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\begin{align*}\Lambda^{AB}_N (\mu, \bar\mu) =\Lambda^{AB}_{N - 1} (\mu, \bar\mu)+ \frac{1}{2 \sqrt{2}} \left( \mu^A_{N - 2} \bar\mu^{B, N - 2}- \mu^B_{N - 2} \bar\mu^{A, N - 2} \right) .\end{align*}

\begin{align*}S_{GB}^{volume} = {1\over 4 \pi^2}\int_{\partial V }\omega^{01}\wedge R^{23} \ .\end{align*}

\begin{align*}\{\{\xi ^a,q^k\},q^l\}=0,\;\;\;\{q^k,q^l\}=0,\;\;\;(k,l=1,\ldots ,n),\end{align*}

\begin{align*} \phi_{i}(\mathcal{Z}(\mathcal{U}( \mathbb{Z}[G]e_{i})))=\mathcal{Z}(\mathcal{U}( \mathbb{Z}[N_{i}]\varepsilon_{i})).\end{align*}

\begin{align*}\mathcal{D}_{G}(a)=N_{G}(a)\exp \left( \int\limits_{0}^{\infty }\frac{dt}{t}\left[ \frac{a^{2}}{2}e^{-2t}-\sum_{\alpha >0}\Psi _{\alpha }(t)\mathcal{F}_{\alpha }(a,t)\right] \right) \end{align*}

\begin{align*}q(t,g)=\frac{a}{2} [\exp(-i(\Omega t+b))+ \frac{\lambda}{8} (1- \frac{21 \lambda}{8}) \exp(-3i(\Omega t+b)) + \frac{\lambda^2}{64} \exp(-5i(\Omega t+b))] + C.C.,\end{align*}

\begin{align*}\, dy^I \,dy^J \, \delta_{IJ} \, = \, dr^2 \, + \, r^2 \, d\Omega_{7}^2\end{align*}

\begin{align*}|g;(N,0)\rangle=U^{(N,0)}(g)|0,0,-\frac{2N}{3};(N,0)\rangle.\end{align*}

\begin{align*}{\cal Z}^{Fermionic} (\beta) =\left. { \det}^{+1} (\omega^2 - \partial_t^2 )\right|_{antiper.} \;.\end{align*}

\begin{align*}H= \sum_{j=1}^{L} \{ J_{x} \sigma_{x,j}\sigma_{x,j+1} + J_{y} \sigma_{y,j}\sigma_{y,j+1} + h \sigma_{z,j} \} \\\end{align*}

\begin{align*}L=P_\mu\Pi^\mu_\tau +Z_{\alpha\beta}\Pi^{\alpha\beta}_\tau +\bar Z_{\dot\alpha\dot\beta}\bar\Pi^{\dot\alpha\dot\beta}_\tau -\lambda_v h_v - \bar\lambda_v \bar h_v -\lambda_u h_u - \bar\lambda_u \bar h_u \, .\end{align*}

\begin{align*}{g_{\tiny\rm UV}^{(a)}\over{g_{\tiny\rm IR}^{(a)}}}={\sin{\pi a\over{k+l+2}}\over{\sin{\pi\over{k+l+2}}}}{\sin{\pi\over{l+2}}\over{\sin{\pi(a-k)\over{l+2}}}}.\end{align*}

\begin{align*}\sigma z^A=\sum\limits_{\Delta =1}^2\left( \left[ z^A,\gamma_i^{(\Delta )a}\right] ^{*}\eta _a^{(\Delta )i}+\left[ z^A,\gamma ^{(\Delta)a}\right] ^{*}\eta _a^{(\Delta )}\right) ,\end{align*}

\begin{align*}\Gamma _q(0)=\Gamma _q,\ \Gamma _q(1)=\tilde \Gamma _q,\end{align*}

\begin{align*}x\hat{K_{i}}+a(1-\hat{K_{i}})\stackrel{\tau}{\mapsto}(\sum_{j=0}^{p_{i}^{n_{i}}-1}x_{j}, \sum_{j=0}^{p_{i}^{n_{i}}-1}x_{j}\zeta_{p_{i}}^{j},...,\sum_{j=0}^{p_{i}^{n_{i}}-1}x_{j}\zeta_{p_{i}^{n_{i}}}^{j},a(1-\hat{K_{i}})),\end{align*}

\begin{align*}1+\sum_{\alpha =1}^{d+1}b_{\alpha } Y_{\alpha}+\sum_{I=d+2}^{k^{\ast }+1}b_{I}\ Y_{I}=0,\end{align*}

\begin{align*}\begin{array}{r c l}\frac{df}{dt}&=&\frac{\partial f}{\partial t}+\{f,H_T\}\\ & & \\&=&\frac{\partial f}{\partial t}+\{f,H_T\}_D+\{f,\chi_s\}\left(\Delta^{-1}\right)^{ss'}\{\chi_{s'},H_T\}\approx\frac{\partial f}{\partial t}+\{f,H_T\}_D\ .\end{array}\end{align*}

\begin{align*}H_{As}=-R^2/2\,(\pi/2)^2\,(1/4\,a_2^2+11/24\,a_4^2+29/60\,a_6^2 -1/3\,a_2\,a_4-1/12\,a_2\,a_6-1/3\,a_4\,a_6).\end{align*}

\begin{align*}{1\over n(k)} \equiv {\omega(k)\over c \; k},\end{align*}

\begin{align*}{\cal S}_{\rho\rho'}=\sum_{ij}Tr_{(b\mbox{ in }\rho)}(uv^{-1}\beta_{i}\alpha_{j})Str_{\rho'}(uv^{-1}\alpha_{i}\beta_{j})\end{align*}

\begin{align*}V_{extr}(r) = 2 e^{- 2 \sqrt 2 r} ( 1- {1 \over 2}e^{- 2 \sqrt 2 r})^2 (1- {3 \over 4}e^{- 2 \sqrt 2 r}) .\end{align*}

\begin{align*}\frac{\partial\phi}{\partial t} = \epsilon\, \dot{\sigma_n} \cos nx,\ \ \\frac{\partial\phi}{\partial x} = - \epsilon\, \sigma_n\, n \sin nx.\end{align*}

\begin{align*} [ w, w u_1 - u_2 ] = 0.\end{align*}

\begin{align*}T(g)\Psi_{P,\{P0\ldots 0\}}(z)=\left[z_{\mu}g^{\mu}_{0}\right]^{P}=<z,\tilde{u}>^{P}, \; \tilde{u}^{\mu}=g^{\mu}_{0}\;,\end{align*}

\begin{align*}w^{(N)}(ab|\alpha \beta ,\, \lambda )=e^{i\omega b}w(ab|\alpha \beta ,\, \lambda ).\end{align*}

\begin{align*}\zeta^{35,35}(z)=\sum_{1\leq j_1+j_2\leq 5} \lambda_{j_1,j_2}^{35,35}z^{j_1}_1z^{j_2}_2 , \ \ \ \ \ \ \ \ \ \ \eta^{35,35}_z(y)=\sum_{1\leq i_1+i_2\leq 5} \omega_{z,i_1,i_2}^{35,35} \ y_1^{i_1}y_2^{i_2},\end{align*}

\begin{align*} [\mathcal{U}(\mathbb{Z}[\zeta_{p_{i}^{n_{i}}}]): O_{(H_{i},K_{i})}]= [\mathcal{U}(\mathbb{Z}[\zeta_{p_{i}^{n_{i}}}]): P_{(H_{i},K_{i})}][ P_{(H_{i},K_{i})} : O_{(H_{i},K_{i})}]\end{align*}

\begin{align*} v_3 = \frac{u_3}l \left( 3\lambda -\frac{d}2 \right)~,\end{align*}

\begin{align*}f = f_1 f_2 f_n = \left (1 + { r^2_1 \over r^2}\right ) \left (1 + { r^2_5 \over r^2}\right ) \left (1 + { r^2_n \over r^2}\right ),~~~h = \left (1 - { r^2_0 \over r^2}\right ).\end{align*}

\begin{align*}\left[ \omega - \sum_{\ell = 1}^{r} \alpha_{\ell} (\kappa_{\ell} )\right]^{-1} \prod_{1}^{r} d_{\ell}(\kappa_{\ell}) ,\end{align*}

\begin{align*}F^C(\beta)={1 \over \beta}\sum_\omega\ln\left(1-e^{-\beta \omega}\right),\end{align*}

\begin{align*}^2S=\int d\rho dz Tr[\rho (J^P)^2-\rho ^{-1}(J^{\Omega})^2].\end{align*}

\begin{align*}I=4\mbox{Tr}(Z\bar Z)^2 -(\mbox{Tr}Z\bar Z)^2+2^4({\it Pf}Z+{\it Pf}\bar Z), \end{align*}

\begin{align*}B_n(\lambda)+B_{n-1}(\lambda)+N\frac{{\rm d}V^{(\alpha)}(\lambda)}{{\rm d} \lambda}=\frac{\lambda}{c_{n-1}}A_{n-1}(\lambda) \ .\end{align*}

\begin{align*}(s_{l}^{M}(t))^{-1}\phi_{l,n,m}^{\prime}=\phi_{l,n,m}.\end{align*}

\begin{align*}\hat{t}_{\mu_1 \ldots \;\mu_n}(k_1,\ldots ,k_n) = {\cal O}(k_1).\end{align*}

\begin{align*}{\cal Z}^{(N_{f})}(\{\mu\}) =\det\left(\begin{array}{ll}A(\{\mu_f\}) & A(\{-\mu_f\})\\A(\{-\mu_f\}) & A(\{\mu_f\})\end{array}\right)\Delta(\{\mu\})^{-1} , A_{ij}= (\mu_i)^{j-1} e^{\mu_i}.\end{align*}

\begin{align*} [P_{(H_{i},K_{i})} : O_{(H_{i},K_{i})}] = p_{i}^{n_{i}-1}{\prod_{\stackrel{1< k < \frac{p_{i}^{n_{i}}}{2}}{(k,p_{i})=1}} o_{p_{i}^{n_{i}}}(k)p_{i}^{n_{i}-1}n_{H_{i},K_{i}}}=p_{i}^{n_{i}-1}\mathfrak{o}_{i}.\end{align*}

\begin{align*}Q=\lim_{C\to\infty} \int_{|x|\in C}d^2x\rho(x)\;.\end{align*}

\begin{align*}R \sqrt{\lambda} \left( \ln {\lambda} + \sigma \right) ' \varepsilon^{\pm}_{ul} = i {\bf P}^{NK}_i \gamma^i \varepsilon^{\mp}_{lu} \pm {1 \over 2} {\bf P}_{ij}^{RF} \gamma^{ij} \varepsilon^{\pm}_{lu} .\end{align*}

\begin{align*}\delta^2{\cal L} = - {\rm Tr} [ (\nabla\epsilon)^2 + ({\cal M}J\nabla\epsilon)^2 - K^\mu (\epsilon\nabla_\mu\epsilon - \nabla_\mu\epsilon\epsilon) ] \ .\end{align*}

\begin{align*} a_1^0(q',q)= -\frac{\alpha}{q'-q} \log \frac{q'}{q},\end{align*}

\begin{align*}V_{eff}(\omega)=\frac{3(2+N_V-N_H)}{64\pi^6 R^4}\, \left[{\rm Li}_5\left(e^{2 i \pi \omega}\right)+{\rm h.c.}\right]\, ,\end{align*}

\begin{align*}S = {k A_H\over4\ell_P^2} + 4\pi {k\over\hbar} \int_H \; J^{\mu\nu\lambda\rho} \; g^\perp_{\mu\lambda} g^\perp_{\nu\rho}\;\sqrt{ {}_2 g} d^2x.\end{align*}

\begin{align*}S_{{\mathrm{eff}}} = \frac 12 \int d\tau \left[\dot{\vartheta}^2 - \frac{(\pi_u - \pi_v)^2}{4\sinh^2\frac{\vartheta}{2}} + \frac{(\pi_u + \pi_v)^2} {4\cosh^2\frac{\vartheta}{2}}\right]\,, \end{align*}

\begin{align*}{\cal M}=\left( \begin{array}{cc}M_1 & M_2 \\M_2 & 0 \end{array} \right).\end{align*}

\begin{align*}L^{(0)} dt=a^{(0)}(\omega^{(0)})-V^{(0)}(\omega^{(0)}) dt\end{align*}

\begin{align*} \tilde W^2(x)=W^2(x)+const.\end{align*}

\begin{align*}[\mathcal{U}(\mathbb{Z}[\zeta_{p_{i}^{n_{i}}}]): P_{(H_{i},K_{i})}] = h_{p_{i}^{n_{i}}}^{+}. \end{align*}

\begin{align*}\ddot{A}=4\pi \ddot{g}_{\theta\theta}= 4 \pi r^2 [- 2 N \dot{K}^\theta_\theta - 2 \dot{N}{K}^\theta_\theta+ 4 N^2 ({K}^\theta_\theta)^2 ] e^{2c}. \end{align*}

\begin{align*}\frac{1}{A^r B^s } = \frac{\Gamma(r{+}s)}{\Gamma(r) \Gamma(s)} \int_0^1 \frac{x^{r-1} (1{-}x)^{s-1} \,dx}{(Ax+B(1{-}x))^{r+s}}\end{align*}

\begin{align*}S =\int d\tau\{ p_a \dot{x}^a+\frac{ms}{(p,n)}\dot{\varphi}-\frac{e(\tau )}{2}(p^2+m^2)\}\, . \end{align*}

\begin{align*}\Delta (\chi_i) = \chi_i \otimes 1 + O_i{}^j \otimes \chi_j,\end{align*}

\begin{align*}\bar{\cal H}^{(n)}_D={\rm diag}(-\frac{d^2}{dx^2}+\sigma_l^2-\frac{2\sigma_n^2}{\cosh^2\sigma_nx}),\ \ \ \ l=\hat{1},2,3,\ldots,\hat{n},\ldots,N ,\end{align*}

\begin{align*}J^{(2)}(2;1,1) = \frac{\mbox{i}\pi}{m_1 m_2} \; \frac{\tau_{12}}{\sin\tau_{12}} ,\end{align*}

\begin{align*}B(t)\approx\sqrt{\frac{1-k}{2-k}}\left(1+2 \;\frac{1-k}{2-k}\;e^{2t}\right),\;\;\;\;t<<0\end{align*}

\begin{align*}Q^- = 2^{3/4} g \int dx^- \mbox{tr} \left\{ ({\rm i}[\phi,\partial_- \phi ] + 2 \psi \psi ) \frac{1}{ \partial_-} \psi \right\}. \end{align*}

\begin{align*} \begin{array}{c} b^-~=~lim_{Q\to q}{{Q^{\pm k N_B \over 2}{(B^-)}^k}\over{([k]!)^{1\over 2}}}\\ b^+~=~lim_{Q\to q}{{{(B^+)}^kQ^{{\pm kN_B\over 2}}\over{([k]!)^{1\over 2}}}} \end{array}\end{align*}

\begin{align*}G_{\mu \nu }^{(1)}-\frac{3}{l^{2}}g_{\mu \nu }^{(1)}=-{\cal T}_{\mu \nu}^{(0)} \end{align*}

\begin{align*}\{\tau \} \equiv \{ \tau_F \} = \bigcap_{n \ge 0} B^n_{v^n}\end{align*}

\begin{align*}F=\left(\begin{array}{ccccc}0 & 0 & 0 & 0 & 0 \\0 & 0 & f & 0 & 0 \\0 & -f & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 & 0\end{array}\right)\end{align*}

\begin{align*}{\cal N}=\int_{\stackrel{x(0)=x}{x(T)=y}}{\cal D} x(\tau) e^{-S}.\end{align*}

\begin{align*}\hat{M}^{(\pm)}_1(k)=\sum_{\ell=2}^{4}\frac{b_{\ell}}{(1\pm ik)^{\ell}};\end{align*}

\begin{align*}S^{-1}_{ab}=-\,\frac{1}{2}\,\epsilon_{ab}\end{align*}

\begin{align*}\frac{\varphi _{n+1}-2\varphi _n+\varphi _{n-1}}{T^2}+\left( \mu ^2-\nabla^2\right) \frac{\left( \varphi _{n+1}+4\varphi _n+\varphi _{n-1}\right) }6 ={}_{{}_{\!\!\!\!\!\!\!c}} \;\, j_n.\end{align*}

\begin{align*}\hat H =i(q-q^{-1})q^{-1/2}(CA-BD+qBA-qCD).\end{align*}

\begin{align*}f^{(2)} = (1/2)\big[(g_1^2/16 \pi^2) \ln^2(q^2/ \mu^2) + (g_2^2/16 \pi^2) \ln^2(q^2/ M^2) \big]^2~.\end{align*}

\begin{align*}I_{1}(x,n)=\int_{x}^{\infty}d\tau\frac{1}{e^{\tau}-1}\tau^{n}=\sum_{q=1}^{\infty}e^{-qx}(\frac{x^{n}}{q}+\frac{nx^{n}}{q^{2}}+...\frac{n!}{q^{n+1}}),\end{align*}

\begin{align*}G(X_1 \ , \dots, X_p) = \prod _{j=1}^p(\rm{sdet}\ [A+ X_j C])^{-n_j}G(X_1^\prime \ , \dots,X^\prime_p).\end{align*}

\begin{align*}{\cal P}_{n+1,m+1} \ = \ \left \langle \begin{array}{c}x^0,x^1,\dots,x^m \\x^0, x^1, \dots,x^n \end{array} \right \rangle\end{align*}

\begin{align*} \Psi_k(w) \equiv \Psi_k^{k+1}(w) = \Psi^{k+1}_v (w + \tau_{k+1}) - \tau_k \end{align*}

\begin{align*}\lim_{t\rightarrow 0}\hat{U}\left( t\right) =\hat{I}_{\mathcal{H}}\end{align*}

\begin{align*}A_4 = \xi A_u \ , \qquad A_1 \equiv \omega_1\ , \qquad A_2 \equiv\omega_2\ , \qquad A_3 \equiv \omega_3\ , \qquad A_5 = \dots = A_8=0.\end{align*}

\begin{align*}\langle g_{xy} \rangle =\left(\begin{array}{cccc}b & 0 & 0 & 0\\0& b &0&0\\0&0&\langle g_{33} \rangle &\langle g_{34}\rangle\\0&0&\langle g_{43} \rangle &\langle g_{44} \rangle\end{array}\right)\end{align*}

\begin{align*}L_n= \displaystyle \oplus_h L_{n}^h\end{align*}

\begin{align*}{\sum_{j=1}^{m+1} \sum_{k=0}^{\infty} E^{(-2k-1)}_{jj}u^{(-2k-1)}_j}\end{align*}

\begin{align*}2m_q\langle \overline{q}q\rangle = -f_{\pi}^2 m_{\pi}^2\end{align*}

\begin{align*}{\rm Res}_\lambda \lambda^j\Psi({\bf u},\lambda)=0\end{align*}

\begin{align*}W_{B} \mbox{ is effective in $B$,} \quad a_{f}\geq0 \mbox{ integer } \end{align*}

\begin{align*}R_1= -{C\over \kappa} + \ln D + O(\kappa),\end{align*}

\begin{align*}\Lambda_{\pm}(q_0)=\Lambda(q_0)\pm\frac{2\kappa\bigg(\frac{2}{\pi}\bigg)^{\frac{1}{2}}}{[\frac{1}{2}(q_0-1)]!}\bigg(\frac{1+k}{1-k}\bigg)^{-\frac{\kappa}{k}}\bigg(\frac{8\kappa}{1-k^2}\bigg)^{\frac{1}{2}q_0}\bigg[1+O\bigg(\frac{1}{\kappa}\bigg)\bigg]\end{align*}

\begin{align*} R^n_k(y) &= R_{k+1}^n(y) + O\big( (-\lambda)^{n-k-1} r_k(y_{k+1}^n) \big), (R^n_k)'(y) &= (R_{k+1}^n)'(y) + O\big( r_k'(y_{k+1}^n) \big) \end{align*}

\begin{align*}y^2=(x^2+\tilde{u})^2-4 \Lambda_2^4.\end{align*}

\begin{align*}\delta B^{(2)}(y)=da(y)\wedge \varphi^{(1)}(y), \qquad \delta a=0.\end{align*}

\begin{align*}{\cal D}_{-} = \bordermatrix{ & {\ } & {\ } \cr {\ } & 0 & D_{-} \cr {\ } & i\partial_{+} & 0 \cr}\end{align*}

\begin{align*}\tilde g_{00} = -N^2 + n^2g \,\,\, , \,\,\,\tilde g_{01} = ng \,\,\, , \,\,\,\tilde g_{11} = g \,\,\, .\end{align*}

\begin{align*}\frac{\partial}{\partial m^2}\Omega_{free}=\frac{1}{12}T^2V+\cdots \end{align*}

\begin{align*}k^{\mu} \equiv \frac{1}{2} ( {\gamma}^{\mu} \otimes {\gamma}^0- {\gamma}^0 \otimes {\gamma}^{\mu} ),\end{align*}

\begin{align*}\psi_{-}(\vec{x}) = \frac{1}{\sqrt{2}} \frac{1}{i\partial_{-} -e V_{-}} \ast \xi (\vec{x}) - \frac{e}{\sqrt{2}}\frac{1}{\Delta_{\perp}} \ast (J_{pzm}^i (x_\perp) - q_{i} )\alpha^i \frac{1}{i\partial_{-} - e V_{-}} \ast\psi_{+} (\vec{x}) \;, \end{align*}

\begin{align*}\frac{1}{T^{\rm on}_{\rm DR}(E)}={\frac{1}{C_{\rm DR}} + i \frac{2 \pi \mu} {\Gamma(d/2) (4 \pi)^{d/2}} (2 \mu E)^{d/2 - 1}}.\end{align*}

\begin{align*}\psi=\frac{1+\gamma^5}{2^{1/4}}\Psi\,,\qquad\chi=\frac{1-\gamma^5}{2^{1/4}}\Psi\,.\end{align*}

\begin{align*}C_{1t}=0,~ ~C_{1+s,t}=\delta_{st}, ~s,t=1,\cdots,k.\end{align*}

\begin{align*} \big| \: \! [\,x + S^n_0(x,y,z) ] - [ \,v_*(x) + S^n_0 (0,y,z) ] \big| = O(\rho^n) .\end{align*}

\begin{align*}X(\alpha_1,\alpha_2,\alpha'_1,\alpha'_2)=\exp \left[ i\alpha_1 q^1{\partial\over\partial q^1}+ i\alpha_2 q^2{\partial\over\partial q^2}+i\alpha'_1 \overline{q}^1{\partial\over\partial \overline{q}^1}+i\alpha'_2 \overline{q}^2{\partial\over\partial \overline{q}^2}\right]\end{align*}

\begin{align*}E(\alpha) = \left(E\cosh\alpha - c(\vec{n}\vec{P})\sinh\alpha\right) {\cal{W}}^{-1}(\alpha, \vec{n}\vec{P},E ) .\end{align*}

\begin{align*}ds^2 = 2 du dv + A_{\mu} (u, x^i ) d x^{\mu} du - g_{ij}(u, x^i ) dx^i d x^j \ , \quad l^\mu A_\mu =0\ ,\end{align*}

\begin{align*}\int_\Gamma\frac{dz}{2\pi i}z^sq(z) = 0 \,,\end{align*}

\begin{align*}\{a_1(x, \lambda) \otimes I~,~ (r +s)_{23}(y,\mu,\eta)\} = H^{r + s}_{1,23}(x, \lambda, \mu, \eta) \delta(x-y).\end{align*}

\begin{align*}p_l(t) = f^l(t) \oint \frac{dz}{2\pi \imath} z^{l-1} V_2'(q(z,t))\end{align*}

\begin{align*}\left( G_{n}^{\alpha }\right) ^{\dagger }=G_{n}^{\alpha },\quad \left(G_{n}^{\beta }\right) ^{\dagger }=G_{n}^{\beta },\end{align*}