harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U[A](x_{0}+\beta,\vec{x};x_{0}^{\prime},\vec{x}^{\prime})=\underbrace{U[A](x_{0}+\beta,\vec{x};x_{0},\vec{x})}_{=L(x)}\,U[A](x_{0},\vec{x};x_{0}^{\prime},\vec{x}^{\prime})\neqU[A](x_{0},\vec{x};x_{0}^{\prime},\vec{x}^{\prime})
S_{\mathrm{int.}}(\Sigma,j_{\mu})=\frac{1}{4\pi^{2}}\varepsilon_{\mu\nu\lambda\rho}\intd^{4}xd^{4}yj_{\mu}(x)\frac{(y-x)_{\nu}}{|y-x|^{4}}\Sigma_{\lambda\rho}(y)
\begin{array}{l}{[P^{I},Q^{J}]=[P^{I},P^{J}]=[Q^{I},Q^{J}]=0,}\\{[P^{I},\overline{{P}}^{J}]=[Q^{I},\overline{{Q}}^{J}]=0,}\\{[P^{I},\overline{{Q}}^{J}]=\delta^{IJ}}\\\end{array}
\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I^{\prime}[\theta,\frac{1}{2}(C+A)\frac{1}{2}(C+A)]=\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}I^{\prime}[\theta,CC]+\pii\int_{0}^{1}dx\Theta[B]
q^{(n-m)(n-j)}\left[\begin{array}{l}{j-m}\\{n-m}\\\end{array}\right]_{q^{2}}^{1/2}\left[\begin{array}{l}{j+n}\\{n-m}\\\end{array}\right]_{q^{2}}^{1/2}P_{j-n}^{n-m,m+n}(-c^{*}c;q^{2})(-c^{*})^{n-m}(a^{*})^{m+n},
\left(\begin{matrix}{t^{{\vec{\Lambda}}_{1}}=p^{{\vec{\Lambda}}_{1}}+\mathrm{i}\,q_{{\vec{\Lambda}}_{1}}\,}\\{{\bar{t}}_{{\vec{\Lambda}}_{1}}=p^{{\vec{\Lambda}}_{1}}-\mathrm{i}\,q_{{\vec{\Lambda}}_{1}}\,}\\\end{matrix}\right)
\begin{array}{rcl}{\delta_{v}{S}([u^{\alpha}(x)];x^{\mu})}&{=}&{\int_{\Omega}d^{n}x\{[{L}_{u^{\alpha}}]\delta_{v}u^{\alpha}+\partial_{\mu}(\frac{\partial{L}}{\partial(\partial_{\mu}u^{\alpha})}\delta_{v}u^{\alpha})\},}\\\end{array}
\int_{{\calF}\timesS^{2}}\omega\(p,\zeta)\;\langlep^{\prime},\zeta^{\prime}|p,\zeta\rangle\langlep,\varsigma|p^{\prime\prime},\zeta^{\prime\prime}\rangle=c\langlep^{\prime},\zeta^{\prime}|p^{\prime\prime},\zeta^{\prime\prime}\rangle
e_{n}H_{\nu}^{(1)\prime}(kr^{\prime})+f_{n}H_{\nu}^{(2)\prime}(kr^{\prime})-d_{n}H_{\nu}^{(1)\prime}(kr^{\prime})=-{\frac{2\nu\Gamma\left({\frac{D-2}{2}}\right)}{4(\pir^{\prime})^{\frac{D}{2}}k}}.
\left\langle\intd^{2}x:\bar{\Psi}(x)T^{3}\Psi(x):\right\rangle=-\frac{1}{2}\mathrm{sgn}(\chi)\left(\sum_{\mathrm{occupied}}\intd^{2}x\psi_{0}^{\dagger}(x)T^{3}\psi_{0}(x)-\sum_{\mathrm{unoccupied}}\intd^{2}x\psi_{0}^{\dagger}(x)T^{3}\psi_{0}(x)\right).
\{I_{(+)},I_{(+)}^{\prime}\}=-\bigl(aG^{+}\frac{\partial_{++}a}{\sqrt{p_{++}}}G^{+}\sqrt{p_{++}}+aP^{+c}\partial_{c}a\sqrt{p_{++}}G^{+}\sqrt{p_{++}}\bigr)\delta(x-x^{\prime})\.
\begin{array}{c}{\psi_{a}\sim\lambda\psi_{a}\,\quad\lambda\in{\bfC}\setminus\{0\}\,}\\{p_{A}\Gamma_{a}^{Ab}\psi_{b}=m\psi_{a}\,\quadZ_{kA}\Gamma_{a}^{Ab}\psi_{b}=0\,\quadk=1,...r}\\\end{array}
\begin{array}{lcl}{e(\boldsymbol{x})}&{=}&{\sum_{a=1,2}\Big(\underbrace{\sum_{N,E(a,N)<0}E(a,N)u^{\dagger}(a,N;\boldsymbol{x})u(a,N;\boldsymbol{x})}_{=:e(a,E<0;\boldsymbol{x})}}\\{}&{}&{-\underbrace{\sum_{N,E(a,N)>0}E(a,N)u^{\dagger}(a,N;\boldsymbol{x})u(a,N;\boldsymbol{x})}_{=:e(a,E>0;\boldsymbol{x})}\Big)}\\\end{array}
\{g,B,A^{(1)},A^{(2)},k,\phi,T,C_{(0)}^{+},C_{(1)}^{+},C_{(2)}^{+},C_{(3)}^{+},C_{(4)}^{+},C_{(0)}^{-},C_{(1)}^{-},C_{(2)}^{-},\}\,,
T^{+}\!=\!a_{\mu}^{+2}a_{\mu}^{1},\;L^{11+}\!=\!\frac{1}{2}a_{\mu}^{1+}a_{\mu}^{1+},\;L^{12+}\!=\!a_{\mu}^{1+}a_{\mu}^{2+},\;L^{22+}\!=\!\frac{1}{2}a_{\mu}^{2+}a_{\mu}^{2+},\;L^{i+}\!=\!p_{\mu}a_{\mu}^{i+}
H_{CSM}=-\frac{1}{2}\sum_{i=1}^{N}\partial_{i}^{2}+\frac{1}{2}\omega^{2}\sum_{i=1}^{N}x_{i}^{2}+\frac{1}{2}\beta(\beta-1)\sum_{{i,j=1}\atop{i\nej}}^{N}\frac{1}{(x_{i}-x_{j})^{2}}\qquad,
p(z)=\sum_{i=1}^{N_{s}}m_{s,i}^{2z}-4\sum_{j=1}^{N_{d}}m_{d,j}^{2z}+3\sum_{k=1}^{N_{v}}m_{v,k}^{2z}~~~,~~~q(z)=\sum_{i=1}^{N_{s}}m_{s,i}^{2z}+2\sum_{j=1}^{N_{d}}m_{d,j}^{2z}-3\sum_{k=1}^{N_{v}}m_{v,k}^{2z}~~~.
F_{\mu}=\delta_{a}^{b^{\prime}}\delta_{a^{\prime}}^{b}\int\bigg\lbracke^{iq\cdot(z-y)}-e^{iq\cdot(z-x)}\bigg\rbrack{\frac{(x_{\mu}-y_{\mu})}{q\cdot(x-y)}}\;\;{\frac{d^{4}q}{(2\pi)^{4}}}+F_{\mu}^{T}\;,
\biggl(-\frac{1}{2}\frac{d^{2}}{dr^{2}}+\frac{(N^{2}-1)(N^{2}-3)}{8r^{2}}+\frac{r^{2}}{2}+\sum_{K=1}^{\infty}\frac{c_{K}g^{K}}{N^{4K-2}}(r^{2})^{2K}\biggr)\rho=\epsilon_{0}\rho.
{\calL}=\left\{\frac{1}{4g^{2}}\,\int\!\mathrm{d}^{2}\theta\,\mathrm{Tr}\,W^{2}+\mathrm{H.c.}\right\}+\frac{1}{4}\int\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{\Phi}\Phi\,\mathrm{Tr}\,e^{V}\,.
\phi_{n,\omega,\rho}=r^{-(n+\omega+\rho)}\sum_{q=0}^{n}C_{i_{1}....i_{n-q}}^{\omega,\rho}x^{i_{1}}...x^{i_{n-q}}(x^{+}x^{-})^{\frac12q}(x^{+})^{\omega}(x^{-})^{\rho}~~~,
\Lambda_{M}=x_{1}(\lambda)^{N}\prod_{a=1}^{M}z(\lambda_{a}-\lambda)+\varepsilon^{M+1}x_{2}(\lambda)^{N}\prod_{a=1}^{M}\frac{z(\lambda-\lambda_{a})}{\omega(\lambda-\lambda_{a})}+x_{3}(\lambda)^{N}\prod_{a=1}^{M}\frac{x_{2}(\lambda-\lambda_{a})}{x_{3}(\lambda-\lambda_{a})}
T_{\parallel,\bot}(\lambda)={\frac{4\sqrt{3}}{\pi\lambda}}\int_{0}^{1}dv(1-v^{2})^{-1}\left[(1-{\frac{1}{3}}v^{2}),({\frac{1}{2}}+{\frac{1}{6}}v^{2})\right]K_{2/3}\left({\frac{4}{\lambda}}{\frac{1}{1-v^{2}}}\right),
\begin{array}{ll}{\hat{M}_{0}(z)Z_{0}=Z_{0}H(z)\,,}&{H(z)=\left(\begin{array}{cc}{\hat{w}(z)-\textstyle{\frac{1}{2}}\hat{\lambda}(z)}&{\sqrt{2}\hat{\rho}^{b}(z)}\\{0}&{\hat{t}{}_{a}{}^{b}(z)}\\\end{array}\right)\,.}\\\end{array}
\left[\frac{2}{\alpha^{\,\prime}}\right]^{\frac12}\frac{ib^{i}\delta\chi_{i}(k_{1})}{2\pi}u_{j}^{*}(k_{2})u^{j}(k_{3})(2\pi)^{p+1}\alpha^{\,\prime\,-\frac{p+1}{2}}\delta(k_{1}+k_{2}+k_{3})\gamma^{\alpha^{\,\prime}k_{1}^{2}+\alpha^{\,\prime}k_{2}^{2}+\alpha^{\,\prime}k_{3}^{2}}.
\sum_{n=-\infty}^{\infty}e^{-{\frac{n^{2}\beta^{2}}{4s}}}=({\frac{4\pis}{\beta^{2}}})^{{\frac{1}{2}}}\sum_{n=-\infty}^{\infty}e^{-{\frac{4\pi^{2}s}{\beta^{2}}}n^{2}}
E_{\mathrm{\scriptsize~in}}((u^{\prime},v^{\prime}),(u,v))=\frac{1}{4}\mathrm{sign}(v-v^{\prime})\:\:\mathrm{whereas}\:\:E_{\mathrm{\scriptsize~out}}((u^{\prime},v^{\prime}),(u,v))=\frac{1}{4}\mathrm{sign}(u-u^{\prime})\:.
\left(\begin{array}{cc}{p}&{p^{\prime}}\\{q}&{q^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{\pm1}&{-kq^{\prime}}\\{\pmk}&{q^{\prime}}\\\end{array}\right)=\left(\begin{array}{cc}{1}&{-k}\\{k}&{1}\\\end{array}\right)\left(\begin{array}{cc}{\pm1}&{0}\\{0}&{q^{\prime}}\\\end{array}\right)
{\calL}_{\mathrm{eff}}^{\beta,\mu}(E)-{\calL}_{\mathrm{eff}}^{\beta,\mu}(0)=\frac{(eE)^{2}}{24\pi^{2}}\int_{m}^{\infty}\frac{d\omega\,\omega}{(\omega^{2}-m^{2})^{1/2}}\frac{d}{d\omega}\left(\frac{1}{e^{\beta(\omega+A_{0}-\mu)}+1}+\frac{1}{e^{\beta(\omega-A_{0}+\mu)}+1}\right)\.
\left.{\frac{\partial{W_{1}(\beta,r_{B})}}{\partial\beta}}\right|_{r_{B}}=\left.{\frac{\partial{\tilde{W}_{1}(\beta,y)}}{\partial\beta}}\right|_{y}+\left.{\frac{\partial{\tilde{W}_{1}(\beta,y)}}{\partialy}}\right|_{\beta}\left.{\frac{\partial{y}}{\partial\beta}}\right|_{r_{B}}~~~,
\Upsilon_{a\alpha}(\vec{k})=\left[\begin{array}{c}{\Upsilon_{11}}\\{\Upsilon_{21}}\\{\Upsilon_{12}}\\{\Upsilon_{22}}\\\end{array}\right]=\left[\begin{array}{c}{c_{k\uparrow}^{\vphantom\dagger}}\\{c_{-k\downarrow}^{\dagger}}\\{c_{k\downarrow}^{\vphantom\dagger}}\\{-c_{-k\uparrow}^{\dagger}}\\\end{array}\right].
r(\varphi,\varphi^{\prime})=\left(\sum_{i\neqj}e^{-\frac{i\alpha_{ij}(X)}{k}(\varphi-\varphi^{\prime})}E_{ij}\otimesE_{ji}+\sum_{i}E_{ii}\otimesE_{ii}\right)\epsilon(\varphi-\varphi^{\prime}),
N_{\ell}\;:=\intd\underline{{x}}\;\widehat{{\mathcalJ}}_{\ell}^{0}\left(t,\underline{{x}}\right)~,~~N_{r}\;:=\intd\underline{{x}}\;\widehat{{\mathcalJ}}_{r}^{0}\left(t,\underline{{x}}\right)~,
\tilde{u}(p)u(p)\deltam={\frac{-i\alpha}{4\pi^{3}}}\tilde{u}(p)\intd^{4}k{\frac{\gamma\cdot(p-k)+m}{[(p-k)^{2}-m^{2}+i\epsilon][k^{2}-\mu^{2}]}}u(p)-[\mu\rightarrow\mu_{1}]
J^{\mu\nu}=-\frac{\mathrm{i}}{2}[d_{0}^{\mu},d_{0}^{\nu}]-\frac{\mathrm{i}}{2}[{\tilde{d}}_{0}^{\mu},{\tilde{d}}_{0}^{\nu}]-\mathrm{i}\sum_{n=1}^{\infty}\left(d_{n}^{\dagger\mu}d_{n}^{\nu}-d_{n}^{\dagger\nu}d_{n}^{\mu}+{\tilde{d}}_{n}^{\dagger\mu}{\tilde{d}}_{n}^{\nu}-{\tilde{d}}_{n}^{\dagger\nu}{\tilde{d}}_{n}^{\mu}\right)
changed\intd^{25-p}q_{\bot}\,(2)^{q_{\bot}}\left(\frac{1+W_{0}^{2}}{k_{0}W_{0}^{2}}\right)^{q_{\bot}^{2}}:c\partialX(i)\cdot\tilde{c}\bar{\partial}X(i)e^{iq_{\bot}\cdotX(i)}:.
u^{\prime}(0)-v^{\prime}(0)+2eEv(0)=(2eE)^{2}\left({\frac{e^{-eEs}}{e^{eEs}-e^{-eEs}}}\right)\int_{0}^{s}ds^{\prime\prime}e^{-2eEs^{\prime\prime}}v(s^{\prime\prime})\;.
{}_{N}\!\langle\psi_{n}^{\prime},\bar{\psi}_{n}^{\prime}|\psi_{n},\bar{\psi}_{n}\rangle_{N}=\prod_{\mu=0}^{p}\delta\left(\mathrm{Re}\psi_{n\mu}^{\prime}-\mathrm{Re}\psi_{n\mu}\right)\delta\left(\mathrm{Im}\psi_{n\mu}^{\prime}-\mathrm{Im}\psi_{n\mu}\right)~.
\zeta_{\mathrm{YM}}(0,M)=\intdx(4\pi)^{-d/2}{\frac{(-1)^{d/2}}{\Gamma(d/2+1)}}\Biggl\{r(d-2)M^{d}+2\sum_{{\alpha}>0}J_{d/2,\,q}\left({\frac{M^{2}}{H}}\right)H^{d/2}\Biggr\}
\chi_{2}^{\prime}\,\chi_{1}^{\prime\prime}=Y_{12}^{\prime}\,\chi_{1}^{\prime}\,\chi_{2}^{\prime\prime}\,,\\\\\,tilde{\chi}_{2}^{\prime}\,\chi_{1}^{\prime\prime}=Z_{12}^{\prime}\,\chi_{1}^{\prime}\,\tilde{\chi}_{2}^{\prime\prime}\,,
D(p_{1},\ldots,p_{m})\equiv\left|\begin{matrix}{p_{1}^{m-1}}&{p_{1}^{m-2}}&{\cdots}&{p_{1}}&{1}\\{p_{2}^{m-1}}&{p_{2}^{m-2}}&{\cdots}&{p_{2}}&{1}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}\\{p_{m}^{m-1}}&{p_{m}^{m-2}}&{\cdots}&{p_{m}}&{1}\\\end{matrix}\right|.
{\tilde{c}}_{\alpha\beta}=-\sum_{\gamma=0}^{\alpha-1}{\tilde{c}}_{\gamma\\alpha+\beta-\gamma}{\frac{\left(\begin{matrix}{\alpha+\beta-\gamma}\\{\beta}\\\end{matrix}\right)\left(\begin{matrix}{\alpha+\beta}\\{\beta+\gamma}\\\end{matrix}\right)}{\left(\begin{matrix}{2\alpha+2\beta}\\{2\beta+2\gamma}\\\end{matrix}\right)}}\.
G_{m}(x_{1},...,x_{m})=[\prod_{i=1}^{m}\Gamma(\alpha_{i})]^{-1}(\prod_{i=1}^{m}\int_{0}^{\infty}dt_{i}\,t_{i}^{\alpha_{i}-1})\,T^{-\frac{D}{2}}e^{-\frac{1}{T}\sum_{i<j}t_{i}t_{j}x_{ij}^{2}}
\sigma^{0}\left(m_{2}^{2}\right)=\frac{12\pi\tilde{\Gamma}_{e,2}\tilde{\Gamma}_{f,2}}{m_{2}^{2}\Gamma_{2}^{2}}\left(1-\frac{\Gamma_{2}^{2}}{m_{2}^{2}}\right)+\sigma_{\mathrm{B}}^{0}\left(m_{2}^{2}\right),
2<\psi_{\alpha}(T)\psi_{\beta}(T)\psi_{\mu}(\tau_{2})\psi_{\nu}(\tau_{2})\psi_{\kappa}(\tau_{1})\psi_{\lambda}(\tau_{1})>_{\psi}={\frac{1}{4}}\delta_{\alpha\beta}[\delta_{\mu\lambda}\delta_{\nu\kappa}-\delta_{\nu\lambda}\delta_{\mu\kappa}][sign(\tau_{1}-\tau_{2})]^{2}
\psi=\left(\begin{array}{c}{\frac{i}{\sqrt{m}}\partial_{0}\varphi}\\{\frac{i}{\sqrt{m}}\partial_{1}\varphi}\\{\frac{i}{\sqrt{m}}\partial_{2}\varphi}\\{\frac{i}{\sqrt{m}}\partial_{3}\varphi}\\{\sqrt{m}\varphi}\\\end{array}\right),\quad(\square+m^{2})\varphi=0.
|R_{1}^{(-)}(\nu,k)|\leq\frac{2(1.1)}{\sqrt{\pi}}\sum_{\ell=1}^{5}(\frac{4}{3})^{\ell}|\sum_{j=0}^{5-\ell}|C^{(2)}(\ell,j)|(-1)^{j}\Gamma(j+\frac{1}{2};\pi)|\equivC_{2}^{\prime},
\langleV_{i_{1}}\dotsV_{i_{n}}\rangle_{g}=\int{\calD}\rho{\calD}Xe^{-S^{*}-g^{i}\int_{\Sigma}d^{2}\sigmaV_{i}+Q^{2}\partial\rho{\overline{\partial}}\rho-Q^{2}\rho\int_{\Sigma}d^{2}\sigma\rhoR^{(2)}}V_{i_{1}}\dotsV_{i_{n}}
\hat{\chi}_{m}^{k}(q)=\hat{\eta}_{m}^{k}\sum_{l=-\infty}^{\infty}q^{kl^{2}+ml}=\hat{\eta}_{m}^{k}\{2k\}_{2k}^{-}\left\{k-m;k+m\right\}_{2k}^{+}.
\begin{array}{l}{e^{+2}\nabla\psi_{+2\dot{q}}^{1-}+e^{-2}\nabla\psi_{-2\dot{q}}^{1-}+e_{p}^{+}\nablah_{+p\dot{q}}^{1-}+e_{p}^{-}\nabla\chi_{p\dot{q}}^{1}+T^{+2}\psi_{+2\dot{q}}^{1-}+T^{-2}\psi_{-2\dot{q}}^{1-}+T_{p}^{+}h_{+p\dot{q}}^{1-}+T_{p}^{-}\chi_{p\dot{q}}^{1}}\\{=\frac12e_{p}^{+}\gamma_{p\dot{q}}^{i}\Omega^{-2i},}\\\end{array}
\bar{Q}_{\mu}\psi_{\nu}^{*}=0,\qquad\bar{Q}_{\mu}A_{\nu}^{*}=\partial_{\mu}\psi_{\nu}^{*},\qquad\bar{Q}_{\mu}\phi^{*}=0,\qquad\bar{Q}_{\mu}C^{*}=\partial_{\mu}\phi^{*},
\Sigma_{a_{1},\ldots,a_{p};\;b_{1},\ldots,b_{q};c}^{\;i_{1},\ldots,i_{p};\;j_{1},\ldots,j_{q}}(u)\equiv\sum_{j=1}^{\infty}\frac{1}{\left(2j\atopj\right)}\frac{u^{j}}{j^{c}}[S_{a_{1}}(j\!-\!1)]^{i_{1}}\ldots[S_{a_{p}}(j\!-\!1)]^{i_{p}}\;[S_{b_{1}}(2j\!-\!1)]^{j_{1}}\ldots[S_{b_{q}}(2j\!-\!1)]^{j_{q}},
\beta(p,k)=\int_{-\infty}^{\infty}dx^{\prime}\frac{q}{2}\frac{A_{p}(0,x^{\prime})}{iq\pi\sqrt{\omega_{p}\omega_{k}}}\left\{\dot{\phi}_{1k}^{+\ast}(t,x){\frac{\partialt}{\partialt^{\prime}}}+\left(\partial_{x}\phi_{1k}^{+\ast}(t,x)\right){\frac{\partialx}{\partialt^{\prime}}}-i\omega_{p}\phi_{1k}^{+\ast}(t,x)\right\}_{t^{\prime}=0}.
\langle{\calO}[\overline{{\psi}},\psi,A,h]\rangle\;=\;\frac{1}{Z}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\langle{\calO}[\overline{{\psi}},\psi,A,h]\Big(S_{M}[\overline{{\psi}},\psi]\Big)^{n}\rangle_{0}\;,
\sum_{|\alpha|\leq\omega}\frac{D^{\alpha}\phi(0)}{\alpha!}\left\langle{^0t},\left(w-w_{\Lambda}\right)x^{\alpha}\right\rangle=-\sum_{|\alpha|\leq\omega}\left(\left(D(\Lambda)-1\right)c\right)^{\alpha}\frac{D^{\alpha}\phi(0)}{\alpha!}.
\begin{array}{cl}{}&{F^{(i)}(\cdots,z_{j+1},z_{j},\cdots)^{\cdots,a_{j-1},a_{j},a_{j+1}\cdots}}\\{=}&{\displaystyle\sum_{a_{j}^{\prime}}W\left(\left.\begin{array}{cc}{a_{j+1}}&{a_{j}}\\{a_{j}^{\prime}}&{a_{j-1}}\\\end{array}\right|z_{1}-z_{2}\right)F^{(i)}(\cdots,z_{j},z_{j+1},\cdots)^{\cdots,a_{j-1},a_{j}^{\prime},a_{j+1}\cdots},}\\\end{array}
\varepsilon_{2}^{\prime\alpha}\sigma_{\alpha\dot{\beta}}^{\mu}\bar{\varepsilon}_{1}^{\prime\dot{\beta}}=-\bar{\varepsilon}_{1\dot{\gamma}}\left(\bar{\sigma}^{\mu}\right)^{\dot{\gamma}\delta}\varepsilon_{2\delta}\,\,\left|A\right|^{2}
H_{c}=\int\mathrm{d}^{2}x\left[\Pi_{\alpha}^{*}\Pi_{\alpha}+\partial_{i}Z_{\alpha}^{*}\partial_{i}Z_{\alpha}-(Z_{\alpha}^{*}\partial_{i}Z_{\alpha})(Z_{\beta}\partial_{i}Z_{\beta}^{*})\right]
\hat{\calL}-\sum_{j=1}^{2}(-)^{j-1}{\calL}(\vec{\varphi}_{j}(x))-\hat{\calL}_{0}-\sum_{j=1}^{2}\frac{\delta\hat{\Gamma}[\vec{\varphi}_{1},\vec{\varphi}_{2}]}{\delta\vec{\varphi}_{j}(x)}\cdot\vec{\phi}_{j}(x).
\begin{array}{ccccc}{\mathrm{0A:}}&{\mathrm{(NS+,NS+)}}&{\mathrm{(NS~-~,NS~-~)}}&{\mathrm{(R+,R~-~)}}&{\mathrm{(R~-~,R+)}}\\{\mathrm{0B:}}&{\mathrm{(NS+,NS+)}}&{\mathrm{(NS~-~,NS~-~)}}&{\mathrm{(R+,R+)}}&{\mathrm{(R~-~,R~-~)}\,.}\\\end{array}
{a}\,{b}\,{{f}_{2}}\,\left(\!\,\mathrm{\&*}\left(\!\,{{f}_{1}},{{Q}_{1}},{{Q}_{2}}\,\!\right)+{{f}_{2}}\,\left(\!\,{{Q}_{1}}\,\mathrm{\&*}\,{{Q}_{2}}\,\!\right)\,\!\right)
|0\rangle\rightarrow|0\rangle_{\beta}=(1+\mathrm{e}^{-\beta\epsilon})^{-1/2}\{|0\rangle_{a}\otimes|0\rangle_{b}+\mathrm{e}^{-\beta\epsilon/2}a^{\dagger}{\tilde{a}}|0\rangle_{a}\otimesb^{\dagger}{\tilde{b}}|0\rangle_{b}\}.
x_{\alpha\dot{\alpha}}=x_{m}\sigma_{\alpha\dot{\alpha}}^{m}=\left(\!\begin{array}{cc}{x_{4}+ix_{3}}&{x_{2}+ix_{1}}\\{-x_{2}+ix_{1}}&{x_{4}-ix_{3}}\\\end{array}\!\right)=\left(\!\begin{array}{cc}{z_{1}}&{z_{2}}\\{-\bar{z}_{2}}&{\bar{z}_{1}}\\\end{array}\!\right),
\begin{array}{ll}{\ell^{-}}&{=0,\frac{1}{2},1,\dots,\frac{1}{2}(k^{-}-1)}\\{}&{}\\{\ell^{+}}&{=0,\frac{1}{2},1,\ldots,\frac{1}{2}(k^{+}-1)}\\\end{array}
\langle\phi(x)^{2}\rangle_{\mathrm{reg}}^{(0)}=(4\pi)^{-2}\left[-(\gamma-1)\bar{\tau}_{1}^{(0)}+\bar{\tau}_{2}^{(0)}(\bar{\tau}_{1}^{(0)})^{-1}+\bar{\tau}_{3}^{(0)}(\bar{\tau}_{1}^{(0)})^{-2}+...\right]
A_{2}^{(0)}(l_{0})=-{\frac{1}{\pisM_{P}^{2}}}e^{\pil_{0}s/2M_{s}^{2}}+{\frac{4}{l_{0}^{2}M_{P}^{2}R^{6}M_{s}^{8}}}E_{3}(-{\frac{\pil_{0}s}{2M_{s}^{2}}})\.
\sum_{k=0}^{n-1}t_{k}^{(i)}z^{k}=-\int_{C_{i}}dw\frac{1}{(2\langleR(w)\rangle-V^{\prime}(w))}\frac{2V^{\prime}(z)V^{\prime}(w)+f(z)+f(w)}{4(z-w)^{2}}
\delta\eta_{(p)\mu_{1}\ldots\mu_{p-2k-1}}^{*}=-\left(\left(p-2k\right)\partial^{\mu}A_{(p)\mu\mu_{1}\ldots\mu_{p-2k-1}}^{*}+\partial_{\left[\mu_{1}\right.}A_{(p)\left.\mu_{2}\ldots\mu_{p-2k-1}\right]}^{*}\right).
\left(\frac{y^{2}}{q}-\frac{(1-q^{2})q^{3}}{1+q^{4}}\right){\hat{\calR}}\left(q^{4}\right)^{-1}-\left(\frac{q}{y^{2}}-\frac{1-q^{2}}{q(1+q^{4})}\right){\hat{\calR}}\left(q^{4}\right)\.
E_{q\bar{q}}^{(Reg)}=\frac{U_{T}}{\pi}\left[1+\sqrt{\frac{a^{2}+1}{2}}\left[\frac{a-1}{2a}K(f_{1}(a))+\frac{a+1}{2a}K(f_{2}(a))-E(f_{1}(a))-E(f_{2}(a))\right]\right],
\begin{array}{rl}{0=\mathrm{sTr}([\omega(g^{-1}F_{R}g)^{q+1}}&{(\omega^{2})^{m-q-2},\omega])=\mathrm{sTr}([\omega,\omega](g^{-1}F_{R}g)^{q+1}(\omega^{2})^{m-q-2})}\\{+}&{(q+1)\mathrm{sTr}(\omega(g^{-1}F_{R}g)^{q}[g^{-1}F_{R}g,\omega](\omega^{2})^{m-q-2})}\\{+}&{(m-q-2)\mathrm{sTr}(\omega(g^{-1}F_{R}g)^{q+1}(\omega^{2})^{m-q-3}[\omega^{2},\omega])}\\\end{array}
Q_{p}\sum_{n=-\infty}^{\infty}\frac{1}{\mid\vec{y}-2\pina\hat{z}\mid^{7-p}}\rightarrow\frac{Q_{p}}{2\pia}\int_{-\infty}^{\infty}\frac{dz}{(r^{2}+z^{2})^{(7-p)/2}}=\frac{\tilde{Q}_{p}}{r^{6-p}},
V_{eff}(x)={\frac{D}{x^{2}}}+{\frac{F(1-2x)}{x^{2}(1-x)}}+{\frac{(1-2x)^{2}}{2x^{2}(1-x)^{2}}}+{\frac{1}{4x^{2}(1-x)^{2}}}\rightarrow{\frac{3}{4(1-x)^{2}}}
\tilde{I}=\frac{1}{2\kappa^{2}}\intd^{D}x\sqrt{-g}(R-\frac{1}{2}(\nabla\phi)^{2}-\frac{1}{2(q+1)!}e^{-\alpha\phi}F_{q+1}^{2}-\frac{1}{2({\tilde{d}}+1)!}e^{\tilde{\beta}\phi}G_{{\tilde{d}}+1}^{2}),
f\astg(x,p)=(2\pi)^{-2n}\intd^{n}u_{1}d^{n}u_{2}d^{n}v_{1}d^{n}v_{2}e^{i(u_{1}\cdotv_{1}+u_{2}\cdotv_{2})}f(x+\frac{hu_{2}}{2},p-\frac{hu_{1}}{2})g(x+v_{1},p+v_{2})\,.
\Gamma_{\mathrm{scal}}[F,V]=\int_{0}^{\infty}\!{\frac{dT}{T}}\;\frac{\mathrm{e}^{-m^{2}T}}{(4\piT)^{D/2}}\;\mathrm{tr}\;\int\!dx_{0}\;\sum_{n=1}^{N}\;\frac{(-T)^{n}}{n!}\;O_{n}[F,V]\,,
e^{iS_{\mathrm{eff}}[A,\Phi]}=e^{i\intd^{4}x{\mathcalL}_{H}}\frac{\int[d\Psi][d\overline{{\Psi}}]e^{i\intd^{4}x{\mathcalL}_{F}}}{\int[d\Psi][d\overline{{\Psi}}]e^{i\intd^{4}x{\mathcalL}_{F}|_{\mathrm{V=0}}}}\,.
<\overline{{\psi}}_{L}^{({\calN})}\ldots\overline{{\psi}}_{L}^{(1)}\psi_{R}^{(1)}\ldots\psi_{R}^{({\calN})}>=(\frac{e^{\gamma}}{4\pi})^{\calN}(\sqrt{\frac{\calN}{\pi}}e_{c})^{\calN}\quad.
\eta^{\mu\nu}\partial_{\mu}V^{A}\partial_{\nu}V_{B}+(\partial_{7}V^{A})^{2}+(\partial_{8}V^{A})^{2}+\left(\nabla_{y}V^{A}-q(x^{7}\partial_{8}-x^{8}\partial_{7})V^{A}\right)^{2}.
V=\sum_{a=1}^{N-1}\left\{\left(\frac{e_{a}^{2}}{2}\sum_{i=1}^{N+1}Q_{ai}|\phi_{i}|^{2}-r_{a}\right)^{2}+\sum_{i=1}^{N-1}Q_{ai}^{2}|\sigma_{a}|^{2}|\phi_{i}|^{2}\right\},
N^{\mu\nu}={\frac{L_{*}{}^{\prime}}{\kappa_{1}}}\eta^{[\mu}{}_{2}X^{\nu]}{}^{\prime}-{\frac{L_{*}\kappa_{3}}{\kappa_{1}}}\eta^{[\mu}{}_{3}X^{\nu]}{}^{\prime}+L_{*}\eta^{[\mu}{}_{1}\eta^{\nu]}{}_{2}\,,
\left.A_{i_{1}\ldotsi_{p}}(x_{0},{\bfx})\partial_{0}A_{i_{1}\ldotsi_{p}}(x_{0},{\bfx})\right|_{x_{0}=\epsilon}\to{\frac{1}{2}}\epsilon^{-2\lambda}\partial_{\epsilon}A_{i_{1}\ldotsi_{p}}^{h}(\epsilon,{\bfx})^{2}.
ds_{{\calM}_{\mathrm{UV}}}^{2}={\frac{1}{8\pi^{2}g_{\mathrm{YM}}^{2}}}[dv^{2}+v^{2}d\Omega_{5}^{2}]\,\qquad\mathrm{with}\quadv={\frac{L}{\alpha^{\prime}}}e^{{r/L}}\,
I_{cs}=\frac{1}{4\pi}Tr\int_{M}\left[\frac{1}{a_{+}}A^{+}\wedge\left(dA^{+}+\frac{2}{3}A^{+}\wedgeA^{+}\right)+\frac{1}{a_{-}}A^{-}\wedge\left(dA^{-}+\frac{2}{3}A^{-}\wedgeA^{-}\right)\right]
\begin{aligned}{[m]\bar{F}_{p+1}=\sum_{i_{k}}\frac{d}{dx^{-}}}&{{}\bar{A}_{i_{1}i_{2}\dotsi_{p}}(2x^{-})a_{i_{1}}(x^{-})^{-1}a_{i_{2}}(x^{-})^{-1}\cdotsa_{i_{p}}(x^{-})^{-1}}\\{}&{{}dx^{-}\wedgedx^{i_{1}}\wedgedx^{i_{2}}\wedge\cdots\wedgedx^{i_{p}}~.}\\\end{aligned}
T(z)=-{\frac{1}{2}}~:\partial_{z}X(z)\partial_{z}X(z):~+{\frac{2}{3}}q^{2}~:\psi_{1}(z)\partial_{z}\psi_{2}(z):~-{\frac{1}{3}}q^{2}~:\partial_{z}\psi_{1}(z)\psi_{2}(z):
\delta_{\hat{\chi}}{\hat{L}}_{{\hat{\mu}}_{1}\dots{\hat{\mu}}_{r}}=r\frac{m}{2}(-1)^{r}(i_{\hat{h}}{\hat{\chi}})_{[{\hat{\mu}}_{1}}(i_{\hat{h}}{\hat{L}})_{{\hat{\mu}}_{2}\dots{\hat{\mu}}_{r}]}\,,
S\;=\;\intdtd^{2}x\,\left[-hb_{0}{\tilde{J}}_{0}-hb_{k}{\tilde{J}}_{k}+\frac{m}{2\rho_{b}}{\vec{J}}_{b}^{2}-\frac{\hbar^{2}}{8m\rho_{b}}\partial_{j}\rho_{b}\partial_{j}\rho_{b}\,-\,V(\rho_{b})\right]\;,
\int\sqrt{g}\\left(\nabla_{\mu_{1}}\cdots\nabla_{\mu_{p_{1}}}\left.\widehat{R}^{\alpha_{1}}\right._{\beta_{1}\gamma_{1}\delta_{1}}\right)\cdots\\left(\nabla_{\mu_{1}}\cdots\nabla_{\mu_{p_{m}}}\left.\widehat{R}^{\alpha_{m}}\right._{\beta_{m}\gamma_{m}\delta_{m}}\right)\
D^{0}(k)D^{0}(k+p)={\frac{1}{\eta\cdotk}}{\frac{1}{\eta\cdot(k+p)}}={\frac{1}{\eta\cdotp}}\biggl[{\frac{1}{\eta\cdotk}}-{\frac{1}{\eta\cdot(k+p)}}\biggr].
T_{\mu\nu}(k)=i\int\frac{d^{2}p}{(2\pi)^{2}}\int_{0}^{1}d\xi_{1}\int_{0}^{1}d\xi_{2}\delta(1-\xi_{1}-\xi_{2})\left[\frac{P(m)}{\left[z-m^{2}+i\varepsilon\right]^{2}}\right]_{z=\sum_{j=1}^{2}\xi_{j}p_{j}^{2}},
+\int\!\!d^{D}\!z_{1}d^{D}\!z_{2}d^{D}\!z_{3}\left.\frac{\delta^{2}W}{\deltaj(x)\deltaj(z_{1})}\,\frac{\delta^{2}W}{\deltaj(x)\deltaj(z_{2})}\,\frac{\delta^{2}W}{\deltaj(x)\deltaj(z_{3})}\,\frac{\delta^{3}\Gamma}{\delta\Phi(z_{1})\delta\Phi(z_{2})\delta\Phi(z_{3})}\right],
[G_{R}Q]\{sI\}\equiv\int_{0}^{s}G_{R}(s,s^{\prime})[Q^{\mu}{}_{\nu}(s^{\prime})]s^{\prime}ds^{\prime}=\int_{0}^{s}(s-s^{\prime})[Q^{\mu}{}_{\nu}(s^{\prime})]s^{\prime}ds^{\prime}.
\varepsilon_{\mu_{1}\ldots\mu_{d}}=e^{-1}e_{\mu_{1}}^{a_{1}}\ldotse_{\mu_{d}}^{a_{d}}\varepsilon_{a_{1}\ldotsa_{d}}\,,\qquad\varepsilon^{\mu_{1}\ldots\mu_{d}}=ee_{a_{1}}^{\mu_{1}}\ldotse_{a_{d}}^{\mu_{d}}\varepsilon^{a_{1}\ldotsa_{d}}\,.
\Gamma_{(0)}^{\prime}=\left(\begin{matrix}{0}&{\hat{\gamma}}\\{\hat{\gamma}^{-1}}&{0}\\\end{matrix}\right)\,\,\qquad\Gamma=\left(\begin{matrix}{0}&{\hat{\gamma}e^{{\hat{a}}}}\\{(\hat{\gamma}e^{{\hat{a}}})^{-1}}&{0}\\\end{matrix}\right)\,,
\gamma_{00}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\quad\gamma_{0i}=\gamma_{i0}=\left(\begin{array}{cc}{0}&{J_{i}}\\{-J_{i}}&{0}\\\end{array}\right),~~~\gamma_{ij}=g_{ij}\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)+\left(\begin{array}{cc}{0}&{\left\{J_{i},J_{j}\right\}}\\{\left\{J_{i},J_{j}\right\}}&{0}\\\end{array}\right)\,,
i{\calD}^{-1}(p^{2})=\mu^{2}-(p^{2}+\omega^{2})+\frac{N\beta^{2}}{2\pi}\left(f\biggl(-\frac{p^{2}}{\beta^{2}\sigma^{2}}\biggr)-f\biggl(\frac{\omega^{2}}{\beta^{2}\sigma^{2}}\biggr)\right)
V_{\mathrm{eff}}\equiv\frac{-A^{2}}{2G}+V(A)=\frac{-A^{2}}{2G}+\frac{\mu}{4M}A^{4}\equiv\frac{\muM^{3}}{4}\left[-2\left(\frac{A^{2}}{M^{2}}\right)+\left(\frac{A^{2}}{M^{2}}\right)^{2}\right]
\not\!pS(p)=1-ie^{2}\int\frac{d^{2}k}{(2\pi)^{2}}\frac{1}{k^{2}(e^{2}/\pi-k^{2}-i\epsilon)}\not\!k\gamma^{5}\left[\gamma^{5}S(p)+S(p+k)\gamma^{5}\right]\;,
\kappa_{q}=\kappa_{p}+\rho^{\mu}\partial_{p^{\mu}}\kappa_{p}+\frac{1}{2}\rho^{\mu}\rho^{\nu}\partial_{p^{\mu}}\partial_{p^{\nu}}\kappa_{p}+\cdots=\kappa_{p}+\rho\cdotu+o(\rho^{2})
H^{1}({\mathbbP}^{1}\timesT^{2},{\calO}_{{\mathbbP}^{1}\timesT^{2}}(6\sigma|_{{\mathbbP}^{1}\timesT^{2}}-2F))|_{c_{-2}^{(1)}}=H^{0}({\calS},{\calO}_{{\calS}})^{*}.
q_{s}^{2}{\frac{\partial^{2}G}{\partialq_{i}\partialq_{j}}}=\left({\frac{\hbar}{q_{s}}}\right)^{2}{\frac{\partial^{2}G}{\partialp_{i}\partialp_{j}}}\,,\quad{\frac{\partial^{2}G}{\partialq_{i}\partialp_{j}}}=-\,{\frac{\partial^{2}G}{\partialq_{j}\partialp_{i}}}\,,
\left\langle{\hat{H}_{\phi}}^{2}\right\rangle\approx\left\langle{\hat{H}_{\phi}}\right\rangle^{2}\quad{\mathrm{o}r}\quad\left\langle({\hat{v}^{\prime}})^{2}\right\rangle\approx(\left\langle{\hat{v}^{\prime}}\right\rangle)^{2}
{\vec{f}}_{i}(x)=\sum_{j=1}^{N}\,\sum_{k=1}^{n}\,\left(1+x\Lambda+\frac{x^{2}}{2!}\Lambda^{2}+\cdots+\frac{x^{n-1}}{(n-1)!}\Lambda^{n-1}\right)_{ik}C_{kj}{\vec{e}}_{j}.
\deltaS=\sum_{\psi}\sum_{j=1}^{2}\left\{\int_{S^{1}\timesU_{j}}\left[\frac{\delta{\calL}}{\delta\psi}-\partial_{\nu}\left(\frac{\delta{\calL}}{\delta\partial_{\nu}\psi}\right)\right]\delta\psi\,d^{3}x+\int_{S^{1}\timesU_{j}}\partial_{\nu}\left(\frac{\delta{\calL}}{\delta\partial_{\nu}\psi}\delta\psi\right)d^{3}x\right\}
{\hat{\vartheta}}\left(x,z\right)=\left(a_{12},a_{21},a_{11},a_{22}\right)\,\cdot\,\left(\begin{matrix}{\xi_{1}({r})\,\xi_{2}({s})}\\{\xi_{2}({r})\,\xi_{1}({s})}\\{\xi_{1}({r})\,\xi_{1}({s})}\\{\xi_{2}({r})\,\xi_{2}({s})}\\\end{matrix}\right)
\sum_{k=1}^{\infty}\frac{J_{\nu}(\gamma_{\nu,k})Y_{\nu}(\sigma\gamma_{\nu,k})-Y_{\nu}(\gamma_{\nu,k})J_{\nu}(\sigma\gamma_{\nu,k})}{J_{\nu}^{2}(\gamma_{\nu,k})/J_{\nu}^{2}(\lambda\gamma_{\nu,k})-1}=\frac{\sigma^{\nu}}{\pi}\frac{(\lambda/\sigma)^{2\nu}-1}{\lambda^{2\nu}-1},\quad\lambda\geq\sigma>1.
g_{IJ}\,\frac{d\phi^{I}}{d\tau}\,\frac{d\phi^{J}}{d\tau}\,\frac{4\pi}{\mathrm{Area}_{H}}\,\tau^{2}\,\stackrel{\tau\to\infty}{\longrightarrow}\,X^{2}\,=\,\mathrm{finite~quantity}
\varphi(0)=\frac{1}{\omega_{d}}\sum_{\gamma\in\Gamma}\int_{S_{\infty}^{d}}d\omega(x^{\prime})\left(\frac{1-\left|\gamma(0)\right|^{2}}{\left|\gamma(0)-x^{\prime}\right|^{2}}\right)^{d}\chi(x^{\prime}),\tag{5.18}
\Upsilon_{I}^{i}=\left(1-\frac{c^{\mathrm{\scriptscriptstyleT}}+c^{\mathrm{\scriptscriptstyleB}}+c^{\mathrm{\scriptscriptstyleL}}}{24}\right)\frac{2b}{a^{2}}\,partial\chi^{i}(z_{I})+\left(\frac{c^{\mathrm{\scriptscriptstyleT}}+c^{\mathrm{\scriptscriptstyleB}}+c^{\mathrm{\scriptscriptstyleL}}}{16}-\frac32\right)\frac{2}{a}\,partial^{2}\chi^{i}(z_{I})\,
\bigg[\prod_{I=1}^{N}:e^{-i2\sqrt{\pi}\omega{(I)}U_{Ia}\Phi^{(I)}(x)}:_{M^{(I)}}\;e^{+i\frac{\theta}{N}}\;+\;\prod_{I=1}^{N}:e^{+i2\sqrt{\pi}\omega{(I)}U_{Ia}\Phi^{(I)}(x)}:_{M^{(I)}}\;e^{-i\frac{\theta}{N}}\bigg]\Bigg\rangle_{K^{\mu}}\;,