harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{\pi_{\zeta}(e_{1})(v_{+},v_{-})=\zeta(0,v_{+}),\quad\pi_{\zeta}(f_{1})(v_{+},v_{-})=\zeta^{-1}(v_{-},0),\quad\pi_{\zeta}(t_{1})(v_{+},v_{-})=(qv_{+},q^{-1}v_{-}),}\\{\pi_{\zeta}(e_{0})(v_{+},v_{-})=\zeta(v_{-},0),\quad\pi_{\zeta}(f_{0})(v_{+},v_{-})=\zeta^{-1}(0,v_{+}),\quad\pi_{\zeta}(t_{0})(v_{+},v_{-})=(q^{-1}v_{+},qv_{-}).}\\\end{array}
I_{\alpha}(v_{1},\ldots,v_{N-1})={\frac{1}{\pi}}\,\int\!d^{2}p\,{\frac{\partial_{\alpha}\tilde{S}(p^{2};\alpha=h_{N},h_{1},\ldots,h_{N-1})}{\tilde{S}(p^{2};\alpha=h_{N},h_{1},\ldots,h_{N-1})}}\cdotp
\left[\deltaB_{1},\,B_{1}^{\dagger}\right]-\left[B_{1},\,\deltaB_{1}^{\dagger}\right]+\left[\deltaB_{2},\,B_{2}^{\dagger}\right]-\left[B_{2},\,\deltaB_{2}^{\dagger}\right]+\deltaII^{\dagger}-I\deltaI^{\dagger}+\deltaJ^{\dagger}J-J^{\dagger}\deltaJ=0\,.
Z_{\mathrm{1-loop}}^{(c)}(M)={\frac{(2\pix)^{{\frac{1}{2}}(\mathrm{dim}H_{c}^{0}-\mathrm{dim}H_{c}^{1})}}{\mathrm{vol}(H_{c})}}\mathrm{e}^{-{\frac{1}{x}}S_{\mathrm{CS}}(A^{(c)})-{\frac{i\pi}{4}}\varphi}{\sqrt{|\tau_{R}^{(c)}|}},
L_{D}=\int_{\calM}{\frac{1}{4\pi}}\mathrm{Tr}\bigg[{\frac{k}{2}}g^{-1}\widetilde{d}gd_{0}(g^{-1}\widetilde{d}g)-{\frac{k}{2}}h^{-1}\widetilde{d}hd_{0}(h^{-1}\widetilde{d}h)+h^{-1}\widetilde{d}\alphahd_{0}(h^{-1}\widetilde{d}h)\bigg],
\begin{aligned}{\int_{-i\infty}^{i\infty}dt\;}&{{}\frac{S_{b}(\beta_{2}-\beta_{3}+t)S_{b}(Q-\beta_{2}-\beta_{3}+t)}{S_{b}(2Q-2\beta_{3}+t)S_{b}(Q+t)}=}\\{}&{{}=\frac{S_{b}(\beta_{2}-\beta_{3})S_{b}(Q-\beta_{2}-\beta_{3})}{S_{b}(2Q-2\beta_{3})}F_{b}(\beta_{2}-\beta_{3},Q-\beta_{2}-\beta_{3};2Q-2\beta_{3};0).\nonumber}\\\end{aligned}
\left(\begin{matrix}{B^{(1)}}\\{B^{(2)}}\\\end{matrix}\right)\to(\Lambda^{T})^{-1}\left(\begin{matrix}{B^{(1)}}\\{B^{(2)}}\\\end{matrix}\right),\\\{\calM}\to\Lambda{\calM}\Lambda^{T};\\\\\Lambda\inSL(2,{\bfR}),
a(\vec{p})=\int{\frac{d^{\nu}x}{(2\pi)^{\frac{\nu}{2}}}}{\frac{e^{-i\vec{p}\cdot\vec{x}}}{\sqrt{2g(\vec{p})}}}\left(\intd^{\nu}y{\frac{1}{2}}G(x,y)\eta(y)+{\frac{\delta}{\delta\eta(\vec{x})}}\right)
E_{M}^{(11)\,A}=\left(\begin{matrix}{e^{-{\frac{\Phi}{3}}}e_{\breve{\mu}}^{(10)\,\breve{\alpha}}}&{e^{{\frac{2}{3}}\Phi}B_{\breve{\mu}}}\\{0}&{e^{{\frac{2}{3}}\Phi}}\\\end{matrix}\right),
0=\frac{\delta_{L}}{\deltaB^{\alpha}}\mathbf{S}_{m}^{a}(\Gamma_{m})=\mathbf{\tilde{Q}}_{m}^{a}\frac{\delta_{L}}{\deltaB^{\alpha}}\Gamma_{m},\qquad0=\frac{\delta_{L}}{\deltaB^{\alpha}}\mathbf{D}_{A}(\Gamma_{m})=\mathbf{\tilde{Q}}_{A}\frac{\delta_{L}}{\deltaB^{\alpha}}\Gamma_{m},
{\calL}_{\mathrm{daughter}}=\frac{1}{2}\sum_{i=1}^{2N}\dot{x}_{i}^{2}-\frac{1}{2}\sum_{i=1}^{2N}\left\{-2Ax_{i}^{2}+x_{i}^{6}\right\}-\frac{1}{2}\sum_{\stackrel{i,j=1}{j\neqi}}^{2N}\frac{1}{(x_{i}-x_{j})^{2}}\,,
C=\sum_{n=0}^{\infty}h^{n}C^{(n)}[c,a_{l}],\quadA_{i}=\sum_{n=0}^{\infty}h^{n}A_{i}^{(n)}[a_{l}],\quadE_{i}{}^{j}=\sum_{n=0}^{\infty}h^{n}E^{(n)}{}_{i}{}^{j}[a_{l}],
\begin{array}{l}{d{\hat{s}}^{2}=2du\,\left(dv-\frac{1}{2}K\,du\right)-F^{-1}\,dx_{I}dx_{I}\qquad,\qquade^{-2\hat{\phi}}=F}\\{\hat{H}_{IJK}=\frac{1}{6}\epsilon_{IJKL}\partial^{L}F^{-1}\,,\qquad\hat{H}_{IJu}=\hat{H}_{IJv}=\hat{H}_{Iuv}=0\,.}\\\end{array}
\zeta(s,d)=\frac{\pi^{s}}{\Gamma(s)}\left\{\frac{1}{s-d/2}-\frac{1}{s}+\int_{1}^{\infty}\!\!dt\,\left(t^{s-1}+t^{d/2-s-1}\right)\,\left[\vartheta^{d}(t)-1\right]\right\}\quad.
\sum_{n\in(\Lambda_{\mathrm{w}}/lP\Lambda_{\mathrm{r}})\backslash{\calM}}f(n,x)=\sum_{t\in\Lambda_{\mathrm{r}}/H\Lambda_{\mathrm{r}}}\int_{C^{r}}f(\beta,x)\mathrm{e}^{-2\pii\beta\cdott}\,d\beta-(2\pii)^{r}\sum_{n\in{\calM}}\mathrm{Res}(h(\beta,x),\beta=n),
\intd^{3}x{F^{a}({\bfr})}^{(l)}{\eta^{a}({\bfr})}_{0}^{(k)}=\frac{1}{a}\intd^{3}xB_{l}^{a}H_{l}^{(ext)}D_{k}\phi^{a}=\frac{1}{3}gH_{k}^{(ext)}.
I_{1}(B)=-\frac{eB}{\pi^{\frac{3}{2}}\beta}\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k}\int_{0}^{\infty}dx\;e^{-\frac{x^{2}}{4}-\frac{\beta^{2}k^{2}m^{2}}{x^{2}}}\frac{1}{e^{\frac{2\beta^{2}eBk^{2}}{x^{2}}}-1}\;.
\tilde{\phi}(l,a)={\frac{1}{25\sqrt{3}\pi}}(\alpha(h)l)^{1/3}(a/5)^{-7/3}e^{-5(\alpha(h)l)^{2}/a}={\frac{1}{25\sqrt{3}\pi}}L^{1/3}A^{-7/3}e^{-L^{2}/A}\,
\langle\Psi|H_{0}+H_{0}^{\prime}|\Psi\rangle={\frac{1}{P}}\int{\frac{dk}{4\pi}}\Biggl[{\frac{m^{2}-e^{2}/\pi}{k}}+{\frac{m^{2}-e^{2}/\pi}{P-k}}+{\frac{2e^{2}}{\pi\epsilon}}\Biggr]|\phi(k)|^{2}\;,
ds^{2}|_{M2}=-4(|y|^{2}+y_{3}^{2})(dx_{+})^{2}-{\frac{2}{i}}(W^{\prime}d(e^{-2ix_{+}}y)-\overline{{W^{\prime}}}d(e^{2ix_{+}}\overline{{y}}))dx_{+}+dyd\overline{{y}}+(dy_{3})^{2}
\frac{1}{|N_{E,q,j}^{\pm}|^{2}}=(E+M)^{2}\int_{D_{r}}\frac{dr}{V}|f_{E,q,j}^{\pm}|^{2}+\int_{D_{r}}dr\,\left(|g_{E,q,j}^{\pm}|^{2}+|h_{E,q,j}^{\pm}|^{2}\right)\,.
\kappa=(u\tilde{u})^{-1}\kappa_{0}=e^{-\frac{1}{2}\phi\gamma^{1}\sigma^{2}}e^{\frac{1}{2}\chi\gamma^{0}\sigma^{2}}e^{-\frac{1}{2}\varphi\gamma^{0}\gamma^{1}\gamma^{2}\sigma^{2}}e^{-\frac{1}{2}(\theta-\frac{\pi}{2})\gamma^{0}\gamma^{1}\gamma^{3}\sigma^{2}}\kappa_{0}\,.
\frac{q^{\frac{r(r-1)}{4}}}{(q,q)_{\infty}}\sum_{j=-\infty}^{\infty}(q^{j[j\nu(\nu+1)+r(\nu+1)-\nu]}-q^{(j\nu+r)[j(\nu+1)+1]})\;;\;\;\;\;\;r=1,2,\ldots,\nu-1,
={\frac{g^{2}}{M^{2}}}\left\{-2ln{\frac{\lambda_{1}^{2}+m_{0}^{2}}{\lambda_{2}^{2}+m_{0}^{2}}}+{\frac{\lambda_{1}^{2}-\lambda_{2}^{2}}{M^{2}}}\left[arctan{\frac{1-g^{2}{\frac{m_{0}^{2}}{M^{2}}}}{g^{2}}}-arctan{\frac{1-g_{2}^{2}{\frac{\Lambda^{2}}{M^{2}}}}{g^{2}}}\right]\right\},
xH(a+{\frac{1}{2}},a+{\frac{1}{2}},{\frac{3}{2}},-x^{2})={\frac{-1}{2(a-{\frac{1}{2}})^{2}}}\frac{d}{dx}H(a-{\frac{1}{2}},a-{\frac{1}{2}},1/2,-x^{2})
1=J_{h}\int\prod_{k=2}^{\calN}d{z^{\prime}}_{k}^{c}d\bar{z^{\prime}}_{k}^{c}D[\delta\sigma_{R}]D[\xi^{0}]e^{-2C(\delta\sigma_{R},\delta\sigma_{R})}e^{-(P\xi^{0},P\xi^{0})}e^{-2\delta{z^{\prime}}_{k}^{c}\delta\bar{z^{\prime}}_{l}^{c}(P\xi^{k},Q_{n})(Q_{n},\bar{Q}_{s})^{-1}(\bar{Q}_{s},P\bar{\xi}^{l})}
P_{R}^{\prime}=(\;\;\;{\rho}\frac{\breve{m}}{2}\;\;\;+(\;\;G_{\bot}\;\;-\;\;B_{\bot}+\frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\breve{n}}-\;\;\frac{1}{2}A_{\bot}^{t}C_{\bot}{\breve{l}},0)=(p_{R}^{\prime},\;0).
\sqrt{g({\vec{x}},t)}\Lambda^{E}=\sqrt{g({\vec{x}},t)}e^{2\Phi({\vec{x}},t)}Q^{2}(t)\to\sqrt{g({\vec{x}},t)}\frac{Q_{0}^{2}}{t^{2}}~,\quadt\to\infty~,
{\bfT}exp\biggl(ie\sum_{a=1}^{N}\int_{0}^{T}dt\,{\dot{\bfx}}_{a}\cdot{\hat{\bfA}}({\bfx}_{a})\biggr)=exp(i\omega)\,exp\biggl(ie\sum_{a=1}^{N}\int_{0}^{T}dt\,{\dot{\bfx}}_{a}\cdot{\hat{\bfA}}({\bfx}_{a})\biggr)\,,
H=\frac{1}{2M}\sum_{m=1}^{D-1}\sum_{i,j=1}^{N}(p_{ij}^{m})^{2}+M^{5}\sum_{m,n=1}^{D-1}\sum_{i,j=1}^{N}|[X^{m},X^{n}]_{ij}|^{2}\.
\delta_{F}(\left[v^{3}\theta^{2}\right])+\delta_{B}(\left[v^{4}\right])+v^{i}\epsilon\left[v^{2}\dot{N}^{i(0)}\right]\theta+\frac{i}{2}\theta\left[v^{3}\dot{M}^{(0)}\right]\epsilon\simeq0~,
b={\frac{8{\calN}^{2}}{er_{0}^{n}r^{2}\left[\left({\frac{r}{r_{0}}}\right)^{\calN}+\left({\frac{r_{0}}{r}}\right)^{\calN}\right]^{2}}}\Re\mathrm{e}\left[g_{n}z^{n-1}\left(n-{\calN}-1+{\frac{2{\calN}}{1+\left({\frac{r}{r_{0}}}\right)^{2{\calN}}}}\right)\right]\,
36\mathrm{Cl}_{2}(\theta_{5})-30\mathrm{Cl}_{2}(2\theta_{5})+4\mathrm{Cl}_{2}(3\theta_{5})+3\mathrm{Cl}_{2}(4\theta_{5})=Z_{20}=20\sum_{k\in\{1,3,7,9\}}\mathrm{Cl}_{2}\left(\frac{2\pik}{20}\right)
S_{\mathrm{em}}=-{\frac{1}{2g_{\mathrm{eff}}^{2}}}\,\int\!d^{2}x\,\Bigl(\partial_{0}\theta\partial_{1}\tilde{\theta}+\partial_{0}\tilde{\theta}\partial_{1}\theta+\bigl(\partial_{1}\theta\bigr)^{2}+\bigl(\partial_{1}\tilde{\theta}\bigr)^{2}\Bigr)+S_{\mathrm{int}}[\theta,\tilde{\theta}]
\sigma_{a^{\dagger}a}^{2}={\frac{1}{4}}\biggl(\mu_{1}^{2}+\mu_{4}^{2}+(m_{0}\omega_{0})^{2}\mu_{3}^{2}+{\frac{\mu_{2}^{2}}{(m_{0}\omega_{0})^{2}}}{\frac{m_{0}\omega_{0}}{\hbar}}\biggr)\,,
S={\frac{a_{H}^{n}\mathrm{Vol}(M^{n})}{4G_{n+2}}},\\\\\\\\\\\\{\calT}={\frac{h^{\prime}(a_{H})}{4\pi}}={\frac{(n+1)a_{H}^{2}+(n-1)kL^{2}}{4\piL^{2}a_{H}}},
G_{l+\nu}(r^{\prime\prime},r^{\prime};E)=G_{l+\nu}^{(0)}(r^{\prime\prime},r^{\prime};E)-\frac{G_{l+\nu}^{(0)}(r^{\prime\prime},a;E)G_{l+\nu}^{(0)}(a,r^{\prime};E)}{G_{l+\nu}^{(0)}(a,a;E)-1/g}\;.
M^{2}=\frac{1}{2}\piNT_{II}\sum_{i=1}^{24}\sum_{n=1}^{\infty}\omega_{n}\left(a_{ni}^{\dagger}a_{ni}+\tilde{a}_{ni}^{\dagger}\tilde{a}_{ni}+2\,c_{ni}^{\dagger}c_{ni}-{\calC}\right),
\tilde{H}_{j-A}=-\,\intd{\bfr}\,\overbrace{g\,{\psi^{\dagger}}({\bfr})\alpha_{i}{\textstyle\frac{\lambda^{h}}{2}}\psi({\bfr})}^{j_{i}^{h}({\bfr})}\,A_{{\sfGI}\,i}^{h}({\bfr})\,.
{\calH}\;=\;-{\frac{1}{2m}}\psi^{\dagger}\nabla^{2}\psi\;+\;{\frac{1}{4}}g(\psi^{\dagger}\psi)^{2}\;+\;{\frac{1}{36}}g_{3}(\psi^{\dagger}\psi)^{3}\;+\;\ldots,
M_{\lambda1}^{-2}(\beta)=\frac{i}{2}\int[k_{1}k_{2}k_{3}]f_{\lambda}\left[\frac{\partial}{\partialp_{3}^{2}}\tilde{\delta}(k_{3\hat{\beta}}-k_{1\hat{\beta}2\hat{\beta}})\right]a_{\lambda1}^{\dagger}a_{\lambda2}^{\dagger}a_{\lambda3}+h.c.\;,
F_{2}({\vec{\theta}})=\prod_{i=2}^{2}\delta^{3}({\vec{\theta}}-{\vec{\theta}}^{i})+\sum_{i=1}^{2}C_{a}^{i}\theta_{a}\delta^{3}({\vec{\theta}}-{\vec{\theta}}^{i})\prod_{j\neqi}\theta_{b}^{j}\theta_{c}^{j}\theta_{d}^{j}\,
\Gamma_{\mu\nu}{}^{\rho}{\calT}_{\rho}{}^{\sigma_{1}\cdots\sigma_{n}}=-\frac{1}{2\cdotn!}\Big(\Gamma_{\mu\nu}{}^{\sigma_{1}\cdots\sigma_{n}}+n(n-1)\,\delta_{\mu}{}^{[\sigma_{1}}\delta_{\nu}{}^{\sigma_{2}}\Gamma^{\sigma_{3}\cdots\sigma_{n}]}\Big)\,.
\Gamma_{\underline{\alpha}\underline{\beta}}^{0}=\left(\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right)\,,\quad\Gamma_{\underline{\alpha}\underline{\beta}}^{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\,,\quad\Gamma_{\underline{\alpha}\underline{\beta}}^{2}=\left(\begin{array}{cc}{-1}&{0}\\{0}&{1}\\\end{array}\right)\,.
\begin{aligned}{I_{1}:}&{{}\,|\Phi_{1}|^{2}=0,\,\,0\leq|\Phi_{2}|^{2}\leqr\,,}\\{I_{2}:}&{{}\,|\Phi_{2}|^{2}=0\,,\,\,0\leq|\Phi_{1}|^{2}\leqr\,,}\\{I_{3}:}&{{}\,|\Phi_{1}|^{2}+|\Phi_{2}|^{2}=r\,,\,\,|\Phi_{1}|,|\Phi_{2}|\geq0\,.}\\\end{aligned}
\{\dot{x}_{i},\dot{x}_{j}\}_{\mathrm{D}}=\frac{1}{\kappa}\,\epsilon_{ij},\quad\{\tilde{p}_{i},\tilde{p}_{j}\}_{\mathrm{D}}=\frac{\kappa}{4}\,\epsilon_{ij},\quad\{\dot{x}_{i},\tilde{p}_{j}\}_{\mathrm{D}}=\frac{1}{2}\,\delta_{ij}
L_{BCL}^{g}=\frac{1}{2e}\dot{x}{}^{2}-\frac{e}{2}m_{-}^{2}+\frac{i}{em_{+}}(\dot{\xi}_{*}+q\epsilon_{\mu\nu\lambda}\dot{x}{}^{\mu}{\calA}^{\nu}\xi^{\lambda})\dot{x}\xi-\frac{i}{2}\xi_{\mu}\dot{\xi}{}^{\mu}+\frac{i}{2}\xi_{*}\dot{\xi}_{*},
ds^{2}=\frac{r^{2}}{R^{2}}(-dx_{0}^{2}+dx_{1}^{2})+\frac{R^{2}}{r^{2}}dr^{2}+R^{2}d\Omega_{3}^{2}+\sqrt{\frac{Q_{1}}{vQ_{5}}}dx^{A}dx^{A}~,
S=\intd^{26}x\left[-\frac{1}{2}\partial_{\mu}\widehat{T}(x)\partial^{\mu}\widehat{T}(x)+\frac{1}{2\alpha^{\prime}}\widehat{T}(x)^{2}+\frac{1}{3}g_{T}\widehat{T}(x)^{3}\right].
\mathrm{~{\cal~R}~}\equiv{\frac{1}{2}}\left(\begin{array}{cccc}{1}&{0}&{-\sqrt{3}}&{0}\\{0}&{1}&{0}&{-\sqrt{3}}\\{\sqrt{3}}&{0}&{1}&{0}\\{0}&{\sqrt{3}}&{0}&{1}\\\end{array}\right).
\intd\etae^{i\biggl(\sqrt{{\frac{\alpha^{\prime}}{2}}}(k\cdot{\bar{X}}+{\bar{k}}\cdotX)+\eta(D_{L}+D_{R})(\zeta\cdot{\bar{X}}+{\bar{\zeta}}\cdotX)\biggr)}|_{\theta_{L}=\theta_{R}}
I(\mu^{2},M^{2},\Lambda^{2})\equiv\frac{\alpha}{2\pi^{2}}\int_{L_{-}}^{L_{+}}\frac{dx}{1-x}\int_{\bbox{q}_{\perp}^{2}\leqL}d^{2}q_{\perp}\frac{\bbox{q}_{\perp}^{2}+(2-x)^{2}M^{2}}{\bbox{q}_{\perp}^{2}+x^{2}M^{2}+(1-x)\mu^{2}}\,,
\int_{\bar{B}}\sqrt{\bar{g}}d^{D-1}x\simeq{\frac{1}{\kappa^{D-1}}}\int_{\Sigma}\int_{0}^{\infty}\rho^{1-D}d\rho\left[1+\frac14\rho^{2}P+O(D-4,\rho^{2})\right]~~~.
H_{SB}(\ell)=O_{\ell}\left(\begin{array}{cccc}{E_{1}}&{0}&{0}&{0}\\{0}&{E_{2}^{\prime}b^{2\ell}}&{0}&{0}\\{0}&{0}&{E_{3}^{\prime}b^{2\ell}}&{0}\\{0}&{0}&{0}&{E_{4}^{\prime}b^{2\ell}}\\\end{array}\right)O_{\ell}^{\dagger}
{\calH}^{(2)}=\frac{2}{l}\gamma_{\mu\nu}\frac{\deltaI^{(2)}}{\delta\gamma_{\mu\nu}}-\frac{\sqrt{-\gamma}}{2\kappa}R+\frac{1}{2}\sqrt{-\gamma}\gamma^{\mu\nu}\partial_{\nu}\phi\partial_{\mu}\phi=0.
\intd^{2}kF_{00}(k)|h_{\lambda}(k)|^{2}\langle0|\deltaB(\lambda)^{2}|0\rangle=\int_{0}^{\infty}\!\!\!\!dm^{2}\rho(m^{2})|f(\lambdam^{2})|^{2}\int_{-\infty}^{\infty}dk_{1}\frac{k_{1}^{2}|g(k_{1}^{2})|^{2}}{2\sqrt{k_{1}^{2}+m^{2}}}
S=\frac{1}{2\pi}\intd^{2}{\xi}\intd\thetad\overline{{\theta}}\left(g_{\mu\nu}+2\pi\alpha^{\prime}B_{\mu\nu}\right)\overline{{D}}{\bfX}^{\mu}({\bfz},\overline{{\bfz}})D{\bfX}^{\nu}({\bfz},\overline{{\bfz}})~,
y(1-y){\calF}^{\prime\prime}(y)+\left[l+\frac{n}{2}-(l+\mu+\frac{3}{2})y\right]{\calF}^{\prime}(y)-\left[\frac{1}{4}l(l+1)+\frac{1}{4}\mu(\mu+1)+\frac{1}{2}l\mu\right]{\calF}(y)=0~.
\left.{\bar{\nabla}}^{k}{\beta}_{k}^{v(1)}\right|_{a=1}=-\frac{1}{2}G^{km}{\beta}_{km}^{b^{\star}(1)},\;\;\;\left.{\bar{\nabla}}^{k}{\beta}_{k}^{w(1)}\right|_{a=1}=-\frac{1}{2}F^{km}{\beta}_{km}^{b^{\star}(1)}.
(-2)n!\prod_{i=1}^{k}d_{i}!{\frac{1}{(n-m)!}}\sum_{\sigma_{2}=\{i,j\}}\left(\begin{matrix}{n-m}\\{2}\\\end{matrix}\right)\left(\begin{matrix}{n-m-2}\\{d_{i}-2}\\\end{matrix}\right)\left(\begin{matrix}{n-m-d_{i}}\\{d_{j}-2}\\\end{matrix}\right)(\cdots),
C=\left(\begin{array}{ccccc}{0}&{l_{12}^{2}}&{l_{13}^{2}}&{l_{1}^{2}}&{1}\\{l_{12}^{2}}&{0}&{l_{23}^{2}}&{l_{2}^{2}}&{1}\\{l_{13}^{2}}&{l_{23}^{2}}&{0}&{l_{3}^{2}}&{1}\\{l_{1}^{2}}&{l_{2}^{2}}&{l_{3}^{2}}&{0}&{1}\\{1}&{1}&{1}&{1}&{0}\\\end{array}\right)
{\calL}=\frac{\theta}{4}\epsilon^{\mu\nu\alpha}F_{\mu\nu}A_{\alpha}+\bar{\psi}(i\not\!\partial-m)\psi+\bar{\varphi}(i\not\!\partial+m)\varphi+e\bar{\psi}\gamma^{\mu}\psiA_{\mu}+e\bar{\varphi}\gamma^{\mu}\varphiA_{\mu},
\{Q_{\alpha}^{i},Q_{\beta}^{j}\}=\sum_{k}\gamma_{[\alpha\beta]}^{[\mu_{1}\cdots\mu_{k}]}Z_{[\mu_{1}\cdots\mu_{k}]}^{0}\Omega^{ij}+\sum_{k}\gamma_{(\alpha\beta)}^{[\mu_{1}\cdots\mu_{k}]}Z_{[\mu_{1}\cdots\mu_{k}]}^{I}\sigma_{I}^{ij}.
X_{I}^{\mu}=\frac{i}{2\sqrt{\piT_{I}}}\sum_{n\neq0}\frac{1}{n}\left[\alpha_{n}^{\mu}(s)e^{i(1+s)n(\sigma-\tau)}+\tilde{\alpha}_{n}^{\mu}(s)e^{-i(1+s)n(\sigma+\tau)}\right],
{\hat{\gamma^{\prime}}}^{(\lambda,\mu,\alpha)}={\hat{F}}_{(\tau,\nu,\beta)}^{(\lambda,\mu,\alpha)}{\hat{\gamma}}^{(\tau,\nu,\beta)},\quad{\hat{\gamma^{\prime}}}_{(\lambda,\mu,\alpha)}={\hat{\gamma}}_{(\tau,\nu,\beta)}{\hat{F}}_{(\lambda,\mu,\alpha)}^{(\tau,\nu,\beta)},\quad(\hat{\gamma^{\prime}}=\hat{F}\hat{\gamma},\quad\bar{\hat{\gamma^{\prime}}}=\bar{\hat{\gamma}}{\bar{\hat{F}}}^{T}),
<n;p_{\lambda}=0;-+|\hat{U}_{\mathrm{BRST}}(t_{2},t_{1})|m;p_{\lambda}=0;-+>=i\delta_{n,m}\int_{-\infty}^{\infty}\frac{d\gamma}{2\pi\hbar}\,e^{-i/\hbar\gamma(\hbar\omega(n+1/2)-\Lambda)}\,
C^{(2)}=\{-\frac{1}{\omega_{A}}(\frac{1}{\omega_{C}}f^{CAB}\varphi_{C}+f^{\lambdaAB}\varphi_{\lambda})+\frac{1}{2\omega_{A}\omega_{B}}f_{C}^{\;AB}\varphi^{C}\}(P_{A}^{a}P_{B}^{b}-P_{A}^{b}P_{B}^{a}).
C_{1}^{\Lambda}=(I\otimes\mathrm{tr})\{[I\otimes\pi_{\Lambda}(q^{2h_{\rho}})]\Gamma\}=\sum_{s,s^{\prime}}\mathrm{tr}(\pi_{\Lambda}(q^{2h_{\rho}}a_{s}b_{s^{\prime}}))b_{s}a_{s^{\prime}}
\int\prod_{i=1}^{3}d^{2}q_{i}\,\prod_{i=1}^{3}\delta^{2}(S_{q}(z_{i})-Z_{i}^{0})\,{\frac{\prod_{i<j=1}^{3}|Z_{i}^{0}-Z_{j}^{0}|^{2}}{\prod_{i<j=1}^{3}|q_{i}^{0}-q_{j}^{0}|^{2}}}=1,
Z_{1}=a_{1}\frac{\prod_{i=1}^{k}(z-b_{1,i})}{\prod_{i=1}^{k}(z-c_{1,i})},\;\;\;\;\;Z_{2}=a_{2}\frac{\prod_{i=1}^{\ell}(z-b_{2,i})}{\prod_{i=1}^{\ell}(z-c_{2,i})}.
M_{6}=\left(\begin{array}{cccccc}{c_{11}}&{c_{13}}&{c_{14}}&{0}&{d_{13}}&{d_{14}}\\{c_{13}}&{c_{33}}&{c_{34}}&{-d_{13}}&{0}&{d_{34}}\\{c_{14}}&{c_{34}}&{c_{44}}&{-d_{14}}&{-d_{34}}&{0}\\{0}&{-d_{13}}&{-d_{14}}&{c_{11}}&{c_{13}}&{c_{14}}\\{d_{13}}&{0}&{-d_{34}}&{c_{13}}&{c_{33}}&{c_{34}}\\{d_{14}}&{d_{34}}&{0}&{c_{14}}&{c_{34}}&{c_{44}}\\\end{array}\right)
S=\intd^{3}x^{\mu}\Big[\partial_{0}A_{i}^{a}\,N_{k}\epsilon^{ij}A_{j}^{a}+A_{0}^{a}\,N_{k}\epsilon^{ij}F_{ij}^{a}-\partial_{i}\left(N_{k}\epsilon^{ij}A_{j}^{a}A_{0}^{a}\right)\Big]\\\,
\{J^{\mu\nu}\}=\left(\begin{array}{cccc}{0}&{0}&{1-x_{1}^{2}}&{-x_{1}x_{2}}\\{0}&{0}&{-x_{1}x_{2}}&{1-x_{2}^{2}}\\{-(1-x_{1}^{2})}&{x_{1}x_{2}}&{0}&{x_{2}p_{1}-x_{1}p_{2}}\\{x_{1}x_{2}}&{-(1-x_{2}^{2})}&{-(x_{2}p_{1}-x_{1}p_{2})}&{0}\\\end{array}\right)
V=\frac{{q^{2}}}{{2\lambda}}\left({1+\frac{{4\pim\Sigma}}{{\lambda^{2}}}}\right)\left({1-e^{-\lambda|y-y{\prime}|}}\right)+\frac{{q^{2}}}{2}\left({1-\frac{{\frac{{e^{2}}}{\pi}}}{{\lambda^{2}}}}\right)|y-y{\prime}|.
Z(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)=\pi^{\zeta/2}e^{\pm\frac{i\pi}{4}(\zeta-\eta)}\widetilde{Z}(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)\,.
{\calA}_{i}(z,\bar{z}){\calA}_{j}(0,0)=\sum_{k}z^{h_{k}-h_{i}-h_{j}}{\bar{z}}^{\tilde{h}_{k}-\tilde{h}_{i}-\tilde{h}_{j}}c^{k}\!_{ij}{\calA}_{k}(0,0).
\Pi_{1B}~=~\frac{2\pi^{2\mu}}{(\mu-1)^{2}\Gamma^{2}(\mu)}\left[1-\frac{\eta_{1}}{N}\left(\frac{2}{(\mu-1)}-\frac{3\mu(2\mu-3)}{2}\left(\Theta+\frac{1}{(\mu-1)^{2}}\right)\right)\right]
\tilde{Q}\;=\;-\frac{\varepsilon_{\rho\mu}\partial_{\rho}}{\triangle}Q_{\mu\nu}\frac{\varepsilon_{\sigma\nu}\partial_{\sigma}}{\triangle}\;=\;\frac{1}{-\triangle+\frac{e^{2}N}{\pi+gN}}\;\frac{-1}{\triangle}\;,
S_{p}={\frac{1}{2\kappa_{D^{\prime}}^{2}}}\intd^{D^{\prime}}x\sqrt{-\hat{G}}\left[{\calR}_{\hat{G}}-{\frac{4}{D^{\prime}-2}}(\partial\varphi)^{2}-{\frac{1}{2\cdot(p+2)!}}e^{2a_{p}\varphi}F_{p+2}^{2}\right].
g_{n}({\sigma}_{0})=exp\left\{J_{3}\sum_{\triangle}{\sigma}_{0}{\sigma}_{1}{\sigma}_{1}+h\sum{\sigma}_{1}\right\}{[g_{n-1}({\sigma}_{1}^{(1)})]}^{\gamma-1}{[g_{n-1}({\sigma}_{1}^{(2)})]}^{\gamma-1}.
\vec{A_{1}}=\left(\begin{array}{c}{x}\\{0}\\{0}\\\end{array}\right)\;\;\;\;\vec{A_{2}}=\left(\begin{array}{c}{0}\\{x}\\{0}\\\end{array}\right)\;\;\;\;\vec{\phi}=\left(\begin{array}{c}{-x}\\{x}\\{0}\\\end{array}\right)\;\;\;\;\dot{\vec{\phi}}=\frac{1}{2}\left(\begin{array}{c}{x}\\{x}\\{0}\\\end{array}\right).
[\hat{D}_{\alpha},\hat{D}_{\beta}]f=-(\partial_{\alpha}A_{\beta}^{\mu}-\partial_{\beta}A_{\alpha}^{\mu})\partial_{\mu}f+(A_{\alpha}^{\mu}\partial_{\mu}A_{\beta}^{\nu}-A_{\beta}^{\mu}\partial_{\mu}A_{\alpha}^{\nu})\partial_{\nu}f\,.
q_{-}^{0}(\tau)=-\frac{\Omega\vartheta_{1}^{\prime}(0)}{2\pi}\frac{e^{\Omega\tau\frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}(\Omegay)}\vartheta_{1}(\Omega(y-\tau))+e^{-\Omega\tau\frac{\vartheta_{1}^{\prime}}{\vartheta_{1}}(\Omegay)}\vartheta_{1}(\Omega(y+\tau))}{\vartheta_{1}(\Omega\tau)\vartheta_{1}(\Omegay)},
d_{ABC}\,h^{A}h^{B}h^{C}=3\Big\{h^{1}\,\big(h^{2}\big)^{2}-h^{1}\,\big(h^{\mu}\big)^{2}-h^{2}\,\big(h^{i}\big)^{2}+\gamma_{\muij}\,h^{\mu}\,h^{i}\,h^{j}\Big\}\.
\int_{0}^{a}dx_{1}\int_{0}^{b}dx_{2}\left<T_{00}(\vec{x})\right>=\int\int_{cavity}\left<T_{00}(\vec{x})\right>=\int\int_{cavity}\left<T_{00}(\vec{x})\right>_{B}+\int\int_{cavity}\left<T_{00}(\vec{x})\right>_{F}
\varphi_{\alpha}(\lambda)\sim{\frac{1}{(4\pi)^{d/2}}}\sum_{n=0}^{\infty}\left[a_{n}~{\frac{\lambda^{d/2-n+\alpha-1}}{\Gamma\left(\fracd2-n+\alpha\right)}}+b_{n}~{\frac{\lambda^{(d-1)/2-n+\alpha-1}}{\Gamma\left({\frac{d-1}{2}}-n+\alpha\right)}}\right],
V=\frac{1}{2}m^{2}(t)\Phi^{2}(t)-\frac{1}{2}\xi(t)\Phi^{2}(t)R+\frac{1}{4!}\lambda(t)\Phi^{4}(t)-\Lambda(t)-\kappa(t)R-a_{1}(t)R^{2}-a_{2}(t)C_{\mu\nu\alpha\beta}^{2}-a_{3}(t)G,
\left(\begin{array}{c}{{\hat{\xi}}^{t}}\\{{\hat{\xi}}^{r}}\\{{\hat{\xi}}^{\varphi}}\\\end{array}\right)={\calG}\left(\begin{array}{c}{{\hat{\Sigma}}^{t}}\\{{\hat{\Sigma}}^{r}}\\{{\hat{\Sigma}}^{\varphi}}\\\end{array}\right);\;\;\;\;\;\;{\calG}=\mathrm{Exp}\left(-\int^{\tau}{\calD}(\tau^{\prime})\;d\tau^{\prime}\right),
\left(\begin{array}{c}{\ddot{C}_{1n}}\\{\ddot{C}_{2n}}\\\end{array}\right)+\left(\begin{array}{cc}{(n+mH\alpha^{\prime})^{2}}&{0}\\{0}&{(n-mH\alpha^{\prime})^{2}}\\\end{array}\right)\left(\begin{array}{c}{{C}_{1n}}\\{{C}_{2n}}\\\end{array}\right)=\left(\begin{array}{c}{0}\\{0}\\\end{array}\right),
Q_{1}(Q_{2})=3D{\frac{1}{{\sqrt{-2\nabla^{2}}}}}[\epsilon_{ij}\partial=_{i}A_{j}\pm{\frac{1}{m}}\Pi_{\theta}];~~P_{1}(P_{2})=3D[{\frac{1}{{\sqrt{-2\nabla^{2}}}}}\epsilon_{ij}\partial_{i}\Pi=_{j}\mp{\frac{m}{2}}{\sqrt{-2\nabla^{2}}}\theta],
G_{\mu\nu}=\left(\begin{array}{cccc}{-g(1-e^{2})}&{0}&{0}&{0}\\{0}&{g(1-e^{2})}&{0}&{0}\\{0}&{0}&{1+g^{2}b^{2}}&{0}\\{0}&{0}&{0}&{1+g^{2}b^{2}}\\\end{array}\right),
S_{B}[\Lambda,a_{\mu},b_{\mu}]=\intd^{2}x[-\frac{1}{2}\partial_{\mu}\Lambda\partial^{\mu}\Lambda+\frac{1}{\sqrt{\pi}}\partial_{\mu}\Lambdab^{\mu}+\frac{1}{\sqrt{\pi}}\partial_{\mu}\Lambda\epsilon^{\mu\nu}a_{\nu}].
{\bfS=}\sum_{i=1}^{N}{{\bfp}^{\bot}}_{i}\times{\bfe}_{i}-\frac12\sum_{i,j=1}^{N}{\phi}_{ij}{\bfe}_{i}\times{\bfe}_{j},\quad\mathrm{tr}\;{\bfS}^{2i}=\mathrm{tr}\;\left(\begin{array}{cc}{0}&{1}\\{-{\hat{\Phi}}}&{{\hat{\phi}}}\\\end{array}\right)^{2i}.
I_{1}=I_{1}\sum_{a=1}^{N}\frac{\lambda_{a}^{N-1}}{f_{a}(\vec{\lambda})}+\eta_{2}\sum_{a=1}^{N}\frac{\lambda_{a}^{N-2}}{f_{a}(\vec{\lambda})}+\dots+\eta_{N}\sum_{a=1}^{N}\frac{1}{f_{a}(\vec{\lambda})}\nonumber
\beta^{-\zeta(0\,;\,T_{+})/2}\,(-\beta)^{-\zeta(0\,;\,-T_{-})/2}=e^{-\frac{i\pi}{4}((\frac{2\theta}{\pi}\mp1)\zeta(0\,;\,|T|)\;\pm\;\eta(0\,;\,T))}\,.
\begin{array}{c}{q_{j}\mapstoq_{j}-\frac{2\pin_{j}}{\mathrm{dim}(E)}}\\{p^{j}\mapstop^{j}}\\{a({\bfk})\mapstoexp(2\pii\tilde{\vartheta}(n_{2}k_{1}-n_{1}k_{2}))a({\bfk})}\\{a^{*}({\bfk})\mapstoexp(-2\pii\tilde{\vartheta}(n_{2}k_{1}-n_{1}k_{2}))a^{*}({\bfk})}\\\end{array}
A=\left(\begin{array}{ccc}{-2M}&{-M\sqrt{2}}&{0}\\{M\sqrt{2}}&{0}&{M\sqrt{2}}\\{0}&{-M\sqrt{2}}&{2M}\\\end{array}\right)\quad,\quadh_{ij}=\delta_{ij}\quad\mathrm{and}\quadf={\frac{1}{r}}.
\langle\sigma_{n_{1},i_{1}}...\sigma_{n_{s},i_{s}}\rangle_{g}=\int_{{\calM}_{g,s}}\inte^{-S}\prod_{k}e^{\alpha_{k}{\tilde{\pi}}(z_{k})}\prod_{a,{\bar{a}}=1}^{3g-3}G(\chi_{a}){\bar{G}}({\bar{\chi}}_{\bar{a}})\prod_{j=1}^{s}\int_{\Sigma}\sigma_{n_{j},i_{j}}^{(2)}
z=\sum_{i=0}^{\infty}\frac{1}{i!}\alpha_{i}(\sigma,\bar{\sigma})\left(\hat{y}\!=\!\varphi\!-\!\hat{\varphi}_{0}(\sigma,\bar{\sigma})\right)^{i}\,,\quad\bar{z}=\sum_{i=0}^{\infty}\frac{1}{i!}\bar{\alpha}_{i}(\sigma,\bar{\sigma})\left(\hat{y}\!=\!\varphi\!-\!\hat{\varphi}_{0}(\sigma,\bar{\sigma})\right)^{i}\,,
=i\left(\begin{array}{cc}{u_{1}\partial/\partialu_{1}+u_{2}\partial/\partialu_{2}}&{-u_{2}\partial/\partial\bar{u}_{1}-u_{1}\partial/\partial\bar{u}_{2}}\\{\bar{u}_{1}\partial/\partialu_{2}-\bar{u}_{2}\partial/\partialu_{1}}&{\bar{u}_{1}\partial/\partial\bar{u}_{1}+\bar{u}_{2}\partial/\partial\bar{u}_{2}}\\\end{array}\right)
\mathcal{H}=\frac{1}{2}\left(a^{\dagger},\,a\right)\left(\begin{matrix}{\alpha^{\dagger}}&{\beta^{\dagger}}\\{\beta^{T}}&{\alpha^{T}}\\\end{matrix}\right)\left(\begin{matrix}{\omega}&{0}\\{0}&{\omega}\\\end{matrix}\right)\left(\begin{matrix}{\alpha}&{\beta^{*}}\\{\beta}&{\alpha^{*}}\\\end{matrix}\right)\left(\begin{matrix}{a}\\{a^{\dagger}}\\\end{matrix}\right).
\frac{1}{m}\oint\frac{\bar{dz}}{2\pii}\bar{\xi}_{t}(z)^{-m}\;(III)=\frac{1}{m}\oint\frac{\bar{dz}}{2\pii}\bar{\xi}_{t}(z)^{-m}\delta^{rt}\frac{\bar{\xi}_{t}^{\prime}(z)}{\bar{\xi}_{t}(z)}=\delta^{rt}\frac{1}{m}\oint\frac{\bar{dw}}{2\pii}w^{-m-1}=0
F_{\tiny\mathrm{div}}^{C}[g,\beta,\mu]=-{\frac{\eta}{48\kappa\beta^{2}}}\int_{\Sigma}\left[bf_{1}+a\left(p_{1}{\frac{4\pi^{2}}{\kappa^{2}\beta^{2}}}{\calP}+p_{2}R+p_{3}{\calQ}\right)\right]~~~.
\frac{\Gamma(\tilde{\sigma}_{i}\tilde{\sigma}\lambda)}{\Gamma(\tilde{\sigma}\lambda)}=\frac{[i(\tilde{\sigma}\lambda)_{i+1}-(\tilde{\sigma}\lambda)_{i}-c]}{[i(\tilde{\sigma}\lambda)_{i+1}-(\tilde{\sigma}\lambda)_{i}+c]},\,\,\frac{\Gamma(\bar{\sigma}_{N}\tilde{\sigma}\lambda)}{\Gamma(\tilde{\sigma}\lambda)}=-1.
n=(n_{-}+n_{+})=\frac{<N>}{V\Lambda^{2}}=\Bigg[(1+\beta^{\prime})^{\frac{1+\beta^{\prime}}{2}}(1-\beta^{\prime})^{\frac{1-\beta^{\prime}}{2}}4\pi\beta\frac{e^{2\gamma_{E}}}{8}\frac{N_{c}}{N_{c}-1}2^{\frac{8(N_{c}-1)^{2}}{\betaN_{c}^{2}}}\Bigg]^{\frac{\betaN_{c}^{2}}{8(N_{c}-1)^{2}-\betaN_{c}^{2}}}\;.
\hat{\omega}_{\mu}=0\,,\qquad\hat{\omega}_{i}=\frac{1}{2}\Gamma_{\underline{{ij}}}\left(\mathrm{Im}(f)^{-1}\partial_{i}\mathrm{Im}(f)-\partial_{i}\mathrm{Re}(g)\right)\qquadi,j=(z,\overline{{z}})\,.
J_{\vec{a}\vec{b}}^{\vec{\alpha}}(w)=\frac{\omega^{-\alpha_{2}(\alpha_{1}+1)}\sigma_{2\vec{\alpha}-\vec{a}+\vec{b}}(0)}{(\sigma_{\vec{\alpha}}\sigma_{\vec{a}-\vec{\alpha}}\sigma_{\vec{a}-\vec{b}-\vec{\alpha}}\sigma_{\vec{b}+\vec{\alpha}})(w/n)}.
L_{\mathrm{BRS}}^{\prime\prime}=\mathrm{}-\frac{1}{2}\partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}-\frac{m^{2}}{2}A_{\mu}A^{\mu}-\frac{1}{2}\partial_{\mu}\theta\partial^{\mu}\theta-\frac{m^{2}}{2}\theta^{2}.
{\frac{1}{A}}{\frac{\partial{\calE}}{\partialm^{2}}}={\frac{1}{2}}\sum_{n=1}^{\infty}\int\int{\frac{dk_{1}dk_{2}}{(2\pi)^{2}}}{\frac{1}{\sqrt{k_{1}^{2}+k_{2}^{2}+{\frac{n^{2}\pi^{2}}{a^{2}}}+m^{2}}}}\,.