pinecone/core-2020-05-10-deduplication huggingface dataset:
Quivers, Line Defects and Framed BPS Invariants

A large class of N=2 quantum field theories admits a BPS quiver description
and the study of their BPS spectra is then reduced to a representation theory
problem. In such theories the coupling to a line defect can be modelled by
framed quivers. The associated spectral problem characterises the line defect
completely. Framed BPS states can be thought of as BPS particles bound to the
defect. We identify the framed BPS degeneracies with certain enumerative
invariants associated with the moduli spaces of stable quiver representations.
We develop a formalism based on equivariant localization to compute explicitly
such BPS invariants, for a particular choice of stability condition. Our
framework gives a purely combinatorial solution of this problem. We detail our
formalism with several explicit examples.Comment: 67 pages, 17 figure
Effective Potentials Generated by Field Interaction in the
  Quasi-Classical Limit

We study the {\it quasi-classical limit} of a quantum system composed of
finitely many non-relativistic particles coupled to a quantized field in
Nelson-type models. We prove that, as the field becomes classical and the
corresponding degrees of freedom are traced out, the effective Hamiltonian of
the particles converges in resolvent sense to a self-adjoint Schr\"{o}dinger
operator with an additional potential, depending on the state of the field.
Moreover, we explicitly derive the expression of such a potential for a large
class of field states and show that, for certain special sequences of states,
the effective potential is trapping. In addition, we prove convergence of the
ground state energy of the full system to a suitable effective variational
problem involving the classical state of the field.Comment: minor revision, Ann. H. Poincar\'e in press, 41 pages, pdfLaTe
Cyclotomic Gaudin models, Miura opers and flag varieties

Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$. Let $\nu
\in \text{Aut}\, \mathfrak{g}$ be a diagram automorphism whose order divides $T
\in \mathbb{Z}_{\geq 1}$. We define cyclotomic $\mathfrak{g}$-opers over the
Riemann sphere $\mathbb{P}^1$ as gauge equivalence classes of
$\mathfrak{g}$-valued connections of a certain form, equivariant under actions
of the cyclic group $\mathbb{Z}/ T\mathbb{Z}$ on $\mathfrak{g}$ and
$\mathbb{P}^1$. It reduces to the usual notion of $\mathfrak{g}$-opers when $T
= 1$.
  We also extend the notion of Miura $\mathfrak{g}$-opers to the cyclotomic
setting. To any cyclotomic Miura $\mathfrak{g}$-oper $\nabla$ we associate a
corresponding cyclotomic $\mathfrak{g}$-oper. Let $\nabla$ have residue at the
origin given by a $\nu$-invariant rational dominant coweight
$\check{\lambda}_0$ and be monodromy-free on a cover of $\mathbb{P}^1$. We
prove that the subset of all cyclotomic Miura $\mathfrak{g}$-opers associated
with the same cyclotomic $\mathfrak{g}$-oper as $\nabla$ is isomorphic to the
$\vartheta$-invariant subset of the full flag variety of the adjoint group $G$
of $\mathfrak{g}$, where the automorphism $\vartheta$ depends on $\nu$, $T$ and
$\check{\lambda}_0$. The big cell of the latter is isomorphic to $N^\vartheta$,
the $\vartheta$-invariant subgroup of the unipotent subgroup $N \subset G$,
which we identify with those cyclotomic Miura $\mathfrak{g}$-opers whose
residue at the origin is the same as that of $\nabla$. In particular, the
cyclotomic generation procedure recently introduced in [arXiv:1505.07582] is
interpreted as taking $\nabla$ to other cyclotomic Miura $\mathfrak{g}$-opers
corresponding to elements of $N^\vartheta$ associated with simple root
generators.
  We motivate the introduction of cyclotomic $\mathfrak{g}$-opers by
formulating two conjectures which relate them to the cyclotomic Gaudin model of
[arXiv:1409.6937].Comment: 59 page
Semiclassical Szeg\"o limit of eigenvalue clusters for the hydrogen atom
  Zeeman Hamiltonian

We prove a limiting eigenvalue distribution theorem (LEDT) for suitably
scaled eigenvalue clusters around the discrete negative eigenvalues of the
hydrogen atom Hamiltonian formed by the perturbation by a weak constant
magnetic field. We study the hydrogen atom Zeeman Hamiltonian $H_V(h,B) =
(1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$, defined on $L^2
(R^3)$, in a constant magnetic field ${\mathbf B}(h) = {\mathbf \nabla} \times
{\mathbf A}(h)=(0,0,\epsilon(h)B)$ in the weak field limit $\epsilon(h)
\rightarrow 0$ as $h\rightarrow{0}$. We consider the Planck's parameter $h$
taking values along the sequence $h=1/(N+1)$, with $N=0,1,2,\ldots$, and
$N\rightarrow\infty$. We prove a semiclassical $N \rightarrow \infty$ LEDT of
the Szeg\"o-type for the scaled eigenvalue shifts and obtain both ({\bf i}) an
expression involving the regularized classical Kepler orbits with energy
$E=-1/2$ and ({\bf ii}) a weak limit measure that involves the component
$\ell_3$ of the angular momentum vector in the direction of the magnetic field.
This LEDT extends results of Szeg\"o-type for eigenvalue clusters for bounded
perturbations of the hydrogen atom to the Zeeman effect. The new aspect of this
work is that the perturbation involves the unbounded, first-order, partial
differential operator $w(h, B) = \frac{(\epsilon(h)B)^2}{8} (x_1^2 + x_2^2) -
\frac{ \epsilon(h)B}{2} hL_3 ,$ where the operator $hL_3$ is the third
component of the usual angular momentum operator and is the quantization of
$\ell_3$. The unbounded Zeeman perturbation is controlled using localization
properties of both the hydrogen atom coherent states $\Psi_{\alpha,N}$, and
their derivatives $L_3(h)\Psi_{\alpha,N}$, in the large quantum number regime
$N\rightarrow\infty$.Comment: 39 page
Quantitative mixing for locally Hamiltonian flows with saddle loops on
  compact surfaces

Given a compact surface $\mathcal{M}$ with a smooth area form $\omega$, we
consider an open and dense subset of the set of smooth closed 1-forms on
$\mathcal{M}$ with isolated zeros which admit at least one saddle loop
homologous to zero and we prove that almost every element in the former induces
a mixing flow on each minimal component. Moreover, we provide an estimate of
the speed of the decay of correlations for smooth functions with compact
support on the complement of the set of singularities. This result is achieved
by proving a quantitative version for the case of finitely many singularities
of a theorem by Ulcigrai (ETDS, 2007), stating that any suspension flow with
one asymmetric logarithmic singularity over almost every interval exchange
transformation is mixing. In particular, the quantitative mixing estimate we
prove applies to asymmetric logarithmic suspension flows over rotations, which
were shown to be mixing by Sinai and Khanin.Comment: 34 pages, 4 figures. Revised version according to the referees'
  suggestion
On the construction of Wannier functions in topological insulators: the
  3D case

We investigate the possibility of constructing exponentially localized
composite Wannier bases, or equivalently smooth periodic Bloch frames, for
3-dimensional time-reversal symmetric topological insulators, both of bosonic
and of fermionic type, so that the bases in question are also compatible with
time-reversal symmetry. This problem is translated in the study, of independent
interest, of homotopy classes of continuous, periodic, and time-reversal
symmetric families of unitary matrices. We identify three $\mathbb{Z}_2$-valued
complete invariants for these homotopy classes. When these invariants vanish,
we provide an algorithm which constructs a "multi-step" logarithm that is
employed to continuously deform the given family into a constant one,
identically equal to the identity matrix. This algorithm leads to a
constructive procedure to produce the composite Wannier bases mentioned above.Comment: 29 pages. Version 2: minor corrections of misprints, corrected proofs
  of Theorems 2.4 and 2.9, added references. Accepted for publication in
  Annales Henri Poicar\'
On Asymptotic Expansions in Spin Boson Models

We consider expansions of eigenvalues and eigenvectors of models of quantum
field theory. For a class of models known as generalized spin boson model we
prove the existence of asymptotic expansions of the ground state and the ground
state energy to arbitrary order. We need a mild but very natural infrared
assumption, which is weaker than the assumption usually needed for other
methods such as operator theoretic renormalization to be applicable. The result
complements previously shown analyticity properties.Comment: 49 page
Effects of boundary conditions on irreversible dynamics

We present a simple one-dimensional Ising-type spin system on which we define
a completely asymmetric Markovian single spin-flip dynamics. We study the
system at a very low, yet non-zero, temperature and we show that for empty
boundary conditions the Gibbs measure is stationary for such dynamics, while
introducing in a single site a $+$ condition the stationary measure changes
drastically, with macroscopical effects. We achieve this result defining an
absolutely convergent series expansion of the stationary measure around the
zero temperature system. Interesting combinatorial identities are involved in
the proofs
The topological classification of one-dimensional symmetric quantum
  walks

We give a topological classification of quantum walks on an infinite 1D
lattice, which obey one of the discrete symmetry groups of the tenfold way,
have a gap around some eigenvalues at symmetry protected points, and satisfy a
mild locality condition. No translation invariance is assumed. The
classification is parameterized by three indices, taking values in a group,
which is either trivial, the group of integers, or the group of integers modulo
2, depending on the type of symmetry. The classification is complete in the
sense that two walks have the same indices if and only if they can be connected
by a norm continuous path along which all the mentioned properties remain
valid. Of the three indices, two are related to the asymptotic behaviour far to
the right and far to the left, respectively. These are also stable under
compact perturbations. The third index is sensitive to those compact
perturbations which cannot be contracted to a trivial one. The results apply to
the Hamiltonian case as well. In this case all compact perturbations can be
contracted, so the third index is not defined. Our classification extends the
one known in the translation invariant case, where the asymptotic right and
left indices add up to zero, and the third one vanishes, leaving effectively
only one independent index. When two translationally invariant bulks with
distinct indices are joined, the left and right asymptotic indices of the
joined walk are thereby fixed, and there must be eigenvalues at $1$ or $-1$
(bulk-boundary correspondence). Their location is governed by the third index.
We also discuss how the theory applies to finite lattices, with suitable
homogeneity assumptions.Comment: 36 pages, 7 figure
Examples of naked singularity formation in higher-dimensional
  Einstein-vacuum spacetimes

The vacuum Einstein equations in 5+1 dimensions are shown to admit solutions
describing naked singularity formation in gravitational collapse from
nonsingular asymptotically locally flat initial data that contain no trapped
surface. We present a class of specific examples with topology
$\mathbb{R}^{3+1} \times S^2$. Thanks to the Kaluza-Klein dimensional
reduction, these examples are constructed by lifting continuously self-similar
solutions of the 4-dimensional Einstein-scalar field system with a negative
exponential potential. The latter solutions are obtained by solving a
3-dimensional autonomous system of first-order ordinary differential equations
with a combined analytic and numerical approach. Their existence provides a new
test-bed for weak cosmic censorship in higher-dimensional gravity. In addition,
we point out that a similar attempt of lifting Christodoulou's naked
singularity solutions of massless scalar fields fails to capture formation of
naked singularities in 4+1 dimensions, due to a diverging Kretschmann scalar in
the initial data.Comment: 34 pages, 5 figures; to match the published version which combines
  this number and arXiv:1509.0795
Uniqueness of Kerr-Newman-de Sitter black holes with small angular
  momenta

We show that a stationary solution of the Einstein-Maxwell equations which is
close to a non-degenerate Reissner-Nordstr\"om-de Sitter solution is in fact
equal to a slowly rotating Kerr-Newman-de Sitter solution. The proof uses the
non-linear stability of the Kerr-Newman-de Sitter family of black holes for
small angular momenta, recently established by the author, together with an
extension argument for Killing vector fields. Our black hole uniqueness result
only requires the solution to have high but finite regularity; in particular,
we do not make any analyticity assumptions.Comment: 10 pages, 1 figur
Pre-big bang geometric extensions of inflationary cosmologies

Robertson-Walker spacetimes within a large class are geometrically extended
to larger cosmologies that include spacetime points with zero and negative
cosmological times. In the extended cosmologies, the big bang is lightlike, and
though singular, it inherits some geometric structure from the original
spacetime. Spacelike geodesics are continuous across the cosmological time zero
submanifold which is parameterized by the radius of Fermi space slices, i.e, by
the proper distances along spacelike geodesics from a comoving observer to the
big bang. The continuous extension of the metric, and the continuously
differentiable extension of the leading Fermi metric coefficient $g_{\tau\tau}$
of the observer, restrict the geometry of spacetime points with pre-big bang
cosmological time coordinates. In our extensions the big bang is two
dimensional in a certain sense, consistent with some findings in quantum
gravity.Comment: 48 pages, 2 figure
The Bisognano-Wichmann property on nets of standard subspaces, some
  sufficient conditions

We discuss the Bisognano-Wichmann property for local Poincar\'e covariant
nets of standard subspaces. We give a sufficient algebraic condition on the
covariant representation ensuring the Bisognano-Wichmann and Duality properties
without further assumptions on the net called modularity condition. It holds
for direct integrals of scalar massive and massless representations. We present
a class of massive modular covariant nets not satisfying the Bisognano-Wichmann
property. Furthermore, we give an outlook in the standard subspace setting on
the relation between the Bisognano-Wichmann property and the Split property.Comment: Final version. To appear in Annales Henri Poincar\'
A mathematical account of the NEGF formalism

The main goal of this paper is to put on solid mathematical grounds the
so-called Non-Equilibrium Green's Function (NEGF) transport formalism for open
systems. In particular, we derive the Jauho-Meir-Wingreen formula for the
time-dependent current through an interacting sample coupled to non-interacting
leads. Our proof is non-perturbative and uses neither complex-time Keldysh
contours, nor Langreth rules of 'analytic continuation'. We also discuss other
technical identities (Langreth, Keldysh) involving various many body Green's
functions. Finally, we study the Dyson equation for the advanced/retarded
interacting Green's function and we rigorously construct its (irreducible)
self-energy, using the theory of Volterra operators.Comment: Annales Henri Poincar\'e 201
Recoupling coefficients and quantum entropies

We prove that the asymptotic behavior of the recoupling coefficients of the
symmetric group is characterized by a quantum marginal problem -- namely, by
the existence of quantum states of three particles with given eigenvalues for
their reduced density operators. This generalizes Wigner's observation that the
semiclassical behavior of the 6j-symbols for SU(2) -- fundamental to the
quantum theory of angular momentum -- is governed by the existence of Euclidean
tetrahedra. As a corollary, we deduce solely from symmetry considerations the
strong subadditivity property of the von Neumann entropy. Lastly, we show that
the problem of characterizing the eigenvalues of partial sums of Hermitian
matrices arises as a special case of the quantum marginal problem. We establish
a corresponding relation between the recoupling coefficients of the unitary and
symmetric groups, generalizing a classical result of Littlewood and Murnaghan.Comment: 25 page
Wegner estimate and disorder dependence for alloy-type Hamiltonians with
  bounded magnetic potential

We consider non-ergodic magnetic random Sch\"odinger operators with a bounded
magnetic vector potential. We prove an optimal Wegner estimate valid at all
energies. The proof is an adaptation of the arguments from [Kle13], combined
with a recent quantitative unique continuation estimate for eigenfunctions of
elliptic operators from [BTV15]. This generalizes Klein's result to operators
with a bounded magnetic vector potential. Moreover, we study the dependence of
the Wegner-constant on the disorder parameter. In particular, we show that
above the model-dependent threshold $E_0(\infty) \in (0, \infty]$, it is
impossible that the Wegner-constant tends to zero if the disorder increases.
This result is new even for the standard (ergodic) Anderson Hamiltonian with
vanishing magnetic field
Coulomb scattering in the massless Nelson model III. Ground state wave
  functions and non-commutative recurrence relations

Let $H_{P,\sigma}$ be the single-electron fiber Hamiltonians of the massless
Nelson model at total momentum $P$ and infrared cut-off $\sigma>0$. We
establish detailed regularity properties of the corresponding $n$-particle
ground state wave functions $f^n_{P,\sigma}$ as functions of $P$ and $\sigma$.
In particular, we show that \[ |\partial_{P^j}f^{n}_{P,\sigma}(k_1,\ldots,
k_n)|, \ \ |\partial_{P^j} \partial_{P^{j'}} f^{n}_{P,\sigma}(k_1,\ldots, k_n)|
\leq \frac{1}{\sqrt{n!}} \frac{(c\lambda_0)^n}{\sigma^{\delta_{\lambda_0}}}
\prod_{i=1}^n\frac{ \chi_{[\sigma,\kappa)}(k_i)}{|k_i|^{3/2}}, \] where $c$ is
a numerical constant, $\lambda_0\mapsto \delta_{\lambda_0}$ is a positive
function of the maximal admissible coupling constant which satisfies
$\lim_{\lambda_0\to 0}\delta_{\lambda_0}=0$ and $\chi_{[\sigma,\kappa)}$ is the
(approximate) characteristic function of the energy region between the infrared
cut-off $\sigma$ and the ultraviolet cut-off $\kappa$. While the analysis of
the first derivative is relatively straightforward, the second derivative
requires a new strategy. By solving a non-commutative recurrence relation we
derive a novel formula for $f^n_{P,\sigma}$ with improved infrared properties.
In this representation $\partial_{P^{j'}}\partial_{P^{j}}f^n_{P,\sigma}$ is
amenable to sharp estimates obtained by iterative analytic perturbation theory
in part II of this series of papers. The bounds stated above are instrumental
for scattering theory of two electrons in the Nelson model, as explained in
part I of this series.Comment: 45 pages, minor revision
The Dirac-Frenkel Principle for Reduced Density Matrices, and the
  Bogoliubov-de-Gennes Equations

The derivation of effective evolution equations is central to the study of
non-stationary quantum many-body sytems, and widely used in contexts such as
superconductivity, nuclear physics, Bose-Einstein condensation and quantum
chemistry. We reformulate the Dirac-Frenkel approximation principle in terms of
reduced density matrices, and apply it to fermionic and bosonic many-body
systems. We obtain the Bogoliubov-de-Gennes and Hartree-Fock-Bogoliubov
equations, respectively. While we do not prove quantitative error estimates,
our formulation does show that the approximation is optimal within the class of
quasifree states. Furthermore, we prove well-posedness of the
Bogoliubov-de-Gennes equations in energy space and discuss conserved
quantities.Comment: 46 pages, 1 figure; v2: simplified proof of conservation of particle
  number, additional references; v3: minor clarification
Stochastic spikes and strong noise limits of stochastic differential
  equations

Motivated by studies of indirect measurements in quantum mechanics, we
investigate stochastic differential equations with a fixed point subject to an
additional infinitesimal repulsive perturbation. We conjecture, and prove for
an important class, that the solutions exhibit a universal behavior when time
is rescaled appropriately: by fine-tuning of the time scale with the
infinitesimal repulsive perturbation, the trajectories converge in a precise
sense to spiky trajectories that can be reconstructed from an auxiliary
time-homogeneous Poisson process. Our results are based on two main tools. The
first is a time change followed by an application of Skorokhod's lemma. We
prove an effective approximate version of this lemma of independent interest.
The second is an analysis of first passage times, which shows a deep interplay
between scale functions and invariant measures. We conclude with some
speculations of possible applications of the same techniques in other areas
Equidistribution for standard pairs in planar dispersing billiard flows

We prove exponential correlation decay in dispersing billiard flows on the
2-torus assuming finite horizon and lack of corner points. With applications
aimed at describing heat conduction, the highly singular initial measures are
concentrated here on 1-dimensional submanifolds (given by standard pairs) and
the observables are supposed to satisfy a generalized H\"older continuity
property. The result is based on the exponential correlation decay bound of
Baladi, Demers and Liverani obtained recentlyfor H\"older continuous
observables in these billiards. The model dependence of the bounds is also
discussed.Comment: 2 figure
Super Quantum Mechanics in the Integral Form Formalism

We reformulate Super Quantum Mechanics in the context of integral forms. This
framework allows to interpolate between different actions for the same theory,
connected by different choices of Picture Changing Operators (PCO). In this way
we retrieve component and superspace actions, and prove their equivalence. The
PCO are closed integral forms, and can be interpreted as super Poincar\'e duals
of bosonic submanifolds embedded into a supermanifold.. We use them to
construct Lagrangians that are top integral forms, and therefore can be
integrated on the whole supermanifold. The $D=1, ~N=1$ and the $D=1,~ N=2$
cases are studied, in a flat and in a curved supermanifold. In this formalism
we also consider coupling with gauge fields, Hilbert space of quantum states
and observables.Comment: 41 pages, no figures. Use birkjour.cls. Minor misprints, moved
  appendix A and B in the main text. Version to be published in Annales H.
  Poincar\'
Matrix product ensembles of Hermite-type and the hyperbolic
  Harish-Chandra-Itzykson-Zuber integral

We investigate spectral properties of a Hermitised random matrix product
which, contrary to previous product ensembles, allows for eigenvalues on the
full real line. We prove that the eigenvalues form a bi-orthogonal ensemble,
which reduces asymptotically to the Hermite Muttalib-Borodin ensemble. Explicit
expressions for the bi-orthogonal functions as well as the correlation kernel
are provided. Scaling the latter near the origin gives a limiting kernel
involving Meijer G-functions, and the functional form of the global density is
calculated. As a part of this study, we introduce a new matrix transformation
which maps the space of polynomial ensembles onto itself. This matrix
transformation is closely related to the so-called hyperbolic
Harish-Chandra-Itzykson-Zuber integral.Comment: 33 pages. To appear in Ann. Henri Poincar