harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{\hphantom{\mu}\nu}^{\mu}=\left(\begin{array}{cc}{\frac{p^{0}}{m}}&{-\frac{p_{n}}{m}}\\{\frac{p^{k}}{m}}&{\delta_{n}^{k}-\frac{\frac{p^{k}}{m}\frac{p_{n}}{m}}{1+\frac{p^{0}}{m}}}\\\end{array}\right),\,\,\,\,L(\hat{p})\left(\begin{array}{c}{1}\\{0}\\{0}\\{0}\\\end{array}\right)=\hat{p}.
\vec{P}^{T}{\calM}_{-}\vec{Q}\,[\vec{Q}^{T}{\calM}_{+}\vec{Q}-\vec{P}^{T}{\calM}_{+}\vec{P}]-\vec{P}^{T}{\calM}_{+}\vec{Q}[\vec{Q}^{T}{\calM}_{-}\vec{Q}-\vec{P}^{T}{\calM}_{-}\vec{P}]=0.
C^{\Lambda,\Lambda_{0}}(x-y)=\frac{1}{\mid\Omega\mid}\sum_{n\in\mathbf{Z}^{d}}\frac{e^{ik(x-y)}}{k^{2}+m^{2}}\left(e^{-\frac{k^{2}+m^{2}}{{\Lambda_{0}}^{2}}}-e^{-\frac{k^{2}+m^{2}}{{\Lambda}^{2}}}\right).
\begin{array}{rclrcl}{\partialF^{(1)m}}&{=}&{0\,,}&{\partialH}&{=}&{{\textstyle\frac{1}{2}}F^{(1)m}F^{(2)}{}_{m}\,,}\\{}&{}&{}&{}&{}&{}\\{\partialF^{(2)}{}_{m}}&{=}&{0\,.}&{}&{}&{}\\\end{array}
Z_{v}^{X}\left(-\frac{1}{\tau}\right)=\vert\Gamma_{\calG}\vert^{-\frac{b_{2}(X)}{2}}\left(\frac{\tau}{i}\right)^{-\frac{\chi(X)}{2}}\sum_{u}\zeta_{|\Gamma_{\calG}|}^{u\cdotv}Z_{u}^{X}(\tau),
a_{1}=\frac{g_{1}g_{2}k^{4}D^{4}}{g_{1}+g_{2}},\qquada_{2}=\frac{g_{2}p_{3}D^{4}}{p_{1}},\qquada_{3}=\frac{p_{3}g_{1}D^{4}k^{\prime4}}{p_{2}},
[\Box,\frac{1}{2}g_{ab}(R_{\;\;\;\;kl}^{ab}+\delta_{\;\;\;\;kl}^{ab})x^{k}x^{l}]=2g_{ab}g^{ab}+2g_{ab}(R_{\;\;\;\;kl}^{ab}+\delta_{\;\;\;\;kl}^{ab})x^{k}\partial^{l}
\widehat{\calL}_{_\mathrm{D}}={\frac{\widehat\Phi}{2(\mathrm{q-1})\Omega^{[q-1]}}}\Big(\widehatR-{\frac{4}{\alpha_{_\mathrm{D}}}}\widehatg{^{\mu\nu}}\phi_{,\mu}\phi_{,\nu}\Big)\,,
\bigg(\langle\langleF_{\mu\nu}(z_{j})F_{\rho\sigma}(z_{j}^{\prime})\rangle\rangle-\langle\langleF_{\mu\nu}(z_{j})\rangle\rangle\,\langle\langleF_{\rho\sigma}(z_{j}^{\prime})\rangle\rangle\bigg)^{\mathrm{SR}}=0.
(H_{\alpha})_{op}=-\frac{\hbar^{2}}{2m}\rho^{-1/2}\alpha(q)^{-1/2}\left(\frac{\partial}{\partialq^{i}}(g^{ij}\alpha)\frac{\partial}{\partialq^{j}}\right)\rho^{+1/2}\alpha(q)^{-1/2}+V(q)
S_{\mathrm{grav}}^{(0)}=\intd^{4}\rho(\frac{1}{2}\partial_{\sigma}\gamma_{\lambda\rho}\diamondsuit\partial^{\sigma}\gamma^{\lambda\rho}-\partial_{\lambda}\gamma^{\rho\kappa}\diamondsuit\partial_{\kappa}\gamma_{\rho}^{\lambda}-\frac{1}{4}\partial_{\rho}\gamma\diamondsuit\partial^{\rho}\gamma-\frac{1}{\alpha}\partial_{\rho}\gamma_{\lambda}^{\rho}\diamondsuit\partial_{\kappa}\gamma^{\kappa\lambda}),
f_{[cd][ef]}^{[ab]}={\frac{1}{2}}\bigg(\eta_{ce}\delta_{d}^{a}\delta_{f}^{b}-\eta_{cf}\delta_{d}^{a}\delta_{e}^{b}+\eta_{df}\delta_{c}^{a}\delta_{e}^{b}-\eta_{de}\delta_{c}^{a}\delta_{f}^{b}-(a\leftrightarrowb)\bigg).
\left(\begin{matrix}{q_{(m)}^{\Lambda}}\\{q_{\Sigma}^{(e)}}\\\end{matrix}\right)=\left(\begin{matrix}{2P^{\Lambda}}\\{2(\mathrm{Re}\,{\calN}_{\Lambda\Sigma})P_{\Lambda}-2(\mathrm{Im}\,{\calN}_{\Lambda\Sigma})Q^{\Lambda}}\\\end{matrix}\right).
\Pi_{i}={\frac{{\sqrt{4m^{2}+\theta^{2}}}}{4m}}\epsilon_{ij}{\frac{\partial_{j}(P_{1}+P_{2})}{{\sqrt{-\nabla^{2}}}}}-m{\frac{\partial_{i}(Q_{1}-Q_{2})}{{\sqrt{-\nabla^{2}}}}},
\stackrel{1}{T}{^a}=\stackrel{1}{D}e^{a}\equivde^{a}+\stackrel{1}{\omega}{_{\,\cdotb}^{\,a}}\wedgee^{b},\;\;\;\;\;\stackrel{1}{R}{^{ab}}=d\stackrel{1}{\omega}{^{ab}}+\stackrel{1}{\omega}{^a_{\cdotc}}\wedge\stackrel{1}{\omega}{^{cb}}-\betae^{a}\wedgee^{b}.
\Gamma_{2}[h]=g^{2}\int\frac{q^{2}dq}{2\pi^{2}}\left[\left(\Pi(-q^{2})-\Pi(m^{2})+m^{2}\Pi^{\prime}(m^{2})\right)\tilde{\chi}(q)^{2}+4v^{2}q^{2}\Pi^{\prime}(-q^{2})\tilde{h}(q)^{2}\right]\.
\langle[{\frac{g^{2}}{32\pi^{2}f_{0}}}F_{\mu\nu}^{a}\,\tilde{F}^{a\mu\nu}](k)\;[{\frac{g^{2}}{32\pi^{2}f_{0}}}F_{\lambda\delta}^{b}\;\tilde{F}^{b\lambda\delta}](-k)\rangle\simm_{a}^{2}\.
\begin{array}{c}{M_{[A(r)}Z_{kB]}=0\,\quadZ_{k}^{A}M_{AA(r-1)}=0\,\quadk=1,...r\,}\\{p^{A}M_{AA(r-1)}=0\,\quadM_{A(r)}\sim\lambda^{2}M_{A(r)}}\\\end{array}
M_{ij}=\left(\begin{matrix}{-I_{p+1}}&{0}\\{0}&{I_{7-p}}\\\end{matrix}\right),\\\M_{\dot{a}b}=i\left(\gamma^{1}\gamma^{2}\dots\gamma^{p+1}\right)_{\dot{a}b},\\\M_{a\dot{b}}=i\left(\gamma^{1}\gamma^{2}\dots\gamma^{p+1}\right)_{a\dot{b}}
E=\left[\begin{array}{ccccc}{1}&{0}&{0}&{0}&{0}\\{-5}&{4}&{-11}&{3}&{-6}\\{10}&{-5}&{35}&{-6}&{24}\\{--10}&{0}&{-45}&{0}&{-36}\\{5}&{5}&{29}&{6}&{24}\\{-1}&{-4}&{-8}&{-3}&{-6}\\\end{array}\right]
\eta_{1}(z)=4\int_{0}^{1}\!\!{\mathrm{d}}\alpha_{1}\,\alpha_{1}(1-\alpha_{1})\!\int_{0}^{1}\!\!{\mathrm{d}}\alpha_{2}\,K_{0}(nz\sqrt{\alpha_{2}}\sqrt{\alpha_{1}(1-\alpha_{1})}),
\sum_{s=1,2,5}g(s)(\varepsilon_{s})^{\mu}(\varepsilon_{s})^{\nu}=\sum_{s=1,2}(\varepsilon_{s})^{\mu}(\varepsilon_{s})^{\nu}-(\varepsilon_{5})^{\mu}(\varepsilon_{5})^{\nu}=g^{\mu\nu}-\frac{1}{\kappa^{2}}k^{\mu}k^{\nu}=g^{\mu\nu}-\frac{1}{k^{2}}k^{\mu}k^{\nu}
\left[\left\{{\frac{Det^{\prime}\left[-(d^{2}/d\tau^{2})+V^{\prime\prime}[x_{c1}(\tau)]\right]}{Det\left[-(d^{2}/d\tau^{2})+81\omega^{2}\right]}}\right\}^{-1/2}\right]^{n_{1}}\\left[\left\{{\frac{Det^{\prime}\left[-(d^{2}/d\tau^{2})+V^{\prime\prime}[x_{c2}(\tau)]\right]}{Det\left[-(d^{2}/d\tau^{2})+36\omega^{2}\right]}}\right\}^{-1/2}\right]^{n_{2}}
I_{1}+I_{3}=\int_{M_{6}}\big({\frac{1}{6!}}E^{a_{0}}\wedgeE^{a_{1}}\wedge...\wedgeE^{a_{p}}\epsilon_{a_{0}a_{1}...a_{p}}(\sqrt{-det(\eta_{ab}+{\calF}_{ab})}-{\frac{i}{4\sqrt{-v_{a}v^{a}}}}{\calF}^{ab}H_{abc}v^{c})
T_{\quad\\nu|\mu}^{A\mu}=0\,\quadT_{\quad\\nu|\mu}^{B\mu}={\frac{\Psi_{|\mu}}{1-\Psi}}T_{\quad\\nu}^{B\mu}-{\frac{1}{2}}{\frac{\Psi_{|\nu}}{1-\Psi}}T^{B}\.
\overline{{\cdot}}:s_{2}\mapsto\overline{{s}}_{2}\,\,,\quad\quad\quad\quad\quad\overline{{s}}_{2}={\frac{1}{2}}{C_{ij}}^{k}c_{k}\pi^{i}\pi^{j}\quad\left(={\frac{1}{2}}{C_{ij}}^{k}c_{k}{\frac{\partial}{\partialc_{i}}}{\frac{\partial}{\partialc_{j}}}\right)\,,
\Phi({\bfr})=U_{1}^{-1}({\bfr})\left(\begin{matrix}{t_{4}-{\frac{1}{2y_{R}}}}&{0}&{0}&{0}\\{0}&{t_{2}+{\frac{1}{2y_{R}}}}&{0}&{0}\\{0}&{0}&{t_{2}-{\frac{1}{2y_{L}}}}&{0}\\{0}&{0}&{0}&{t_{1}+{\frac{1}{2y_{L}}}}\\\end{matrix}\right)U_{1}({\bfr})+\cdots
\mu^{2}\equiv\frac{1}{2}(1+\sqrt{1-4H^{2}L^{2}}),\;\;\;\;\;\nu^{2}\equiv\frac{1}{2}(1-\sqrt{1-4H^{2}L^{2}}),\;\;\;\;\;k\equiv\frac{\sqrt{\mu^{2}-\nu^{2}}}{\mu}.
E(n_{b},m_{b};a^{(b)}):=\Big\langle\prod_{b=1}^{N}\prod_{i=1}^{n_{b}}\overline{{\psi}}^{(b)}({x}_{i}^{(b)})P_{+}\psi^{(b)}({x}_{i}^{(b)})\prod_{j=1}^{m_{b}}\overline{{\psi}}^{(b)}({y}_{j}^{(b)})P_{-}\psi^{(b)}({y}_{j}^{(b)})e^{ie\sum_{c=1}^{N}(a_{\mu}^{(c)},j_{\mu}^{(c)})}\Big\rangle_{0}^{\theta}\;.
\mathrm{d}s^{2}=\left(\begin{matrix}{\mathrm{d}x_{+},}&{\mathrm{d}x_{-}}\\\end{matrix}\right)\left(\begin{matrix}{0}&{1}\\{1}&{0}\\\end{matrix}\right)\left(\begin{matrix}{\mathrm{d}x_{+}}\\{\mathrm{d}x_{-}}\\\end{matrix}\right)
\dot{\overline{{\rho_{\chi0}}}}+3H_{\mathrm{h}}(t)(\overline{{\rho_{\chi0}}}+\overline{{p_{\chi0}}})-\frac{\partial\overline{{V_{\chi}(\varphi_{\mathrm{h}},\tilde{\chi})}}}{\partial\varphi_{\mathrm{h}}}\dot{\varphi}_{\mathrm{h}}=0\,.
\hat{h}=\left(\begin{array}{ccccc}{1}&{0}&{0}&{\cdots}&{0}\\{0}&{\omega}&{0}&{\cdots}&{0}\\{0}&{0}&{\omega^{2}}&{\ddots}&{\vdots}\\{\vdots}&{\vdots}&{\ddots}&{\ddots}&{0}\\{0}&{0}&{\cdots}&{0}&{\omega^{n-1}}\\\end{array}\right).
G_{00}=\frac{1}{2}\nu(\nu-1)n^{2}\!\left(\frac{\dot{a}^{2}}{n^{2}a^{2}}-\frac{{a^{\prime}}^{2}}{a^{2}}+\frac{k}{a^{2}}\right)-\nun^{2}\frac{a^{\prime\prime}}{a},
W=w+{\frac{2i}{3}}\eta^{+}e^{-3w}\partial_{++}(e^{3w}\psi_{L})+{\frac{2i}{3}}\eta^{-}e^{-3w}\partial_{--}(e^{3w}\psi_{R})+4\eta^{+}\eta^{-}\psi_{L}\psi_{R}\partial_{++}\partial{--}w,
\frac{\partial^{2}\varphi_{L}(t)}{\partialt^{2}}+3\Bigl(\frac{\partiala(t)/\partialt}{a(t)}\Bigr)\frac{\partial\varphi_{L}(t)}{\partialt}+\frac{\delta^{2}V(\hat{\phi})}{\delta\hat{\phi}^{2}}\varphi_{L}(t)=0.
\Pi^{\mu\nu}=A\,\left(\hat{\eta}^{\mu\nu}-\frac{\hat{p}^{\mu}\hat{p}^{\nu}}{\hat{p}^{2}}\right)+B\,\frac{p^{2}}{\hat{p}^{2}}\,\bar{u}^{\mu}\bar{u}^{\nu}+C\,\frac{\tilde{p}^{\mu}\tilde{p}^{\nu}}{\tilde{p}^{2}}+D\,(\bar{u}^{\mu}\tilde{p}^{\nu}+\bar{u}^{\nu}\tilde{p}^{\mu})
\begin{array}{c}{S=(-1)^{s}\int\phi^{2}\left(-\nabla^{\mu}\tilde{h}^{\mu_{1}\ldots\mu_{s}}\nabla_{\mu}\tilde{h}_{\mu_{1}\ldots\mu_{s}}+\right.}\\{\left.+\left[{\frac{(d-1)^{2}}{4}}+s^{2}+(d-6)s-2(d-3)\right]\tilde{h}^{\mu_{1}\ldots\mu_{s}}\tilde{h}_{\mu_{1}\ldots\mu_{s}}\right)}\\\end{array}
{\calI}_{\mathrm{1-loop}}^{I}=-\frac{V^{(8)}}{2}\sum_{\sigma,ij}\rho_{\sigma}\int\frac{dt}{t}\Big[\epsilon_{\sigma}^{4}\sum_{a_{ij}^{\sigma}+\Gamma_{2}}e^{-\pitp_{I}G^{IJ}p_{J}/2}\Big]
G_{n}=\mathrm{Tr}\;\langle{\mathcalO}_{1}\ldots{\mathcalO}_{n}\rangle={\frac{1}{Z}}\int{\mathcalD}q{\mathcalD}V\;e^{iS_{N=4\;SYM}}\;{\mathcalO}_{1}\ldots{\mathcalO}_{n}\;.
\left[-\frac{d^{2}}{d\rho^{2}}+\frac{4s^{2}-1}{4\rho^{2}}+m^{2}+\frac{\lambda}{2}\phi_{\mathrm{wall}}^{2}(\rho-\bar{\rho})+\frac{\sigma}{4!}\phi_{\mathrm{wall}}^{4}(\rho-\bar{\rho})\right]\chi_{n,s}(\rho)=E_{\mathrm{wall}}^{2}(n,s)\chi_{n,s}(\rho)\;.
V(y)=\textstyle{\frac{D-2p-1}{4}}{\calW}^{-1}{\calW}^{\prime\prime}+\textstyle{\frac{(D-2p-1)(D-2p-5)}{16}}{\calW}^{-2}({\calW}^{\prime})^{2}+m^{2}{\calW}^{-1}.
{\Bigl(\partial^{2}-2ieF\partial-{\frac{B}{\bar{T}}}\Bigl)}^{-1}={\Bigl(\partial^{2}-2ieF\partial\Bigr)}^{-1}+{\Bigl(\partial^{2}-2ieF\partial\Bigr)}^{-1}{\frac{B_{ab}}{\bar{T}}}{\Bigl(\partial^{2}-2ieF\partial\Bigr)}^{-1}+\cdots,
A_{1}\sim\frac{1}{\alpha^{\prime}}\left(1+(1-\alpha^{\prime}\lambda/2)\right)=\frac{1}{\alpha^{\prime}}\left(2-\alpha^{\prime}\lambda/2\right)\.A_{2}\sim\frac{1}{\alpha^{\prime}}(1-(1-\alpha^{\prime}\lambda/2))=\frac{\lambda}{2}\.
{\calS}_{MTM}=\intd^{2}x\left\{i\overline{{\psi}}\gamma^{\nu}\partial_{\nu}\psi-M\\overline{{\psi}}\psi-\frac{g}{2}\left(\overline{{\psi}}\gamma^{\nu}\psi\right)\left(\overline{{\psi}}\gamma_{\nu}\psi\right)\right\},
S=\intd^{4}x\;\mathrm{Tr}\;\left\{-{\frac{1}{8}}\nabla^{\alpha\alpha^{\prime}}\phi^{ab}\nabla_{\alpha\alpha^{\prime}}\phi_{ab}+{\frac{1}{2}}\chi_{\alpha^{\prime}}^{a}\nabla^{\alpha\alpha^{\prime}}\chi_{\alphaa}+{\frac{1}{6}}G^{\alpha^{\prime}\beta^{\prime}}F_{\alpha^{\prime}\beta^{\prime}}-\phi^{ab}\chi_{a}^{\alpha}\chi_{\alphab}\right\}\.
a_{1}(\lambda)\equiva_{12}\lambda^{2}=\frac{9}{4\pi^{2}}(V_{c}+1)\sum_{N=1}^{\infty}\frac{[C_{N}^{\frac{3}{2}}(cos{\theta_{4}^{0}})]^{2}}{N^{2}(N+3)^{2}}\frac{K^{2}}{R}\lambda^{2}
P(x_{i}^{\mu}\rightarrowx_{f}^{\mu})=\int_{(\infty)}\frac{d^{D}P_{2}d^{D}x_{2}}{(2\pi)^{D}}\frac{d^{D}P_{1}d^{D}x_{1}}{(2\pi)^{D}}\,<x_{f}^{\mu}|P_{2},x_{2}><P_{2},x_{2}|{\calE}|P_{1},x_{1}><P_{1},x_{1}|x_{i}^{\mu}>\\\,
Z_{1}(u,a)=e^{-2a}\langleu|_{\tau=-t}u\rangle_{\tau=0}=(N_{0})^{2}\cdot(e^{-a}Z_{0}(u))^{2}\cdot\frac{\sqrt{u/\pi}}{\sqrt{2+ut}}\prod_{m=1}^{\infty}\frac{1}{1-e^{-2mt}\left(\frac{m-u}{m+u}\right)^{2}}\,.
\Big\langle\frac{1}{4}\intd^{4}x\,d^{4}\theta\,\Big(\phi^{*}e^{2V}\phi+\tilde{\phi}^{*}e^{-2V}\tilde{\phi}\Big)\Big\rangle=-\frac{1}{16\pi^{2}}\mathrm{Re}\intd^{4}x\,d^{2}\theta\,W_{a}C^{ab}W_{b}.
{\tilde{G}}_{3\xi}({\hat{\xi}}_{3}{\eta}_{3};{{\hat{\xi}}_{3}}^{\prime}{{\eta}_{3}}^{\prime})={\hat{G}}({{\hat{\xi}}_{3}}{\hat{\eta}}_{3};{{\hat{\xi}}_{3}}^{\prime}{{\hat{\eta}}_{3}}^{\prime})\timesF(p_{3},{p_{3}}^{\prime})
\tilde{\nabla}_{i}=i\sqrt{V}P_{i}+\frac{i}{2}V\sqrt{V}\varepsilon_{ijk}\Sigma_{j}^{*}B_{k}\,,\quad\tilde{\nabla}_{5}=\frac{i}{\sqrt{V}}P_{5}-\frac{i}{2}V\sqrt{V}\vec{\Sigma}^{*}\cdot\vec{B}\,.
\tilde{L}^{(1)}=\pi_{i}\dot{a}_{i}+(a_{i}a_{i}-1)\dot{\rho}-M-\frac{1}{8\lambda}\pi_{i}\pi_{i}+\frac{1}{4\lambda}(a_{i}\pi_{i})\theta-\frac{1}{8\lambda}a_{i}^{2}\theta.
\bar{X}={\bfu}^{*}\cdot\bar{{\bfX}}=\left(\begin{array}{c}{f^{1}}\\{C^{1}}\\\end{array}\right)\left(\begin{array}{cc}{f^{2T}}&{C^{2}}\\\end{array}\right)-\left(\begin{array}{c}{f^{2}}\\{C^{2}}\\\end{array}\right)\left(\begin{array}{cc}{f^{1T}}&{C^{1}}\\\end{array}\right).
O_{0}^{(1)}(z_{1})=\zeta_{\mu}^{(1)}[\frac{1}{2}c(\partialX_{\mu}+ik^{(1)}\cdot\psi\psi_{\mu})-\frac{1}{4}\etae^{\phi}\psi_{\mu}]e^{ik^{(1)}X}(z_{1}),
\Omega_{3}={\frac{1}{3R^{2}}}\varepsilon_{\hat{m}\hat{n}\hat{p}\hat{q}}X^{\hat{m}}dX^{\hat{n}}dX^{\hat{p}}dX^{\hat{q}}\,,\qquad\tilde{\Omega}_{3}={\frac{1}{3R^{2}}}\varepsilon_{\hat{m}^{\prime}\hat{n}^{\prime}\hat{p}^{\prime}\hat{q}^{\prime}}X^{\hat{m}^{\prime}}dX^{\hat{n}^{\prime}}dX^{\hat{p}^{\prime}}dX^{\hat{q}^{\prime}}\,,
H_{3}=\hat{H}_{3}-\left(d\hat{z}+{\frac{\hat{E}^{{\#}(-)}}{\hat{E}_{\hat{z}}^{{\#}}}}\right)\wedgei_{\hat{z}}\hat{H}_{3}\;+\left(dy+{\frac{{E}^{{*}(-)}}{{E}_{y}^{{*}}}}\,\right)\wedgei_{y}H_{3}\;,
h_{pq}=\frac1{\tau_{2}}\left(\begin{array}{cc}{|\tau|^{2}}&{\tau_{1}}\\{\tau_{1}}&{1}\\\end{array}\right)\quad,\quad\tilde{h}_{pq}=\frac1{\tilde{\tau}_{2}}\left(\begin{array}{cc}{|\tilde{\tau}|^{2}}&{\tilde{\tau}_{1}}\\{\tilde{\tau}_{1}}&{1}\\\end{array}\right).
\delta_{\eta}q_{i}^{a}=\{q_{i}^{a},\eta\phi\}=\eta\epsilon^{ab}q_{i}^{b}\\,\\\delta_{\eta}p_{i}^{a}=\{p_{i}^{a},\eta\phi\}=\eta\epsilon^{ab}p_{i}^{b}\\,\\\delta_{\eta}\lambda=\dot{\eta}\\,
e^{{\frac{i}{\hbar}}S_{\mathrm{IF}}[x_{c}]}=\intdqdq_{i}dq_{i}^{\prime}\int_{q_{i}}^{q}{\calD}q\int_{q_{i}^{\prime}}^{q}{\calD}q^{\prime}\;e^{{\frac{i}{\hbar}}\int^{\calC}ds_{c}\left\{{\calL}_{Q}(q_{c}(s))+{\calL}_{\mathrm{int}}(x_{c}(s),q_{c}(s))\right\}}\rho_{Q}(q_{i},q_{i}^{\prime};t_{i}),
\mathrm{case~1:}\qquadS=\frac{1}{2\kappa^{2}}\int{d^{4}x\sqrt{-g}\left[\muR-\sum_{i}\left\{\left(\nabla\chi_{i}\right)^{2}-\left(\nabla\psi_{i}\right)^{2}\right\}-V(\chi_{1},\ldots,\chi_{k};\psi_{1},\ldots,\psi_{k})\right]}.
\delta_{\lambda^{\dagger}}\widehat\Lambda^{\dagger}(\lambda^{\dagger},A^{\dagger})-\delta_{\eta^{\dagger}}\widehat\Lambda^{\dagger}(\lambda^{\dagger},A^{\dagger})-i[\widehat\Lambda^{\dagger}(\lambda^{\dagger},A^{\dagger})\stackrel{*}{,}\widehat\Lambda^{\dagger}(\eta^{\dagger},A^{\dagger})]=\widehat\Lambda^{\dagger}(-i[\lambda^{\dagger},\eta^{\dagger}],A^{\dagger}).
\bigg[Me_{\nu}(0,h)\bigg]_{h^{2}\rightarrow0}=c_{0}^{\nu}(0)\bigg[1-\frac{3}{2+i\sqrt{5}}\bigg]=c_{0}^{\nu}(0)\bigg[\frac{i\sqrt{5}-1}{2+i\sqrt{5}}\bigg]=\bigg[Me_{-\nu}(0,h)\bigg]_{h^{2}\rightarrow0}
q(r)=\frac{\pi^{2}}{4(2\pi)^{N+1}a^{3}}\sum_{{\bf\alpha}}\frac{\sigma_{n,l}^{3}T_{n}^{ab}(\sigma_{n,l})}{\sqrt{\sigma_{n,l}^{2}+k^{2}a^{2}+m^{2}a^{2}}}f_{n}^{(q)}[g_{n}(\sigma_{n,l},\sigma_{n,l}r/a)],\quadq=\varepsilon,p_{i},
{V}_{\mathrm{I}}^{\mathrm{gauge}}(r)=-\,\frac{g_{\mathrm{YM}}^{2}}{2}~\left(\delta^{AE}\,\delta^{FD}-\delta^{AF}\,\delta^{DE}\right)~\frac{1}{7\,\Omega_{8}\,r^{7}}~~,
\partial_{\theta}\mathrm{Tr}\left[e^{-\xi^{2}P_{\theta}}\right]=\mathrm{Tr}\left[2\xi^{2}f_{\theta}(x)\Omega_{\theta}^{-2}Pe^{-\xi^{2}P_{\theta}}\right]=-\xi\partial_{\xi}\mathrm{Tr}\left[f_{\theta}(x)e^{-\xi^{2}P_{\theta}}\right],
\ddot{f}+\frac{1}{\tau}\dot{f}+\left[\frac{\alpha_{\tau}P^{2}}{\alpha_{r}}\frac{1}{\tau^{2}}+\alpha_{\tau}\tau^{n-3}\left(M^{2}+\frac{K^{2}}{\alpha_{\omega}^{2}}\right)\right]f=0\,,
z=C\Bigg(\frac{1}{e^{2}}\Bigg)^{N_{c}}e^{-8\pi^{2}/e^{2}}M^{3N_{c}-N_{f}}\Bigg(\frac{m_{\lambda}^{2/3}}{B^{11/12}}\Bigg)^{N_{c}}\Big(\mathrm{det}(m_{q})_{i}^{j}\Big)^{2/3}\Big(\mathrm{det}(m_{sq})_{i}^{j}\Big)^{1/3},
{\frac{\partial}{\partialr}}\left[(r^{2}+A^{2}-2Mr){\frac{\partialG}{\partialr}}\right]+{\frac{1}{sin\theta}}{\frac{\partial}{\partial\theta}}\left[sin\theta{\frac{\partialG}{\partial\theta}}\right]=-\delta(r-r_{o})\delta(cos\theta-cos{\theta}_{o})\delta(\varphi-{\varphi}_{o})
Z_{W}^{\gamma=0}={\frac{\mathrm{Im}~\tau^{-5/2}}{\eta^{5}{\bar{\eta}}^{5}}}{\frac{1}{8}}|\theta_{3}^{4}-\theta_{4}^{4}-\theta_{2}^{4}|^{2}\sum_{L}|\chi_{k}^{L}|^{2}
c^{(b)}\;=\;\Big(\frac{M^{(1)}e^{\gamma}}{2}\Big)^{\frac{\pi}{\pi+gN}\frac{1}{N}}\;\Big(\frac{Me^{\gamma}}{2}\Big)^{1-\frac{1}{N}}\;=:\;c\;\;\;\;\;\;\forall\;b=1,2\;...\;N\;\;,
\Xi_{g=1}(\kappa)={\frac{1}{4\sqrt{\pi}}}\int_{0}^{\infty}{\frac{ds}{s^{3/2}}}\left(k_{H}(s)\Theta_{T}({\frac{s}{\kappa}})-\sqrt{\frac{\pi}{s}}(({\frac{1}{3}}-{\frac{s}{20}})\kappa)\right)e^{-{s/4}}~~.
S_{D}=-T_{p}\int_{W}d^{p+1}\xi\bigg\{e^{-\Phi}\sqrt{det\big(G_{ab}+B_{ab}+2\pi\alpha^{\prime}F_{ab}\big)}+i\mu_{p}\int_{W}\sum_{p}C_{(p+1)}Tr\bigg(e^{2\pi\alpha^{\prime}(F+B)}\bigg)\bigg\}
E=\left[\begin{array}{cccccc}{1}&{0}&{0}&{0}&{0}&{0}\\{1}&{7}&{0}&{14}&{0}&{7}\\{-6}&{-21}&{-14}&{-63}&{-7}&{-35}\\{15}&{35}&{49}&{119}&{28}&{70}\\{-20}&{-35}&{-70}&{-119}&{-42}&{-70}\\{15}&{21}&{49}&{63}&{28}&{35}\\{-6}&{-7}&{-14}&{-14}&{-7}&{-7}\\\end{array}\right]
T_{a}=\left(\begin{array}{c}{\Omega_{1}^{+}-\Omega_{6}^{+}}\\{g_{1}+g_{2}}\\{\Omega_{2}^{+}-\Omega_{5}^{+}}\\\end{array}\right),\qquadT_{\dot{a}}=\left(\begin{array}{c}{\Omega_{3}^{+}-\Omega_{8}^{+}}\\{g_{1}-g_{2}}\\{\Omega_{4}^{+}-\Omega_{7}^{+}}\\\end{array}\right).
A_{t}=\frac1{e_{-}^{t}}(-e_{-}^{\phi}A_{\phi}+K_{-})\qquad,\qquad\tilde{A}_{t}=\frac1{e_{+}^{t}}(-{e_{+}^{\phi}}\tilde{A}_{\phi}+\tilde{K}_{+})\qquad.
{\int\mathrm{str}\left\{\frac{\kappa_{M}}{2}\mu\left(\frac{1}{I_{M}^{+}}D_{M}-D_{G}\frac{1}{I_{G}}\right)\eta\right\}}=-\int\mathrm{str}\left\{\mu\frac{1}{I_{M}^{+}}\mathrm{ad}(W_{M}+W_{G})\frac{1}{I_{G}}\eta\right\}\,,
iD_{\perp\mu\nu}^{\underline{{a}}\underline{{b}}}(k)=\frac{i\delta_{\underline{{a}}\underline{{b}}}}{k^{2}[1+\pi_{R}(k^{2})]-m_{G}^{2}}\left[-g_{\mu\nu}+\frac{k_{\mu}k_{\nu}}{k^{2}}\right]\;;
\eta(q(\tau_{i}))=\frac{u_{i}}{(2\pi)^{3/2}\sqrt{\beta}}e^{ik_{i}\cdotq(\tau_{i})}\equiv\frac{u_{i}}{(2\pi)^{3/2}\sqrt{\beta}}e^{i\left[{\bfk}_{i}{\bfq}(\tau_{i})+(\frac{2\pi}{\beta})(n_{i}+\frac12)q_{4}(\tau_{i})\right]}.
D_{\mu\nu}=\left[\begin{matrix}{p^{2}-p_{0}^{2}+\frac{1}{\alpha\beta^{2}}}&{-p_{0}p_{1}}&{-p_{0}p_{2}}&{-p_{0}p_{3}}\\{-p_{0}p_{1}}&{p^{2}-p_{1}^{2}}&{-p_{1}p_{2}}&{-p_{1}p_{3}}\\{-p_{0}p_{2}}&{-p_{1}p_{2}}&{p^{2}-p_{2}^{2}}&{-p_{2}p_{3}}\\{-p_{0}p_{3}}&{-p_{1}p_{3}}&{-p_{2}p_{3}}&{p^{2}-p_{3}^{2}}\\\end{matrix}\right]\beta^{2}.
\{z^{\delta},z^{\sigma}\}=i{\chi}(x){\theta}^{{\mu}{\nu}}\frac{\partial{z^{\delta}}}{\partial{x^{\mu}}}\frac{\partial{z^{\sigma}}}{\partial{x^{\nu}}}-\frac{ia}{2}{\chi}^{2}(x){\theta}^{{\alpha}{\beta}}{\eta}^{{\mu}{\nu}}\frac{{\partial}^{2}{z^{\delta}}}{\partial{x^{\mu}}\partial{x^{\alpha}}}\frac{{\partial}^{2}{z^{\sigma}}}{\partial{x^{\nu}}\partial{x^{\beta}}}.
R_{F}(t,\star;\rho,w,\rho^{\prime},w^{\prime},\bullet,\bullet^{\prime};\alpha)=F(t,\star;\rho,w,\rho^{\prime},w^{\prime},\bullet,\bullet^{\prime})+\i\alpha|t|^{\frac{1}{2}}\times\cdots\\\(\mathrm{A}.2)\notag
\frac{W_{+}(z)W_{+}(z^{\prime})}{e^{i\hbar\gamma^{2}h^{+}(z-z^{\prime})}}-\frac{W_{+}(z^{\prime})W_{+}(z)}{e^{-i\hbar\gamma^{2}h^{-}(z-z^{\prime})}}=\frac{i\hbar}{2\gamma^{2}}\left(\eta^{2}-1+\frac{\gamma^{2}\hbar}{\pi}\right):\!e^{2i\gamma\varphi(z)}e^{2i\gamma\varphi(z^{\prime})}\!:\!\partial_{z}\delta_{2\pi}(z-z^{\prime}),
\beta(\tau)={\frac{2\pii\left(\Theta_{4}^{4}-\Theta_{2}^{4}\right)}{(\Theta_{2}^{8}-\Theta_{2}^{4}\Theta_{3}^{4})\left[\large{\int_{-1}^{1}}dx\sqrt{{\frac{1}{(x^{2}-1)(x\Theta_{3}^{4}+\Theta_{2}^{4}-\Theta_{4}^{4})}}}\right]^{2}}},
A=\left(\frac{1}{2}a^{2}\delta^{\prime}\right);\B=\left(3a\delta^{\prime}+a^{2}\delta^{\prime\prime}-\frac{4}{3}\frac{\delta^{\prime}a^{2}}{1+\delta}\right);\C=\frac{3}{2}\frac{H_{0}^{2}\Omega_{\mathrm{NR}}}{a^{3}}\delta(1+\delta)
\zeta_{R}(s\midq)=\sum_{n=0}^{\infty}\frac{1}{(q+n)^{s}}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{t^{s-1}e^{-qt}}{1-e^{-t}}dt,\;\;\;\;\;\;\;\mathrm{Re}s>0.
M=\left(\begin{array}{cc}{e^{i\phi}r}&{ie^{t}\sqrt{1-r^{2}}}\\{ie^{-t}\sqrt{1-r^{2}}}&{e^{-i\phi}r}\\\end{array}\right)~~~\Longrightarrow~~~ds^{2}=-(1-r^{2})dt^{2}+\frac{1}{1-r^{2}}dr^{2}+r^{2}d\phi^{2}\,.
R=\left(\begin{array}{ccccc}{\phantom{-}A(\theta)}&{B(\theta)}&{0}&{0}&{\cdots}\\{-B(\theta)}&{A(\theta)}&{0}&{0}&{\cdots}\\{\phantom{-}0}&{0}&{R_{0}(\theta)}&{0}&{\cdots}\\{\phantom{-}0}&{0}&{0}&{R_{0}(\theta)}&{\cdots}\\{\phantom{-}\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}\\\end{array}\right)\.\
\psi(\vec{x},0)=\frac{1}{\sqrt{V}}\sum_{p}\;\mathrm{e}^{i\vec{p}\cdot\vec{x}}\left\{\left(\begin{array}{r}{\chi_{_{p,L}}a_{p,L}^{}}\\{\chi_{_{p,R}}a_{p,R}^{}}\\\end{array}\right)\;+\;\left(\begin{array}{r}{\chi_{_{p,R}}b_{-p,R}^{\dagger}}\\{\;-\;\chi_{_{p,L}}b_{-p,L}^{\dagger}}\\\end{array}\right)\right\}
\epsilon_{K}(t,x)=\sum_{n}\left[\frac{e^{-i\omega_{n}t}}{\sqrt{2\omega_{n}}}a_{n}\epsilon_{n}(x)~+~\frac{e^{i\omega_{n}t}}{\sqrt{2\omega_{n}}}a_{n}^{\dagger}\epsilon_{n}^{\dagger}(x)\right]
W_{\mathrm{eff}}^{\prime}=\left(N_{f}-N_{c}+2\right)\left[\frac{\mathrm{det}\left(T\right)-BT^{N_{f}-N_{c}}B}{\Lambda^{3N_{c}-N_{f}-6}}\right]^{1/(N_{f}-N_{c}+2)}+\sum_{i=1}^{N_{f}}m_{i}T^{ii}.
S_{\mathrm{\Lambda}}=(n-1)\frac{V}{4G_{n+1}l_{n+1}},~~S_{\mathrm{D}}=\sqrt{|S_{\mathrm{H}}^{2}-S_{\mathrm{\Lambda}}^{2}|},~~T_{\mathrm{D}}=-\frac{\dot{H}}{2\pi\sqrt{|1/l^{2}-H^{2}|}}.
({\frac{{A_{1}^{0}}^{2}-{B_{1}^{0}}^{2}}{A_{1}^{0}A_{2}^{0}-B_{1}^{0}B_{2}^{0}}}dt-d\theta)^{2}+{\frac{(A_{1}^{0}A_{2}^{0}-B_{1}^{0}B_{2}^{0})^{2}}{{A_{1}^{0}}^{2}-{B_{1}^{0}}^{2}}}d\theta^{2}-d\rho^{2}
{\calL}_{R}^{(2)}=\frac{\alpha^{3}}{\pim^{4}}\left[\frac{16}{81}(\eta^{2}-\epsilon^{2})^{2}+\frac{263}{162}(\eta\epsilon)^{2}\right]+\ldots;\quad\frac{e\eta}{m^{2}},\:\frac{e\epsilon}{m^{2}}\ll1.
\begin{array}{rl}{\tau}&{\frac{3}{2}\mathrm{Tr}\,\theta=\frac{3}{2}(\theta^{\,1}{}_{1}+\theta^{\,2}{}_{2}+\theta^{\,3}{}_{3})\,}\\{\gamma^{\,i}{}_{j}{}^{\,k}{}_{l}}&{\frac{1}{6}(z^{\,i}{}_{l}z^{\,k}{}_{j}-\frac{1}{3}z^{\,i}{}_{j}z^{\,k}{}_{l})\.}\\\end{array}
[p_{i},x_{j}]=\delta_{ij}\,e^{-2p_{0}/\kappa}-\frac{1}{\kappa^{2}}\left(\vec{p}\,{}^{2}\,\delta_{ij}-2p_{i}p_{j}\right),\quad[p_{0},x_{i}]=-\frac{2}\kappa\,p_{i}
\frac{d}{dt}\phi_{t}^{*}(w)|_{t=t_{0}}=lim_{s\rightarrow0}\frac{\phi_{t_{0}+s}^{*}w-\phi_{t_{0}}^{*}w}{s}=\phi_{t_{0}}^{*}lim_{s\rightarrow0}\frac{\phi_{s}^{*}w-w}{s}=\phi_{t_{0}}^{*}({\calL}_{X}w)=0.
R=6C^{2}e^{2\phi_{0}}e^{6\tau-\chi-\psi}\Bigl(\frac{\ddot{\chi}+\ddot{\psi}}{2}+\frac{\dot{\chi}^{2}+\dot{\psi}^{2}-\dot{\chi}\dot{\psi}}{2}-\ddot{\chi}-\ddot{\psi}\Bigr)
\sum_{\ell=0}^{\infty}C_{\ell}^{(-2/3)}\left\{\epsilon_{m-\ell\pm{\frac{q}{3}}}^{\mu\pm}\epsilon_{n+\ell+{\frac{2\pmq}{3}}}^{\nu\pm}-\epsilon_{n-\ell\pm{\frac{q}{3}}}^{\nu\pm}\epsilon_{m+\ell+{\frac{2\pmq}{3}}}^{\mu\pm}\right\}={\varepsilon^{\mu\nu}}_{\rho}\epsilon_{m+n+{\frac{2\pm2q}{3}}}^{\rho\mp}
T_{B}^{(i)}(z)=\displaystyle\sum_{\mu=0}^{n-1}G_{a+\bar{\varepsilon}_{\mu}}\phi^{*(a,a+\bar{\varepsilon}_{\mu})}(z)K^{(i)}\left(\left.a+\bar{\varepsilon}_{\mu}\begin{array}{c}{a}\\{a}\\\end{array}\right|z\right)\phi^{(a+\bar{\varepsilon}_{\mu},a)}(z).
j_{+}{}^{n}|_{\mathrm{reg}}=\mathrm{lim}_{s\rightarrow0}\sum_{m=-\infty}^{\infty}\frac{1}{\lambda_{m,s}}a_{m}^{\dagger}a_{m+n},\quadj_{-}{}^{n}|_{\mathrm{reg}}=\mathrm{lim}_{s\rightarrow0}\sum_{m=-\infty}^{\infty}\frac{1}{\lambda_{m,s}}b_{m+n}^{\dagger}b_{m}.
\check{R}_{bb}^{bb}\check{R}_{bb}^{*bb}-\check{R}_{bb}^{f_{1}f_{2}}\check{R}_{bb}^{*f_{1}f_{2}}-\check{R}_{bb}^{f_{2}f_{1}}\check{R}_{bb}^{*f_{2}f_{1}}=1
ds_{\mathrm{BTZ}}^{2}=-{\frac{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}{\rho^{2}R^{2}}}dt^{2}+\rho^{2}(d\varphi-{\frac{J}{2\rho^{2}}}dt)^{2}+{\frac{\rho^{2}R^{2}}{(\rho^{2}-\rho_{+}^{2})(\rho^{2}-\rho_{-}^{2})}}d\rho^{2}.
\begin{array}{rl}{\partial_{\tau}\Theta}&{=-iD\left(D\Theta\right),}\\{{\calE}_{\tau}^{\underline{a}}}&{=\partial_{\tau}X^{\underline{a}}-i\partial_{\tau}\bar{\Theta}\Gamma^{\underline{a}}\Theta}\\{}&{=-iD\left(DX^{\underline{a}}-iD\bar{\Theta}\Gamma^{\underline{a}}\Theta\right)-D\bar{\Theta}\Gamma^{\underline{a}}D\Theta.}\\\end{array}
\begin{array}{l}{L={\frac{1}{g_{YM}^{2}}}\int\mathrm{tr}\left\{-(\partial_{\mu}A_{\nu})^{2}+(\partial_{\mu}\phi^{I})^{2}+\bar{\psi}\hat{\partial}\psi\right.+}\\{+\left.2A_{\mu}(\phi^{I}\stackrel{\leftrightarrow}{\partial}\phi^{I})+\bar{\psi}[\hat{A},\psi]+\bar{\psi}\Gamma^{I}[\phi^{I},\psi]+{\frac{1}{2}}[\phi^{I},\phi^{J}]^{2}+{\frac{1}{2}}[A_{\mu},A_{\nu}]^{2}\right\}}\\\end{array}
S_{\chi}=\pi^{2}\chi_{0}^{2}\int_{0}^{\infty}dr\,r^{3}\left\{[f^{\prime}(r)]^{2}+m_{\chi}^{2}[f(r)]^{2}-\mu_{0}^{2}\theta(r-r_{0})[f(r)]^{2}\right\}
\int\int_{cavity}\left<T_{00}(\vec{x})\right>_{F}=-\frac{ab}{32\pi^{2}}Z(2|a,b)+\frac1{32\pi}\zeta(3)\left(\frac1{a^{2}}+\frac1{b^{2}}\right)+\int_{ext}\left<T_{00}(\vec{x})\right>_{B},
A^{\prime\prime}+{\frac{D_{1}}{2}}{A^{\prime}}^{2}+{\frac{D_{2}}{2}}A^{\prime}B^{\prime}={\frac{4\kappa^{2}}{D-2}}\left(-\Lambda+{\frac{F^{2}}{4g^{2}}}\mathrm{e}^{-2B}\right)~,
\tilde{H}=\frac{\lambdaL}{2}\left[\sum_{n>0}a_{n}^{\dagger}a_{n}+\frac{2}{N}\sum_{n,m>0}(a_{n+m}^{\dagger}a_{n}a_{m}+a_{n}^{\dagger}a_{m}^{\dagger}a_{n+m})-\frac{1}{N^{2}}(\sum_{n>0}a_{n}^{\dagger}a_{n})^{2}\right].
S=\intd^{4}x[-\frac{1}{4}F_{\mu\nu}^{a}F_{\mu\nu}^{a}+\frac{1}{2\alpha}(\partial_{\mu}A_{\mu})^{2}++\bar{c}^{a}\partial_{\mu}(\delta^{ab}\partial_{\mu}-gt^{abc}A_{\mu}^{c})c^{b}]