n}^0, e^{(\\alpha-Q)c}]$ (and similarly for $\\tilde{{\\bf L}}_n^{0,\\alpha}$) \nis viewed as an unbounded operator acting on $L^2(\\Omega_\\mathbb{T})$ (i.e only in the $(x_n,y_n)_n$ variables but not on $c$) and well defined on $\\mathcal{S}\\subset L^2(\\Omega_\\mathbb{T})$; it also satisfies $({\\bf L}_n^{0,\\alpha})^*={\\bf L}_{-n}^{0,2Q-\\bar{\\alpha}}$ on $L^2(\\Omega_\\mathbb{T})$. By \\eqref{descendant_general}\n\\begin{equation}\\label{Ln0alpha}\n \\sum_{\\nu'\\in \\mathcal{T}}\\ell_{n}^{\\alpha}(\\nu,\\nu')\\mathcal{Q}_{\\alpha,\\nu',\\tilde{\\nu}}={\\bf L}_n^{0,\\alpha}\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}, \\qquad \n\\sum_{\\tilde{\\nu}'\\in \\mathcal{T}}\\ell_{n}^{\\alpha}(\\tilde{\\nu},\\tilde{\\nu}')\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}'}=\\tilde{{\\bf L}}_n^{0,\\alpha}\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}.\n\\end{equation}\nFollowing \\cite[Section 6.4]{GKRV}, we define the Hilbert space \n\\[ \\mathcal{H}_{\\mathcal{T}}=\\bigoplus_{n=0}^\\infty \\mathbb{C}^{d_n} , \\quad \\langle v,v'\\rangle_{\\mathcal{T}}=\\sum_{n=0}^\\infty \\langle v_n,v_n'\\rangle_{d_n}\\] \nwhere $d_n=\\sharp \\mathcal{T}_n$, and $\\langle \\cdot,\\cdot\\rangle_{d_n}$ is the standard Hermitian product on $\\mathbb{C}^{d_n}$. \nIf $(e_{\\nu})_{\\nu \\in \\mathcal{T}}$ denotes \nthe canonical Hilbert basis of $\\mathcal{H}_{\\mathcal{T}}$, we see that for $\\alpha\\notin Q-\\frac{\\gamma}{2}\\mathbb{N}-\\frac{2}{\\gamma}\\mathbb{N}$\n\\[ \\mathcal{I}_{\\alpha}: \\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}} \\to L^2(\\Omega_\\mathbb{T}), \\quad e_{\\nu}\\otimes e_{\\tilde{\\nu}} \\mapsto \\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}\\]\nis an isomorphism, but it is not a unitary map since $(\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}})_{\\nu,\\tilde{\\nu}}$ is not orthonormal. \nBy \\cite[Proposition 4.9]{GKRV} we have for $P>0$\n\\[ \\langle \\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}, \\mathcal{Q}_{Q+iP,\\nu',\\tilde{\\nu}'}\\rangle_{L^2(\\Omega_\\mathbb{T})}=F_{Q+iP}(\\nu,\\nu')F_{Q+iP}(\\tilde{\\nu},\\tilde{\\nu}') \\]\nwhere $(F_{Q+iP}(\\nu,\\nu'))_{\\nu,\\nu'\\in \\mathcal{T}}$ are the so-called \\emph{Shapovalov matrices}, which satisfy $F_{Q+iP}(\\nu,\\nu')=0$ if $|\\nu|\\not=|\\nu'|$ and so that \n$(F_{Q+iP}(\\nu,\\nu'))|_{\\nu,\\nu'\\in \\mathcal{T}_n}$ are invertible for all $n\\in \\mathbb{N}$; the inverse of $F_{Q+iP}$ is denoted $F_{Q+iP}^{-1}$. \nTherefore, if we equip $\\mathcal{H}_{\\mathcal{T}}$ with the following scalar product\n\\[ \\langle u,u'\\rangle_{Q+iP}:= \\sum_{\\nu,\\nu'\\in\\mathcal{T}} F_{Q+iP}(\\nu,\\nu') u_\\nu \\bar{u}_{\\nu'} ,\\quad \\textrm{for } u=\\sum_{\\nu} u_\\nu e_\\nu, \\,\\, u'=\\sum_{\\nu} u'_\\nu e_\\nu ,\\]\nthe map $\\mathcal{I}_{Q+iP}: (\\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}}, \\langle\\cdot,\\cdot\\rangle_{Q+iP}\\otimes \\langle\\cdot,\\cdot\\rangle_{Q+iP}) \\to L^2(\\Omega_\\mathbb{T})$ becomes a unitary isomorphism of Hilbert spaces. \nLet us consider the linear maps for $\\alpha\\notin Q-\\frac{\\gamma}{2}\\mathbb{N}-\\frac{2}{\\gamma}\\mathbb{N}$\n\\begin{equation}\\label{def:ell_nalpha}\n\\ell_n^{\\alpha}:\\mathcal{H}_{\\mathcal{T}}\\to \\mathcal{H}_{\\mathcal{T}},\\quad \n\\ell_n^{\\alpha}(e_{\\nu})=\\sum_{\\nu'\\in \\mathcal{T}}\\ell_n^{\\alpha}(\\nu,\\nu')e_{\\nu'} \n\\end{equation}\nand we also call $\\ell_n^\\alpha$ (resp. $\\tilde{\\ell}_n^\\alpha$) the action of $\\ell_n^\\alpha$ on $\\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}}$ on the left variable \n(resp. on $\\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}}$ on the right variable).\nNotice that \n$\\mathcal{I}_{\\alpha}\\ell_n^{\\alpha}={\\bf L}^{0,\\alpha}_n\\mathcal{I}_{\\alpha}$ and $\\mathcal{I}_{\\alpha}\\tilde{\\ell}_n^{\\alpha}=\\tilde{{\\bf L}}^{0,\\alpha}_n\\mathcal{I}_{\\alpha}$.\nFrom \\eqref{Ln0alpha} and the fact that $({\\bf L}_n^{0,Q+iP})^*={\\bf L}_{-n}^{0,Q+iP}$ for $P>0$, we conclude that $(\\ell_n^{Q+iP})^*=\\ell_{-n}^{Q+iP}$ on $(\\mathcal{H}_{\\mathcal{T}}, \\langle\\cdot,\\cdot\\rangle_{Q+iP})$ and \n\\begin{equation}\\label{ellnadjoint}\n(\\ell_n^{Q+iP})^*=\\ell_{-n}^{Q+iP}, \\quad (\\tilde{\\ell}_n^{\\, Q+iP})^*=\\tilde{\\ell}_{-n}^{\\, Q+iP} \\textrm{ on }(\\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}}, \\langle\\cdot,\\cdot\\rangle_{Q+iP})\\otimes \\langle\\cdot,\\cdot\\rangle_{Q+iP}).\n\\end{equation}\n\nLet us now define the vector spaces $\\mathcal{V}_\\alpha^0, \\overline{\\mathcal{V}}_\\alpha^0$ by \n\\[ \\begin{gathered}\n\\mathcal{V}_\\alpha^0:= {\\rm span}\\{ \\Psi^0_{\\alpha,\\nu,0}\\,|\\, \\nu \\in \\mathcal{T} \\} \\subset e^{(|{\\rm Re}(\\alpha)-Q|+\\epsilon)|c|}L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T}), \\\\\n\\overline{\\mathcal{V}}_\\alpha^0:= {\\rm span}\\{ \\Psi^0_{\\alpha,0,\\tilde{\\nu}}\\,|\\, \\nu \\in \\mathcal{T} \\} \\subset e^{(|{\\rm Re}(\\alpha)-Q|+\\epsilon)|c|}L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})\n\\end{gathered}\\]\nIn the langage of representation theory (see \\cite[Chapter 6]{Schottenloher}), if $\\alpha\\notin Q-\\frac{\\gamma}{2}\\mathbb{N}-\\frac{2}{\\gamma}\\mathbb{N}$ the state $\\Psi_{\\alpha}^0$ is a \\emph{highest weight vector} associated to the highest weight representation of Virasoro algebra ${\\rm Vir}(c_L)$ with central charge $c_L=1+6Q^2$ generated by the operators ${\\bf L}_n^0$ for $n\\in \\mathbb{Z}$ and $c_L$, \nand $\\mathcal{V}_\\alpha^0$ is an associated \\emph{Verma module} (and similarly for $\\overline{\\mathcal{V}}_\\alpha^0$):\n\\[ \\left \\{ \\begin{array}{lll}\n {\\bf L}_n^{0}\\Psi_\\alpha^0=0, \\,\\, \\forall n>0, \\\\\n{\\bf L}_0^0 \\Psi_{\\alpha}^0= \\Delta_{\\alpha} \\Psi_{\\alpha}^0\n \\end{array}\\right. ,\\qquad \n\\left \\{ \\begin{array}{lll}\n \\tilde{{\\bf L}}_n^{0}\\Psi_\\alpha^0=0, \\,\\, \\forall n>0, \\\\\n\\tilde{{\\bf L}}_0^0 \\Psi^0_{\\alpha}= \\Delta_{\\alpha} \\Psi^0_{\\alpha} \n \\end{array}\\right..\n \\]\n For $\\alpha=Q+iP$ with $P>0$, we equip $\\mathcal{V}_{Q+iP}^{0}$ and $\\overline{\\mathcal{V}}_{Q+iP}^{0}$ with the scalar product \n\\begin{equation}\\label{scalarproduct}\n\\langle F,F'\\rangle_{\\mathcal{V}^0_{Q+iP}}:= \\langle e^{-iPc}F, e^{-iPc}F'\\rangle_{L^2(\\Omega_\\mathbb{T})},\n\\end{equation} \nand the two representations ${\\bf L}_n^0$ in $\\mathcal{V}_{Q+iP}^{0}$ and $\\tilde{\\bf L}_n^0$ in $\\overline{\\mathcal{V}}_{Q+iP}^{0}$ are then unitary. \nThe direct sum ${\\rm Vir}(c_L)\\oplus {\\rm Vir}(c_L)$ also has a representation into \n\\[\\mathcal{W}^{0}_{\\alpha}:={\\rm span}\\{ \\Psi^0_{\\alpha,\\nu,\\tilde{\\nu}}\\,|\\, \\nu,\\tilde{\\nu} \\in \\mathcal{T} \\}\\] \ngiven by the action of $({\\bf L}_n^0)_n$ and $(\\tilde{\\bf L}_n^0)_n$ on the basis elements $\\Psi^0_{\\alpha,\\nu,\\tilde{\\nu}}$; \nnotice that the representation of $({\\bf L}_n^0)_n$ on $\\mathcal{W}^{0}_{\\alpha}$ produces canonically the representation of \n$({\\bf L}_n^0)_n$ on $\\mathcal{V}_\\alpha^0$ by using the inclusion $\\mathcal{V}_\\alpha^0\\subset \\mathcal{W}_\\alpha^0$. \nWe use again the expression \\eqref{scalarproduct} for the scalar product on $\\mathcal{W}^{0}_{Q+iP}$.\nWe see that \n\\[ \\left\\{\\begin{array}{rcl} \n(\\mathcal{H}_{\\mathcal{T}}, \\langle\\cdot,\\cdot\\rangle_{Q+iP})& \\to & (\\mathcal{V}_{Q+iP}^{0},\\langle \\cdot,\\cdot \\rangle_{\\mathcal{V}^0_{Q+iP}}) \\\\\n e_{\\nu}& \\mapsto & \\Psi^0_{Q+iP,\\nu,0}\n\\end{array}\\right.,\\quad \n\\left\\{\\begin{array}{rcl} \n(\\mathcal{H}_{\\mathcal{T}}, \\langle\\cdot,\\cdot\\rangle_{Q+iP})& \\to & (\\mathcal{V}_{Q+iP}^{0},\\langle \\cdot,\\cdot \\rangle_{\\mathcal{V}^0_{Q+iP}}) \\\\\n e_{\\tilde{\\nu}}& \\mapsto & \\Psi^0_{Q+iP,0,\\tilde{\\nu}}\n\\end{array}\\right.\n\\]\nare unitary isomorphisms, which conjugate respectively ${\\bf L}_{n}^0$ and $\\tilde{\\bf L}_{n}^0$ with $\\ell_{n}^{Q+iP}$. \nOne also has a unitary isomorphism \n\\[\\left\\{\\begin{array}{rcl} \n(\\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}}, \\langle\\cdot,\\cdot\\rangle_{Q+iP}\\otimes \\langle\\cdot,\\cdot\\rangle_{Q+iP})& \\to & (\\mathcal{W}_{Q+iP}^0,\\langle \\cdot,\\cdot \\rangle_{\\mathcal{V}^0_{Q+iP}}) \\\\\n e_{\\nu}\\otimes e_{\\tilde{\\nu}}& \\mapsto & \\Psi^0_{Q+iP,\\nu,\\tilde{\\nu}}\n\\end{array}\\right.\\]\nwhich allows us to identify \n\\[\\mathcal{V}_{Q+iP}^{0}\\otimes \\overline{\\mathcal{V}}_{Q+iP}^{0} \\simeq \\mathcal{W}^{0}_{Q+iP}\\]\n\nSince for $\\alpha\\notin Q-\\frac{\\gamma}{2}\\mathbb{N}-\\frac{2}{\\gamma}\\mathbb{N}$, $\\ell_{n}^{\\alpha}$ maps a finite sum $\\sum_{\\nu,\\tilde{\\nu}\\in \\mathcal{T}_N}a_{\\nu,\\tilde{\\nu}}e_{\\nu}\\otimes e_{\\tilde{\\nu}}$ to a finite sum of the same form with $N$ replaced by $N-n$, we can compose $\\ell_n^\\alpha \\ell_m^\\alpha$ on such finite sums, and their commutators are given by the Virasoro commutation laws \n\\[ [ \\ell^{\\alpha}_n,\\ell^{\\alpha}_m]=(n-m)\\ell^{\\alpha}_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}, \\quad [\\tilde{\\ell}^{\\alpha}_n,\\tilde{\\ell}^{\\alpha}_m]=(n-m)\\tilde{\\ell}^{\\alpha}_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}.\\]\nFor each $P>0$, the $(\\ell_n^{Q+iP})_n$ thus forms a unitary representations of the Virasoro algebra into \n$\\mathcal{H}_{\\mathcal{T}}$, and $(\\ell_n^{Q+iP}, \\tilde{\\ell}_m^{\\, Q+iP})_{n,m}$ is a representation of ${\\rm Vir}(c_L)\\oplus {\\rm Vir}(c_L)$ into $(\\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}}, \\langle\\cdot,\\cdot\\rangle_{Q+iP}\\otimes \\langle\\cdot,\\cdot\\rangle_{Q+iP})$, with each copy being unitary.\n\n\n\n\n\n\n\\subsubsection{Liouville descendants}\n\nThe Liouville descendant states $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ are defined using the Poisson operator of Proposition \\ref{poissonprop} (the square root below is defined so that $\\sqrt{Re^{i\\theta}}=\\sqrt{R}e^{i\\theta\/2}$ for $R>0$ and $\\theta \\in (-\\pi,\\pi)$)\n: \n\\begin{proposition}[Proposition 6.26 and Proposition 6.9 of \\cite{GKRV}]\\label{constPoisson}\nFor $\\nu,\\tilde{\\nu}\\in \\mathcal{T}$, let $\\ell:=|\\nu|+|\\nu'|$, and set for $\\alpha=Q+iP$ with ${\\rm Re}(\\alpha)\\leq Q$\n\\[ \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}:=\\mathcal{P}_\\ell(Q+i\\sqrt{P^2+2\\ell})\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}},\\]\nand $\\Psi_{\\alpha}:=\\Psi_{\\alpha,0,0}$. The function $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ are well defined, analytic in $\\alpha$, as an element in $e^{- \\beta \\rho}\\mathcal{D}(\\mathcal{Q})$ for all $\\beta>Q-{\\rm Re}(\\alpha)$ such that $\\alpha$ belongs to the set \n\\begin{equation}\\label{defWell}\nW_\\ell:= \\Big\\{ \\alpha \\in \\mathbb{C} \\setminus \\mathcal{D}_0 , |\\, \n{\\rm Re}(\\alpha)\\leq Q, \\,\\, {\\rm Im}(\\sqrt{P^2+2\\ell})>{\\rm Im}(P)-\\gamma\/2\\Big\\}\n\\end{equation}\nwhere $\\mathcal{D}_0 :=\\bigcup_{j\\geq 0}\\{Q\\pm i\\sqrt{2j}\\}$ is a discrete set where \n$\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ is continuous in $\\alpha$ with at most square root singularities. Moreover $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ \nis an eigenfunction of ${\\bf H}$ with eigenvalue \n$\\frac{Q^2+P^2}{2}+\\ell$ if $\\alpha=Q+iP$.\nThe set $W_\\ell$ is a connected subset of the half-plane $\\{{\\rm Re}(\\alpha)\\leq Q\\}$, containing $(Q+i\\mathbb{R})\\setminus \\mathcal{D}_0$ and the real half-line $(-\\infty,Q-\\frac{2\\ell}{\\gamma}-\\frac{\\gamma}{4})$. Moreover, for $\\alpha\\ll -1$ real valued, $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ is given by \n\\begin{equation}\\label{defdescendantsbis}\n\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\lim_{t \\to +\\infty}e^{t(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|)}e^{-t{\\bf H}}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\n\\end{equation}\nwhere the limit holds in a weighted space $e^{-\\beta\\rho}L^2$ for $\\beta>0$ large enough. \n\\end{proposition}\n\nWe first upgrade the statement \\eqref{defdescendantsbis} to a limit in $e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q})$ for $\\alpha\\ll -1$ real valued and $\\beta>Q-\\alpha$:\n\\begin{equation}\\label{defdescendantsbisstrong}\n\\lim_{t\\to \\infty}\\|\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}-e^{t(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|)}e^{-t{\\bf H}}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\\|_{e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q})}=0\n\\end{equation} \n To prove this, we note that for some small $\\epsilon>0$, there is $C_\\epsilon>0$ independent of $t$ such that\n \\begin{align*} \n & \\|\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}-e^{t(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|)}e^{-t{\\bf H}}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\\|_{e^{-\\beta\\rho}\\mathcal{D}(\\mathcal{Q})} \\\\\n&= \\|e^{-\\epsilon{\\bf H}}e^{\\epsilon(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|)}(\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}-e^{(t-\\epsilon)(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|)}e^{-(t-\\epsilon){\\bf H}}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0)\\|_{e^{-\\beta\\rho}\\mathcal{D}(\\mathcal{Q})}\\\\\n & \\leq C_\\epsilon \\|\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}-e^{(t-\\epsilon)(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|)}e^{-(t-\\epsilon){\\bf H}}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0)\\|_{e^{-\\beta \\rho} L^2}\n \\end{align*}\nwhere we used \\cite[Lemma 6.5]{GKRV} which states that for all $t>0$ and all $\\beta \\in \\mathbb{R}$\n\\[\ne^{-t{\\bf H}}: e^{-\\beta \\rho}L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T}) \\to e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q}), \\quad e^{-t{\\bf H}}: e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})\\to e^{-\\beta \\rho}L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})\n\\]\nare bounded. Note that, writing $e^{-t{\\bf H}}=e^{-\\frac{t}{2}{\\bf H}}e^{-\\frac{t}{2}{\\bf H}}$, this also implies boundedness of the operator\n\\begin{equation}\\label{boundednesspropag}\ne^{-t{\\bf H}}: e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})\\to e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q}).\n\\end{equation}\n\\begin{corollary}\\label{cor:cts}\nLet $\\mathbf{v}$ be a Markovian vector field which satisfies the conditions of Theorem \\ref{theoremfreefield}. If $\\alpha\\ll -1$ real valued and $\\beta>Q-\\alpha$ then the following limit holds \n\\[ \\lim_{t\\to +\\infty}\\|\\mathbf{H}_\\mathbf{v}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}-e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)}\\mathbf{H}_\\mathbf{v} e^{-t\\mathbf{H}}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\\|_{e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})}=0.\\]\n\\end{corollary}\n\\begin{proof}\nThis is an immediate consequence of \\eqref{defdescendantsbisstrong} and the continuity of \n\\begin{equation}\\label{boundednessHvweight}\n\\mathbf{H}_\\mathbf{v}: e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q})\\to e^{-\\beta\\rho}\\mathcal{D}'(\\mathcal{Q}).\n\\end{equation}\nThis last fact can be checked by considering the expression \\eqref{defHv}: one has \n${\\bf L}_n^0e^{-\\beta \\rho}= e^{-\\beta \\rho}({\\bf L}_n^0-i\\beta\\rho'{\\bf A}_n)$ (and similarly for $\\tilde{{\\bf L}}_n^0e^{-\\beta \\rho}$) and \n ${\\bf L}_n^0,{\\bf A}_n$ are bounded as maps $\\mathcal{D}(\\mathcal{Q})\\to \\mathcal{D}'(\\mathcal{Q})$, while $e^{\\beta \\rho}{\\bf H}e^{-\\beta \\rho}=\\mathbf{H} -\\tfrac{\\beta^2}{2}(\\rho'(c))^2+\\tfrac{\\beta}{2} \\rho''(c)+\\beta \\rho'(c)\\partial_{c}$ and all terms in the RHS are bounded from $\\mathcal{D}(\\mathcal{Q})$ to $\\mathcal{D}'(\\mathcal{Q})$.\n\\end{proof}\n\nNext, we show that the action of $\\mathbf{H}_\\mathbf{v}$ on Liouville descendants can be written as a Poisson transform of the action of ${\\bf H}_{\\mathbf{v}}^0$ on descendants of the free field. \n\\begin{lemma}\\label{prop:intertwine_reps}\nUnder the assumptions of Corollary \\ref{cor:cts}, we have the following limit in $e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})$\n\\[\\mathbf{H}_\\mathbf{v}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)} e^{-t\\mathbf{H}}\\mathbf{H}^0_{\\mathbf{v}_t}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0.\\]\nwhere $\\mathbf{v}_t=e^tv(e^{-t}z)\\partial_z$. Similarly, if $\\mathbf{v}$ is a polynomial then \n\\[\\mathbf{H}_\\mathbf{v}^*\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)} e^{-t\\mathbf{H}}(\\mathbf{H}^0_{\\mathbf{v}_{-t}})^*\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0.\\]\nIn particular, for all $n\\in\\mathbb{Z}$,\n\\begin{align*}\n&\\mathbf{L}_n\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|-n)} e^{-t\\mathbf{H}}\\mathbf{L}_n^0\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\\\\\n&\\tilde{\\mathbf{L}}_n\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|-n)} e^{-t\\mathbf{H}}\\tilde{\\mathbf{L}}_n^0\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0.\n\\end{align*}\n\\end{lemma}\n\n\n\\begin{proof}\nWe begin with the case of a Markovian vector field $\\mathbf{v}$ and we introduce the notation $v_t(z)=e^tv(e^{-t}z)$. First, we remark that both $\\mathbf{H}_\\mathbf{v} e^{-t\\mathbf{H}}$ and $e^{-t\\mathbf{H}}\\mathbf{H}_{\\mathbf{v}_t}^0$ are well-defined as continuous operators $e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q})\\to e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})$ for all $\\beta\\in \\mathbb{R}$ by \\eqref{boundednesspropag} and \\eqref{boundednessHvweight} since for all $t$ the vector field $\\mathbf{v}_t$ satisfies the conditions of Theorem \\ref{theoremfreefield}. By definition, we have $\\mathbf{H}_\\mathbf{v} e^{-t\\mathbf{H}}=-\\frac{\\d}{\\d\\varepsilon}_{|\\varepsilon=0}e^{-\\varepsilon\\mathbf{H}_\\mathbf{v}}e^{-t\\mathbf{H}}$. \nBy the Markov property of the GFF, we have for $g_{t,\\epsilon}(z)=e^{\\varepsilon\\mathbf{v}}e^{t\\mathbf{v}_0}(z)$ and $F\\in \\mathcal{C}_{\\rm exp}$\n\\[ e^{-\\varepsilon\\mathbf{H}_\\mathbf{v}}e^{-t\\mathbf{H}}F(c,\\varphi)=(1+ o(\\epsilon))e^{-\\frac{Q^2t}{2}}\\mathbb{E}_{\\varphi}\\Big[ F\\Big(\\Big(c+P\\varphi\\circ g_{t,\\epsilon} +X_\\mathbb{D} \\circ g_{t,\\epsilon}+Q\\log \\frac{|g_{t,\\epsilon}'|}{|g_{t,\\epsilon}|}\\Big)\\Big|_{\\mathbb{T}}\\Big) e^{- \\mu e^{\\gamma c} \\int_{\\mathbb{D}\\setminus g_{t,\\epsilon}(\\mathbb{D})} \\frac{e^{\\gamma X(x)}} {|x| ^{\\gamma Q} } \\text{\\rm d} x } \\Big]. \\]\nLet us study the conformal map $g_{t,\\epsilon}$ as $\\epsilon \\to 0$\n\\[e^{\\varepsilon\\mathbf{v}}e^{t\\mathbf{v}_0}(z)=e^{-t}z+\\varepsilon v(e^{-t}z)+o(\\varepsilon).\\]\nNow, we want to commute $e^{\\varepsilon\\mathbf{v}}$ with $e^{t\\mathbf{v}_0}$. We have $e^{-t}z+\\varepsilon v(e^{-t}z)+o(\\varepsilon)=e^{-t}(z+\\varepsilon v_t(z)+o(\\varepsilon))$ with $v_t(z)=e^{t}v(e^{-t}z)$. If we define the vector field $\\mathbf{v}_t=e^tv(e^{-t}z)\\partial_z$, \n we have $e^{\\varepsilon\\mathbf{v}}e^{t\\mathbf{v}_0}=e^{t\\mathbf{v}_0}e^{\\varepsilon\\mathbf{v}_t}+o(\\varepsilon)$. Differentiating at $\\varepsilon=0$ gives the identity of vector fields\n $\\mathbf{v}(e^{t\\mathbf{v}_0}(z))=(de^{t\\mathbf{v}_0}).\\mathbf{v}_t(z)$ and \n \\[e^{-\\varepsilon\\mathbf{H}_\\mathbf{v}}e^{-t\\mathbf{H}}F(c,\\varphi)=e^{-t\\mathbf{H}}e^{-\\varepsilon\\mathbf{H}_{\\mathbf{v}_t}}F(c,\\varphi)+o(\\varepsilon).\\]\nWe deduce by differentiating at $\\varepsilon=0$\n\\begin{equation} \\label{eq:BCH}\n \\mathbf{H}_\\mathbf{v} e^{-t\\mathbf{H}}=e^{-t\\mathbf{H}}\\mathbf{H}_{\\mathbf{v}_t}\n \\end{equation}\n as continuous operators $e^{-\\beta \\rho}\\mathcal{D}(\\mathcal{Q})\\to e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})$ for all $\\beta\\in \\mathbb{R}$. \nTherefore, by Corollary \\ref{cor:cts}, we have the following limit in $e^{-\\beta \\rho} \\mathcal{D}'(\\mathcal{Q})$ as $t\\to\\infty$:\n\\begin{align*}\n\\mathbf{H}_\\mathbf{v}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}\n=\\underset{t\\to\\infty}{\\lim}\\,e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)}\\mathbf{H}_\\mathbf{v} e^{-t\\mathbf{H}}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\n&=\\underset{t\\to\\infty}{\\lim}\\,e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)}e^{-t\\mathbf{H}}\\mathbf{H}_{\\mathbf{v}_t}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\n\\end{align*}\nNow, we write $\\mathbf{H}_{\\mathbf{v}_t}=\\mathbf{H}_{\\mathbf{v}_t}^0+\\mu e^{\\gamma c}V_{\\mathbf{v}_t}$ and we need to show that \n\\[\\underset{t\\to\\infty}{\\lim}\\,e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)}e^{-t\\mathbf{H}}e^{\\gamma c}V_{\\mathbf{v}_t}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0=0\\]\nin $e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})$. We can use the probabilistic representation of the semigroup \\eqref{FeynmanKac}. Writing $X=X_\\mathbb{D}+P\\varphi$, $\\varphi_t(e^{i\\theta})=X(e^{-t+i\\theta})-B_t$ and $c_t=c+B_t$, using that $|v_t|\\leq C$ for some uniform $C>0$, we have for all $c\\in\\mathbb{R}$ and $t>0$ \n\n \\begin{align*}\n&\\left|\\mathbb{E}_\\varphi \\left[e^{-t\\mathbf{H}}( e^{\\gamma c}V_{\\mathbf{v}_t}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0)\\right]\\right|\\\\\n&\\leq C e^{-\\frac{Q^2}{2}t}e^{(\\gamma+\\alpha-Q)c}\\mathbb{E}_\\varphi\\left[e^{(\\alpha-Q)B_t}|\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi_t)|\\int_0^{2\\pi}e^{\\gamma X(e^{-t+i\\theta})}\\d\\theta e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_t^c}\\frac{e^{\\gamma X(x)}}{|x|^{\\gamma Q}} \\text{\\rm d} x }\\right]\\\\\n&=C |1-e^{-t}|^{\\frac{\\gamma^2}{2}}e^{-\\frac{Q^2}{2}t+\\frac{1}{2}(\\gamma+\\alpha-Q)^2t}e^{(\\gamma+\\alpha-Q)c}\\int_0^{2\\pi}e^{\\gamma P\\varphi (e^{-t+i \\theta})} \\mathbb{E}_\\varphi \\left[|\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi_t)|e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_t^c}\\frac{ e^{\\gamma X(x)} }{|x|^{\\gamma\\alpha}|x-e^{-t+i\\theta}|^{\\gamma^2}} \\text{\\rm d} x}\\right]\\d\\theta\\\\\n&\\leq C e^{-2t\\Delta_\\alpha}e^{\\gamma(\\alpha-Q)t}e^{(\\gamma+\\alpha-Q)c} e^{\\gamma \\sup_{\\theta} P\\varphi (e^{-t+i \\theta})} \\mathbb{E}_\\varphi [|\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi_t)|]\n\\end{align*}\nwhere we used the Girsanov transform in the third line. Since $\\P_\\mathbb{T}$ is stationary for the Ornstein-Uhlenbeck process, we have $\\mathbb{E}[|\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi_t)|^4]=\\mathbb{E}[|\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi)|^4]<\\infty$ uniformly in $t$. Hence for $t \\geq 1$ and by using Jensen on the conditional expectation $\\mathbb{E}_\\varphi[\\cdot]$\n\\begin{align*}\n\\mathbb{E}[ | e^{\\gamma \\sup_{\\theta} P\\varphi (e^{-t+i \\theta})} \\mathbb{E}_\\varphi [|\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi_t)|] |^2] & \\leq \\mathbb{E}[ | e^{4 \\gamma \\sup_{\\theta} P\\varphi (e^{-t+i \\theta})} | ]^{1\/2} \\mathbb{E}[ \\mathbb{E}_\\varphi [|\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi_t)|] ^4] ]^{1\/2} \\\\\n& \\leq \\mathbb{E}[ | e^{4 \\gamma \\sup_{\\theta} P\\varphi (e^{-t+i \\theta})} | ]^{1\/2} \\mathbb{E}[ |\\mathcal{Q}_{\\alpha,\\nu,\\tilde{\\nu}}(\\varphi_t)|^4 ] ^{1\/2} \\\\\n& \\leq C \n\\end{align*}\nMoreover, we have $\\int_\\mathbb{R} e^{2(\\gamma+\\alpha-Q)c}e^{2\\beta\\rho(c)}\\d c<\\infty$ since $\\beta>Q-\\alpha$ and $\\alpha-Q+\\gamma<0$ so there is $C>0$ such that\n\\[e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)}\\|e^{-t\\mathbf{H}}(e^{\\gamma c}V_{\\mathbf{v}_t}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0)\\|_{e^{-\\beta\\rho}L^2}\\leq Ce^{(\\gamma(\\alpha-Q)+|\\nu|+|\\tilde{\\nu}|)t} \\]\nwhich converges to 0 for $\\alpha < Q-\\frac{|\\nu|+|\\tilde{\\nu}|}{\\gamma} $. This proves that the limit\n\\[\\mathbf{H}_\\mathbf{v}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)} e^{-t\\mathbf{H}}\\mathbf{H}^0_{\\mathbf{v}_t}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\\]\nholds in $e^{-\\beta \\rho}\\mathcal{D}'(\\mathcal{Q})$.\n\nIn fact, identity \\eqref{eq:BCH} holds for all vector field $v$ (not just Markovian) which admits a holomorphic extension in a neighorhood of $\\mathbb{D}$. We use \\eqref{eq:BCH} and the fact that $e^{-t\\mathbf{H}}:\\mathcal{D}'(\\mathcal{Q})\\to\\mathcal{D}(\\mathcal{Q})$ is continuous to obtain for all $F,G\\in\\mathcal{D}(\\mathcal{Q})$\n\\begin{align*}\n\\langle\\mathbf{H}_{\\mathbf{v}_t}^*e^{-t\\mathbf{H}}F,G\\rangle_2\n=\\langle e^{-t\\mathbf{H}}F,\\mathbf{H}_{\\mathbf{v}_t}G\\rangle_2\n&=\\langle F,e^{-t\\mathbf{H}}\\mathbf{H}_{\\mathbf{v}_t}G\\rangle_2\\\\\n&=\\langle F,\\mathbf{H}_\\mathbf{v} e^{-t\\mathbf{H}}G\\rangle_2\\\\\n&=\\langle\\mathbf{H}_\\mathbf{v}^*F,e^{-t\\mathbf{H}}G\\rangle_2\\\\\n&=\\langle e^{-t\\mathbf{H}}\\mathbf{H}_\\mathbf{v}^*F,G\\rangle_2,\n\\end{align*}\nwhich implies the equality of bounded operators $\\mathcal{D}(\\mathcal{Q})\\to\\mathcal{D}'(\\mathcal{Q})$\n\\begin{equation}\\label{staridentity}\n\\mathbf{H}_{\\mathbf{v}_t}^*e^{-t\\mathbf{H}}=e^{-t\\mathbf{H}}\\mathbf{H}_\\mathbf{v}^*.\n\\end{equation}\nIf $\\mathbf{v}$ is a polynomial then we can apply identity \\eqref{staridentity} with $\\mathbf{v}_{-t}$ in place of $\\mathbf{v}$ and using similar arguments as in the previous case we deduce that\n$\\mathbf{H}_\\mathbf{v}^*\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)} e^{-t\\mathbf{H}}(\\mathbf{H}^0_{\\mathbf{v}_{-t}})^*\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0$ provided that $\\alpha < Q-\\frac{N-1+|\\nu|+|\\tilde{\\nu}|}{\\gamma} $ where $N$ is the degree of the polynomial (this is due to the fact that $|e^{-t}v(e^{t}z)| \\leq e^{(N-1)t}$).\n\n\n\\end{proof}\n\n\n\\begin{proposition}\\label{intertwining_for_descendants}\nLet $\\ell\\in \\mathbb{N}$, and $\\nu,\\tilde{\\nu}\\in\\mathcal{T}$ such that $|\\nu|+|\\tilde{\\nu}|=\\ell$. For $n\\in \\mathbb{Z}$, and $\\alpha$ in the set $W_\\ell\\cap W_{\\ell-n}$ defined by \\eqref{defWell} and $\\alpha\\notin Q-\\frac{\\gamma}{2}\\mathbb{N}-\\frac{2}{\\gamma}\\mathbb{N}$, for $\\beta>Q-{\\rm Re}(\\alpha)$, \nwe have the following identities in $e^{-\\beta\\rho}\\mathcal{D}(\\mathcal{Q})$ \n\\begin{equation}\\label{eq:intertwine_reps_ctd}\n \\mathbf{L}_{n}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\sum_{\\nu'\\in \\mathcal{T}}\\ell_{n}^\\alpha(\\nu,\\nu') \\Psi_{\\alpha,\\nu',\\tilde{\\nu}},\\qquad \n \\tilde{\\mathbf{L}}_{n}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\sum_{\\tilde{\\nu}\\in \\mathcal{T}}\\tilde{\\ell}_{n}^{\\alpha}(\\tilde{\\nu},\\tilde{\\nu}')\\Psi_{\\alpha,\\nu,\\tilde{\\nu}'}\n \\end{equation}\n where $\\ell_{\\alpha,n}(\\nu,\\nu')$ is defined by \\eqref{descendant_general}. \nIn particular, if $n>0$, $\\nu=(\\nu_1,\\dots,\\nu_k)\\in \\mathcal{T}$ and $\\tilde{\\nu}=(\\tilde{\\nu}_1,\\dots,\\tilde{\\nu}_{k'})$,\n\\begin{equation}\\label{eq:intertwine_reps_ctd2} \n\\forall n\\leq \\nu_k, \\,\\, \\mathbf{L}_{-n}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\Psi_{\\alpha,(\\nu,n),\\tilde{\\nu}}, \\qquad \n\\forall n\\leq \\tilde{\\nu}_{k'} ,\\, \\, \\mathbf{L}_{-n}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\Psi_{\\alpha,\\nu,(\\tilde{\\nu},n)}.\n\\end{equation}\n\\end{proposition}\n\\begin{proof} Consider the vector field ${\\bf v}=\\omega {\\bf v}_0+{\\bf v}_n=(-\\omega z+z^{n+1})\\partial_z=:v(z)\\partial_z$ for $n\\geq 1$ and $\\omega>0$ large enough so that ${\\bf v}$ is Markovian. The vector field ${\\bf v}_t=e^{t}v(e^{-t})\\partial_z$ can be written \n\\[ {\\bf v}_t= -\\omega {\\bf v}_0+e^{-nt}{\\bf v}_n.\\]\nBy \\eqref{defHv} (applied to the free-field case), one has ${\\bf H}^0_{{\\bf v}_t}=\\omega {\\bf H}^0+e^{-nt}{\\bf L}_n^0$. \nBy Lemma \\ref{prop:intertwine_reps}, we then get \n\\[\\begin{split} \n{\\bf H}_{\\mathbf{v}} \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=& \n\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)} e^{-t\\mathbf{H}}\\mathbf{H}^0_{\\mathbf{v}_t}\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\\\\\n=& \\omega(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|) \\Psi_{\\alpha,\\nu,\\tilde{\\nu}} + \n\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|-n)} \\sum_{\\nu'\\in\\mathcal{T}}\\ell_{n}^\\alpha(\\nu,\\nu')\\Psi_{\\alpha,\\nu',\\tilde{\\nu}}^0\\\\\n=& \\omega(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|) \\Psi_{\\alpha,\\nu,\\tilde{\\nu}} + \\sum_{\\nu'\\in\\mathcal{T}}\\ell_{n}^\\alpha(\\nu,\\nu')\\Psi_{\\alpha,\\nu',\\tilde{\\nu}}\n\\end{split}\\] \nSince \\eqref{defHv} tells us that $\\mathbf{H}_\\mathbf{v}=\\omega \\mathbf{H}+{\\bf L}_n$, we see that \\eqref{eq:intertwine_reps_ctd} holds for $n\\leq -1$ when $\\alpha$ is real valued. Using the holomorphic extension of \n$\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ with respect to $\\alpha$ in Proposition \\ref{constPoisson}, one obtains the result in $W_{\\ell}\\cap W_{\\ell+n}$.\n\nTo deal with the ${\\bf L}_{-n}$ case with $n>0$, we apply the same type of argument: \none has $({\\bf H}^0_{{\\bf v}_{-t}})^*=\\omega {\\bf H}^0+e^{nt}{\\bf L}_{-n}^0$ and \n\\[\\begin{split} \n{\\bf H}_{\\mathbf{v}}^* \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=& \n\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)} e^{-t\\mathbf{H}}(\\mathbf{H}^0_{\\mathbf{v}_{-t}})^*\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}^0\\\\\n=& \\omega(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|) \\Psi_{\\alpha,\\nu,\\tilde{\\nu}} + \n\\underset{t\\to\\infty}{\\lim}\\, e^{t(2\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|+n)} e^{-t\\mathbf{H}}\\sum_{\\nu'\\in\\mathcal{T}}\\ell_{n}^\\alpha(\\nu,\\nu')\\Psi_{\\alpha,\\nu'\\tilde{\\nu}}^0\\\\\n=& \\omega(\\Delta_{\\alpha}+|\\nu|+|\\tilde{\\nu}|) \\Psi_{\\alpha,\\nu,\\tilde{\\nu}} + \\sum_{\\nu'\\in\\mathcal{T}}\\ell_{n}^\\alpha(\\nu,\\nu')\\Psi_{\\alpha,\\nu',\\tilde{\\nu}}.\n\\end{split}\\] \nSince ${\\bf H}_{\\mathbf{v}}^*=\\omega \\mathbf{H}+{\\bf L}_{-n}$, we obtain the desired result, since the same argument works exactly the same for $\\tilde{\\bf L}_n\\Psi_{\\alpha,\\nu,\\nu'}$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\\section{Scattering coefficient, analyticity of the eigenstates and Verma modules of Liouville}\n\nWe can define the scattering matrices as in \\cite[Definition 6.23]{GKRV}: \n\\begin{definition} \nFor each $\\ell\\in \\mathbb{N}_0$, the scattering matrix ${\\bf S}_\\ell(Q+ip):E_\\ell\\to E_\\ell$, defined for $p\\in \\mathbb{R}\\setminus[-\\sqrt{2\\ell},\\sqrt{2\\ell}]$ \nis the linear map defined by: $\\forall F_j\\in \\ker ({\\bf P}-j)$ with $j\\leq \\ell$\n\\begin{equation}\\label{scatmatrix}\n{\\bf S}_\\ell(Q+ip)\\sum_{j=0}^{\\ell}F_j:= \\sum_{j=0}^{\\ell}F_j^+(Q+ip)\n\\end{equation}\nwhere $F_j^+(Q+ip)\\in \\ker ({\\bf P}-j)$ are the functions appearing in the asymptotic expansion \\eqref{expansionP} of $\\mathcal{P}_\\ell(Q+ip)\\sum_{j=0}^\\ell F_j$. The matrices ${\\bf S}_0(\\alpha)$ admit an analytic extension from $Q+i\\mathbb{R}$ to ${\\rm Re}(\\alpha)\\in(Q-\\epsilon,Q]$ for some $\\epsilon>0$.\n\\end{definition}\nFrom \\eqref{egalitepoissonell}, we also see that for $j<\\ell$\n\\[ {\\bf S}_\\ell(Q+ip)|_{\\ker ({\\bf P}-j)}={\\bf S}_j(Q+ip)|_{\\ker ({\\bf P}-j)}.\\]\nIt is convenient to use a change of complex parameter $p=\\sqrt{P^2+2\\ell}$ and define \n\\[ \\tilde{\\bf S}_\\ell(Q+iP):={\\bf S}_\\ell(Q+i\\sqrt{P^2+2\\ell})\\]\nBy \\cite[Corollary 6.24]{GKRV}, $\\tilde{\\bf S}_\\ell(Q+iP)$ is analytic in \n\\[\\{P\\in \\mathbb{C}\\,|\\, {\\rm Im}(P)\\in [0,\\gamma\/2), \n{\\rm Im}\\sqrt{P^2+2\\ell}\\geq 0\\} \\setminus \\bigcup_{j\\geq \\ell}\\pm\\sqrt{2(j-\\ell)}.\\]\n\n\nIn particular we see from \\eqref{expansionP} that for $P \\in \\mathbb{R}\\setminus \\{0\\}$, \n$\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}$ have asymptotic expansion for $\\ell:=|\\nu|+|\\tilde{\\nu}|$\n\\begin{equation}\\label{asymptoticdescendants}\n\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}(c,\\varphi)=e^{iPc}\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}(\\varphi)+ \\sum_{j=0}^\\ell \\Pi_{\\ker ({\\bf P}-j)}(\\tilde{{\\bf S}}_\\ell(Q+iP)\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}})(\\varphi)e^{-ic\\sqrt{P^2+2(\\ell-j)}}+G_{Q+iP,\\nu,\\tilde{\\nu}}(c,\\varphi)\n\\end{equation}\nwhere $G_{Q+iP,\\nu,\\tilde{\\nu}}\\in \\mathcal{D}(\\mathcal{Q})$, and they are the unique solution of $({\\bf H}-\\frac{Q^2+P^2}{2}+\\ell)u=0$ with such an asymptotic expansion where the coefficient of $e^{iPc}$ is \ngiven by $\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}(\\varphi)$.\n\n\n\n\\subsection{Computation of the reflection coefficient for primary fields} \nIn this section we compute the scattering matrix ${\\bf S}_0(\\alpha)$, also called reflection coefficient.\n \nThe primary fields are $\\Psi_{\\alpha}:=\\Psi_{\\alpha,0,0}=\\mathcal{P}_0(\\alpha)1$, by Proposition \\ref{poissonprop} they are analytic in $\\{{\\rm Re}(\\alpha)\\leq Q\\}\\setminus \\mathcal{D}_0$ and satisfy in ${\\rm Re}(\\alpha)\\in (Q-\\gamma\/2,Q]$\n\\begin{equation}\\label{asymptpsialpha}\n \\Psi_{\\alpha}=e^{(\\alpha-Q)c}+ e^{(Q-\\alpha)c}{\\bf S}_0(\\alpha)1+G_\\alpha, \\quad \\textrm{with } G_{\\alpha}\\in \\mathcal{D}(\\mathcal{Q}).\n \\end{equation}\n\\begin{proposition}\\label{coefreflection}\nFor ${\\rm Re}(\\alpha)\\in (Q-\\gamma\/2,Q]$, the scattering coefficient $R(\\alpha):={\\bf S}_0(\\alpha)1$ is given by the explicit expression \n\\begin{equation}\\label{Ralpha} \nR(\\alpha)=-\\Big(\\pi \\mu \\frac{\\Gamma(\\frac{\\gamma^2}{4})}{\\Gamma(1-\\frac{\\gamma^2}{4})}\\Big)^{2\\frac{(Q-\\alpha)}{\\gamma}}\\frac{\\Gamma(-\\frac{\\gamma(Q-\\alpha)}{2})\\Gamma(-\\frac{2(Q-\\alpha)}{\\gamma})}{\\Gamma(\\frac{\\gamma(Q-\\alpha)}{2})\\Gamma(\\frac{2(Q-\\alpha)}{\\gamma})}.\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof} \n\nFirst we prove the result for $\\alpha\\in (Q-\\gamma\/2,Q)$ and then we use the fact that both ${\\bf S}_0(\\alpha)1$ and the RHS of \\eqref{Ralpha} extend analytically in $\\{{\\rm Re}(\\alpha)\\in (Q-\\epsilon,Q]\\}\\setminus \\mathcal{D}_0$ for some $\\epsilon>0$ small. \nFor $ \\alpha0}$ of the Dirac mass $\\delta_0$. First let us show that the contribution of the integral outside of any disk \n$\\mathbb{D}_r=\\{|x|\\leq r\\}$ does not not play a part in the asymptotics.\nUsing the inequality $1-e^{-\\mu x}\\leq \\mu x$ in the second line below, we get\n\\begin{align*}\n| \\mathbb{E}_\\varphi [ e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}} |x|^{-\\gamma\\alpha }e^{\\gamma X(x)} \\text{\\rm d} x}] -\\mathbb{E}_\\varphi [e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_r} |x|^{-\\gamma\\alpha }e^{\\gamma X(x)} \\text{\\rm d} x}]| &\n\\leq \\mathbb{E}_\\varphi[1-e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}\\setminus \\mathbb{D}_r} |x|^{-\\gamma\\alpha }e^{\\gamma X(x)} \\text{\\rm d} x } ]\n\\\\\n\\leq & \\mu e^{\\gamma c}r^{-\\alpha\\gamma} \\mathbb{E}_\\varphi [\\int_\\mathbb{D} e^{\\gamma P\\varphi}(1-|x|^2)^{\\frac{\\gamma^2}{2}}M_{\\gamma,\\mathbb{D}}(\\text{\\rm d} x)]\\\\\n\\leq & \\mu e^{\\gamma c}r^{-\\alpha\\gamma} \\int_\\mathbb{D} e^{\\gamma P\\varphi(x)}(1-|x|^2)^{\\frac{\\gamma^2}{2}}\\text{\\rm d} x.\n\\end{align*}\nFor fixed $r$, this quantity is $e^{\\gamma c}\\mathcal{O}(1)$, where $\\mathcal{O}(1)$ means bounded in $L^2(\\Omega_\\mathbb{T})$ as $c\\to-\\infty$. Therefore we can focus on evaluating what happens inside $\\mathbb{D}_r$. We want to make use of Kahane's convexity inequalities (see \\cite{review}) to estimate the contribution coming from $\\mathbb{D}_r$. Let us set $m_r:=-\\inf_{x,x'\\in\\mathbb{D}_r}\\log |1-x\\bar x'| $. For $x,x'\\in \\mathbb{D}_r$ \n\\begin{equation}\n\\ln \\frac{1}{|x-x'|}-m_r\\leq G_\\mathbb{D}(x,x')\\leq \\log \\frac{1}{|x-x'|}.\n\\end{equation}\nLet us consider a centered Gaussian field $\\tilde{X}$ with covariance $\\mathbb{E}[\\tilde{X}(x)\\tilde{X}(x')]=\\ln \\frac{1}{|x-x'|}$ inside $\\mathbb{D}_r$, a standard Gaussian random variable $Z$ and the GFF on the circle $\\varphi$, all of them independent of each other under $\\P$ (with expectation $\\mathbb{E}$). Let $M_{\\gamma,\\tilde{X}}$ be the GMC measure of $\\tilde{X}$ and set $E_r:= e^{\\gamma m_r^{1\/2}Z-\\frac{\\gamma^2m_r}{2}}$. Using Kahane's inequalities, we get\n\\begin{equation}\\label{eq:def_D}\n\\begin{aligned}\nD(c,\\varphi)\n&:=\\mathbb{E}_\\varphi [ 1-e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma} e^{\\gamma P\\varphi(x)}(1-|x|^2)^{\\frac{\\gamma^2}{2}}M_{\\gamma,\\tilde{X}}(\\text{\\rm d} x)}] \\\\\n&\\geq\\mathbb{E}_\\varphi\\Big[1-e^{-\\mu e^{\\gamma c}E_r\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma}e^{\\gamma P\\varphi(x)}(1-|x|^2)^\\frac{\\gamma^2}{2}M_{\\gamma,\\mathbb{D}}(\\d x)}\\Big]\n\\end{aligned}\n\\end{equation}\nand \n\\begin{align*}\nd(c,\\varphi):=\n&\\mathbb{E}_\\varphi\\Big[ 1-e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma} e^{\\gamma P\\varphi(z)}(1-|x|^2)^{\\frac{\\gamma^2}{2}} M_{\\gamma,\\tilde{X}}(\\text{\\rm d} x)}\\Big]\\\\\n&\\leq\\mathbb{E}_\\varphi\\Big[1-e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma}e^{\\gamma P\\varphi(x)}(1-|x|^2)^\\frac{\\gamma^2}{2}M_{\\gamma,\\mathbb{D}}(\\d x)}\\Big].\n\\end{align*}\nNext we show that, in $L^2(\\mathbb{R}\\times\\Omega_\\mathbb{T})$, \n\\begin{align}\n&\\limsup_{c\\to-\\infty}e^{2(\\alpha-Q)c} D(c,\\varphi)\n\\leq-\\mu^{\\frac{2(Q-\\alpha)}{\\gamma}} \\tfrac{2(Q-\\alpha)}{\\gamma}\\bar{R}(\\alpha)\\Gamma\\big(-\\tfrac{2(Q-\\alpha)}{\\gamma}\\big)\\label{limsup}\\\\\n&\\liminf_{c\\to-\\infty}e^{2(\\alpha-Q)c} d(c,\\varphi)\n\\geq-\\mu^{\\frac{2(Q-\\alpha)}{\\gamma}} \\tfrac{2(Q-\\alpha)}{\\gamma}\\bar{R}(\\alpha)\\Gamma\\big(-\\tfrac{2(Q-\\alpha)}{\\gamma}\\big)\\label{liminf}\n\\end{align}\nwith $\\bar{R}(\\alpha)$ the unit volume reflection coefficient (see \\cite[Equations (3.10) \\& (3.12)]{dozz}).\nAs the proof is quite similar for both of them, we will only focus on the first one.\n\nIt was proven in \\cite[Lemma 3.1]{dozz} that\n$$\\big|\\P\\Big(\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma} (1-|x|^2)^{\\frac{\\gamma^2}{2}} M_{\\gamma,\\tilde{X}}(\\text{\\rm d} x)>u\\Big)-u^{-\\frac{2}{\\gamma}(Q-\\alpha)}\\bar R(\\alpha)\\big|\\leq C_ru^{-\\frac{2}{\\gamma}(Q-\\alpha)-\\eta}$$\nfor some constants $C_r>0$ and $\\eta>0$. Therefore, setting $S_r:=\\sup_{x\\in\\mathbb{D}_r}e^{\\gamma P\\varphi(x)}$ and using for the \nrandom variable $Y\\geq 0$ the relation $\\mathbb{E}[1-e^{-aY}]=\\int_0^\\infty a\\P(Y>y)e^{-ay}\\,\\d y$, we get\n\\begin{align*}\n\\mathbb{E}_\\varphi\\Big[ 1-&\\exp\\big(-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_r} |x|^{-\\alpha\\gamma} e^{\\gamma P\\varphi(x)}(1-|x|^2)^{\\frac{\\gamma^2}{2}} M_{\\gamma,\\tilde{X}}(\\text{\\rm d} x))\\Big]\\\\\n\\leq & \\mathbb{E}_\\varphi\\Big[ 1-\\exp\\big(-\\mu e^{\\gamma c}S_r\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma} (1-|x|^2)^{\\frac{\\gamma^2}{2}} M_{\\gamma,\\tilde{X}}(\\text{\\rm d} x)\\big)\\Big]\\\\\n= &\\mathbb{E}_\\varphi\\Big[\\int_0^\\infty\\P \\Big(\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma} (1-|x|^2)^{\\frac{\\gamma^2}{2}} M_{\\gamma,\\tilde{X}}(\\text{\\rm d} x)>u\\Big)\\mu e^{\\gamma c}S_re^{-u\\mu e^{\\gamma c}S_r}\\,du\\Big]\\\\\n\\leq &\\mathbb{E}_\\varphi\\Big[\\int_0^\\infty \\Big( u^{-\\frac{2}{\\gamma}(Q-\\alpha)}\\bar R(\\alpha)+C_ru^{-\\frac{2}{\\gamma}(Q-\\alpha)-\\eta}\\Big)\\mu e^{\\gamma c}S_re^{-u \\mu e^{\\gamma c}S_r }\\,du\\Big]\\\\\n=&\\bar R(\\alpha)\\Gamma\\Big(1-\\frac{2}{\\gamma}(Q-\\alpha)\\Big)\\mu^{\\frac{2}{\\gamma}(Q-\\alpha)}e^{2(Q-\\alpha)c}S_r^{\\frac{2}{\\gamma}(Q-\\alpha)} \\\\\n&+C_r \\mu^{\\frac{2}{\\gamma}(Q-\\alpha)+\\eta}e^{2(Q-\\alpha)c+\\gamma\\eta c}S_r^{\\frac{2}{\\gamma}(Q-\\alpha)+\\eta}.\n\\end{align*}\nAs $\\lim_{r\\to 0}S_r^{\\beta}=1$ in $L^2(\\Omega_\\mathbb{T})$ for any fixed $\\beta>0$, we deduce that\n\\begin{align*}\n\\limsup_{c\\to-\\infty}e^{2(\\alpha-Q)c} &\\mathbb{E}_\\varphi\\Big[ 1-\\exp\\big(-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma} e^{\\gamma P\\varphi(x)}(1-|x|^2)^{\\frac{\\gamma^2}{2}}M_{\\gamma,\\tilde{X}}(\\text{\\rm d} x)\\big)\\Big]\\\\\n\\leq &\\bar R(\\alpha)\\Gamma\\big(1-\\frac{2}{\\gamma}(Q-\\alpha)\\big)\\mu^{\\frac{2}{\\gamma}(Q-\\alpha)}.\n\\end{align*}\nTo conclude, we need to get rid of the $E_r$ appearing in the last line of \\eqref{eq:def_D}. Along the same lines as previously and using the relation $\\mathbb{E}[1-e^{-aY}]=\\int_0^\\infty a\\P(Y>y)e^{-ay}\\,\\d y$ for a variable $Y \\geq 0$ \n\\begin{align*}\n&\\underset{r \\to 0}{\\lim}\\underset{c\\to-\\infty}{\\limsup}\\,\\Big|\\mathbb{E}_\\varphi\\Big[e^{-\\mu e^{\\gamma c}E_r\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma}e^{\\gamma P\\varphi(x)}(1-|x|^2)^\\frac{\\gamma^2}{2}M_{\\gamma,\\mathbb{D}}(\\d x)}-e^{-\\mu e^{\\gamma c}\\int_{\\mathbb{D}_r}|x|^{-\\alpha\\gamma}e^{\\gamma P\\varphi(x)}(1-|x|^2)^\\frac{\\gamma^2}{2}M_{\\gamma,\\mathbb{D}}(\\d x)}\\Big]\\Big|=0\\\\\n\\end{align*}\nCombining \\eqref{eq:def_D}, \\eqref{limsup} and \\eqref{liminf} yields the desired result.\n\\end{proof} \n\n\n\n\n\\subsection{Reflection coefficients for descendants}\n\nWe first recall from \\cite[Proposition 7.2]{GKRV} the link between $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ and $\\Psi^0_{\\alpha,\\nu,\\tilde{\\nu}}$:\n\\[ \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}=\\lim_{t\\to +\\infty}e^{t(\\Delta_\\alpha+|\\nu|+|\\tilde{\\nu}|)}e^{-t{\\bf H}}\\Psi^0_{\\alpha,\\nu,\\tilde{\\nu}}\\]\nfor all $\\alpha< \\min(Q-\\frac{\\gamma}{4}-\\frac{2(|\\nu|+|\\tilde{\\nu}|)}{\\gamma},Q-\\gamma)$, with $\\Delta_\\alpha=\\frac{\\alpha}{2}(Q-\\frac{\\alpha}{2})$ if $|\\nu|+|\\tilde{\\nu}|>0$, while this holds for all $\\alpha0$ fixed. Then by \\eqref{assumptionpsi} and \\eqref{formulaLn}, we have if $F$ is supported in $\\{c\\leq 0\\}$\n\\begin{equation}\\label{splitting}\n \\begin{split}\n\\langle \\Psi_{Q+iP,\\nu',\\tilde{\\nu}},{\\bf L}_n F\\rangle_{2} = &\\langle e^{iPc}\\mathcal{Q}_{Q+iP,\\nu',\\tilde{\\nu}}+R(Q+iP)e^{-iPc}\\mathcal{Q}_{Q-iP,\\nu',\\tilde{\\nu}}, {\\bf L}^0_nF\\rangle_{2} \\\\\n& +\\langle G_{Q+iP,\\nu',\\tilde{\\nu}},{\\bf L}_nF\\rangle_{2} \\\\\n& + \\frac{\\mu}{2}\\langle e^{iPc}\\mathcal{Q}_{Q+ip,\\nu',\\tilde{\\nu}}+R(Q+iP)e^{-iPc}\\mathcal{Q}_{Q-iP,\\nu',\\tilde{\\nu}}, e^{\\gamma c} V_n F\\rangle_{2}. \n\\end{split}\\end{equation}\nWe consider some particular choice of $F$ depending on a parameter $T$, namely\n\\[ F_T(c,\\varphi)=\\chi_T(c) e^{iP'c}h(\\varphi) , \\quad \\textrm{ with }h\\in E_{\\ell}, \\quad \\chi_T(c)=\\chi(c+T)\\]\nfor $P'\\in \\mathbb{R}$, and $\\chi \\in C_0^\\infty(\\mathbb{R};[0,1])$ is satisfies $\\chi=1$ near $0$. We also choose $\\tilde{\\chi}$ with the same property but with $\\tilde{\\chi}\\chi=\\chi$ and we let $\\tilde{\\chi}_T(c)=\\tilde{\\chi}(c+T)$.\nNote that \n${\\bf L}_n^0(e^{iP'c}h(\\varphi))= e^{iP'c}h'$ for some $h'\\in E_{\\ell-n}$.\n\nWe will show that, as $T\\to \\infty$, the first line in the RHS of \\eqref{splitting} will have an explicit asymptotic behaviour while the second and third line will go to $0$. This will allow us to isolate the scattering coefficient in the expression $\\langle \\Psi_{Q+iP,\\nu',\\tilde{\\nu}},{\\bf L}_n(\\chi_T e^{iP'c}h)\\rangle_2$.\n\nWe start with the term $\\langle G_{Q+iP,\\nu',\\tilde{\\nu}},{\\bf L}_nF_T\\rangle_{L^2(\\mathbb{R}^-\\times \\Omega_\\mathbb{T})}$ and decompose ${\\bf L}_n={\\bf L}_n^0+\\frac{\\mu}{2}e^{\\gamma c}V_n$.\nFirst, we will show that $\\langle G_{Q+iP,\\nu',\\tilde{\\nu}},e^{\\gamma c}V_nF_T\\rangle \\to 0$.\n Since $\\| \\chi_Te^{iP'c}h\\|_{\\mathcal{D}(\\mathcal{Q})}\\leq C$ for some $C>0$ uniform with respect to $T>1$,\nthen $\\chi_T e^{\\gamma c}V_ne^{iP'c}h\\in \\mathcal{D}'(\\mathcal{Q})$ with uniform norm with respect to $T>1$.\nUsing that $\\psi G_{Q+iP,\\nu',\\tilde{\\nu}}\\in \\mathcal{D}(\\mathcal{Q})$ for any $\\psi \\in C^\\infty(\\mathbb{R};[0,1])$ with support \nin $\\mathbb{R}_-$, there is $C>0$ such that as $T\\to \\infty$\n\\begin{equation}\\label{firstboundG}\n |\\langle G_{Q+iP,\\nu',\\tilde{\\nu}},e^{\\gamma c}\\chi_T V_ne^{iP'c}h\\rangle_{2}|\\leq C\\|\\tilde{\\chi}_TG_{Q+iP,\\nu',\\tilde{\\nu}}\\|_{\\mathcal{D}(\\mathcal{Q})} \\|\\chi_Te^{iP'c}h\\|_{\\mathcal{D}(\\mathcal{Q})}\n\\end{equation}\nwhere we used the boundedness $e^{\\gamma c}V_n:\\mathcal{D}(\\mathcal{Q})\\to \\mathcal{D}'(\\mathcal{Q})$.\nOne has \n\\[ \\|\\chi_Te^{iP'c}h\\|_{\\mathcal{D}(\\mathcal{Q})}^2\\leq \\frac{1}{2}\\int_{\\mathbb{R}} (|-iP'\\chi_T+\\chi'_T|^2+(Q^2+2\\ell)\\chi^2_T) \\|h\\|_{L^2(\\Omega_\\mathbb{T})}^2+\\chi_T(c)^2\\mathcal{Q}_{e^{\\gamma c}V}(h) \\text{\\rm d} c \\]\nwhich is uniformly bounded as $T\\to \\infty$ and \n\\[ \\begin{split}\n\\|\\tilde{\\chi}_TG_{Q+iP,\\nu',\\tilde{\\nu}}\\|_{\\mathcal{D}(\\mathcal{Q})}^2\\leq & \\frac{1}{2}\\int_{\\mathbb{R}} (|\\tilde{\\chi}'_T|^2+Q^2) \\|G_{Q+iP,\\nu',\\tilde{\\nu}}\\|^2\n+ \\tilde{\\chi}_T^2 \\|\\partial_c G_{Q+iP,\\nu',\\tilde{\\nu}}\\|^2 \\text{\\rm d} c\\\\\n& + \\int_{\\mathbb{R}} \\tilde{\\chi}_T^2 \\|{\\bf P}^{\\frac{1}{2}}G_{Q+iP,\\nu',\\tilde{\\nu}}\\|^2+\\tilde{\\chi}_T^2 \\mathcal{Q}_{e^{\\gamma c}V}(G_{Q+iP,\\nu',\\tilde{\\nu}})\\text{\\rm d} c\n\\end{split}\n\\]\nconverges to $0$ as $T\\to \\infty$ since $G_{Q+iP,\\nu',\\tilde{\\nu}}\\in \\mathcal{D}(\\mathcal{Q})$ and $\\mathrm{supp}\\, \\tilde{\\chi}_T\\subset [-T-1,-T+1]$. Consequently \\eqref{firstboundG} goes to $0$ as $T\\to \\infty$.\n\nNext we show that $\\langle G_{Q+iP,\\nu',\\tilde{\\nu}},{\\bf L}_n^0F_T\\rangle_2 \\to 0$. First, we compute \n\\begin{equation}\\label{commutationLnchi} \n[{\\bf L}_n,\\chi_T]=[{\\bf L}^0_n,\\chi_T]= i\\chi_T'{\\bf A}_n,\n\\end{equation}\n\\begin{equation}\\label{Ln0Psi_T}\n\\begin{split} \n {\\bf L}_n^0 (\\chi_Te^{iP'c}h) = ie^{iP'c}\\chi_T'{\\bf A}_nh+ \n \\chi_T {\\bf L}_{n}^0(e^{iP'c}h).\n \\end{split}\n \\end{equation}\n Notice that ${\\bf A}_nh\\in E_{\\ell-n}$ since $n>0$, and that $e^{-iP'c}{\\bf L}_{n}^0(e^{iP'c}h)\\in E_{\\ell-n}$. In particular one can bound \n$\\|{\\bf L}_n^0 (\\chi_Te^{iP'c}h)\\|_{L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})}\\leq C$ for some $C>0$ uniform in $T$. We then have \n \\[ |\\langle G_{Q+iP,\\nu',\\tilde{\\nu}},{\\bf L}_n^0F_T\\rangle_2|\\leq \\|\\tilde{\\chi}_TG_{Q+iP,\\nu',\\tilde{\\nu}}\\|_{L^2}\\|{\\bf L}_n^0F_T\\|_{L^2}\\leq C\\|\\tilde{\\chi}_TG_{Q+iP,\\nu',\\tilde{\\nu}}\\|_{L^2} \\to 0\\]\n as $T\\to 0$, thus the second line of \\eqref{splitting} goes to $0$ as $T\\to \\infty$.\n \nThe third line of \\eqref{splitting} also goes to $0$ as $T\\to \\infty$: \n\\begin{equation}\\label{3rdline}\n |\\langle e^{\\pm iPc}\\mathcal{Q}_{Q\\pm iP,\\nu',\\tilde{\\nu}}, e^{\\gamma c}V_n F_T\\rangle_{2}|\\leq C \\Big| \\int_{-T-1}^{-T+1}e^{\\gamma c}\\text{\\rm d} c\\Big|\\times \n|\\langle V_n h,\\mathcal{Q}_{Q\\pm iP,\\nu',\\tilde{\\nu}}\\rangle_{L^2(\\Omega_\\mathbb{T})}| \\to 0\n\\end{equation}\nwhere $\\langle V_n h,\\mathcal{Q}_{Q+iP,\\nu',\\tilde{\\nu}}\\rangle_{L^2(\\Omega_\\mathbb{T})}$ makes sense \nsince $\\mathcal{Q}_{Q+iP,\\nu',\\tilde{\\nu}},h\\in \\mathcal{S}$.\n\nFinally, we deal with the first line of \\eqref{splitting}. Since $({\\bf L}_n^0)^*={\\bf L}_{-n}^0$,\n\\begin{align*}\n & \\langle e^{iPc}\\mathcal{Q}_{Q+iP,\\nu',\\tilde{\\nu}}+R(Q+iP)e^{-iPc}\\mathcal{Q}_{Q-iP,\\nu',\\tilde{\\nu}}, {\\bf L}^0_n F_{T}\\rangle_2=\\\\\n& \\langle \\chi _T(c){\\bf L}^0_{-n}(e^{iPc}\\mathcal{Q}_{Q+iP,\\nu',\\tilde{\\nu}}+R(Q+iP)e^{-iPc}\\mathcal{Q}_{Q-iP,\\nu',\\tilde{\\nu}}), e^{iP'c}h\\rangle_2.\n\\end{align*}\nWe compute for all $P\\in \\mathbb{R}$\n\\[\n{\\bf L}^0_{-n}(e^{\\pm iPc}\\mathcal{Q}_{Q\\pm iP,\\nu',\\tilde{\\nu}})= e^{\\pm iPc}\\mathcal{Q}_{Q\\pm iP,(\\nu',n),\\tilde{\\nu}}\n\\]\nso that \n\\[\\begin{split} \n\\langle e^{\\pm iPc}\\mathcal{Q}_{Q\\pm iP,\\nu',\\tilde{\\nu}}, {\\bf L}^0_n (\\chi_Te^{ iP'c}h)\\rangle_2=& \\int \\chi(c+T)e^{ic(\\pm P-P')}\\text{\\rm d} c \n\\langle\\mathcal{Q}_{Q\\pm iP,(\\nu',n),\\tilde{\\nu}} , h\\rangle_{L^2(\\Omega_\\mathbb{T})}\\\\\n=&e^{iT(P'\\mp P)} \\hat{\\chi}(P'\\mp P)\\langle\\mathcal{Q}_{Q\\pm iP,(\\nu',n),\\tilde{\\nu}} , h \\rangle_{L^2(\\Omega_\\mathbb{T})}\n\\end{split}\\]\nwhere $\\hat{\\chi}$ denotes the Fourier transform. We therefore have, choosing $P'=-P$ and $\\chi$ so that $\\int \\chi=1$, that as $T\\to +\\infty$\n\\begin{equation}\\label{asymptoticPsiLnF}\n\\begin{split}\n\\langle \\Psi_{Q+iP,\\nu',\\tilde{\\nu}},{\\bf L}_n F_T\\rangle_{2} = & e^{-2iTP}\\hat{\\chi}(-2P)\\langle\\mathcal{Q}_{Q+iP,(\\nu',n),\\tilde{\\nu}} , h\\rangle_{L^2(\\Omega_\\mathbb{T})}\\\\\n& +R(Q+ip)\\langle\\mathcal{Q}_{Q- iP,(\\nu',n),\\tilde{\\nu}} , h\\rangle_{L^2(\\Omega_\\mathbb{T})}+o(1)\n\\end{split}\n\\end{equation}\nNext, we use \\eqref{asymptoticdescendants} and the same arguments as above to obtain, as $T\\to \\infty$,\n\\begin{align*} \n& \\langle \\Psi_{Q+iP,(\\nu',n),\\tilde{\\nu}}, F_{T}\\rangle_2= \\\\ \n&\\qquad e^{-2iTP}\\hat{\\chi}(-2P)\\langle\\mathcal{Q}_{Q+iP,(\\nu',n),\\tilde{\\nu}} , h\\rangle_{L^2(\\Omega_\\mathbb{T})}+ \\langle \n\\Pi_{\\ker ({\\bf P}-\\ell)}(\\tilde{{\\bf S}}_\\ell(Q+iP)\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}), h\\rangle_{L^2(\\Omega_\\mathbb{T})}\n\\\\\n&\\qquad+\\sum_{j<\\ell} e^{iT(P-P_{j\\ell})}\\hat{\\chi}(P-P_{j\\ell}) \n\\langle \\Pi_{\\ker ({\\bf P}-j)}(\\tilde{{\\bf S}}_\\ell(Q+iP)\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}), h\\rangle_{L^2(\\Omega_\\mathbb{T})}+o(1)\n\\end{align*}\nwith $P_{j\\ell}:=\\sqrt{P^2+2(\\ell-j)}$.\nCombining this asympotic expansion with \\eqref{asymptoticPsiLnF} and the fact that we can choose $h$ arbitrary in $E_\\ell$,\nwe deduce that \n\\[ \\Pi_{\\ker ({\\bf P}-\\ell)}(\\tilde{{\\bf S}}_\\ell(Q+iP)\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}}) =R(Q+iP) \\mathcal{Q}_{Q-iP,\\nu,\\tilde{\\nu}}.\\]\nApplying the same arguments but choosing $P'=-P_{j\\ell}$ for $j<\\ell$, we also get \n\\[ \\Pi_{\\ker ({\\bf P}-j)}(\\tilde{{\\bf S}}_\\ell(Q+iP)\\mathcal{Q}_{Q+iP,\\nu,\\tilde{\\nu}})=0.\\]\nWe have thus proved \\eqref{assumptionpsi} and therefore $\\mathcal{I}(\\ell)$ holds. This ends the proof.\n\\end{proof}\n\n\\subsection{Analyticity and functional equation for the Liouville eigenstates $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$.}\n\nFirst, let us prove \n\\begin{lemma}\\label{analyticity}\nThe function $\\Psi_{\\alpha}$ extends analytically from $\\{{\\rm Re}(\\alpha)0$), then \n $\\gamma_j^*\\Psi_\\alpha=\\Psi_{\\alpha}$. Since $\\gamma_j^*r_k=-r_k$ for $k\\geq j$ and $\\gamma_j^*r_k=r_k$ if $k0$ and $\\alpha=Q+iP$\n \\[ \\gamma_j^*\\Psi_{\\alpha}(c,\\varphi)=e^{iPc}+e^{-iPc}R(\\alpha)+\\gamma_j^*G_{\\alpha}\\]\nbut then \n\\[ (\\gamma_j^*\\Psi_{\\alpha}-\\Psi_{\\alpha})\\in \\mathcal{D}(\\mathcal{Q}).\\]\nSince ${\\bf H}$ does not have $L^2$-eigenvalues by \\cite[Lemma 6.2]{GKRV}, we deduce that $\\gamma_j^*\\Psi_{\\alpha}=\\Psi_{\\alpha}$ and, by analytic continuation, this shows that the meromorphic extension of $\\Psi_{\\alpha}$ in $U$ descends to $\\pi(U)$. The functional equation also comes from the uniqueness of $\\Psi_{Q+iP}$ having prescribed asymptotic term $e^{iPc}$ and analyticity come from the fact that $R(\\alpha)$ has only poles in ${\\rm Re}(\\alpha)|{\\rm Re}(\\alpha)-Q|$. They satisfy the functional equation \n\\[ \\Psi_{2Q-\\alpha,\\nu,\\tilde{\\nu}}=R(2Q-\\alpha)\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}\\]\nwhere $R(\\alpha)$ is the reflection coefficient \\eqref{Ralpha}. The zeros of $\\alpha\\mapsto \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$ are located exactly at $(Q+\\frac{\\gamma}{2}\\mathbb{N}) \\cup(Q+\\frac{2}{\\gamma}\\mathbb{N})$\n\\end{theorem}\n\\begin{proof} First, for $\\Psi_\\alpha$, the analytic extension is given by the functional equation $\\Psi_{\\alpha}=R(\\alpha)\\Psi_{2Q-\\alpha}$ \nand the fact that $\\Psi_\\alpha$ is analytic in $\\{{\\rm Re}(\\alpha)\\leq Q\\}$. Next for the descendants $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$, we just have to use \nProposition \\ref{intertwining_for_descendants} inductively and the result for $\\Psi_{\\alpha}$.\n\\end{proof}\n\nAs a consequence of this result, we can extend the validity of Proposition \\ref{intertwining_for_descendants} to all $\\alpha\\in \\mathbb{C}\\setminus Q\\pm (\\frac{\\gamma}{2}\\mathbb{N}+\\frac{2}{\\gamma}\\mathbb{N})$. \n\n\\subsection{Verma modules for the Liouville CFT and representation of Virasoro algebra}\\label{VermaLiouville}\n\nDefine for $\\alpha\\notin Q\\pm (\\frac{2}{\\gamma}\\mathbb{N}+\\frac{\\gamma}{2}\\mathbb{N})$ the vector space (for $\\epsilon>0$)\n\\[\\begin{gathered}\n\\mathcal{V}_\\alpha:= {\\rm span}\\{\\Psi_{\\alpha,\\nu,0} \\, | \\, \\nu,\\in \\mathcal{T} \\}\\subset e^{(|{\\rm Re}(\\alpha)-Q|+\\epsilon)c_-}L^2(\\mathbb{R}\\otimes \\Omega_\\mathbb{T}),\\\\\n\\overline{\\mathcal{V}}_\\alpha:= {\\rm span}\\{\\Psi_{\\alpha,0,\\tilde{\\nu}} \\, | \\, \\tilde{\\nu}\\in \\mathcal{T} \\}\\subset e^{(|{\\rm Re}(\\alpha)-Q|+\\epsilon)c_-}L^2(\\mathbb{R}\\otimes \\Omega_\\mathbb{T})\n\\end{gathered}\\]\nwhere, for now, elements are simply finite linear combinations of elements $\\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$. We also define \n\\[ \\mathcal{W}_\\alpha:={\\rm span}\\{\\Psi_{\\alpha,\\nu,\\tilde{\\nu}} \\, | \\, \\nu,\\tilde{\\nu}\\in \\mathcal{T} \\}\\simeq \\mathcal{V}_\\alpha\\otimes \\overline{\\mathcal{V}}_\\alpha \\]\nwhere the right identification is done by the map $\\Psi_{\\alpha,\\nu,0}\\otimes \\Psi_{\\alpha,0,\\tilde{\\nu}}\\mapsto \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$.\nProposition \\ref{intertwining_for_descendants} then implies that ${\\bf L}_n$ preserve $\\mathcal{V}_\\alpha$, $\\tilde{\\bf L}_n$ preserves $\\overline{\\mathcal{V}}_\\alpha$ and both preserve $\\mathcal{W}_\\alpha$. This space $\\mathcal{W}_\\alpha$ can be identified with $\\mathcal{H}_{\\mathcal{T}}\\otimes \\mathcal{H}_{\\mathcal{T}}$ by the linear isomorphism $e_{\\nu}\\otimes e_{\\tilde{\\nu}}\\mapsto \\Psi_{\\alpha,\\nu,\\tilde{\\nu}}$\nand the maps \n${\\bf L}_n$ and $\\tilde{\\bf L}_n$ are then conjugated to $\\ell_n^{\\alpha}$ and $\\tilde{\\ell}_n^{\\alpha}$, which implies that the following commutations hold\n\\[ [\\mathbf{L}_n,\\mathbf{L}_m]=(n-m)\\mathbf{L}_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}, \\quad [\\widetilde{\\mathbf{L}}_n,\\widetilde{\\mathbf{L}}_m]=(n-m)\\widetilde{\\mathbf{L}}_{n+m}+\\frac{c_L}{12}(n^3-n)\\delta_{n,-m}\\] \non respectively $\\mathcal{V}_\\alpha$ and $\\overline{\\mathcal{V}}_\\alpha$, and both also hold on $\\mathcal{W}_\\alpha$ where in addition $[{\\bf L}_n,\\tilde{\\bf L}_m]=0$.\n \nWhen $\\alpha=Q+iP$ with $P>0$, we define the scalar product on $\\mathcal{W}_{Q+iP}$\n\\[ \\langle \\Psi_{Q+iP,\\nu,\\tilde{\\nu}}, \\Psi_{Q+iP,\\nu',\\tilde{\\nu}'} \\rangle_{Q+iP}:= F_{Q+iP}(\\nu,\\nu')F_{Q+iP}(\\tilde{\\nu},\\tilde{\\nu}').\n\\]\nBy combining Proposition \\ref{intertwining_for_descendants}, Theorem \\ref{extensionPsialpha} and \\cite[Corollary 6.5]{GKRV1}, we obtain the\n\\begin{theorem}\\label{vermamodule}\nFor $\\alpha\\notin Q\\pm (\\frac{2}{\\gamma}\\mathbb{N}+\\frac{\\gamma}{2}\\mathbb{N})$, the operators $({\\bf L}_n)_{n\\in \\mathbb{Z}}$ acting on the vector space $\\mathcal{V}_\\alpha$ give a representation of the Virasoro algebra ${\\rm Vir}(c_L)$ with central charge $c_L=1+6Q^2$, $\\mathcal{V}_\\alpha$ is \na Verma module associated to the highest weight state $\\Psi_{\\alpha}$, and this representation is equivalent to the representation \nof $(\\ell_n^\\alpha)_n$ on $\\mathcal{H}_{\\mathcal{T}}$ given in Section \\ref{Vermafreefield}. \nThe same holds for $(\\tilde{\\bf L}_n)_{n\\in \\mathbb{Z}}$ acting on $\\overline{\\mathcal{V}}_\\alpha$. \nMoreover, for $\\alpha=Q+iP$ with $P>0$, these representations are unitary representations of the Virasoro algebra and are unitarily equivalent to the representation of $(\\ell_n^{Q+iP})_n$ and $(\\tilde{\\ell}_n^{Q+iP})_n$ on $\\mathcal{H}_T$. The family $({\\bf L}_n, \\tilde{\\bf L}_m)_{n,m\\in \\mathbb{Z}}$ gives\n a representation of ${\\rm Vir}(c_L)\\oplus {\\rm Vir}(c_L)$ into $\\mathcal{W}_{\\alpha}$, whose restriction to both ${\\rm Vir}(c_L)\\oplus \\{0\\}$ and $ \\{0\\}\\oplus {\\rm Vir}(c_L)$ is unitary when $\\alpha=Q+iP$ for $P>0$.\nFinally, one has the direct integral decomposition \n\\[ L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})= \\frac{1}{2\\pi}\\int_{0}^\\infty \\mathcal{V}_{Q+iP}\\otimes \\overline{\\mathcal{V}}_{Q+iP} \\, \\text{\\rm d} P=\\frac{1}{2\\pi}\\int_{0}^\\infty \\mathcal{W}_{Q+iP} \\, \\text{\\rm d} P\\]\n in the sense that for all $u,u'\\in L^2(\\mathbb{R}\\times \\Omega_\\mathbb{T})$\n \\[ \\langle u,u'\\rangle_2=\\frac{1}{2\\pi} \\sum_{\\nu,\\nu',\\tilde{\\nu},\\tilde{\\nu}'\\in\\mathcal{T}}\\int_0^\\infty \\langle u,\\Psi_{Q+iP,\\nu,\\tilde{\\nu}}\\rangle_2 \\langle \\Psi_{Q+iP,\\nu',\\tilde{\\nu}'},u'\\rangle_2 F_{Q+iP}^{-1}(\\nu,\\nu')F_{Q+iP}^{-1}(\\tilde{\\nu},\\tilde{\\nu}')\\, \\text{\\rm d} P.\\]\n\\end{theorem}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nAlthough theoretically possible, it is not yet clear whether building a sizeable quantum computer is feasible. The main obstacle is to find ways of dealing with decoherence and other quantum noise. When a quantum system interacts with its environment quantum information is leaked out and it begins to act in a probabilistic manner. In effect, parts of the system are being measured by the environment. Extraneous operations may also appear randomly during the computation, which corrupts the state and ultimately the result. Although research is focused on building quantum computers that are less likely to interact with their environment, it is impossible to completely isolate a system and therefore decoherence and quantum noise are inevitable. However, a number of techniques have been developed by the quantum software community that promise to reduce their effects on the system even further.\n\nQuantum error correction encapsulates software related techniques for reducing the impact of decoherence and other quantum noise. Some of the techniques that have so far been developed are inspired by coding theory and techniques already used within classical error correction. The basic principle is to encode information in such a way that the existence of errors can be detected and the nature of those errors identified. It is then possible to apply recovery operators before the information is decoded into its original form. However, checking for errors in a quantum computer is more problematic and there are new, entirely non classical, errors to contend with.\n\n\\section{The Quantum IO Monad}\n\nThe Quantum IO Monad is a library of functions written in Haskell that provides an interface to define and simulate quantum computations\\cite{Altenkirch}. It is used within the current work to demonstrate the use of encoding schemes. In functional programming computation is considered to be the application of functions as opposed to the manipulation of some global state. The following is a simple example of a QIO program.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{3}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\Varid{example}\\mathbin{::}\\Conid{QIO}\\;\\Conid{Bool}{}\\<[E]%\n\\\\\n\\>[B]{}\\Varid{example}\\mathrel{=}\\mathbf{do}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{q1}\\leftarrow \\Varid{mkQbit}\\;\\Conid{False}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{applyU}\\mathbin{\\$}\\Varid{unot}\\;\\Varid{q1}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{b}\\leftarrow \\Varid{measQbit}\\;\\Varid{q1}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{return}\\;\\Varid{b}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nThis function is written using \\textbf{do} notation, which resembles a more imperative style, but is in fact functional underneath. Such seemingly impure computations may be defined thanks to the use of monads that are built upon some polymorphic type. The function \\textit{mkQbit} creates a new qubit initialised in the state given by the boolean argument, with False and True representing the base states $ | 0 \\rangle $ and $ | 1 \\rangle $ respectively. Unitary operations are then applied using \\textit{applyU}, the quantum NOT gate, or the Pauli X gate, in this case. The \\textit{unot} operation is defined as a \\textit{Rotation}, as are all single qubit operations. Other unitaries defined in QIO include \\textit{swap}, \\textit{cond,} for controlled operations, and \\textit{ulet}, which is used to declare and encapsulate the use of ancillary qubits. Finally, qubits are measured using \\textit{measQbit} and a boolean value is returned randomly depending on the superposition of the qubit at the time. The qubit itself is also collapsed into the base state corresponding to this boolean value.\n\n\\section{Implementing Quantum Error Correction in QIO}\n\nAlthough it may be some time before real quantum computers are realised, work is already being carried out by computer scientists to develop good quantum programming languages. The development of the Quantum IO Monad is an example of such work. Although reliable error correction techniques will be crucial, in order to allow programmers to concentrate on the matter at hand some form of automated conversion into equivalent but error resilient programs would be extremely advantageous. The following describes an attempt to do just this within the Quantum IO Monad.\n\n\\subsection{Introducing an Encoded Qubit Type}\n\nThe first step towards being able to convert a QIO program into an equivalent that incorporates quantum error correction is to declare a class called \\textit{EnQubit}, instances of which will represent encoded qubits. By defining such a class the main program can be written in a generic fashion without the need to understand how encoded qubits are represented\\cite{Barratt}. As long as the instances of this class provide definitions of the functions declared here there will be no need to alter the main program when switching schemes, since the appropriate functions will be called at run time.\n\nEach \\textit{EnQbit} is thought to have a \"parent\" qubit, the one from which the encoded state is produced. The function \\textit{getQbit} returns this particular qubit and is required by \\textit{measEnQbit}, defined below.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{5}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{18}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\mathbf{class}\\;\\Conid{EnQbit}\\;\\Varid{a}\\;\\mathbf{where}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{mkEnQbit}\\mathbin{::}\\Conid{Bool}\\to \\Conid{QIO}\\;\\Varid{a}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{getQbit}\\mathbin{::}\\Varid{a}\\to \\Conid{Qbit}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{measEnQbit}\\mathbin{::}\\Varid{a}\\to \\Conid{QIO}\\;\\Conid{Bool}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{measEnQbit}\\;\\Varid{eq}\\mathrel{=}\\mathbf{do}{}\\<[E]%\n\\\\\n\\>[5]{}\\hsindent{13}{}\\<[18]%\n\\>[18]{}\\Varid{applyU}\\mathbin{\\$}\\Varid{decode}\\;\\Varid{eq}{}\\<[E]%\n\\\\\n\\>[5]{}\\hsindent{13}{}\\<[18]%\n\\>[18]{}\\Varid{b}\\leftarrow \\Varid{measQbit}\\;(\\Varid{getQbit}\\;\\Varid{eq}){}\\<[E]%\n\\\\\n\\>[5]{}\\hsindent{13}{}\\<[18]%\n\\>[18]{}\\Varid{applyU}\\mathbin{\\$}\\Varid{encode}\\;\\Varid{eq}{}\\<[E]%\n\\\\\n\\>[5]{}\\hsindent{13}{}\\<[18]%\n\\>[18]{}\\Varid{return}\\;\\Varid{b}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nThe class also provides flexibility in the error correcting codes that may be employed. By providing different \\textit{encode} and \\textit{correct} functions a different encoding scheme may be used with the same representation. The function \\textit{decode} is in fact defined here as the inverse of \\textit{encode}, since it is unitary.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{5}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{encode}\\mathbin{::}\\Varid{a}\\to \\Conid{U}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{decode}\\mathbin{::}\\Varid{a}\\to \\Conid{U}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{decode}\\;\\Varid{eq}\\mathrel{=}\\Varid{urev}\\mathbin{\\$}\\Varid{encode}\\;\\Varid{eq}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{correct}\\mathbin{::}\\Varid{a}\\to \\Conid{U}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nThe following functions define \\textit{EnQbit} versions of the standard unitary operators of QIO. These are implemented to simply decode the given \\textit{EnQbit} and call the appropriate standard unitary, passing the \"parent\" qubit. Before returning, the qubits are encoded again. This approach only provides protection in between the application of operations, while qubits are perhaps being stored or transmitted, as suggested by Shor\\cite{Shor}. Being able to perform operations on the actual encoded qubits themselves would make decoding unnecessary until the end of the computation and thus provide greater protection. This is possible for certain operations in a bitwise fashion, those with the property of transversality, but it does depend on the code being used\\cite{Zurek}. So although these functions are defined here as a standard, they may be overridden by instances of the class, if such encoded operations are available.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{5}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{9}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{rotEnQbit}\\mathbin{::}\\Varid{a}\\to \\Conid{Rotation}\\to \\Conid{U}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{swapEnQbit}\\mathbin{::}\\Varid{a}\\to \\Varid{a}\\to \\Conid{U}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{condEnQbit}\\mathbin{::}\\Varid{a}\\to (\\Conid{Bool}\\to \\Conid{U})\\to \\Conid{U}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{uletEnQbit}\\mathbin{::}\\Conid{Bool}\\to (\\Conid{Qbit}\\to \\Varid{a}\\to \\Conid{U})\\to \\Conid{U}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nThe \\textit{uletEnQbit} function is not defined here but simply declared as this depends entirely on the number of qubits being used in the encoding, to determine how many ancillas to create.\n\nIn terms of encoded qubit representations, the following type could be used for the three qubit bit flip code, with \\textit{EnQbit} functions defined as appropriate.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{5}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\mathbf{newtype}\\;\\Conid{EQ3}\\mathrel{=}\\Conid{EQ3}\\;(\\Conid{Qbit},\\Conid{Qbit},\\Conid{Qbit}){}\\<[E]%\n\\\\[\\blanklineskip]%\n\\>[B]{}\\mathbf{instance}\\;\\Conid{EnQbit}\\;\\Conid{EQ3}\\;\\mathbf{where}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{mkEnQbit}\\mathrel{=}\\Varid{mkEQ3}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{getQbit}\\mathrel{=}\\Varid{fstEQ3}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{encode}\\mathrel{=}\\Varid{encode3}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{correct}\\mathrel{=}\\Varid{correct3}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{uletEnQbit}\\mathrel{=}\\Varid{uletEQ3}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nHowever, this representation could also be used for the three qubit phase flip code by simply altering the implementations of \\textit{encode} and \\textit{correct}. An instance has also been implemented for Steane's code\\cite{Steane} using a 7-tuple of qubits and appropriate encoding and correction procedures.\n\n\\subsection{Converting a QIO program}\n\nIn order to manipulate an existing QIO program, the function \\textit{ecQIO'} pattern matches on it and inserts new function calls depending on the construct found, exploiting the monadic structure of the \\textit{QIO} type. It is defined recursively with the base case being \\textit{QReturn}, signifying the end of the computation.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\Varid{ecQIO'}\\mathbin{::}\\Conid{EnQbit}\\;\\Varid{a}\\Rightarrow \\Conid{QIO}\\;\\Varid{t}\\to [\\mskip1.5mu \\Varid{a}\\mskip1.5mu]\\to \\Conid{QIO}\\;\\Varid{t}{}\\<[E]%\n\\\\\n\\>[B]{}\\Varid{ecQIO'}\\;(\\Conid{QReturn}\\;\\Varid{a})\\;\\kern0.06em \\vbox{\\hrule\\@width.5em} \\mathrel{=}\\Conid{QReturn}\\;\\Varid{a}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nWhen a qubit is being created, measured or manipulated the appropriate \\textit{EnQbit} function is substituted in to replace the original. As such we obtain a program that consists of extra operations and qubits, together encapsulating the logical state of the single qubit referred to. Whatsmore, a list is passed on in the recursive call and threaded through the whole computation as a register of the encoded qubits. In the case of \\textit{MkQbit} the newly created \\textit{EnQbit} is appended to the list before passing it on. Here \\textit{MkQbit} is one of the constructors of the \\textit{QIO} type, which the convenience function \\textit{mkQbit} seen earlier relies upon.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{3}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\Varid{ecQIO'}\\;(\\Conid{MkQbit}\\;\\Varid{b}\\;\\Varid{f})\\;\\Varid{eqs}\\mathrel{=}\\mathbf{do}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{eq}\\leftarrow \\Varid{mkEnQbit}\\;\\Varid{b}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{ecQIO'}\\;(\\Varid{f}\\;(\\Varid{getQbit}\\;\\Varid{eq}))\\;(\\Varid{eq}\\mathbin{:}\\Varid{eqs}){}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nThe interesting case in \\textit{ecQIO'} is the \\textit{ApplyU} constructor, where it is necessary to pattern match further on the \\textit{U} data type representing unitary operations. This takes place in the function \\textit{extendU}, where we plan to extend the operation's influence to include the rest of the qubits in the encoded qubit. \n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{3}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\Varid{ecQIO'}\\;(\\Conid{ApplyU}\\;\\Varid{u}\\;\\Varid{f})\\;\\Varid{eqs}\\mathrel{=}\\mathbf{do}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{applyU}\\mathbin{\\$}\\Varid{extendU}\\;\\Varid{u}\\;\\Varid{eqs}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{3}{}\\<[3]%\n\\>[3]{}\\Varid{ecQIO'}\\;\\Varid{f}\\;\\Varid{eqs}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nBy pattern matching on the \\textit{U} data type being applied, \\textit{extendU} calls the appropriate function implementing the \\textit{EnQbit} version of the unitary. The case of a rotation is given here.\n\\smallskip\n\\begingroup\\par\\noindent\\advance\\leftskip\\mathindent\\(\n\\begin{pboxed}\\SaveRestoreHook\n\\column{B}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{5}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{9}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\column{E}{@{}>{\\hspre}l<{\\hspost}@{}}%\n\\>[B]{}\\Varid{extendU}\\mathbin{::}\\Conid{EnQbit}\\;\\Varid{a}\\Rightarrow \\Conid{U}\\to [\\mskip1.5mu \\Varid{a}\\mskip1.5mu]\\to \\Conid{U}{}\\<[E]%\n\\\\\n\\>[B]{}\\Varid{extendU}\\;(\\Conid{Rot}\\;\\Varid{q}\\;\\Varid{r}\\;\\Varid{u})\\;\\Varid{eqs}\\mathrel{=}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\mathbf{let}\\;\\Varid{eq}\\mathrel{=}(\\Varid{fromJust}\\;(\\Varid{getEnQbit}\\;\\Varid{q}\\;\\Varid{eqs}))\\;\\mathbf{in}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{rotEnQbit}\\;\\Varid{eq}\\;\\Varid{r}\\mathbin{`\\Varid{mappend}`}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{correctAll}\\;\\Varid{eqs}\\mathbin{`\\Varid{mappend}`}{}\\<[E]%\n\\\\\n\\>[B]{}\\hsindent{5}{}\\<[5]%\n\\>[5]{}\\Varid{extendU}\\;\\Varid{u}\\;\\Varid{eqs}{}\\<[E]%\n\\ColumnHook\n\\end{pboxed}\n\\)\\par\\noindent\\endgroup\\resethooks\n\nSince a standard QIO program performs operations on individual qubits, a mechanism for identifying their associated \\textit{EnQbit} is needed when faced with measurement and operator constructs. A function \\textit{getEnQbit} performs this task by making a comparison between the qubit given and the \"parent\" qubit of each \\textit{EnQbit} in our list.\n\nThe function \\textit{correctAll} is called upon, and the returned unitary appended to the end of the rotation, to perform the error correction procedure on all encoded qubits created so far, which should take place periodically in order to prevent a build up of errors that is unmanageable for the code. If this were to happen the decoding procedure would yield an incorrect value.\n\n\\section{Conclusions}\n\nIn terms of providing error correction to QIO programs, the components that are responsible for converting a program appear to fulfill their task of inserting new function calls at appropriate points. Flexibility is also provided on the implementation of codes used, as the same converting functions can work with any coding scheme.\n\nTwo encoding schemes have been implemented, the first being the three qubit bit flip code, with Steane's code as the second. Unfortunately, it appears that programs encoded with Steane's code are not evaluated efficiently, preventing any real observation of adding Steane's code to even the simplest of programs. As more qubits and\/or unitary operations are added the performance degrades very quickly. This is not such an issue for the three qubit bit flip code, which is evaluated fairly quickly even for larger programs using error correction.\n\nThe applicability of this kind of encoding automation has been demonstrated however, and if these evaluations were to be taking place on a real quantum computer then these problems would not be present.\n\n\n\n\\section*{Acknowledgements}\n\nI would like to thank my MSc dissertation supervisor, Thorsten Altenkirch, for his guidance throughout the course of this work, and also Alexander Green, for his assistance.\n\nI would also like to thank my Mother and Father for their unwavering support, and for simply putting up with me.\n\nLastly, I would like to dedicate this paper to the memory of my brother, Bill Barratt (1972-2010).\n\n\\bibliographystyle{eptcs}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nLow-dimensional structures are now under scrutiny in\nnonperturbative QCD, cosmology, high-energy physics,\nand condensed matter physics. Properties of particles\nplaced into such structures are usually described by\nconsidering quantum theory in two dimensions. However,\nthere is no doubt that real space remains three\ndimensional, which may lead to qualitative differences in\nsome observables.\n\nThis especially concerns the particle spin properties,\nwhich are crucially different at two and three spatial\ndimensions (see, e.g., Refs.\n\\cite{Yip,DiracRep,DiracRe}). Thus, transition to (2+1)-dimensional spacetimes leads to losses of a significant part of such properties.\nAt the same time, in the two-dimensional space, anyons \\cite{Chen:1989xs}\nmay appear.\n\nIn the present work, we investigate the problem of transformation of the spin properties under the compactification of some spatial dimension.\nThis problem is generally very difficult because the spin dynamics depends on many factors.\nTo extract some common properties, we consider the toy model\n\\cite{Fiziev1} of the curved space of \\emph{variable} dimensionality\nsmoothly changing from three to two. A great preference of the model used is a possibility to obtain \\emph{exact} quantum-mechanical solutions.\n\nWe use the conventional Dirac equation for a consistent description of spin-1\/2\nparticle motion in the curved space and take into\naccount relativistic effects. While such effects are not too important in condensed\nmatter physics (except for graphene), we keep in mind their further applications to the processes at\nLarge Hadron Collider in the case \\cite{Mureika:LHC,StojkovicReview} of variable\n(momentum) space dimension. We use the \\emph{relativistic} method \\cite{JMP} of\nthe Foldy-Wouthuysen (FW) transformation \\cite{FW} to derive exact quantum-mechanical equations of motion\nand obtain their classical limit.\n\nIn this work, we focus our attention on the spin properties.\nWe show that, in contrast to a ``naive'' estimation, the\nspin in an effectively two-dimensional space may precess\nabout the noncompactified dimensions and therefore a\n``flick'' may appear in the remnant space once or twice\nduring the period. \n\n\\section{Hermitian Hamiltonians\nfor the metric admitting the effective dimensional\nreduction}\\label{Hamiltonian}\n\n\nLet us start with the following metric proposed by Fiziev \\cite{Fiziev1}:\n\\begin{equation}\nds^2=c^2dt^2-\\rho_1(z)^2d\\Phi_1^2-\\rho_2(z)^2d\\Phi_2^2-\\rho_3(z)^2dz^2,\n\\label{gmetric}\n\\end{equation}\nwhere $\\rho_3(z)^2=1+\\rho'_1(z)^2+\\rho'_2(z)^2$, the primes define derivatives\nwith respect to $z$, and $\\rho_i$ are the functions of $z$.\nThe spatial coordinates vary in the limits $-\\infty < z < \\infty, ~ 0 < \\Phi_{1,2} < 2 \\pi$.\n\nWe suppose $\\rho_i(z)$ to be positive.\nThe (3+1)-dimensional manifold defining this metric\nis a hypersurface in a flat pseudo-Euclidean (5+1)-dimensional space.\nThe tetrad $e_{0}^{\\widehat{0}}=1,~e_{i}^{\\widehat{j}}=\\delta^{ij}\\sqrt{g_{ii}}$ allows us to define the local\nLorentz (tetrad) frame. This considerably simplifies an\nanalysis of results from possibly using the rescaled Cartesian coordinates $dX=\\rho_1(z)d\\Phi_{1}, ~dY=\\rho_2 (z) d\\Phi_{2},~dZ=\\rho_3(z)dz$\nin the neighborhood of any point.\n\nTaking the limit $\\rho_1(z) \\rightarrow 0$ or the limit $\\rho_2(z) \\rightarrow 0$\nmay lead to the reduction of dimension of\nthe physical space from $d = 3$ to $d = 2$. We consider the case when the compactification\nof the $\\bm e_1~(\\bm e_2)$ direction results in the confinement of the particle in\na narrow interval of $\\Phi_1~(\\Phi_2)$ angles.\n\nThe transverse part of the metric (if $z$ is assumed to be a\nlongitudinal coordinate) has the structure of the Clifford\ntorus, which is the product of two unit circles in the fourdimensional\nEuclidean space: \n\\begin{equation}\ny_1^2+y_2^2=y_3^2+y_4^2=1.\n\\label{tmetric}\n\\end{equation}\nThe Clifford tori are used for analyzing twisted\nmaterials \\cite{clifford09} and vesicles \\cite{cliffordva,cliffordve,cliffordvf}.\nThere is also some qualitative\nsimilarity to projection of a tube in a six-dimensional space\nonto a three-dimensional space, which was used for the\nconstruction of the quasicrystals theory\n\\cite{Kalugin}.\n\nWe consider Clifford tori as a toy model of dimensional reduction.\nWe are not necessarily assigning the physical sense to all of the\nintermediate values of $z$ except the asymptotics for $z \\to\\pm \\infty$\ncorresponding to the three- and\ntwo-dimensional spaces. Here, varying the dimension plays\nthe same role as varying the coupling constant for the case\nof an adiabatic switch on the interaction.\n\n\n\nTo describe the spin-1\/2 particles, we use the conventional covariant Dirac equation (see Ref. \\cite{OSTgrav} and references therein).\nTo find the\nHamiltonian form of this equation,\none can substitute the given metric into the general equation\nfor the Hermitian Dirac\nHamiltonian (Eq. (2.21) in Ref. \\cite{OSTRONG}). For the metric (\\ref{gmetric}), the Hermitian Dirac\nHamiltonian was first derived in Ref. \\cite{GorNeznArXiv}. It can be presented in the form\n\\begin{eqnarray}\n{\\cal H}_D =\\beta mc^2 - \\frac {i\\hbar c}{ \\rho_1}\\alpha_1\n\\frac{\\partial}{\\partial\\Phi_1}- \\frac{ i\\hbar c}{ \\rho_2}\\alpha_2\n\\frac{\\partial}{\\partial\\Phi_2}- \\frac {i\\hbar c}{ 2}\\alpha_3\\left\\{{\\frac\n1 \\rho_3}, \\frac{\\partial}{\\partial z}\\right\\},\n\\label{Hamilton2}\\end{eqnarray} where $\\{\\dots,\\dots\\}$ denotes an\nanticommutator.\n\nWe transform this Hamiltonian to the FW representation by the method elaborated in Ref. \\cite{JMP}\nwhich was earlier applied in our previous works\n\\cite{OSTgrav,OSTRONG,OST}.\nAfter the {\\it exact} FW transformation, we get the result\n\\begin{eqnarray}\n{\\cal H}_{FW} =\\beta\n\\sqrt{a+{\\hbar}\\bm\\Sigma\\cdot\\bm b},\n\\label{HamiltonFW}\\end{eqnarray}\nwhere\n\\begin{eqnarray}\na=m^2c^4+\\frac{c^2p_1^2}{\\rho^2_1}+\\frac{c^2p_2^2}{\\rho^2_2} + \\frac{c^2}{4}\n\\left\\{{\\frac 1 \\rho_3},\np_3\\right\\}^2,~~~ \\bm{b}=b_1\\bm e_1+b_2\\bm e_2=\\frac {c^2\\rho'_2}{\\rho^2_2\\rho_3}p_2\\bm e_1-\\frac\n{c^2\\rho'_1}{\\rho^2_1\\rho_3}p_1\\bm e_2,\n\\label{denot}\\end{eqnarray}\nand $(p_1,p_2,p_3)\n=\\left(-i\\hbar\\frac{\\partial}{\\partial\\Phi_1},\n-i\\hbar\\frac{\\partial}{\\partial\\Phi_2},-i\\hbar\\frac{\\partial}{\\partial\nz}\\right)$ is the generalized momentum operator. Primes denote derivatives with respect to $z$.\nThe $\\bm e_1,\\bm e_2,\\bm e_3$ vectors form the spatial part of the orthonormal basis\ndefining the local Lorentz (tetrad) frame.\nFor the given time-independent metric,\nthe operators ${\\cal H}_{FW},~p_1,$ and $p_2$ are\nintegrals of motion.\n\nNeglecting a noncommutativity of the $a$ and $\\bm b$ operators allows us to omit anticommutators and results in\n\\begin{eqnarray}\n{\\cal H}_{FW} =\\frac \\beta\n2\\left(\\sqrt{a+\\hbar b}+\\sqrt{a-\\hbar b}\\right)+\\frac{\\bm\\Pi\\cdot \\bm\nb}{2b}\\left(\\sqrt{a+\\hbar b}-\\sqrt{a-\\hbar b}\\right),\n\\label{HFW}\\end{eqnarray} where $\\bm\\Pi=\\beta\\bm\\Sigma$ is the spin polarization operator. It can be proven that extra terms appearing from the above noncommutativity are of order of\n$|\\hbar\/(p_zl)|^3$, where $p_z$ is the particle momentum and $l$ is the characteristic\nsize of the nonuniformity region of the external\nfield (in the $z$ direction). With this accuracy,\n\\begin{eqnarray}\n{\\cal H}_{FW} =\\beta \\left(\\sqrt{a}-\\frac{\\hbar^2b^2}{8a^{3\/2}}\\right)+\\hbar\\frac{\\bm\\Pi\\cdot\\bm b}{2\\sqrt{a}}.\n\\label{HFWb}\\end{eqnarray}\nThe second term proportional to $\\hbar^2$ is important even when it is relatively small. This term contributes to the difference between gravitational interactions of spinning and spinless particles and therefore violates the weak equivalence principle. Its importance relative to the main term is defined by the ratio $(\\hbar b\/a)^2$. The weak equivalence principle is also violated by the spin-dependent Mathisson force (see Refs. \\cite{OSTgrav,Plyatsko:1997gs} and references therein) defined by the third term in Eq. (\\ref{HFWb}). While the third term is usually much bigger than the second one, it vanishes for unpolarized spinning particles. The second term proportional to $(\\bm\\Pi\\cdot\\bm b)^2$ is always nonzero. An analysis of Eqs. (\\ref{denot}) and (\\ref{HFWb}) leads to the conclusion that this term can be comparable with the main one (proportional to $\\sqrt{a}$) when $l\\sim\\lambda_B$, where $\\lambda_B$ is the de Broglie wavelength.\nThe existence of the term proportional to $\\hbar^2$ is not a specific property of the toy model used. The appearance of such terms in the FW Hamiltonians describing a Dirac particle in Riemannian spacetimes was noticed in several works \\cite{OST,DH,Jentschura},\nwhereas its relation to the spin-originated effect leading to the violation of the weak equivalence principle was never mentioned.\n\nThe equation of spin motion is given by\n\\begin{eqnarray}\n{\\frac {d{\\bm \\Pi}}{dt}} = \\bm \\Omega\\times{\\bm \\Pi}, ~~~ \\bm \\Omega=\n\\beta\\frac {\\bm b}{\\sqrt{a}}.\n\\label{finalOmegase}\n\\end{eqnarray}\n\nAs a result, the spin rotates\n\\emph{relative to} $\\bm e_i$ \\emph{vectors} $(i=1,2,3)$ with the angular velocity\n$\\bm \\Omega$. Its motion relative to the Cartesian axes is much more complicated.\n\nIt has been proven in Ref. \\cite{JINRLet1} that finding a classical limit of \\emph{relativistic}\nquantum mechanical equations reduces to the replacement of operators by\nrespective classical quantities when the condition of the Wentzel-Kramers-Brillouin approximation,\n$\\hbar\/|pl|\\ll1$, is satisfied.\nIt has also been shown that the classical limit of the FW Hamiltonians for Dirac \\cite{OSTgrav,OSTRONG,OST} and scalar\n\\cite{Honnefscalar} particles in Riemannian spacetimes coincides with the corresponding purely\nclassical Hamiltonians.\n\n\n\n\n\\section{Motion of particle at variable dimensions}\\label{Region}\n\nLet us first study the motion of the particle by neglecting the influence of the spin onto its trajectory.\nSince $p_1$ and $p_2$ are integrals of motion,\nthey can be replaced with the eigenvalues $\\mathcal{P}_1$ and $\\mathcal{P}_2$, respectively.\nLet us choose the $\\bm\ne_1$ axis as the compactified dimension and suppose that $\\rho_{1}(z)$ is a decreasing function ($\\rho_{1}(z)\\rightarrow0$ when $z\\rightarrow\\infty$).\nWe can neglect a dependence of $\\rho_{2}$ on $z$, assuming that this function changes much more slowly.\nWe denote initial values of all parameters by additional zero indices and consider the general case\nwhen the initial value of the metric component, $\\rho_{10}\\equiv\\rho_{1}(z_0)$, is\nnot small.\n\n The classical limit of the Hamiltonian is given by\n\\begin{eqnarray}\n{\\cal H}\n=\\sqrt{m^2c^4+\\frac{c^2\\mathcal{P}_1^2}{\\rho^2_1}+\\frac{c^2\\mathcal{P}_2^2}{\\rho^2_2}+\\frac{c^2p_3^2}{\\rho^2_3}}.\n\\label{HamiltonC}\\end{eqnarray}\n\n\n\nThe possibility of making general conclusions with the special model used is based on the fact that the\nHamiltonian of a particle in an arbitrary static spacetime is given by\n\\begin{eqnarray}\n{\\cal H}\n=\\sqrt{\\frac{c^2\\left(m^2c^2+g^{ij}p_ip_j\\right)}{g^{00}}}, ~~~ i,j=1,2,3.\n\\label{Hamiltong}\\end{eqnarray}\nEquation (\\ref{Hamiltong}) covers spinless \\cite{Cogn} and spinning \\cite{OSTRONG,OSTgrav} particles in classical gravity as well as the classical limit of the corresponding quantum-mechanical Hamiltonians for scalar \\cite{Honnefscalar}\nand Dirac \\cite{OSTgrav} particles. For spinning particles, the term $\\bm s\\cdot\\bm\\Omega$ should be added to this Hamiltonian \\cite{OSTRONG,OSTgrav}. When the metric is diagonal, $g^{ii}=1\/{g_{ii}}$ and Eq. (\\ref{Hamiltong}) takes the same form as Eq. (\\ref{HamiltonC}).\n\n\n\nTo describe the compactification, we can introduce the compactification radius $\\delta$ so that the\n``compactification point'' $z_c$ can be defined by $\\rho_1(z_c)=\\delta$. Due to the energy $E$ conservation, the\nparticle can reach this point if\n\\begin{eqnarray}\nE\n\\geq \\sqrt{m^2c^4+\\frac{c^2\\mathcal{P}_1^2}{\\delta^2}+\\frac{c^2\\mathcal{P}_2^2}{\\rho^2_2(z_c)}}.\n\\label{HamiltonE}\\end{eqnarray}\nNote that the decrease of compatification radius $\\delta$ while $E$ remains finite implies the corresponding decrease of $\\mathcal P_1$.\n\n\nThe particle velocity is equal to\n\\begin{equation}\nv_z\\equiv\\frac {dz}{dt} =\\frac{\\partial{\\cal H}}{\\partial p_3}=\n\\frac{c^2p_3}{E\\rho^2_3}\\\\=\nc \\frac{\\sgn{(p_3)}}{E\\rho_3(z)}\\sqrt{E^2-m^2c^4- c^2 R(z)}, ~~~ R(z)=\\frac{\\mathcal{P}_1^2}{\\rho^2_1(z)} + \\frac{\\mathcal{P}_2^2}{\\rho^2_2(z)}.\n\\label{finalmw}\n\\end{equation}\nDifferent signs correspond to the two different directions of the\nlongitudinal particle motion.\nNote that the arrival to the compactification point with zero velocity ($z_c =z_f$ being the final point of particle trajectory) corresponds to\nthe equality sign in Eq. (\\ref{HamiltonE}).\n\nA tedious but simple calculation allows us to obtain the longitudinal component of the particle acceleration:\n\\begin{eqnarray}\na_z\\equiv\\frac {d^2z}{dt^2} =\n- \\frac{c^4}{E^2 \\rho_3^2}\\left(\n\\frac{R'}{2}+\\frac{p_3^2\\rho'_3}{\\rho^3_3}\n\\right).\n\\label{vd}\n\\end{eqnarray}\nIt is obvious that $p_3(z_f)=0,~ R'(z_f) \\geq 0$ (for monotonic continuously differentiable $R(z)$), so that $a_z(z_f) \\leq 0$. Therefore, $z_f$ is the turning (if $R'(z_f) > 0$) or attracting\n(if $R'(z_f) = 0$) point. For nonmonotonic $R(z)$ there is a possibility of passage to the region $z > z_f$ due to possible growth of $\\rho_2(z)$. The particle motion is then limited by the point\n$\\tilde z_f$ corresponding to the neglect of the motion in the $\\bm e_2$ direction\n\\begin{eqnarray}\nE\n= \\sqrt{m^2c^4+\\frac{c^2\\mathcal{P}_1^2}{\\rho_1(\\tilde z_f)}}.\n\\label{tilde}\\end{eqnarray}\n\n\n\n\n\nThe important particular case of Eq. (\\ref{HamiltonC}) corresponds to $\\mathcal{P}_1=0$. The particle penetrates into the region of the effective dimensional reduction ($z\\rightarrow\\infty$)\nand does not reverse the direction of its motion.\n\nIn this study, as was mentioned above, we consider that\nthe smooth adiabatic transition from the three-dimensional\nspace to the effectively two-dimensional one does not\nnecessarily attribute the physical sense to all intermediate\npoints in particle motion. At the same time, the true change\nof the dimensionality was discussed in cosmology (see\nRefs. \\cite{Mureika:2011bv,StojkovicReview,Fiziev:2010je,Sotiriou:2011xy})\nand in connection with experiments at the LHC\n(see Refs. \\cite{StojkovicReview,Mureika:LHC,SCarlip,Sotiriou:LHC}). Our analysis\ncan also be applicable at the LHC.\n\nNote also that the motion in the opposite direction of increasing dimension does not impose any conditions for the\ninitial state of the particle. One may say that the region of lower dimension is ``repulsive'' whereas the region of higher dimension is ``attractive'',\nimplying a sort of \\emph{irreversibility} in the particle dynamics. This property emerges because of the appearance of $\\rho_1$\nin the expression for the Hamiltonian in the denominator. Such a situation is a general one that can be seen from Eq. (\\ref{Hamiltong}) in the case of diagonal metric.\nThis may give additional support to the hypothesis \\cite{Mureika:2011bv,Fiziev:2010je}\nthat such a transition from the lower dimensionality to the higher one leaded to the evolution\nof the Universe.\n\n\n\n\n\\section{Spin evolution at variable dimensions}\n\nIn the classical limit, the angular velocity of spin precession is given by\n\\begin{eqnarray}\n\\bm \\Omega=\n\\frac {\\bm b}{E}=\\frac{c^2}{E\\rho_3}\\left(\n\\frac{\\mathcal{P}_2\\rho'_2}{\\rho^2_2}\\bm e_1-\\frac\n{\\mathcal{P}_1\\rho'_1}{\\rho^2_1}\\bm e_2\\right).\n\\label{finalOmegacl}\n\\end{eqnarray}\nBecause $d\\bm s\/dt=v_z(z)(d\\bm s\/dz)$, Eqs. (\\ref{finalmw}) and (\\ref{finalOmegacl})\ndefine an easily solvable system of first-order homogeneous linear differential equations.\n\nEquation (\\ref{finalOmegacl}) is rather informative about details of the compactification. Only the $\\Omega_2$ component contains parameters of the compactified dimension.\nAlthough $|\\mathcal{P}_1|\/|\\mathcal{P}_2|\\ll1$, the presence of additional factors\ndoes not allow for\nneglecting $\\Omega_2$ as compared with $\\Omega_1$ (under the condition that $\\mathcal{P}_1\\neq0$).\n\nWhen\n$\\rho_2(z)=const$, $\\Omega_1=0$ and the spin rotates about the\n$\\bm e_2$ axis, the spin projection onto the $\\bm e_2\\bm e_3$ surface, which\nis the spatial part of the (2+1)-dimensional spacetime,\noscillates. The spin appears in this surface only once (in the\nspecial case when the cone of spin precession is tangent to\nthis surface) or twice per rotation period. Evidently, the\norigin of this spin ``flickering'', as well as the appearance of\npseudovector, is completely unexplainable in terms of the\ntwo-dimensional space.\n\n\nThe model used allows to obtain an exact analytical description of the spin evolution.\nIt is characterized by a change of the angle $\\varphi$ defining the direction\nof the spin in the plane orthogonal to $\\bm\\Omega$:\n\\begin{eqnarray}\n \\Delta\\varphi(z) =\\int{\\Omega(t)dt}=\\int_{z_{0}}^{z}{\\frac{\\Omega(y)}{v_z(y)}dy}.\n\\label{varph}\n\\end{eqnarray}\n\nThe problem of spin evolution at the effective dimensional reduction can be solved in a general form.\nTo simplify the\nanalysis, let us consider the case of $\\rho_2(z)=\\rho_{20}=const$.\nIn this case, the \\emph{exact} value of the integral is\n\\begin{eqnarray}\n\\Delta\\varphi(z)=\n\\arcsin{\\frac{c\\mathcal{P}_1}{A\\rho_1(z)}}-\n\\arcsin{\\frac{c\\mathcal{P}_1}{A\\rho_{10}}},\n\\label{varin}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray} A=\\sqrt{E^2-m^2c^4-\\frac{c^2\\mathcal{P}_2^2}{\\rho^2_{20}}}=\nc \\sqrt{\\frac{p_{30}^2}{\\rho^2_{30}}+\\frac{\\mathcal{P}_1^2}{\\rho^2_{10}}}.\n\\label{eqna}\n\\end{eqnarray}\nSince\n\\begin{eqnarray}\n\\rho_{1}(z_f)=c\\left|\\mathcal{P}_{1}\\right|\\left(E^2-m^2c^4-\\frac{c^2\\mathcal{P}_2^2}{\\rho^2_{20}}\\right)^{-1\/2},\n\\label{zetf}\n\\end{eqnarray}\nthe total spin turn ($z=z_f$)\nis given by\n\\begin{eqnarray}\n\\Delta\\varphi=\\sgn{(\\mathcal{P}_1)}\\cdot\\frac\\pi 2-\n\\arctan{\\frac{\\mathcal{P}_1\\rho_{30} }{\\rho_{10}p_{30}}}.\n\\label{varif}\n\\end{eqnarray}\nThe passage of the particle to the region of compactification implies, as was discussed above,\nthe relative smallness of the second term so that\nthe spin rotates by about $90^\\circ$.\n\n\nIf $\\mathcal{P}_1=0$,\nthe spin projection onto the $\\bm e_1$ direction is always conserved. The spin can, however, rotate about the $\\bm e_1$ direction if $\\rho_2$ depends on $z$. In this case, the angle of the spin turn is equal to\n\\begin{eqnarray}\n\\Delta\\phi(z)=-\\arcsin{\\frac{c\\mathcal{P}_2}{B\\rho_2(z)}}+\n\\arcsin{\\frac{c\\mathcal{P}_2}{B\\rho_{20}}}, ~~~\nB=\\sqrt{E^2-m^2c^4}=\n\\sqrt{\\frac{c^2p_{30}^2}{\\rho^2_{30}}+\\frac{c^2\\mathcal{P}_2^2}{\\rho^2_{20}}}.\n\\label{varfi}\n\\end{eqnarray}\n\nThe total spin turn ($z=z_f$) is given by\n\\begin{eqnarray}\n\\Delta\\phi=\n\\arctan{\\frac{\\mathcal{P}_2\\rho_{30} }{\\rho_{20}p_{30}}}-\\sgn{(\\mathcal{P}_2)}\\cdot\\frac\\pi 2.\n\\label{varff}\n\\end{eqnarray}\n\n\\section{Conclusions and outlook}\nWe considered the Dirac fermion dynamics in the curved\nspace model of variable dimension. The advantage of the\ntoy model used is the possibility of performing the exact\nFW transformation of the Dirac equation and then\nobtaining the exact solutions of the equations of motion\nfor momentum and spin in the classical limit. At the same\ntime, the obtained Hamiltonian (\\ref{HamiltonC}) is similar to the generic one (\\ref{Hamiltong}) so that one can expect that\nqualitative features of spin and momentum dynamics will persist for other compactification-related metrics as well.\n\n\nThe analysis of particle momentum evolution allows us to describe the motion at the boundary between the regions of space\nhaving different dimensions.\nThe passage to the region of lower dimension is more natural in the special case when the generalized momentum in the compactified direction $\\mathcal{P}_1=0$.\nAt the same time, the transition to the region of higher dimension (considered in Refs. \\cite{Mureika:2011bv,Fiziev:2010je} as a possible way of the evolution of the Universe) does not impose the constraints for its initial state, manifesting a sort of irreversibility.\n\nThe particle motion (especially near the turning point) is characterized by the three main properties which cannot be naturally explained from the point of view of\nobserver residing in the compactified spacetime:\n\\emph{i)} a reversion of the direction of motion; \\emph{ii)} a rather quick motion along the compactified direction, which may be seen as a sort of ``zitterbewegung'';\n\\emph{iii)} the appearance of a pseudovector of spin in the compactified (2+1)-dimensional space and its\nrotation or flickering [when the spin pseudovector crosses\nthe remnant (2+1)-dimensional layer]. \n\nThe experimental tests of the emerging spin effects may be performed by studies of spin polarizations of $\\Lambda$ (and, probably, also $\\Lambda_c$) hyperons produced in the high-energy collisions where the compactification \\cite{Mureika:LHC,StojkovicReview}\ntakes place. This may bear a resemblance to the recently proposed \\cite{Baznat:2013zx}\ntests of the vorticity in heavy-ion collisions, although a detailed analysis is required.\n\n\n\n\n\n\n\nWe can finally conclude that the transition to (2+1)-dimensional spacetime\nleads to the nontrivial behavior of spin which, generally speaking, cannot be adequately described from the point of view of an observer residing at (2+1) dimensions.\n\n\n\n\n\n\n\\section*{Acknowledgments}\n\nWe are indebted to P.P. Fiziev, V.P. Neznamov, and D.V. Shirkov for stimulating discussions.\nThis work was supported in part by the RFBR (Grants No. 11-02-01538 and 12-02-91526) and BRFFR\n(Grant No. $\\Phi$12D-002).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Deninger's method}\n\nIn this section we express the Mahler measure of $P$ in terms of the integral of a differential $1$-form on the modular curve $X_1(13)$, following Deninger's method \\cite{deninger:mahler}.\n\nWe view $P$ as a polynomial in $h$:\n\\begin{equation*}\nP(H,h) = -H + (-H^2+2H+1)h+(H^2+H-1)h^2-Hh^3.\n\\end{equation*}\nNote that the constant term of $P$ is given by $P^*(H)=-H$.\n\nLet $Z \\subset \\mathbf{G}_m^2$ be the curve defined by the equation $P=0$. Then $Z$ identifies with an affine open subscheme of $X_1(13)$ by \\cite[p. 56]{lecacheux:13}. In particular $Z$ is smooth.\n\n\nLooking at the resultant of the polynomials $P(H,h)$ and $H^2 h^3 P(\\frac{1}{H},\\frac{1}{h})$ with respect to $h$, it can be checked that $P$ doesn't vanish on the torus $T^2 = \\{(H,h) \\in \\mathbf{C} : |H|=|h|=1\\}$. Moreover, we check numerically that for each $H \\in T$, there exists a unique $h(H) \\in \\mathbf{C}$ such that $P(H,h(H))=0$ and $0<|h(H)|<1$. The map $H \\in T \\mapsto h(H)$ defines a closed cycle $\\gamma_P$ in $H_1(Z(\\mathbf{C}),\\mathbf{Z})$. We call $\\gamma_P$ the \\emph{Deninger cycle} associated to $P$. We give $\\gamma_P$ the canonical orientation coming from $T$.\n\nSince $P^*$ doesn't vanish on $T$, the polynomial $P$ satisfies the assumptions \\cite[3.2]{deninger:mahler}, so that the discussion in \\emph{loc. cit.} applies. Consider the differential form $\\eta = \\log |h| \\frac{dH}{H}$ on $Z(\\mathbf{C})$. Using Jensen's formula, and noting that $m(P^*)=0$, we have \\cite[(23)]{deninger:mahler}\n\\begin{equation*}\nm(P) = -\\frac{1}{2\\pi i} \\int_{\\gamma_P} \\eta.\n\\end{equation*}\nNow we may express this as an integral of a closed differential form. By \\cite[Prop. 3.3]{deninger:mahler}, we get\n\\begin{equation*}\nm(P) = -\\frac{1}{2\\pi i} \\int_{\\gamma_P} \\log |H| \\cdot (\\partial-\\overline{\\partial}) \\log |h| - \\log |h| \\cdot (\\partial-\\overline{\\partial}) \\log |H|.\n\\end{equation*}\n\nWe now introduce a standard notation.\n\n\\begin{definition}\nFor any two meromorphic functions $u,v$ on a Riemann surface, define\n\\begin{equation*}\n\\eta(u,v) : = \\log |u| \\operatorname{darg}(v) - \\log |v| \\operatorname{darg}(u).\n\\end{equation*}\n\\end{definition}\n\nThe $1$-form $\\eta(u,v)$ is well-defined outside the set of zeros and poles of $u$ and $v$. It is closed, so we may integrate it over cycles. Moreover, we have $\\operatorname{darg}(u) = -i (\\partial-\\overline{\\partial}) \\log |u|$. Thus we have proved the following proposition.\n\n\n\\begin{pro}\\label{pro deninger}\nWe have $m(P)=\\frac{1}{2\\pi} \\int_{\\gamma_P} \\eta(h,H)$.\n\\end{pro}\n\n\\begin{lem}\nLet $c$ denote complex conjugation on $Z(\\mathbf{C})$. We have $c_* \\gamma_P = -\\gamma_P$.\n\\end{lem}\n\n\\begin{proof}[Proof]\nFor every $H \\in T$, we have $h(\\overline{H})=\\overline{h(H)}$. It follows that $c_* \\gamma_P=-\\gamma_P$.\n\\end{proof}\n\n\n\\section{Determining Deninger's cycle}\n\nIn this section, we determine $\\gamma_P$ explicitly in terms of modular symbols.\n\nThe space $S_2(\\Gamma_1(13))$ of cusp forms of weight $2$ and level $13$ has dimension $2$ over $\\mathbf{C}$. Let $\\varepsilon : (\\mathbf{Z}\/13\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$ be the unique Dirichlet character satisfying $\\varepsilon(2)=\\zeta_6 := e^{\\frac{2\\pi i}{6}}$. It is even and has order $6$. A basis of $S_2(\\Gamma_1(13))$ is given by $(f_\\varepsilon,f_{\\overline{\\varepsilon}})$, where $f_\\varepsilon$ (resp. $f_{\\overline{\\varepsilon}}$) is a newform having character $\\varepsilon$ (resp. ${\\overline{\\varepsilon}}$). The Fourier coefficients of $f_\\varepsilon$ and $f_{\\overline{\\varepsilon}}$ belong to the field $\\mathbf{Q}(\\zeta_6)$ and are complex conjugate to each other. We define $f=f_\\varepsilon+f_{\\overline{\\varepsilon}}$.\n\nWe denote by $\\langle d \\rangle$ the diamond automorphism of $X_1(13)$ associated to $d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times\/\\pm 1$.\n\nLet $\\hat{\\mathcal{H}}=H_1(X_1(13)(\\mathbf{C}),\\{\\mathrm{cusps}\\},\\mathbf{Z})$ be the homology group of $X_1(13)(\\mathbf{C})$ relative to the cusps. Let $E_{13}$ be the set of non-zero vectors of $(\\mathbf{Z}\/13\\mathbf{Z})^2$. For any $x \\in E_{13}$, we let $\\xi(x) = \\{g_x 0, g_x \\infty\\}$, where $g_x \\in \\SL_2(\\mathbf{Z})$ is any matrix whose bottom line is congruent to $x$ modulo $13$. Using Manin's algorithm \\cite{manin} and its implementation in Magma \\cite{magma}, we find that a $\\mathbf{Z}$-basis of $\\mathcal{H}=H_1(X_1(13)(\\mathbf{C}),\\mathbf{Z})$ is given by\n\\begin{align*}\n\\gamma_1 & = \\xi(1,-5)-\\xi(2,5)-\\xi(1,-2) = \\left\\{\\frac15,\\frac25\\right\\}\\\\\n\\gamma_2 & = \\langle 2 \\rangle_* \\gamma_1 = \\xi(2,3)-\\xi(4,-3)-\\xi(2,-4)\\\\\n\\gamma_3 & = \\xi(1,-3)-\\xi(1,3) = \\left\\{\\frac13,-\\frac13\\right\\}\\\\\n\\gamma_4 & = \\langle 2 \\rangle_* \\gamma_3 = \\xi(2,-6)-\\xi(2,6).\n\\end{align*}\n\nConsider the pairing\n\\begin{align*}\n\\langle \\cdot,\\cdot \\rangle : \\hat{\\mathcal{H}} \\times S_2(\\Gamma_1(13)) & \\to \\mathbf{C}\\\\\n(\\gamma,f) & \\mapsto \\int_\\gamma 2\\pi i f(z)dz.\n\\end{align*}\n\n\\begin{definition}\nLet $\\mathcal{H}^- := \\{ \\gamma \\in \\mathcal{H} : c_* \\gamma=-\\gamma\\}$. We define the map\n\\begin{align*}\n\\iota : \\mathcal{H}^- & \\to \\mathbf{C}\\\\\n\\gamma & \\mapsto \\langle \\gamma,f_\\varepsilon \\rangle.\n\\end{align*}\n\\end{definition}\n\n\\begin{lem} \\label{iota injective}\nThe map $\\iota$ is injective.\n\\end{lem}\n\n\\begin{proof}[Proof]\nIf $\\iota(\\gamma)=0$ then $\\langle \\gamma,f_{\\overline{\\varepsilon}} \\rangle = \\overline{\\langle c_* \\gamma, f_\\varepsilon \\rangle} = -\\overline{\\langle \\gamma,f_\\varepsilon \\rangle} = 0$. Thus $\\gamma$ is orthogonal to $S_2(\\Gamma_1(13))$, which implies $\\gamma=0$.\n\\end{proof}\n\n\\begin{lem}\\label{iota lattice}\nThe image of $\\iota$ is the hexagonal lattice generated by $\\iota(\\gamma_3)$ and $\\iota(\\gamma_4) = \\zeta_6 \\iota(\\gamma_3)$.\n\\end{lem}\n\n\\begin{proof}[Proof]\nThe action of complex conjugation on $\\mathcal{H}$ is given by\n\\begin{align*}\nc_*(\\gamma_1) & = \\gamma_1+\\gamma_4\\\\\nc_*(\\gamma_2) & = \\gamma_2 - \\gamma_3 + \\gamma_4\\\\\nc_*(\\gamma_3) & = -\\gamma_3\\\\\nc_*(\\gamma_4) & = -\\gamma_4.\n\\end{align*}\nFrom these formulas, it is clear that a $\\mathbf{Z}$-basis of $\\mathcal{H}^-$ is given by $(\\gamma_3,\\gamma_4)$. By Lemma \\ref{iota injective}, we have $\\iota(\\gamma_3) \\neq 0$. Then\n\\begin{equation*}\n\\iota(\\gamma_4) = \\langle \\langle 2 \\rangle_* \\gamma_3, f_\\varepsilon \\rangle = \\langle \\gamma_3, f_\\varepsilon | \\langle 2 \\rangle \\rangle = \\varepsilon(2) \\iota(\\gamma_3) = \\zeta_6 \\iota(\\gamma_3).\n\\end{equation*}\n\\end{proof}\n\nWe have $\\gamma_3 = \\{\\frac13,-\\frac13\\} = \\{\\frac13,g_1 \\left(\\frac13\\right)\\}$ with $g_1 = \\begin{pmatrix} 14 & -5 \\\\ -39 & 14 \\end{pmatrix} \\in \\Gamma_1(13)$. Let us choose $z_0 = \\frac{14+i}{39}$. Then \n$g_1 (z_0) = \\frac{-14+i}{39}$. We have\n\\begin{equation*}\n\\langle \\gamma_3,f_\\varepsilon \\rangle = \\int_{z_0}^{g_1 z_0} 2\\pi i f_\\varepsilon(z) dz = \\sum_{n=1}^{\\infty} \\frac{a_n(f_\\varepsilon)}{n} \\left(e^{\\frac{-28\\pi i n}{39}}-e^{\\frac{28\\pi i n}{39}}\\right) e^{-\\frac{2\\pi n}{39}}.\\end{equation*}\nUsing Magma, we get numerically\n\\begin{equation*}\n\\langle \\gamma_3,f_\\varepsilon \\rangle \\sim 1.06759 - 2.60094i.\n\\end{equation*}\n\n\\begin{pro} \\label{pro gamma0}\nLet $\\gamma_P \\in \\mathcal{H}^-$ be Deninger's cycle. We have $\\gamma_P=\\gamma_3$.\n\\end{pro}\n\n\\begin{proof}[Proof]\nA $\\mathbf{Q}$-basis of $\\Omega^1(X_1(13))$ is given by $(\\omega, h\\omega)$ where\n\\begin{equation*}\n\\omega = \\frac{(h^2 - h)H - h^3 + h^2 + 2h - 1}{h^4 - 2h^3 + 3h^2 - 2h + 1}dH.\n\\end{equation*}\nUsing Magma, we compute the Fourier expansion of $\\omega$ and $h\\omega$ at infinity, and deduce\n\\begin{equation}\\label{feps omega}\n2\\pi i f_\\varepsilon(z)dz = \\alpha \\omega + \\beta h\\omega\n\\end{equation}\nwith\n\\begin{equation*}\n\\alpha \\sim 0.71163+0.70256i \\qquad \\beta \\sim 0.25262-0.96757i.\n\\end{equation*}\nNote that $\\alpha$ and $\\beta$ are algebraic numbers, but we won't need an explicit formula for them. With Pari\/GP \\cite{pari273}, we find\n\\begin{equation}\\label{int gammaP}\n\\int_{\\gamma_P} \\omega \\sim - 3.21731i \\qquad \\int_{\\gamma_P} h\\omega \\sim - 1.23275i.\n\\end{equation}\nFrom (\\ref{feps omega}) and (\\ref{int gammaP}), it follows that\n\\begin{equation*}\n\\langle \\gamma_P, f_\\varepsilon \\rangle \\sim 1.06759-2.60094i \\sim \\langle \\gamma_3, f_\\varepsilon \\rangle.\n\\end{equation*}\nSince the image of $\\iota$ is a lattice by Lemma \\ref{iota lattice}, we may ascertain that $\\gamma_P=\\gamma_3$.\n\\end{proof}\n\nWe will also need to make explicit the action of the Atkin-Lehner involution $W_{13}$ on $\\gamma_P$.\n\n\\begin{pro} \\label{pro W13gamma0}\nWe have $W_{13} \\gamma_P = \\gamma_4-\\gamma_3$.\n\\end{pro}\n\n\\begin{proof}[Proof]\nBy \\cite[Thm 2.1]{Atkin-Li}, we have $W_{13} f_\\varepsilon = w \\cdot f_{\\overline{\\varepsilon}}$ with\n\\begin{equation}\\label{eq w}\nw = \\frac{3\\zeta_6-4}{13} \\tau(\\varepsilon) \\sim -0.96425+0.26501i.\n\\end{equation}\nWe deduce\n\\begin{equation*}\n\\iota(W_{13} \\gamma_P) = \\langle \\gamma_P, W_{13} f_\\varepsilon \\rangle = w \\langle \\gamma_P, f_{\\overline{\\varepsilon}} \\rangle = w \\overline{\\langle c_* \\gamma_P, f_\\varepsilon \\rangle} = -w \\overline{\\langle \\gamma_P, f_\\varepsilon \\rangle} \\sim 1.71869+2.22503i.\n\\end{equation*}\nMoreover, we have\n\\begin{equation*}\n\\iota(\\gamma_4) = \\zeta_6 \\iota(\\gamma_3) \\sim 2.78628-0.37591i \\sim \\iota(W_{13} \\gamma_P) + \\iota(\\gamma_3).\n\\end{equation*}\nUsing Lemma \\ref{iota lattice} again, we conclude that $W_{13} \\gamma_P = \\gamma_4-\\gamma_3$.\n\\end{proof}\n\n\n\\section{Beilinson's theorem}\n\nWe now recall the explicit version of Beilinson's theorem on the modular curve $X_1(N)$ \\cite{brunault:smf}. Let $\\mathbf{C}(X_1(N))$ be the function field of $X_1(N)$. The \\emph{regulator map} on $X_1(N)$ is defined by\n\\begin{align*}\nr_N : K_2(\\mathbf{C}(X_1(N))) & \\to \\Hom_\\mathbf{C}(S_2(\\Gamma_1(N)),\\mathbf{C})\\\\\n\\{u,v\\} & \\mapsto \\left(f \\mapsto \\int_{X_1(N)(\\mathbf{C})} \\eta(u,v) \\wedge \\omega_f \\right)\n\\end{align*}\nwhere $\\omega_f := 2\\pi i f(z) dz$. After tensoring with $\\mathbf{C}$, we get a linear map\n\\begin{equation*}\nr_N : K_2(\\mathbf{C}(X_1(N))) \\otimes \\mathbf{C} \\to \\Hom_\\mathbf{C}(S_2(\\Gamma_1(N)),\\mathbf{C}).\n\\end{equation*}\nFor any even non-trivial Dirichlet character $\\chi : (\\mathbf{Z}\/N\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$, there exists a modular unit $u_{\\chi} \\in \\mathcal{O}^{\\times}(Y_1(N)(\\mathbf{C})) \\otimes \\mathbf{C}$ satisfying\n\\begin{equation*}\n\\log |u_{\\chi}(z)| = \\frac{1}{\\pi} \\lim_{\\substack{s \\rightarrow 1\\\\ \\real(s)>1}} \\left(\\sideset{}{'}\\sum_{(m,n) \\in \\mathbf{Z}^2} \\frac{\\chi(n) \\cdot \\imag(z)^s}{\\abs{Nmz+n}^{2s}}\\right) \\qquad (z \\in \\mathfrak{H}),\n\\end{equation*}\nwhere $\\sideset{}{'}\\sum$ denotes that we omit the term $(m,n) = (0,0)$ (see \\cite[Prop 5.3]{brunault:smf}).\n\n\\begin{remark}\nWe are working with the model of $X_1(N)$ in which the $\\infty$-cusp is not defined over $\\mathbf{Q}$, but rather over $\\mathbf{Q}(\\zeta_N)$. Therefore, the modular unit $u_\\chi$ is not defined over $\\mathbf{Q}$ but rather over $\\mathbf{Q}(\\zeta_N)$.\n\\end{remark}\n\n\\begin{thm}\\cite[Thm 1.1]{brunault:smf}\\label{explicit beilinson}\nLet $f \\in S_2(\\Gamma_1(N),\\psi)$ be a newform of weight $2$, level $N$ and character $\\psi$. For any even primitive Dirichlet character $\\chi : (\\mathbf{Z}\/N\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$, with $\\chi \\neq \\overline{\\psi}$, we have\n\\begin{equation}\\label{explicit beilinson formula}\nL(f,2) L(f,\\chi,1) = \\frac{N \\pi \\tau(\\chi)}{2 \\phi(N)} \\bigl\\langle r_N(\\{u_{\\overline{\\chi}},u_{\\psi\\chi}\\}), f \\bigr\\rangle\n\\end{equation}\nwhere $L(f,\\chi,s):=\\sum_{n=1} a_n(f) \\chi(n) n^{-s}$ denotes the $L$-function of $f$ twisted by $\\chi$, $\\tau(\\chi):=\\sum_{a \\in (\\mathbf{Z}\/N\\mathbf{Z})^\\times} \\chi(a) e^{\\frac{2\\pi ia}{N}}$ denotes the Gauss sum of $\\chi$, and $\\phi(N)$ denotes Euler's function.\n\\end{thm}\n\nWe will also need the following lemma.\n\n\\begin{lem}\\label{lem ceta}\nLet $c$ denote complex conjugation on $Y_1(N)(\\mathbf{C})$. For any even non-trivial Dirichlet characters $\\chi,\\chi' : (\\mathbf{Z}\/N\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$, we have $c^* \\eta(u_\\chi,u_{\\chi'}) = -\\eta(u_\\chi,u_{\\chi'})$.\n\\end{lem}\n\n\\begin{proof}[Proof]\nRecall that $c$ is given by $c(z)=-\\overline{z}$ on $\\mathfrak{H}$. We have $c^* \\log |u_\\chi| = \\log |u_\\chi|$, and $c^*$ exchanges the holomorphic and anti-holomorphic parts of $\\operatorname{dlog} |u_\\chi|$. Since $\\operatorname{darg}(u_\\chi) = -i (\\partial-\\overline{\\partial}) \\log |u_\\chi|$, we get $c^* \\operatorname{darg}(u_\\chi) = -\\operatorname{darg}(u_\\chi)$, and thus $c^* \\eta(u_\\chi,u_{\\chi'}) = -\\eta(u_\\chi,u_{\\chi'})$.\n\\end{proof}\n\n\\begin{remark}\nBy \\cite[Prop. 5.4 and Prop. 6.1]{brunault:smf}, we have $\\{u_\\chi,u_{\\chi'}\\} \\in K_2(X_1(N)(\\mathbf{C})) \\otimes \\mathbf{C}$. This implies that for $\\gamma \\in H_1(Y_1(N)(\\mathbf{C}),\\mathbf{Z})$, the integral $\\int_\\gamma \\eta(u_{\\chi},u_{\\chi'})$ depends only on the image of $\\gamma$ in $H_1(X_1(N)(\\mathbf{C}),\\mathbf{Z})$ (see for example the discussion in \\cite[\\S 3]{dokchitser-dejeu-zagier}). Therefore, we have a well-defined map\n\\begin{equation*}\n\\int \\eta(u_{\\chi},u_{\\chi'}) : H_1(X_1(N)(\\mathbf{C}),\\mathbf{Z}) \\to \\mathbf{C}.\n\\end{equation*}\nIt can be extended by linearity to $H_1(X_1(N)(\\mathbf{C}),\\mathbf{C})$.\n\\end{remark}\n\n\\begin{remark}\\label{rem int eta}\nSince $c^* \\eta(u_\\chi,u_{\\chi'}) = -\\eta(u_\\chi,u_{\\chi'})$ by Lemma \\ref{lem ceta}, we have $\\int_\\gamma \\eta(u_\\chi,u_{\\chi'}) = \\int_{\\gamma^-} \\eta(u_\\chi,u_{\\chi'})$ with $\\gamma^- = \\frac12(\\gamma-c_* \\gamma)$.\n\\end{remark}\n\n\\section{Merel's formula}\n\nIn this section, we express the regulator integral appearing in the right hand side of (\\ref{explicit beilinson formula}) as a linear combination of periods. In order to do this, we use an idea of Merel to express the integral over $X_1(N)(\\mathbf{C})$ as a linear combination of products of $1$-dimensional integrals.\n\nLet $N \\geq 1$ be an integer. Let $E_N$ be the set of vectors $(u,v) \\in (\\mathbf{Z}\/N\\mathbf{Z})^2$ such that $(u,v,N)=1$. For any $f \\in S_2(\\Gamma_1(N))$ and any $x \\in E_N$, we define the \\emph{Manin symbol}\n\\begin{equation*}\n\\xi_f(x) = -\\frac{1}{2\\pi} \\langle \\xi(x),f \\rangle = -i \\int_{g_x 0}^{g_x \\infty} f(z) dz,\n\\end{equation*}\nwhere $g_x \\in \\SL_2(\\mathbf{Z})$ is any matrix whose bottom row is congruent to $x$ modulo $N$.\n\nLet $\\rho = e^{\\frac{\\pi i}{3}}$ and $\\sigma = \\begin{pmatrix} 0 & -1 \\\\ 1 & 0\\end{pmatrix}$, $\\tau= \\begin{pmatrix} 0 & -1 \\\\ 1 & -1 \\end{pmatrix}$, $T=\\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix} \\in \\SL_2(\\mathbf{Z})$.\n\nThe following theorem is a variant of a theorem of Merel which expresses the Petersson scalar product of two cusp forms $f$ and $g$ of weight 2 as a linear combination of products of Manin symbols of $f$ and $g$ \\cite[Th\u00e9or\u00e8me 2]{merel:symbmanin}.\n\n\\begin{thm}\\label{thm reg eta}\nLet $f \\in S_2(\\Gamma_1(N))$ be a cusp form of weight 2 and level $N$, and let $u,v \\in \\mathcal{O}^\\times(Y_1(N)(\\mathbf{C}))$ be two modular units. We have\n\\begin{equation}\\label{eq reg eta}\n\\int_{X_1(N)(\\mathbf{C})} \\eta(u,v) \\wedge \\omega_f = \\frac{\\pi}{2} \\sum_{x \\in E_N} \\left(\\int_{g_x \\rho}^{g_x \\rho^2} \\eta(u,v) \\right) \\xi_f(x).\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}[Proof]\nLet $\\mathcal{F}$ be the standard fundamental domain of $\\SL_2(\\mathbf{Z}) \\backslash \\mathfrak{H}$:\n\\begin{equation*}\n\\mathcal{F} = \\{z \\in \\mathfrak{H} : |\\mathrm{Re}(z)| \\leq \\frac12, |z| \\geq 1\\}.\n\\end{equation*}\nIts boundary $\\partial \\mathcal{F}$ is the hyperbolic triangle with vertices $\\rho^2,\\rho,\\infty$. Define\n\\begin{equation*}\nF_x(z) = \\int_\\infty^z \\omega_f | g_x \\qquad (x \\in E_N, z \\in \\mathfrak{H}).\n\\end{equation*}\nWe have\n\\begin{equation*}\n\\int_{X_1(N)(\\mathbf{C})} \\eta(u,v) \\wedge \\omega_f = \\sum_{x \\in E_N\/\\pm 1} \\int_{\\mathcal{F}} (\\eta(u,v) \\wedge \\omega_f) | g_x.\n\\end{equation*}\nSince $\\eta(u,v)$ is closed, we have $(\\eta(u,v) \\wedge \\omega_f) | g_x = -d(F_x \\cdot (\\eta(u,v)|g_x))$ and Stokes' formula gives\n\\begin{align}\n\\nonumber \\int_{X_1(N)(\\mathbf{C})} \\eta(u,v) \\wedge \\omega_f & = - \\sum_{x \\in E_N\/\\pm 1} \\int_{\\partial \\mathcal{F}} F_x \\cdot (\\eta(u,v)|g_x)\\\\\n\\label{eq int} & = - \\sum_{x \\in E_N\/\\pm 1} \\left(\\int_{\\rho^2}^\\rho + \\int_\\rho^\\infty + \\int_\\infty^{\\rho^2}\\right) F_x \\cdot (\\eta(u,v)|g_x).\n\\end{align}\nThe matrix $T$ fixes $\\infty$ and maps $\\rho^2$ to $\\rho$. We have\n\\begin{equation*}\nF_x(Tz) = \\int_\\infty^{Tz} \\omega_f | g_x = \\int_\\infty^{z} \\omega_f | g_x T = F_{xT}(z).\n\\end{equation*}\nIt follows that\n\\begin{align*}\n\\sum_{x \\in E_N\/\\pm 1} \\int_\\rho^\\infty F_x \\cdot (\\eta(u,v)|g_x) & = \\sum_{x \\in E_N\/\\pm 1} \\int_{\\rho^2}^\\infty F_x | T \\cdot (\\eta(u,v)|g_x T)\\\\\n& = \\sum_{x \\in E_N\/\\pm 1} \\int_{\\rho^2}^\\infty F_{xT} \\cdot (\\eta(u,v)|g_{xT})\\\\\n& = \\sum_{x \\in E_N\/\\pm 1} \\int_{\\rho^2}^\\infty F_x \\cdot (\\eta(u,v)|g_x).\n\\end{align*}\nHence (\\ref{eq int}) simplifies to\n\\begin{equation*}\n\\int_{X_1(N)(\\mathbf{C})} \\eta(u,v) \\wedge \\omega_f = \\sum_{x \\in E_N\/\\pm 1} \\int_{\\rho}^{\\rho^2} F_x \\cdot (\\eta(u,v)|g_x).\n\\end{equation*}\nSimilarly, let us use the matrix $\\sigma$, which exchanges $\\rho$ and $\\rho^2$, as well as $0$ and $\\infty$. Since $F_x(\\sigma z) = F_{x \\sigma}(z)+2\\pi \\xi_f(x)$, we get\n\\begin{equation*}\n\\int_{\\rho}^{\\rho^2} F_x \\cdot (\\eta(u,v)|g_x) = \\int_{\\rho^2}^{\\rho} F_{x\\sigma} \\cdot (\\eta(u,v)|g_{x\\sigma}) + 2\\pi \\xi_f(x) \\int_{\\rho^2}^{\\rho} \\eta(u,v) | g_x.\n\\end{equation*}\nSumming over $x$ and using the fact that $\\xi_f(x\\sigma)=-\\xi_f(x)$, we get\n\\begin{align*}\n\\int_{X_1(N)(\\mathbf{C})} \\eta(u,v) \\wedge \\omega_f & = \\frac12 \\sum_{x \\in E_N\/\\pm 1} 2\\pi \\xi_f(x) \\int_{\\rho^2}^{\\rho} \\eta(u,v) | g_{x\\sigma}\\\\\n& = \\pi \\sum_{x \\in E_N\/\\pm 1} \\xi_f(x) \\int_\\rho^{\\rho^2} \\eta(u,v) | g_{x}.\n\\end{align*}\n\\end{proof}\n\n\\begin{remark}\nIt can be shown that if $\\{u,v\\}$ defines an element in $K_2(X_1(N)(\\mathbf{C})) \\otimes \\mathbf{Q}$, then the cycle $\\sum_{x \\in E_N} \\left(\\int_{g_x \\rho}^{g_x \\rho^2} \\eta(u,v)\\right) \\xi(x)$ is \\emph{closed}. This follows from the fact that if $\\gamma_P$ denotes a small loop around a cusp $P$ of $X_1(N)(\\mathbf{C})$, then $\\int_{\\gamma_P} \\eta(u,v) = 2\\pi \\log |\\partial_P(u,v)|$, where $\\partial_P(u,v)$ denotes the tame symbol of $\\{u,v\\}$ at $P$ (see for example \\cite[\\S 4, Lemma]{rodriguez:modular}).\n\\end{remark}\n\n\\begin{definition}\nLet $f \\in S_2(\\Gamma_1(N))$ be a cusp form of weight 2 and level $N$. Consider the following relative cycle on $Y_1(N)(\\mathbf{C})$:\n\\begin{equation*}\n\\gamma_f := \\sum_{x \\in E_N} \\xi_f(x) \\{g_x \\rho, g_x \\rho^2\\}.\n\\end{equation*}\nFurthermore, let us define $\\gamma_f^- := \\frac12 (\\gamma_f-c_* \\gamma_f)$.\n\\end{definition}\n\nCombining Theorem \\ref{explicit beilinson}, Theorem \\ref{thm reg eta} and Remark \\ref{rem int eta}, we get the following result.\n\n\\begin{thm}\\label{explicit beilinson 2}\nLet $f \\in S_2(\\Gamma_1(N),\\psi)$ be a newform of weight $2$, level $N$ and character $\\psi$. For any even primitive Dirichlet character $\\chi : (\\mathbf{Z}\/N\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$, with $\\chi \\neq \\overline{\\psi}$, we have\n\\begin{equation}\\label{explicit beilinson formula 2}\nL(f,2) L(f,\\chi,1) = \\frac{N \\pi^2 \\tau(\\chi)}{4 \\phi(N)} \\int_{\\gamma_f} \\eta(u_{\\overline{\\chi}},u_{\\psi\\chi}) = \\frac{N \\pi^2 \\tau(\\chi)}{4 \\phi(N)} \\int_{\\gamma_f^-} \\eta(u_{\\overline{\\chi}},u_{\\psi\\chi}).\n\\end{equation}\n\\end{thm}\n\nWe will also need an explicit expression of $\\gamma_f$ in terms of Manin symbols. For any $f \\in S_2(\\Gamma_1(N))$ and any $x=(u,v) \\in E_N$, let us define $x^c=(-u,v)$ and\n\\begin{equation*}\n\\xi_f^+(x) = \\frac12 (\\xi_f(x)+\\xi_f(x^c)) = \\frac12 (\\xi_f(x)+\\overline{\\xi_{f^*}(x)}),\n\\end{equation*}\nwhere $f^*$ denotes the cusp form with complex conjugate Fourier coefficients.\n\n\\begin{pro}\\label{pro gammaf}\nLet $f \\in S_2(\\Gamma_1(N))$ be a cusp form of weight $2$ and level $N$. The cycle $\\gamma_f$ is closed, and its image in $H_1(X_1(N)(\\mathbf{C}),\\mathbf{Z})$ can be expressed as follows:\n\\begin{equation}\\label{eq gammaf}\n\\gamma_f = -\\frac13 \\sum_{x \\in E_N} \\left(\\xi_f(x)+2\\xi_f(x \\tau)\\right) \\xi(x).\n\\end{equation}\nMoreover, we have\n\\begin{equation}\\label{eq gammaf 2}\n\\gamma_f^- = -\\frac13 \\sum_{x \\in E_N} \\left(\\xi_f^+(x)+2\\xi_f^+(x \\tau)\\right) \\xi(x).\n\\end{equation}\n\\end{pro}\n\n\\begin{proof}[Proof]\nLet us compute the boundary of $\\gamma_f$. Since $\\sigma(\\rho)=\\rho^2$ and $\\xi_f(x\\sigma)=-\\xi_f(x)$, we have\n\\begin{align*}\n\\partial \\gamma_f & = \\sum_{x \\in E_N} \\xi_f(x) ([g_x \\rho^2]-[g_x \\rho])\\\\\n& = \\sum_{x \\in E_N} \\xi_f(x) ([g_{x\\sigma} \\rho]-[g_x \\rho])\\\\\n& = -2 \\sum_{x \\in E_N} \\xi_f(x) [g_x \\rho].\n\\end{align*}\nSince $\\tau(\\rho)=\\rho$ and because of Manin's relation $\\xi_f(x)+\\xi_f(x\\tau)+\\xi_f(x\\tau^2)=0$, we get\n\\begin{align*}\n\\partial \\gamma_f & = -\\frac23 \\sum_{x \\in E_N} \\xi_f(x) ([g_x \\rho] + [g_{x \\tau} \\rho]+[g_{x \\tau^2} \\rho])\\\\\n& = -\\frac23 \\sum_{x \\in E_N} (\\xi_f(x)+\\xi_f(x \\tau)+\\xi_f(x \\tau^2)) [g_x \\rho] = 0.\n\\end{align*}\nOn the other hand, we have\n\\begin{align*}\n\\gamma_f & = \\sum_{x \\in E_N} \\xi_f(x) (\\{g_x \\rho, g_x \\infty\\}+ \\{g_x \\infty, g_x \\rho^2\\})\\\\\n& = \\sum_{x \\in E_N} \\xi_f(x) (\\{g_x \\rho, g_x \\infty\\} - \\sum_{x \\in E_N} \\xi_f(x) \\{g_x 0, g_x \\rho\\})\\\\\n& = 2 \\sum_{x \\in E_N} \\xi_f(x) \\{g_x \\rho, g_x \\infty\\} - \\sum_{x \\in E_N} \\xi_f(x) \\xi(x).\n\\end{align*}\nUsing the matrix $\\tau$, we get\n\\begin{align*}\n\\gamma_f & = \\frac23 \\sum_{x \\in E_N} \\left(\\xi_f(x) \\{g_x \\rho, g_x \\infty\\}+\\xi_f(x\\tau) \\{g_{x\\tau} \\rho, g_{x\\tau} \\infty\\}+\\xi_f(x\\tau^2) \\{g_{x\\tau^2} \\rho, g_{x\\tau^2} \\infty\\}\\right) - \\sum_{x \\in E_N} \\xi_f(x) \\xi(x)\\\\\n& = \\frac23 \\sum_{x \\in E_N} \\left(\\xi_f(x) \\{g_x \\rho, g_x \\infty\\}+\\xi_f(x\\tau) \\{g_{x} \\rho, g_{x} 0\\}+\\xi_f(x\\tau^2) \\{g_{x} \\rho, g_{x} 1\\}\\right) - \\sum_{x \\in E_N} \\xi_f(x) \\xi(x)\\\\\n& = \\frac23 \\sum_{x \\in E_N} \\left(\\xi_f(x\\tau) \\{g_{x} \\infty, g_{x} 0\\}+\\xi_f(x\\tau^2) \\{g_{x} \\infty, g_{x} 1\\}\\right) - \\sum_{x \\in E_N} \\xi_f(x) \\xi(x)\\\\\n& = \\frac23 \\sum_{x \\in E_N} \\left(-\\xi_f(x\\tau) \\xi(x) +\\xi_f(x\\tau^2) \\{g_{x\\tau^2} 0, g_{x\\tau^2} \\infty\\}\\right) - \\sum_{x \\in E_N} \\xi_f(x) \\xi(x)\\\\\n& = \\frac23 \\sum_{x \\in E_N} \\left(-\\xi_f(x\\tau) \\xi(x) +\\xi_f(x) \\xi(x)\\right) - \\sum_{x \\in E_N} \\xi_f(x) \\xi(x)\\\\\n& = \\frac13 \\sum_{x \\in E_N} (\\xi_f(x) - 2 \\xi_f(x\\tau)) \\xi(x).\n\\end{align*}\nThis gives (\\ref{eq gammaf}). The action of complex conjugation on $\\gamma_f$ is given by\n\\begin{align*}\nc_* \\gamma_f & = \\sum_{x \\in E_N} \\xi_f(x) \\{c(g_x \\rho),c(g_x \\rho^2)\\}\\\\\n& = \\sum_{x \\in E_N} \\xi_f(x) \\{g_{x^c} \\rho^2,g_{x^c} \\rho\\}\\\\\n& = - \\sum_{x \\in E_N} \\xi_f(x^c) \\{g_x \\rho,g_x \\rho^2\\}.\n\\end{align*}\nIt follows that\n\\begin{equation*}\n\\gamma_f^- = \\sum_{x \\in E_N} \\xi_f^+(x) \\{g_x \\rho,g_x \\rho^2\\}.\n\\end{equation*}\nSince the quantities $\\xi_f^+(x)$ satisfy the Manin relations, the same proof as above gives (\\ref{eq gammaf 2}).\n\\end{proof}\n\n\\section{Proof of the main theorem}\n\nLet us return to the case $N=13$. Using Theorem \\ref{explicit beilinson 2} with $f=f_\\varepsilon$, $\\psi=\\varepsilon$ and $\\chi=\\varepsilon^3$, we get\n\\begin{equation}\\label{formula 1}\nL(f_\\varepsilon,2) L(f_\\varepsilon,\\varepsilon^3,1) = \\frac{13 \\pi^2 \\tau(\\varepsilon^3)}{48} \\int_{\\gamma_{f_\\varepsilon}^-} \\eta(u_{\\varepsilon^3},u_{{\\overline{\\varepsilon}}^2}).\n\\end{equation}\nWe are going to make explicit each term in this formula. Note that $\\tau(\\varepsilon^3)=\\sqrt{13}$.\n\n\\begin{definition}\nFor any Dirichlet character $\\psi : (\\mathbf{Z}\/13\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$, let us denote $\\mathcal{H}(\\psi)$ (resp. $\\hat{\\mathcal{H}}(\\psi)$) the $\\psi$-isotypical component of $\\mathcal{H} \\otimes \\mathbf{C}$ (resp. $\\hat{\\mathcal{H}} \\otimes \\mathbf{C}$) with respect to the action of diamond operators $\\langle d \\rangle_*$, $d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times$. For any $\\gamma \\in \\hat{\\mathcal{H}} \\otimes \\mathbf{C}$, let $\\gamma^\\psi$ denote its $\\psi$-isotypical component. Moreover, let us define $\\hat{\\mathcal{H}}^{\\pm}(\\psi) = (\\hat{\\mathcal{H}}^\\pm \\otimes \\mathbf{C}) \\cap \\hat{\\mathcal{H}}(\\psi)$ and $\\mathcal{H}^{\\pm}(\\psi) = (\\mathcal{H}^\\pm \\otimes \\mathbf{C}) \\cap \\mathcal{H}(\\psi)$.\n\\end{definition}\n\n\\begin{lem}\\label{lem Hpsi}\nLet $\\psi=\\varepsilon$ or $\\overline{\\varepsilon}$. Then $\\mathcal{H}^{\\pm}(\\psi)$ has dimension $1$, and a generator is given by\n\\begin{align*}\n\\gamma_\\psi^+ & := \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\xi(1,a)^\\psi\\\\\n\\gamma_\\psi^- & := \\xi(1,-3)^\\psi - \\xi(1,3)^\\psi.\n\\end{align*}\nMoreover, we have $W_{13} \\gamma_\\psi^+ = \\psi(2) \\gamma_{\\overline{\\psi}}^+$.\n\\end{lem}\n\n\\begin{proof}[Proof]\nThe pairing $\\langle \\cdot,\\cdot \\rangle$ induces a perfect pairing\n\\begin{equation*}\n\\mathcal{H}^{\\pm}(\\psi) \\times S_2(\\Gamma_1(13),\\psi) \\to \\mathbf{C}.\n\\end{equation*}\nSince $S_2(\\Gamma_1(13),\\psi)$ is $1$-dimensional, we get $\\dim_\\mathbf{C} \\mathcal{H}^{\\pm}(\\psi)=1$. From the definition, it is clear that $\\gamma_\\psi^+ \\in \\hat{\\mathcal{H}}^+(\\psi)$ and $\\gamma_\\psi^- \\in \\hat{\\mathcal{H}}^-(\\psi)$. Moreover, since $\\gamma_\\psi^- = \\gamma_3^\\psi$, we have $\\gamma_\\psi^- \\in \\mathcal{H}^-(\\psi)$.\n\nLet us compute the boundary of $\\gamma_\\psi^+$. For any $u,v \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times$, we have $\\partial \\xi(u,v) = P_u - P_v$ with $P_d:=\\langle d \\rangle(0)$. Moreover, for any $x \\in E_{13}$, we have\n\\begin{equation*}\n\\xi(x)^\\psi = \\frac{1}{12} \\sum_{d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\overline{\\psi}(d) \\langle d \\rangle_* \\xi(x) = \\frac{1}{12} \\sum_{d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\overline{\\psi}(d) \\xi(d x).\n\\end{equation*}\nIt follows that\n\\begin{align*}\n\\partial \\gamma_\\psi^+ & = \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\partial(\\xi(1,a)^\\psi)\\\\\n& = \\frac{1}{12} \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\sum_{d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\overline{\\psi}(d) \\partial \\xi(d,da)\\\\\n& = \\frac{1}{12} \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\sum_{d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\overline{\\psi}(d) (P_d-P_{da})\\\\\n& = \\frac{1}{12} \\sum_{d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\left(\\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a)-\\varepsilon^3\\psi(a)\\right) \\overline{\\psi}(d) \\cdot P_d = 0.\n\\end{align*}\nHence $\\gamma_\\psi^+ \\in \\mathcal{H}^+(\\psi)$. By \\cite[Lemme 5]{rebolledo}, the elements $\\xi(1,0)^\\psi, \\xi(1,2)^\\psi, \\xi(1,3)^\\psi, \\xi(1,-3)^\\psi$ form a basis of $\\hat{\\mathcal{H}}(\\psi)$, and we can express $\\gamma_\\psi^+$ in terms of this basis. This gives\n\\begin{equation}\\label{eq gammapsi}\n\\gamma_\\psi^+ = (2-4\\psi(2)) \\xi(1,2)^\\psi + \\xi(1,3)^\\psi + \\xi(1,-3)^\\psi.\n\\end{equation}\nIn particular $\\gamma_\\psi^+$ and $\\gamma_\\psi^-$ are nonzero, and thus they generate $\\mathcal{H}^{\\pm}(\\psi)$.\n\nIt remains to compute the action of $W_{13}$ on $\\gamma_\\psi^+$. In view of (\\ref{eq gammapsi}), it is enough to determine the action of $W_{13}$ on $\\xi(1,2)$ and $\\xi(1,3)$. We have\n\\begin{align*}\nW_{13} \\xi(1,2) & = \\left\\{\\frac{2}{13},\\infty\\right\\} = \\left\\{\\frac{2}{13},\\frac{1}{6}\\right\\}+\\left\\{\\frac{1}{6},0\\right\\}+\\{0,\\infty\\}\\\\\n& = -\\xi(0,-6)+\\xi(1,-6)+\\xi(0,1).\n\\end{align*}\nHence, using \\cite[Lemme 5]{rebolledo} again, we get\n\\begin{align*}\nW_{13} (\\xi(1,2)^\\psi) & = -\\xi(0,-6)^{\\overline{\\psi}}+\\xi(1,-6)^{\\overline{\\psi}}+\\xi(0,1)^{\\overline{\\psi}}\\\\\n& = (\\overline{\\psi}(6)-1) \\xi(1,0)^{\\overline{\\psi}} - \\overline{\\psi}(6) \\xi(1,2)^{\\overline{\\psi}}.\n\\end{align*}\nSimilarly, we find\n\\begin{align*}\nW_{13} (\\xi(1,3)^\\psi) & = (\\overline{\\psi}(4)-1) \\xi(1,0)^{\\overline{\\psi}} - \\overline{\\psi}(4) \\xi(1,-3)^{\\overline{\\psi}}\\\\\nW_{13} (\\xi(1,-3)^\\psi) & = (\\overline{\\psi}(4)-1) \\xi(1,0)^{\\overline{\\psi}} - \\overline{\\psi}(4) \\xi(1,3)^{\\overline{\\psi}}.\n\\end{align*}\nSince we know that $W_{13} \\gamma_\\psi^+$ is a multiple of $\\gamma_{\\overline{\\psi}}^+$, we deduce $W_{13} \\gamma_\\psi^+ = -\\overline{\\psi}(4) \\gamma_{\\overline{\\psi}}^+ = \\psi(2) \\gamma_{\\overline{\\psi}}^+$.\n\\end{proof}\n\n\\begin{pro}\\label{pro Lfeps3}\nWe have $L(f_\\varepsilon,\\varepsilon^3,1)=\\frac{\\overline{\\varepsilon}(2)}{\\sqrt{13}} \\langle \\gamma_\\varepsilon^+, f_\\varepsilon \\rangle$.\n\\end{pro}\n\n\\begin{proof}[Proof]\nBy \\cite[Thm 4.2.b)]{manin}, we have\n\\begin{equation*}\nL(f_\\varepsilon,\\varepsilon^3,1) = \\frac{1}{\\sqrt{13}} \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\int_{a\/13}^{\\infty} \\omega_{f_\\varepsilon}.\n\\end{equation*}\nLet us compute the cycle $\\theta = \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\{\\frac{a}{13},\\infty\\}$ in terms of Manin symbols. We have\n\\begin{equation*}\nW_{13} (\\theta^\\varepsilon) = (W_{13} \\theta)^{\\overline{\\varepsilon}} = \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\left\\{-\\frac{1}{a},0\\right\\}^{\\overline{\\varepsilon}} = \\sum_{a \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} \\varepsilon^3(a) \\xi(1,a)^{\\overline{\\varepsilon}} = \\gamma_{\\overline{\\varepsilon}}^+.\n\\end{equation*}\nBy Lemma \\ref{lem Hpsi}, it follows that\n\\begin{equation*}\n\\langle \\theta, f_\\varepsilon \\rangle = \\langle \\theta^\\varepsilon, f_\\varepsilon \\rangle = \\langle W_{13}(\\gamma_{\\overline{\\varepsilon}}^+), f_\\varepsilon \\rangle = \\overline{\\varepsilon}(2) \\langle \\gamma_\\varepsilon^+, f_\\varepsilon \\rangle.\n\\end{equation*}\n\\end{proof}\n\n\\begin{pro}\\label{pro gammafeps}\nWe have $\\gamma_{f_\\varepsilon}^- = \\frac{1-2\\zeta_6}{\\pi} \\langle \\gamma_\\varepsilon^+, f_\\varepsilon \\rangle \\cdot \\gamma_{\\overline{\\varepsilon}}^-$.\n\\end{pro}\n\n\\begin{proof}[Proof]\nBy Proposition \\ref{pro gammaf}, we have\n\\begin{equation*}\n\\gamma_{f_\\varepsilon}^- = - \\frac13 \\sum_{x \\in E_{13}} (\\xi_{f_\\varepsilon}^+(x)+2\\xi_{f_\\varepsilon}^+(x\\tau)) \\xi(x).\n\\end{equation*}\nThis sum involves 168 terms, but we may reduce it to 14 terms by considering the action of diamond operators. Let $\\mathcal{E}$ be the set of 2-tuples $(0,1)$ and $(1,v)$, $v \\in \\mathbf{Z}\/13\\mathbf{Z}$. We have\n\\begin{align*}\n\\gamma_{f_\\varepsilon}^- & = - \\frac13 \\sum_{x \\in \\mathcal{E}} \\sum_{d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} (\\xi_{f_\\varepsilon}^+(dx)+2\\xi_{f_\\varepsilon}^+(dx\\tau)) \\xi(dx)\\\\\n& = - \\frac13 \\sum_{x \\in \\mathcal{E}} \\sum_{d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times} (\\xi_{f_\\varepsilon}^+(x)+2\\xi_{f_\\varepsilon}^+(x\\tau)) \\cdot \\varepsilon(d) \\langle d \\rangle_* \\xi(x)\\\\\n& = -4 \\sum_{x \\in \\mathcal{E}} (\\xi_{f_\\varepsilon}^+(x)+2\\xi_{f_\\varepsilon}^+(x\\tau)) \\xi(x)^{\\overline{\\varepsilon}}.\n\\end{align*}\nA simple computation shows that the terms $x=(0,1)$ and $x=(1,0)$ cancel each other. Hence\n\\begin{equation*}\n\\gamma_{f_\\varepsilon}^- = -4 \\sum_{v \\in (\\mathbf{Z}\/13\\mathbf{Z})^{*}} \\bigl(\\xi^{+}_{f_{\\varepsilon}}(1,v)+2 \\varepsilon(v) \\xi^{+}_{f_{\\varepsilon}}(1,1+\\frac{1}{v})\\bigr) \\cdot \\xi(1,v)^{{\\overline{\\varepsilon}}}.\n\\end{equation*}\nUsing \\cite[Lemme 5]{rebolledo}, we may express $\\xi_{f_\\varepsilon}^+(1,v)$, $v \\neq 0$ in terms of $\\xi_{f_\\varepsilon}^+(1,2)$ and $\\xi_{f_\\varepsilon}^+(1,3)$. We find $\\xi^{+}_{f_{\\varepsilon}}(1,-v) = \\xi^{+}_{f_{\\varepsilon}}(1,v)$ and\n\\begin{align*}\n\\xi^{+}_{f_{\\varepsilon}}(1,1) & = 0 & \\xi^{+}_{f_{\\varepsilon}}(1,4) & = (1-\\zeta_{6})\n\\xi^{+}_{f_{\\varepsilon}}(1,3)\\\\\n\\xi^{+}_{f_{\\varepsilon}}(1,5) & = (\\zeta_{6}-1) \\bigl(\\xi^{+}_{f_{\\varepsilon}}(1,2) -\n\\xi^{+}_{f_{\\varepsilon}}(1,3)\\bigr) & \\xi^{+}_{f_{\\varepsilon}}(1,6)& = (\\zeta_{6}-1) \\xi^{+}_{f_{\\varepsilon}}(1,2).\n\\end{align*}\nMoreover, also by \\cite[Lemme 5]{rebolledo}, the cycles $\\xi(1,v)^{\\overline{\\varepsilon}}$, $v \\neq 0$, are linear combinations of $\\xi(1,2)^{\\overline{\\varepsilon}}$, $\\xi(1,3)^{\\overline{\\varepsilon}}$ and $\\xi(1,-3)^{\\overline{\\varepsilon}}$. Thus the same is true for $\\gamma_{f_\\varepsilon}^-$. But we know that $\\gamma_{f_\\varepsilon}^-$ is a multiple of $\\gamma^{-}_{{\\overline{\\varepsilon}}} = \\xi(1,3)^{{\\overline{\\varepsilon}}}-\\xi(1,-3)^{{\\overline{\\varepsilon}}}$. It is thus enough to compute the coefficient in front of $\\xi(1,3)^{{\\overline{\\varepsilon}}}$, which leads to the identity\n\\begin{equation*}\n\\gamma_{f_\\varepsilon}^- = \\left(12\\xi^{+}_{f_{\\varepsilon}}(1,2)+(8\\zeta_{6}-4) \\xi^{+}_{f_{\\varepsilon}}(1,3)\\right) \\cdot \\gamma^-_{\\overline{\\varepsilon}}.\n\\end{equation*}\nUsing (\\ref{eq gammapsi}) with $\\psi=\\varepsilon$, we get the proposition.\n\\end{proof}\n\nConsider the modular units $x=W_{13}(h)$ and $y=W_{13}(H)$.\n\n\\begin{pro}\\label{pro etaxy}\nWe have $\\int_{\\gamma_{\\overline{\\varepsilon}}^-} \\eta(x,y) = \\frac{13^2 \\sqrt{13}}{48}(1+\\zeta_6) \\tau(\\varepsilon^2) \\int_{\\gamma_{\\overline{\\varepsilon}}^-} \\eta(u_{\\varepsilon^3},u_{{\\overline{\\varepsilon}}^2})$.\n\\end{pro}\n\n\\begin{proof}[Proof]\nSince $h$ and $H$ are supported in the cusps above $0 \\in X_0(13)(\\mathbf{Q})$, it follows that $x$ and $y$ are supported in the cusps above $\\infty \\in X_0(13)(\\mathbf{Q})$, namely the cusps $\\langle d \\rangle \\infty$, $d \\in (\\mathbf{Z}\/13\\mathbf{Z})^\\times\/\\pm 1$. The method of proof is simple : we decompose the divisors of $x$ and $y$ as linear combinations of Dirichlet characters.\n\nLet us write $\\begin{pmatrix} n_1 & n_2 & \\cdots & n_6 \\end{pmatrix}$ for the divisor $\\sum_{d=1}^6 n_d \\cdot \\langle d \\rangle \\infty$. By \\cite[p. 56]{lecacheux:13}, we have\n\\begin{align*}\n\\dv(x) & = \\begin{pmatrix} 0 & 1 & 1 & -1 & 0 & -1 \\end{pmatrix}\\\\\n\\dv(y) & = \\begin{pmatrix} 1 & -1 & 1 & 1 & -1 & -1 \\end{pmatrix}.\n\\end{align*}\nThe divisors of $u_{\\varepsilon^3}$ and $u_{{\\overline{\\varepsilon}}^2}$ are given by \\cite[Prop 5.4]{brunault:smf}. We have\n\\begin{equation*}\n\\dv (u_{\\varepsilon^{3}}) = -\\frac{L(\\varepsilon^{3},2)}{\\pi^{2}} \\cdot \\begin{pmatrix} 1 & -1 & 1 & 1 & -1 & -1 \\end{pmatrix} = -\\frac{4 \\sqrt{13}}{13^{2}} \\dv (y).\n\\end{equation*}\nSince the divisor of $x$ is invariant under the diamond operator $\\langle 5 \\rangle$, it is a linear combination of $\\dv(u_{\\varepsilon^2})$ and $\\dv(u_{{\\overline{\\varepsilon}}^2})$. We find explicitly\n\\begin{align*}\n\\dv(x) & = \\frac{1-2\\zeta_{6}}{3} \\Bigl(\\frac{\\dv(u_{\\varepsilon^{2}})}{L(\\varepsilon^{2},2)\/\\pi^{2}} - \\frac{\\dv(u_{{\\overline{\\varepsilon}}^{2}})}{L({\\overline{\\varepsilon}}^{2},2)\/\\pi^{2}}\\Bigr)\\\\\n& = \\frac{13}{12} \\bigl((2-\\zeta_{6}) \\tau({\\overline{\\varepsilon}}^{2}) \\dv(u_{\\varepsilon^{2}}) + (1+\\zeta_{6})\n\\tau(\\varepsilon^{2}) \\dv(u_{{\\overline{\\varepsilon}}^{2}})\\bigr).\n\\end{align*}\nHere we have used the classical formula \\cite[(1.80) and (3.87)]{cartier}\n\\begin{equation*}\n\\frac{L(\\chi,2)}{\\pi^2} = \\frac{\\tau(\\chi)}{N} \\sum_{a=0}^{N-1} {\\overline{\\chi}}(a) B_2\\left(\\frac{a}{N}\\right)\n\\end{equation*}\nwhere $\\chi$ is an even non-trivial Dirichlet character modulo $N$, and $B_2(x)=x^2-x+\\frac16$ is the second Bernoulli polynomial.\n\nConsidering $u_{\\varepsilon^3}$ and $u_{{\\overline{\\varepsilon}}^2}$ as elements of $\\mathcal{O}^{*}(Y_{1}(13)(\\mathbf{C})) \\otimes\n\\mathbf{C}$ and following the notations of \\cite[(65)]{brunault:smf}, we have $\\widehat{u_{\\varepsilon^3}}(\\infty)=\\widehat{u_{{\\overline{\\varepsilon}}^2}}(\\infty)=1$ by \\cite[Prop. 5.3]{brunault:smf}. Moreover, looking at the behaviour of $x$ and $y$ at $\\infty$, we find $x(\\infty)=1$ and $\\widehat{y}(\\infty)=-1$. Hence $x \\otimes 1$ can be expressed as a linear combination of $u_{\\varepsilon^2}$ and $u_{{\\overline{\\varepsilon}}^2}$ in $\\mathcal{O}^{*}(Y_{1}(13)(\\mathbf{C})) \\otimes\n\\mathbf{C}$, while $y \\otimes 1$ is proportional to $u_{\\varepsilon^3}$. Thus\n\\begin{equation*}\n\\eta(x,y) = -\\frac{13^2}{4 \\sqrt{13}} \\cdot \\frac{13}{12} \\left((2-\\zeta_{6}) \\tau({\\overline{\\varepsilon}}^{2}) \\eta(u_{\\varepsilon^{2}},u_{\\varepsilon^3}) + (1+\\zeta_{6}) \\tau(\\varepsilon^{2}) \\eta(u_{{\\overline{\\varepsilon}}^{2}},u_{\\varepsilon^3})\\right).\n\\end{equation*}\nSince the differential form $\\eta(u_{\\varepsilon^2},u_{\\varepsilon^3})$ has character $\\varepsilon$, we have $\\int_{\\gamma_{{\\overline{\\varepsilon}}}^-} \\eta(u_{\\varepsilon^2},u_{\\varepsilon^3}) = 0$, and the proposition follows.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{main thm}]\nCombining (\\ref{formula 1}) with Propositions \\ref{pro Lfeps3}, \\ref{pro gammafeps}, \\ref{pro etaxy}, we get\n\\begin{equation}\\label{formula 2}\nL(f_\\varepsilon,2) = \\frac{\\pi}{\\sqrt{13}} \\cdot \\frac{1-\\zeta_6}{\\tau(\\varepsilon^2)} \\int_{\\gamma_{{\\overline{\\varepsilon}}}^-} \\eta(x,y).\n\\end{equation}\nFormula (\\ref{formula 2}) simplifies if we use the functional equation of $L(f_\\varepsilon,s)$. Recall that $W_{13}(f_\\varepsilon) = w f_{{\\overline{\\varepsilon}}}$. Let $\\Lambda(f,s):=13^{s\/2} (2\\pi)^{-s} \\Gamma(s) L(f,s)$. Then the functional equation of $L(f_\\varepsilon,s)$ reads\n\\begin{equation*}\n\\Lambda(f_\\varepsilon,s) = -w \\Lambda(f_{{\\overline{\\varepsilon}}},2-s).\n\\end{equation*}\nUsing (\\ref{eq w}), we deduce that\n\\begin{equation*}\nL(f_\\varepsilon,2) = \\frac{4\\pi^2}{13^2} (4-3\\zeta_6) \\tau(\\varepsilon) L'(f_{\\overline{\\varepsilon}},0).\n\\end{equation*}\nReplacing in (\\ref{formula 2}) and using $\\tau(\\varepsilon^2) \\tau(\\varepsilon) = (4\\zeta_6-3) \\sqrt{13}$, we get\n\\begin{equation}\\label{formula 3}\n\\int_{\\gamma_{\\overline{\\varepsilon}}^-} \\eta(x,y) = 4\\pi (\\zeta_6-1) L'(f_{\\overline{\\varepsilon}},0).\n\\end{equation}\nTaking complex conjugation, and since $\\overline{\\eta(x,y)}=\\eta(x,y)$, we obtain\n\\begin{equation}\\label{formula 4}\n\\int_{\\gamma_\\varepsilon^-} \\eta(x,y) = -4\\pi \\zeta_6 L'(f_\\varepsilon,0).\n\\end{equation}\nWe have a direct sum decomposition $\\mathcal{H}^- \\otimes \\mathbf{C} = \\mathcal{H}^-(\\varepsilon) \\oplus \\mathcal{H}^-({\\overline{\\varepsilon}})$. Write $\\gamma_3 = \\gamma_3^\\varepsilon + \\gamma_3^{{\\overline{\\varepsilon}}}$. Then $\\gamma_4 = \\langle 2 \\rangle_* \\gamma_3 = \\varepsilon(2) \\gamma_3^\\varepsilon + {\\overline{\\varepsilon}}(2) \\gamma_3^{\\overline{\\varepsilon}}$. By Proposition \\ref{pro W13gamma0}, we deduce\n\\begin{equation*}\nW_{13} \\gamma_P = \\gamma_4 - \\gamma_3 = (\\zeta_6-1) \\gamma_3^\\varepsilon + (\\overline{\\zeta_6}-1) \\gamma_3^{\\overline{\\varepsilon}} = (\\zeta_6-1) \\gamma_\\varepsilon^- + (\\overline{\\zeta_6}-1) \\gamma_{\\overline{\\varepsilon}}^-.\n\\end{equation*}\nBy (\\ref{formula 3}) and (\\ref{formula 4}), we then have\n\\begin{align*}\n\\int_{W_{13} \\gamma_P} \\eta(x,y) & = (\\zeta_6-1) \\int_{\\gamma_\\varepsilon^-} \\eta(x,y) + (\\overline{\\zeta_6}-1) \\int_{\\gamma_{\\overline{\\varepsilon}}^-} \\eta(x,y)\\\\\n& = 4\\pi (L'(f_\\varepsilon,0) + L'(f_{\\overline{\\varepsilon}},0)).\n\\end{align*}\nBy Proposition \\ref{pro deninger}, we conclude that\n\\begin{equation*}\nm(P)=\\frac{1}{2\\pi} \\int_{\\gamma_P} \\eta(h,H) = \\frac{1}{2\\pi} \\int_{W_{13} \\gamma_P} \\eta(x,y) = 2 L'(f,0).\n\\end{equation*}\n\\end{proof}\n\n\\begin{remark}\nThere may have been a quicker way to proceed. Starting from Theorem \\ref{explicit beilinson} in the particular case $N=13$, probably all we need is a symplectic basis of $H_1(X_1(13)(\\mathbf{C}),\\mathbf{Z})$ with respect to the intersection pairing (see the formula \\cite[A.2.5]{bost}). But this is less canonical than Theorem \\ref{thm reg eta}.\n\\end{remark}\n\n\\begin{remark}\nAnother way of proving Theorem \\ref{main thm} would be to use the main formula of \\cite{zudilin}. We have not worked out the details of this computation.\n\\end{remark}\n\n\\begin{question}\nLet $g = f | \\langle 2 \\rangle = \\zeta_6 f_\\varepsilon+ \\overline{\\zeta_6} f_{\\overline{\\varepsilon}}$. Then $(f,g)$ is a basis of the space $S_2(\\Gamma_1(13),\\mathbf{Q})$ of cusp forms with rational Fourier coefficients. Is there a polynomial $Q \\in \\mathbf{Z}[x,y]$ such that $m(Q)$ is proportional to $L'(g,0)$?\n\\end{question}\n\n\\section{Examples in higher level}\n\nWe note that the functions $H$ and $h$ used in the proof of Theorem \\ref{main thm} are modular units on $X_1(13)$ and that $P$ is their minimal polynomial. There is a similar story for the modular curve $X_1(11)$ \\cite[Cor 3.3]{brunault:cras} and we may try to generalize this phenomenon.\n\nLet $N \\geq 1$ be an integer, and let $u$ and $v$ be two modular units on $X_1(N)$. Let $P \\in \\mathbf{C}[x,y]$ be an irreducible polynomial such that $P(u,v)=0$. Then the map $z \\mapsto (u(z),v(z))$ is a modular parametrization of the curve $C_P : P(x,y)=0$ and we have a natural map $Y_1(N) \\to C_P$. Assuming $P$ satisfies Deninger's conditions, we may express $m(P)$ in terms of the integral of $\\eta(u,v)$ over a (non necessarily closed) cycle $\\gamma_P$.\n\nThe most favourable case is when the curve $C_P$ intersects the torus $T^2 = \\{|x|=|y|=1\\}$ only at cusps. In this case $\\gamma_P$ is a modular symbol and we may use \\cite{zudilin} to compute $\\int_{\\gamma_P} \\eta(u,v)$ in terms of special values of $L$-functions.\n\nIn this section, we work out this idea for some examples of increasing complexity. We work with the modular units provided by \\cite{yang}. These modular units are supported on the cusps above $\\infty \\in X_0(N)$, so that \\cite[Prop 6.1]{brunault:smf} implies that $P$ is automatically tempered.\n\nIn all examples below, we found that $\\gamma_P$ can be written as the sum of a closed path $\\gamma_0$ and a path $\\gamma_1$ joining cusps. The integral of $\\eta(u,v)$ over $\\gamma_1$ can be computed using \\cite[Thm 1]{zudilin}. The integral of $\\eta(u,v)$ over $\\gamma_0$ can be dealt with using either \\cite[Thm 1]{zudilin} or the explicit version of Beilinson's theorem -- we have not carried out the details of the computation. So in order to establish the identities below rigorously, it only remains to express $\\gamma_0$ in terms of modular symbols and to compute $\\int_{\\gamma_0} \\eta(u,v)$ using the tools explained above.\n\nIt would be interesting to understand when the identities obtained involve cusp forms (like (\\ref{eq mP16})), are of Dirichlet type (like (\\ref{eq mP18})), or of mixed type (like (\\ref{eq mP25})). In the general case, it would be also interesting to find conditions on the modular units $u$ and $v$ so that the boundary of $\\gamma_P$ consists of cusps or other interesting points.\n\n\\subsection{$N=16$}\n\nThe modular curve $X_1(16)$ has genus 2 and has been studied in \\cite{lecacheux:16}. Let $u$ and $v$ be the following modular units:\n\\begin{align*}\nu & = q \\prod_{\\substack{n \\geq 1\\\\ n \\equiv \\pm 1,\\pm 5 (16)}} (1-q^n) \/ \\prod_{\\substack{n \\geq 1 \\\\ n \\equiv \\pm 3,\\pm 7 (16)}} (1-q^n)\\\\\nv & = q \\prod_{\\substack{n \\geq 1\\\\ n \\equiv \\pm 14 (16)}} (1-q^n) \/ \\prod_{\\substack{n \\geq 1 \\\\ n \\equiv \\pm 10(16)}} (1-q^n).\n\\end{align*}\nTheir minimal polynomial is given by\n\\begin{equation*}\nP_{16} = y-x-xy-xy^2+x^2y+xy^3.\n\\end{equation*}\nThis polynomial vanishes on the torus at the points $(x,y)=(1,1)$, $(1,\\pm i)$, $(-1,-1)$, but the Deninger cycle $\\gamma_{P_{16}}$ is \\emph{closed}. So we may expect that $m(P_{16})$ is equal to $L'(f,0)$ for some cusp form $f$ of level $16$ with rational coefficients. Indeed, we find numerically\n\\begin{equation}\\label{eq mP16}\nm(P_{16}) \\stackrel{?}{=} L'(f,0)\n\\end{equation}\nwhere $f$ is the trace of the unique newform of weight $2$ and level $16$, having coefficients in $\\mathbf{Z}[i]$.\n\n\\subsection{$N=18$}\n\nThe modular curve $X_1(18)$ has genus $2$ and has been studied in \\cite{lecacheux:18}. It has 3 cusps above $\\infty$, so we may form essentially two modular units supported on these cusps. Let $u$ and $v$ be the following modular units:\n\\begin{align*}\nu & = q^3 \\prod_{\\substack{n \\geq 1\\\\ n \\equiv \\pm 1,\\pm 2 (18)}} (1-q^n) \/ \\prod_{\\substack{n \\geq 1 \\\\ n \\equiv \\pm 7,\\pm 8 (18)}} (1-q^n)\\\\\nv & = q^2 \\prod_{\\substack{n \\geq 1\\\\ n \\equiv \\pm 1,\\pm 4 (18)}} (1-q^n) \/ \\prod_{\\substack{n \\geq 1 \\\\ n \\equiv \\pm 5, \\pm 8 (18)}} (1-q^n).\n\\end{align*}\nTheir minimal polynomial is given by\n\\begin{equation*}\nP_{18} = -x^2+y^3+xy^2-x^2y+x^2y^2-x^3y^2.\n\\end{equation*}\nThis polynomial vanishes on the torus at the points $(x,y)=(1,\\pm 1)$, $(-1,\\pm 1)$, $(\\zeta_6^2,\\zeta_6)$ and $(\\overline{\\zeta_6}^2,\\overline{\\zeta_6})$ with $\\zeta_6=e^{2\\pi i\/6}$. The points $(\\zeta_6^2,\\zeta_6)$ and $(\\overline{\\zeta_6}^2,\\overline{\\zeta_6})$ correspond respectively to the cusps $\\frac16$ and $-\\frac16$, and the Deninger cycle $\\gamma_{P_{18}}$ is given by $\\gamma_0+\\{-\\frac16,\\frac16\\}$, where $\\gamma_0$ is a closed cycle. Using \\cite[Thm 1]{zudilin}, we find\n\\begin{equation*}\n\\int_{-1\/6}^{1\/6} \\eta(u,v) = \\frac{1}{4\\pi} L(F,2)\n\\end{equation*}\nwhere $F$ is a modular form of weight 2 and level (at most) $18^2$. Actually $F$ has level $18$ and \\cite[Thm 1]{zudilin} simplifies if we use the functional equation $L(F,2)=-\\frac{2\\pi^2}{9} L'(W_{18} F,0)$. In fact \\cite[Lemma 2]{zudilin} guarantees that $W_{18} F$ will be a modular form with \\emph{integral} Fourier coefficients. In this case, we find\n\\begin{equation*}\nW_{18} F = -36 E_2^{\\psi}\n\\end{equation*}\nwhere $E_2^{\\psi} = \\sum_{n = 1}^\\infty (\\sum_{d | n} d) \\psi(n) q^n$ is an Eisenstein series of level $9$, and $\\psi : (\\mathbf{Z}\/3\\mathbf{Z})^\\times \\to \\{\\pm 1\\}$ is the unique Dirichlet character of conductor $3$. Since $L(E_2^{\\psi},s)=L(\\psi,s) L(\\psi,s-1)$, we may expect that $m(P_{18})$ involves $L$-values of Dirichlet characters. Indeed, we find numerically\n\\begin{equation}\\label{eq mP18}\nm(P_{18}) \\stackrel{?}{=} 2 L'(\\psi,-1).\n\\end{equation}\n\n\n\\subsection{$N=25$}\n\nThe modular curve $X_1(25)$ has genus 12 and the quotient $X=X_1(25)\/\\langle 7 \\rangle$ has genus $4$. The curve $X$ and its modular units have been studied by Lecacheux \\cite{lecacheux:25} and Darmon \\cite{darmon:X1_25}. Consider the following modular units:\n\\begin{align*}\nu & = q \\prod_{\\substack{n \\geq 1\\\\ n \\equiv \\pm 3,\\pm 4 (25)}} (1-q^n) \/ \\prod_{\\substack{n \\geq 1 \\\\ n \\equiv \\pm 2,\\pm 11 (25)}} (1-q^n)\\\\\nv & = q^{-1} \\prod_{\\substack{n \\geq 1\\\\ n \\equiv \\pm 9,\\pm 12 (25)}} (1-q^n) \/ \\prod_{\\substack{n \\geq 1 \\\\ n \\equiv \\pm 6,\\pm 8 (25)}} (1-q^n).\n\\end{align*}\nTheir minimal polynomial is given by\n\\begin{equation*}\nP_{25}=y^2 x^4 + (y^3 + y^2) x^3 + (3y^3 - y^2 - 2y)x^2 + (y^4 - 4y^2 + y - 1)x - y^3.\n\\end{equation*}\nThis polynomial vanishes on the torus at the points $(x,y)=(\\zeta,-\\zeta)$ for each primitive $5$-th root of unity $\\zeta$. These points are cusps: letting $\\zeta_5=e^{2\\pi i\/5}$, we have\n\\begin{align*}\nu(1\/5) & =\\zeta_5^2=-v(1\/5) & u(-1\/5) & =\\zeta_5^{-2} = -v(-1\/5)\\\\\nu(2\/5) & =\\zeta_5=-v(2\/5) & u(-2\/5) & =\\zeta_5^{-1} = -v(-2\/5).\n\\end{align*}\nThe Deninger cycle associated to $P_{25}$ is given by $\\gamma_{P_{25}}=\\gamma_0+\\gamma_1$ where $\\gamma_0$ is a closed cycle and $\\gamma_1 = \\left\\{\\frac15,-\\frac15\\right\\}+ \\left\\{-\\frac25,\\frac25\\right\\}$. Using \\cite[Thm 1]{zudilin}, we get\n\\begin{equation*}\n\\int_{\\gamma_1} \\eta(u,v) = \\frac{1}{4\\pi} L(F,2)\n\\end{equation*}\nwhere $F$ is a modular form of weight 2 and level 25. This time $F$ is a linear combination of newforms and Eisenstein series. Let $\\varepsilon : (\\mathbf{Z}\/25\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$ be the unique Dirichlet character such that $\\varepsilon(2)=\\zeta_5$. A basis of eigenforms of $\\Omega^1(X) \\otimes \\mathbf{C}$ is given by newforms $(f_a)_{a \\in (\\mathbf{Z}\/5\\mathbf{Z})^\\times}$ having Fourier coefficients in $\\mathbf{Q}(\\zeta_5)$ and forming a single Galois orbit. The newform $f_a$ has character $\\varepsilon^a$ and for any $\\sigma \\in \\Gal(\\mathbf{Q}(\\zeta_5)\/\\mathbf{Q})$, we have $\\sigma(f_a) = f_{\\chi(\\sigma) a}$ where $\\chi$ is the cyclotomic character. Moreover, let $\\psi : (\\mathbf{Z}\/5\\mathbf{Z})^\\times \\to \\mathbf{C}^\\times$ be the Dirichlet character defined by $\\psi(2)=i$. Then $W_{25} F$ has integral coefficients and is given by\n\\begin{equation*}\nW_{25} F = -10 \\operatorname{Tr}_{\\mathbf{Q}(\\zeta_5)\/\\mathbf{Q}} (\\lambda f_1) - 25(1+i) E_2^{\\psi,\\overline{\\psi}} - 25 (1-i) E_2^{\\overline{\\psi},\\psi}\n\\end{equation*}\nwhere $\\lambda = 2\\zeta_5+\\zeta_5^{-1}+2\\zeta_5^{-2}$ and $E_2^{\\psi,\\overline{\\psi}}$ is the Eisenstein series defined by\n\\begin{equation*}\nE_2^{\\psi,\\overline{\\psi}} = \\sum_{m,n=1}^\\infty m \\overline{\\psi}(m) \\psi(n) q^{mn}.\n\\end{equation*}\nWe may therefore expect $m(P_{25})$ being a linear combination of $L'(\\psi,-1)$, $L'(\\overline{\\psi},-1)$ and $L'(f,0)$, where $f$ is a cusp form with rational Fourier coefficients. Indeed, we find numerically\n\\begin{equation}\\label{eq mP25}\nm(P_{25}) \\stackrel{?}{=} L'(f,0) + \\frac{1+2i}{5} L'(\\overline{\\psi},-1) + \\frac{1-2i}{5} L'(\\psi,-1)\n\\end{equation}\nwhere\n\\begin{equation*}\nf =\\frac15 \\operatorname{Tr}((2+\\zeta_5+2\\zeta_5^{-2}) f_1) = q + q^2-q^3-q^4-3q^5-2q^9+3q^{10}+4q^{11}+O(q^{12}).\n\\end{equation*}\n\n\\bibliographystyle{smfplain} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Preliminaries}\n\nA linear code of length $n$ over a finite field $\\mathbb{F}$ is a subspace\nof $\\mathbb{F}^{n}.$ An Additive code of length $n$ over a finite field $%\n\\mathbb{F}$ is a subgroup of $\\mathbb{F}^{n}.$ Additive codes over the\nfinite field $\\mathbb{F}_{4}=\\left\\{ 0,1,\\alpha ,\\alpha ^{2}\\right\\} $ where \n$\\alpha ^{2}+\\alpha +1=0$ were introduced in \\cite{Calderbank1998} because\nof their applications in quantum computing. Define the mapping $T:\\mathbb{F}%\n_{4}^{n}\\rightarrow \\mathbb{F}_{4}^{n}$ by $T\\left( \\left(\nc_{0},c_{1},\\ldots ,c_{n-1}\\right) \\right) =\\left(\nc_{n-1},c_{0},c_{1},\\ldots ,c_{n-2}\\right) .$ Cyclic additive codes over $%\n\\mathbb{F}_{4}^{n}$ are additive codes over $\\mathbb{F}_{4}^{n}$ such that\nif $c=\\left( c_{0},c_{1},\\ldots ,c_{n-1}\\right) \\in C,$ then $T\\left(\nc\\right) \\in C.$\n\nLet $C$ be a linear code over a finite field $\\mathbb{F}.$ Right and left\nlinear polycyclic codes over finite fields were introduced in \\cite%\n{Sergio2009}. These codes are generalization of cyclic codes over finite\nfields. Their properties and structures are studied in details in (\\cite%\n{Sergio2009},\\cite{Matsuoka2012},\\cite{Adel2016},\\cite{Huffman2007}).\n\nIn this paper, we are interested in studying the structure and the properties\nof additive right and left polycyclic codes induced by a binary vector $a=\\left( a_{0},a_{1},\\ldots ,a_{n-1}\\right) \\in \\mathbb{F}_{2}^{n}.$ In this work, we consider the important special case where the vector $a$ has all binary entries. Useful results are obtained in this special case both in terms of the structure of the codes and obtaining codes with good parameters.\nWe give the definition of these codes, study their properties and find\ntheir generator polynomials and their cardinality. We also\nstudy different duals of these codes and show that if $C$ is a right\npolycyclic code induced by a vector $a\\in \\mathbb{F}_{2}^{n}$, then the\nHermitian dual of $C$ is a sequential code induced by $a.$ As an application\nof our study, we present examples of codes with good parameters. We have\nthree sets of examples. One is a set of additive polycyclic codes that\ncontain more codewords (twice as many) than comparable optimal linear codes\nover $\\mathbb{F}_{4}$ with the same length and the minimum distance. Another\none is a set of best known binary linear codes, most of which are also\noptimal, that are obtained from additive right polycyclic codes over $%\n\\mathbb{F}_{4}$ via certain maps. And the third is a set of optimal quantum\ncodes according to the database \\cite{database} obtained from\nadditive polycyclic codes. \n\n\\section{Introduction}\n\nConsider the finite filed $\\mathbb{F}_{4}=\\left\\{ 0,1,\\alpha ,\\alpha\n^{2}\\right\\} $ where $\\alpha ^{2}+\\alpha +1=0.$ An additive code $C$ of\nlength $n$ over $\\mathbb{F}_{4}$ is a subgroup of $(\\mathbb{F}_{4}^{n},+).$\n\n\\begin{definition}\n\\label{Defn Right polycyclic}Let $C$ be an additive code over $\\mathbb{F}_{4}\n$ and let $a=\\left( a_{0},a_{1},\\ldots ,a_{n-1}\\right) \\in \\mathbb{F}_{2}^{n}$. $C$ is called\nan additive right polycyclic code induced by $a$, if for any $c=\\left( c_{0},c_{1},\\ldots ,c_{n-1}\\right) \\in C$, we have \n\\[\n\\left( 0,c_{0},c_{1},\\ldots ,c_{n-2}\\right) +c_{n-1}\\left(\na_{0},a_{1},\\ldots ,a_{n-1}\\right) \\in C.\n\\]\n\\end{definition}\n\n\\begin{definition}\n\\label{Defn Left polycyclic}Let $C$ be an additive code over $\\mathbb{F}_{4}\n$ and let $a=\\left( a_{0},a_{1},\\ldots ,a_{n-1}\\right) \\in \\mathbb{F}_{2}^{n}$. $C$ is called\nan additive left polycyclic code induced by $a$, if for any $c=\\left( c_{0},c_{1},\\ldots ,c_{n-1}\\right) \\in C$, we have \n\\[\n\\left( c_{1},c_{2},\\ldots ,c_{n-1},0\\right) +c_{0}\\left( a_{0},a_{1},\\ldots\n,a_{n-1}\\right) \\in C.\n\\]\n\\end{definition}\n\nNotice that if $a=\\left( 1,0,0,\\ldots ,0\\right) ,$ then right polycyclic\ncodes induced by $a$ are just the familiar cyclic codes.\n\nSuppose that $a=\\left( a_{0},a_{1},\\ldots ,a_{n-1}\\right) \\in \\mathbb{F}%\n_{2}^{n}.$ As in the case of additive cyclic codes over finite fields, it is\n useful to have polynomial representations of additive (right or\nleft) polycyclic codes.\n\nLet $a=\\left( a_{0},a_{1},\\ldots ,a_{n-1}\\right) \\in \\mathbb{F}_{2}^{n}.$\nThe vector $a$ can be represented as $a\\left( x\\right) =a_{0}+a_{1}x+\\ldots\n+a_{n-1}x^{n-1}\\in \\mathbb{F}_{2}\\left[ x\\right] $. Consider the ring $R_{n}=%\n\\mathbb{F}_{4}\\left[ x\\right] \/\\left\\langle x^{n}-a\\left( x\\right)\n\\right\\rangle$, which is an $\\mathbb{F}_{2}\\left[ x\\right] $%\n-module. Let $c\\left( x\\right) =c_{0}+c_{1}x+\\ldots +c_{n-1}x^{n-1}\\in \n\\mathbb{F}_{4}\\left[ x\\right] \/\\left\\langle x^{n}-a\\left( x\\right)\n\\right\\rangle .$ Then,%\n\\begin{eqnarray*}\nxc\\left( x\\right) &=&c_{0}x+c_{1}x^{2}+\\ldots +c_{n-2}x^{n-1}+c_{n-1}x^{n}\n\\\\\n&=&c_{0}x+c_{1}x^{2}+\\ldots +c_{n-2}x^{n-1}+c_{n-1}\\left(\na_{0}+a_{1}x+\\ldots +a_{n-1}x^{n-1}\\right) \\\\\n&=&c_{n-1}a_{0}+x\\left( c_{0}+c_{n-1}a_{1}\\right) +\\ldots +x^{n-1}\\left(\nc_{n-2}+c_{n-1}a_{n-1}\\right) .\n\\end{eqnarray*}%\nThe polynomial representation of $xc\\left( x\\right) $ is $\\left(\nc_{n-1}a_{0},c_{0}+c_{n-1}a_{1},\\ldots ,c_{n-2}+c_{n-1}a_{n-1}\\right)\n=\\left( 0,c_{0},c_{1},\\ldots ,c_{n-2}\\right) +c_{n-1}\\left(\na_{0},a_{1},\\ldots ,a_{n-1}\\right) .$\n\nSimilarly, let $c=\\left( c_{0},c_{1},\\ldots ,c_{n-1}\\right) \\in \\mathbb{F}%\n_{4}^{n}$ be represented by the polynomial $c\\left( x\\right)\n=c_{n-1}+c_{n-2}x+\\cdots +c_{0}x^{n-1}\\in \\mathbb{F}_{4}\\left[ x\\right]\n\/\\left\\langle x^{n}-a\\left( x\\right) \\right\\rangle ,$ where $a=\\left(\na_{0},a_{1},\\ldots ,a_{n-1}\\right) \\in \\mathbb{F}_{2}^{n}$ and $a\\left(\nx\\right) =a_{n-1}+a_{n-2}x+\\ldots +a_{0}x^{n-1}\\in \\mathbb{F}_{2}\\left[ x%\n\\right] $. Then,%\n\\begin{eqnarray*}\nxc\\left( x\\right) &=&c_{n-1}x+c_{n-2}x^{2}+\\ldots +c_{1}x^{n-1}+c_{0}x^{n}\n\\\\\n&=&c_{n-1}x+c_{n-2}x^{2}+\\ldots +c_{1}x^{n-1}+c_{0}\\left(\na_{n-1}+a_{n-2}x+\\ldots +a_{0}x^{n-1}\\right) \\\\\n&=&c_{0}a_{n-1}+x\\left( c_{n-1}+c_{0}a_{n-2}\\right) +\\ldots +x^{n-1}\\left(\nc_{1}+c_{0}a_{0}\\right) .\n\\end{eqnarray*}%\nThe polynomial representation of $xc\\left( x\\right) $ is $\\left(\nc_{1}+c_{0}a_{0},\\ldots ,c_{n-1}+c_{0}a_{n-2},c_{0}a_{n-1}\\right) =\\left(\nc_{1},c_{2},\\ldots ,c_{n-1},0\\right) +c_{0}\\left( a_{0},a_{1},\\ldots\n,a_{n-1}\\right) .$ Hence, we get the following lemmas.\n\n\\begin{lemma}\n\\label{submodule-right}$C$ is an additive right polycyclic code induced by $%\na $ if and only if $C$ is an $\\mathbb{F}_{2}\\left[ x\\right] $-submodule of $%\nR_{n}.$\n\\end{lemma}\n\n\\begin{lemma}\n\\label{submodule-left}$C$ is an additive left polycyclic code induced by $a$\nif and only if $C$ is an $\\mathbb{F}_{2}\\left[ x\\right] $-submodule of $%\nR_{n}.$\n\\end{lemma}\n\nLet $C$ be an additive right polycyclic code induced by $\\mathbf{a}.$ Then $%\nC $ is invariant under right multiplication by the square matrix \n\\[\nD=\\left[ \n\\begin{array}{ccccc}\n0 & 1 & 0 & 0 & \\ldots \\\\ \n0 & 0 & 1 & 0 & \\ldots \\\\ \n\\vdots & & & \\ddots & \\\\ \n0 & 0 & \\ldots & 0 & 1 \\\\ \na_{0} & a_{1} & a_{2} & \\ldots & a_{n-1}%\n\\end{array}%\n\\right] . \n\\]%\nSimilarly, an additive left polycyclic code induced by $\\mathbf{d}=\\left(\nd_{0},d_{1},\\ldots ,d_{n-1}\\right) $ is invariant under right multiplication\nby the square matrix \n\\[\nE=\\left[ \n\\begin{array}{ccccc}\nd_{0} & d_{1} & d_{2} & \\ldots & d_{n-1} \\\\ \n1 & 0 & 0 & \\ldots & 0 \\\\ \n0 & 1 & 0 & \\ldots & 0 \\\\ \n\\vdots & & \\ddots & 0 & 1 \\\\ \n0 & 0 & \\ldots & 1 & 0%\n\\end{array}%\n\\right] . \n\\]\n\n\\begin{theorem}\n\\label{right-left1}Let $C$ be an additive right polycyclic code induced by $%\n\\mathbf{a}=\\left( a_{0},a_{1},\\ldots ,a_{n-1}\\right) $ with $a_{0}\\neq 0$.\nThen $C$ is an additive left polycyclic code induced by $\\mathbf{d}=\\left(\nd_{0},d_{1},\\ldots ,d_{n-1}\\right) $ where $d_{j}=\\frac{-{a_{j+1}}}{a_{0}}$\nfor $j