harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calG}_{\mathrm{rel}}=\frac{4\pi\kappa}{e^{2}}\left[d\rho^{2}+\rho^{2}\frac{\sum_{s=1}^{4N-1}\sigma_{s}\sigma_{s}}{4}\right]=\frac{4\pi\kappa}{e^{2}}\left[d\rho^{2}+\rho^{2}d\Omega_{4N-1}^{2}\right]=\frac{4\pi\kappa}{e^{2}}\,{\calG}_{\mathrm{flat}}\,,
{\frac{I-{\bfA}\cdot{\bfB}-i({\bfA}+{\bfB}+{\bfA}\times{\bfB})\cdot{\bf\sigma}}{\sqrt{1+A^{2}}\sqrt{1+B^{2}}}}=U_{3}U_{4}={\frac{I+i{\not\!\!C}}{\sqrt{1+C^{2}}}},
2q_{j-1}^{\prime}+q_{j}^{\prime\prime}+\mathrm{C}_{n}^{j}\,V^{(n-j)}+\sum_{i=j+1}^{n-1}\,q_{i}\,\mathrm{C}_{i}^{j}\,V^{(i-j)}-2q_{n-1}^{\prime}\,q_{j}=0,
S_{3/2}=\int_{M}d^{4}x\sqrt{g}\biggr[{\frac{1}{2}}\tilde{\kappa}_{\mu}{\widehat\nabla}\kappa^{\mu}+\tilde{\epsilon}{\widehat\nabla}\epsilon+{\frac{1}{2}}\tilde{\eta}{\widehat\nabla}\eta\biggr]\;,
d(\vec{z}_{\bot})=-\frac{1}{L^{2}}\sum_{\vec{n}\neq\vec{0}}\frac{1}{q_{n}^{2}}e^{i\vec{q}_{n}\vec{z}_{\bot}}\,\\\\\vec{q}_{n}=\frac{2\pi}{L}(n_{1},n_{2})\.
\hat{{\calG}}^{E}[\Lambda]=i\hbar\sum_{v\inB}\Lambda(v)\left(\sum_{e(1)=v}X_{L}^{3}(h_{e})-\sum_{e(0)=v}X_{R}^{3}(h_{e})+X_{L}^{3}(h_{v})-X_{R}^{3}(h_{v})\right)
I_{AB}(\hat{{\calX}}_{12})=\textstyle{\frac{1}{4}}\mathrm{tr}(\gamma_{A}\hat{{\calX}}_{12}\gamma_{B}\hat{{\calX}}_{21})=\textstyle{\frac{1}{4}}\mathrm{tr}(\tilde{\gamma}_{A}\hat{{\calX}}_{21}^{-1}\tilde{\gamma}_{B}\hat{{\calX}}_{12}^{-1})
S_{\mathrm{YM}}=\int\mathrm{d}t\,\mathrm{d}^{3}x\;\mathrm{Tr}\Big[-{\scriptstyle{\frac{1}{4}}}F_{\mu\nu}\,F^{\mu\nu}-{\scriptstyle{\frac{1}{2}}}{D}_{\mu}\phi\,{D}^{\mu}\phi\Big]-V(\phi)\,.
\begin{array}{c}{p_{a}=\pi_{a}-\frac{i}{2}\partial_{a}{\bfg},\quad\chi_{i}^{m}=\mathrm{e}_{b}^{m}\eta_{i}^{b}:}\\{\Omega=dp_{a}\wedgedz^{a}+d{\bar{p}}_{\bar{a}}\wedged{\bar{z}}^{\bar{a}}+d\chi_{i}^{m}\wedged{\bar{\chi}}_{i}^{\bar{m}},}\\\end{array}
G\left(\tau,\tau^{\prime};\tau_{2},\tau_{1}\right)\equiv\frac12|\tau-\tau^{\prime}|-\frac{\tau_{1}+\tau_{2}}{2\left(\tau_{1}-\tau_{2}\right)}\left(\tau+\tau^{\prime}\right)+\frac{\tau\tau^{\prime}}{\tau_{1}-\tau_{2}}+\frac{\tau_{1}\tau_{2}}{\tau_{1}-\tau_{2}}.
\frac{ds^{2}}{\alpha^{\prime}\sqrt{4\pig_{s}N}}=(1+\delta_{1})\frac{d\rho^{2}}{\rho^{2}-\frac{b^{2}}{\rho^{2}}}+(1+\delta_{2})(\rho^{2}-\frac{b^{4}}{\rho^{2}})dt^{2}+\rho^{2}dx_{i}^{2}+d\Omega_{5}^{2},
\Omega_{X\,jB}{}^{iA}\equivf_{jB}^{Y}\left(\partial_{X}f_{Y}^{iA}-\Gamma_{XY}^{Z}f_{Z}^{iA}\right)=-\omega_{Xj}{}^{i}\delta_{B}{}^{A}-\omega_{XB}{}^{A}\delta_{j}{}^{i}\,,
2{\calL}=-\frac{1}{4}\left(F_{\mu\nu}\right)^{2}+\frac{1}{2}\left[\bar{\psi},(i\gamma^{\mu}D_{\mu})\psi\right]+\frac{G}{2}\left[(\bar{\psi}\psi)^{2}+(\bar{\psi}i\gamma_{5}\psi)^{2}\right],
\begin{array}{l}{m_{A}(x)={\hat{A}}^{*\mu}A_{\mu}\,,}\\{n_{A}(x)=A_{\mu}\left({\displaystyle{\frac{\delta\Sigma}{\deltaA_{\mu}}}}+eg^{\mu\nu}\partial_{\nu}b\right)-{\hat{A}}^{*\mu}{\displaystyle{\frac{\delta\Sigma}{\delta{\hat{A}}^{*\mu}}}}\,,}\\\end{array}
A_{\mu\nu\gamma}\,\Pi^{\mu\mu^{\prime}}\,\Pi^{\nu\nu^{\prime}}\,\Pi^{\gamma\gamma^{\prime}}\,\left(i\epsilon_{\mu^{\prime}\nu^{\prime}\gamma^{\prime}\alpha}\,(X\partial^{\alpha}X^{\dagger}-X^{\dagger}\partial^{\alpha}X)\,\phi^{N}\right)
{{M_{T}}^{\alpha\beta}}_{\rho\sigma}=\delta_{\rho}^{(\alpha}\delta_{\sigma}^{\beta)}(-\Box-P)-g^{\alpha\beta}g_{\rho\sigma}Q-2a_{2}\delta_{(\rho}^{(\alpha}\nabla^{\beta)}\nabla_{\sigma)}+a_{2}g_{\rho\sigma}\nabla^{(\alpha}\nabla^{\beta)}
\Gamma=\frac{1}{8\pi^{2}}\int\mathrm{d}^{8}z\int_{0}^{\infty}\mathrm{d}t\,t\,\mathrm{e}^{-t}\frac{W^{2}\bar{W}^{2}}{M^{2}}\,\omega(t\Psi,t\bar{\Psi}),
\tau(r_{i})=\frac{1}{2\kappa^{2}}\frac{H^{\prime}}{H}=\frac{1}{2\kappa^{2}}\frac{(p-7)+\frac{Q^{2}}{(3-p)\rhor_{i}^{3-p}}}{1-\frac{Q^{2}}{\rho(3-p)(10-2p)r_{i}^{3-p}}}\frac{1}{r_{i}}\,sim\,frac{1}{2\kappa^{2}}\Bigg(\frac{p-7}{r_{i}}+\frac{Q^{2}}{(10-2p)\rho}\frac{1}{r_{i}^{4-p}}\Bigg)
\sum_{I=1}^{2}\:e_{\mu}^{I}e_{\nu}^{I}=-g_{\mu\nu}^{tr}\equiv-g_{\mu\nu}-\left[\frac{1}{2}\:(k_{\mu}\bar{k}_{\nu}+k_{\nu}\bar{k}_{\mu})+k^{2}\eta_{\mu}\eta_{\nu}\right]\;\frac{1}{(k\cdot\eta)^{2}-k^{2}}\quad,
\mathrm{Z}_{\mathrm{orb.}}(\alpha^{\prime1/2})=\mathrm{Z}_{\mathrm{circ.}}({\frac{\alpha^{\prime1/2}}{2}})=\mathrm{Z}_{\mathrm{circ.}}(2\alpha^{\prime1/2})\equiv\mathrm{Z}_{KT}\quad.
{\calL}_{\mathrm{odd/odd}}^{1}=\epsilon^{r_{1}\cdotsr_{9}m}\epsilon^{s_{1}\cdotss_{9}}{}_{m}W_{r_{1}r_{2}s_{1}s_{2}}W_{r_{3}r_{4}s_{3}s_{4}}W_{r_{5}r_{6}s_{5}s_{6}}\Big(H_{r_{7}r_{8}s_{9}}H_{s_{7}s_{8}r_{9}}-\frac{1}{9}H_{r_{7}r_{8}r_{9}}H_{s_{7}s_{8}s_{9}}\Big)\,,
{\frac{M_{4}^{2}}{M_{D}^{2}}}=\left({\frac{M_{D}}{2\pi}}\right)^{D-4}\intd^{D-4}y\sqrt{g_{D-4}}e^{2A}\equiv\left({\frac{M_{D}}{2\pi}}\right)^{D-4}V_{w}\,.
\int_{-\infty}^{\infty}dz_{1}\int_{-\infty}^{\infty}dz_{2}\;\psi_{z_{1},z_{2}}^{\ast\;\nu,\mu}(u^{0},x^{\prime},y^{\prime})\psi_{z_{1},z_{2}}^{\nu,\mu}(u^{0},x,y)=\delta\left(x-x^{\prime}\right)\delta\left(y-y^{\prime}\right)\;.
[{\hat{M}}^{\mu\nu},{\hat{J}}^{\rho}(x)]=-\imath\{(x^{\mu}\partial^{\nu}-x^{\nu}\partial^{\mu}){\hat{J}}^{\rho}(x)+g^{\mu\rho}{\hat{J}}^{\nu}(x)-g^{\nu\rho}{\hat{J}}^{\mu}(x)\}
\begin{array}{c}{<j_{1}-m_{1}\j_{2}-m_{2}|j_{1}+j_{2}\-m_{1}-m_{2}>_{q}=}\\{=<j_{2}m_{2}\j_{1}m_{1}|j_{1}+j_{2}\m_{1}+m_{2}>_{q}=}\\{=<j_{1}m_{1}\j_{2}m_{2}|j_{1}+j_{2}\m_{1}+m_{2}>_{q^{-1}}}\\\end{array}
\sum_{\{n\}}\Psi_{P,\{n\}}(z^{\prime})\overline{{\Psi_{P,\{n\}}(z)}}=\sum_{\{n\}}\Psi_{P,\{n\}}(z^{\prime})\Psi_{P,\{n\}}(\tilde{z})=<z^{\prime},\tilde{z}>^{P}\:,
H=\sum_{n=1}^{N-1}h_{n,n+1}+\frac{1}{2}(K_{1}^{-}(0)^{-1})\dot{K}_{1}^{-}(0)+\frac{tr_{0}[K_{0}^{+t}(-\eta)h_{N0}]}{tr[K^{+}(-\eta)]}\;\;\;,
\frac{1}{2}\int_{\epsilon^{2}R^{2}}^{\infty}\frac{dt}{t^{3}}\sum_{k=-\infty}^{\infty}e^{-(k+1/2)^{2}\pit/R^{2}}=R^{-4}{\calJ}(\epsilon^{2},\alpha=-2)=\frac{1}{R^{4}}\left\{\frac{-15}{16}\frac{3}{4\pi^{2}}\zeta(5)+\frac{1}{5}\epsilon^{-5}\right\}
V_{(+1/2)}^{f}\sim\biggl\{e^{c/2}[\partialX_{\mu}+i/4\(K\cdot\psi)\psi_{\mu}]\gamma_{\alpha\beta}^{\mu}S^{\beta}+1/2\e^{3/2\c}\etabS_{\alpha}\biggr\}e^{iK\cdotX}\.
|\phi(a^{I})\rangle=\rho^{s}\sum_{s^{\prime}=0}^{s}(1+\zeta^{2})^{s-s^{\prime}}C_{s-s^{\prime}}^{\frac{d-4}{2}+s^{\prime}}(t)|\phi_{s^{\prime}}(\zeta^{i})\rangle\,,\quad\zeta\partial_{\zeta}|\phi_{s^{\prime}}\rangle=s^{\prime}|\phi_{s^{\prime}}\rangle\,,\quad\partial_{\zeta^{i}}^{2}|\phi_{s^{\prime}}\rangle=0\,,
G_{D}({\bfr^{\prime\prime}},{\bfr^{\prime}};E)=\left(r^{\prime\prime}r^{\prime}\right)^{-(D-1)/2}\,\sum_{l=0}^{\infty}\sum_{m=1}^{d_{l}}Y_{lm}({\bf\Omega^{\prime\prime}})Y_{lm}^{*}({\bf\Omega^{\prime}})G_{l+\nu}(r^{\prime\prime},r^{\prime};E)\;
a\circb=\sum_{k=0}^{\infty}(-\frac{ih}{2})^{k}\frac{1}{k!}\omega^{{\mu_{1}}{\nu_{1}}}\ldots\omega^{{\mu_{k}}{\nu_{k}}}.{\frac{\partial^{k}a}{\partialy^{\mu_{1}}\ldots\partialy^{\mu_{k}}}}{\frac{\partial^{k}b}{\partialy^{\nu_{1}}\ldots\partialy^{\nu_{k}}}}
y^{2}=\prod_{a=1}^{\tilde{n}_{c}}(x-\phi_{a})^{2}\prod_{k=1}^{n_{c}-\tilde{n}_{c}}(-\Lambda\omega^{2k})^{2}+4\Lambda^{2n_{c}-n_{f}}\prod_{i=1}^{n_{f}}(x+m_{i}),
g=\frac{\Lambda^{2}}{(1+rAx)^{2}}\biggl\{-F(r)\,ds^{2}+F(r)^{-1}dr^{2}+r^{2}\,G(x)^{-1}dx^{2}+r^{2}\,G(x)\,\Lambda^{-4}d\phi^{2}\biggr\}.
S(A,\phi^{3})=\intd^{3}x[\frac{1}{4g^{2}}F_{\mu\nu}^{2}-\frac{2\theta^{2}}{\eta^{2}}\intd^{3}x^{\prime}\Box^{-1}(x-x^{\prime})\partial_{\mu}\tilde{F}_{\mu}(x^{\prime})\partial_{\nu}\tilde{F}_{\nu}(x)]+2i\thetaq
\rho_{\mathrm{S}}(x)=\left(\frac{\pi\rho(0)}{2}\right)^{2}x\left(J_{\alpha+\frac12}^{2}+J_{\alpha-\frac12}^{2}-J_{\alpha+\frac12}J_{\alpha-\frac32}-J_{\alpha-\frac12}J_{\alpha+\frac32}\right)\left({\pi\rho(0)x}\right).
v=\frac{1}{5}\left[\frac{1}{12}v_{0}^{\prime\prime\prime}+\frac{1}{12}u_{0}^{\prime\prime}v_{0}^{\prime}+\frac{1}{4}u_{0}^{\prime}v_{0}^{\prime\prime}+\frac{1}{6}\left(u_{0}^{\prime}\right)^{2}v_{0}^{\prime}-\frac{8}{27}\left(v_{0}^{\prime}\right)^{3}\right]\quadetc\quad,
S_{1}=k\intd^{5}x\verte\verte_{A}^{[i}e_{B}^{j]}F_{ij}^{AB}=k\intd^{5}x\verte\vert(e_{a}^{[i}e_{b}^{j]}F_{ij}^{ab}+2e_{[5}^{[i}e_{b]}^{j]}F_{ij}^{5b})
M_{I}^{2}=e^{K}(K^{\varphi\bar{\varphi}})^{2}|\lambdaA|^{2}=\frac{|\lambdaA|^{2}e^{G/3}}{9(s+\bar{s}-bG)}\left(1+\frac{b}{s+\bar{s}-bG}\right)^{-2}.
\begin{array}{rcl}{H_{\{0\}}H_{\{4\}}}&{=}&{1}\\{-H_{\{1\}}H_{\{3\}}}&{=}&{{\calA}_{\{1\}}}\\{H_{\{2\}}H_{\{2\}}}&{=}&{{\calA}_{\{2\}}}\\{-H_{\{3\}}H_{\{1\}}}&{=}&{{\calA}_{\{3\}}}\\{H_{\{4\}}H_{\{0\}}}&{=}&{{\calA}_{\{4\}}}\\\end{array}
\xi_{4}(x,y)=\frac{1}{4}\int_{-y}^{y}h_{44}(x,y^{\prime})dy^{\prime}-\frac{1}{4C}\int_{-y}^{y}F(y^{\prime})dy^{\prime}\int_{-\infty}^{\infty}h_{44}(x,y^{\prime})dy^{\prime},
(x_{13}^{2})^{-2r}(x_{24}^{2})^{-2}u^{-2}\negthickspace\negthickspace\sum_{D\in\{84,105,175\}}\sum_{\substack{a\,s\inS_{r}}}\frac{1}{N^{2}}P_{D}^{I_{a}I_{r+1},J_{s(a)}J_{r+1}}(\lambda_{D}\phi+\mu_{D})(u,v)\;\prod_{\substack{i=1\,\neqa}}^{r}\delta^{I_{i}J_{s(i)}}
\hat{F}_{(\hat{d})}{}^{\hat{\mu}\hat{\rho}_{1}\cdots\hat{\rho}_{d}}\hat{F}_{(\hat{d})}{}^{\hat{\nu}}{}_{\hat{\rho}_{1}\cdots\hat{\rho}_{d}}-{\textstyle\frac{1}{2\hat{d}}}\hat{g}^{\hat{\mu}\hat{\nu}}\hat{F}_{(\hat{d})}^{2}
{\calL}_{\mathrm{spin}}^{(2)}[B]={(4\pi)}^{-D}e^{2}\int_{0}^{\infty}{\frac{dT}{T}}e^{-m^{2}T}T^{2-D}\int_{0}^{\infty}d\hat{T}\int_{0}^{1}du_{a}\,J(z,u_{a},\hat{T},D)\,.\nonumber\,
ds_{(\mu,0)}^{2}=-\left(1-\frac{\tilde{r}^{2}}{l^{2}}\right)\left(d\tilde{t}\right)^{2}+\frac{\left(d\tilde{r}\right)^{2}}{1-\frac{\tilde{r}^{2}}{l^{2}}}+\tilde{r}^{2}\left(d\tilde{\phi}\right)^{2},
f_{jl}^{ab(-1)}={\left(\begin{matrix}{\epsilon_{ab}\epsilon_{jlm}{\frac{\partial_{m}}{\nabla^{2}}}}&{{-\delta_{ab}}{\frac{\partial_{j}}{\nabla^{2}}}}\\{{-\delta_{ab}}{\frac{\partial_{l}}{\nabla^{2}}}}&{0}\\\end{matrix}\right)}\delta(x-y)
{\frac{d\zeta}{d\varphi}}=A[1-\varphi^{2}-{\frac{\epsilon}{2}}{\frac{1-\zeta^{2}}{1-\varphi^{2}}}+{\frac{\epsilon^{2}}{8}}{\frac{(1-\zeta^{2})^{2}}{(1-\varphi^{2})^{3}}}+\cdots].
{\calV}=m^{2}g_{R}^{2}\frac{1}{(2\pi)^{8-2\epsilon}}\pi^{4-\epsilon}\frac{2}{\epsilon}\Gamma(\epsilon)\frac{1}{1-\epsilon}[\Gamma(1-\frac{\epsilon}{2})]^{3}\frac{1}{\Gamma(2-3\epsilon/2)}\Biggr(\frac{\mu^{2}}{p^{2}}{\Biggr)}^{\epsilon}\quad.\quad
\sqrt{2}{\calF}_{tr}^{i}=4G_{N}{\frac{e^{2\Phi^{\prime}(r)}\lambda(r)}{r^{2}}}M_{ij}(\alpha_{j}+\Psi\beta_{j}),\\\\\\\\sqrt{2}{\calF}_{\theta\phi}^{i}=L_{ij}\beta_{j}\,\mathrm{sin}\,\theta,
\hat{\Phi}=\Phi^{(1)}\otimes\left(\begin{matrix}{1}&{0}\\{0}&{0}\\\end{matrix}\right)+\phi\otimes\left(\begin{matrix}{0}&{0}\\{1}&{0}\\\end{matrix}\right)+\phi^{*}\otimes\left(\begin{matrix}{0}&{1}\\{0}&{0}\\\end{matrix}\right)+\Phi^{(2)}\otimes\left(\begin{matrix}{0}&{0}\\{0}&{1}\\\end{matrix}\right).
\int\frac{d^{4}s}{(2\pi)^{4}}\frac{(\delta_{mn}s^{2}-s_{m}s_{n})}{s^{2}(s^{2}+s_{0}^{2})(s^{2}+2sq+q^{2})}=\int_{0}^{1}dx_{1}\int_{0}^{1-x_{1}}dx_{2}\Gamma(3)\int\frac{d^{4}\tilde{s}}{(2\pi)^{4}}\frac{(\delta_{mn}\tilde{s}^{2}-\tilde{s}_{m}\tilde{s}_{n})}{[\tilde{s}^{2}+\tilde{s}_{0}^{2}x_{2}+\tilde{M}^{2}]^{3}},
V_{2}(R_{2})=kR_{2}^{2}\left[3(1+\sigma_{0})^{1/2}(1+\sigma_{1})-\frac{3}{2}\left(\frac{3-d}{12\pi}\right)^{2/3}(\sigma_{0}+2\sigma_{1})k^{1/3}\rho^{2}+\frac{(d-3)^{2}}{48\pi^{2}}\rho^{6}\right],
{\delta}A_{\mu}^{p}=0,\quad{\delta}H_{\mu\nu}^{p}={\calD}_{\mu}^{pq}{\xi}_{\nu}^{q}-{\calD}_{\nu}^{pq}{\xi}_{\mu}^{q}\equiv{\calR}_{\mu\nu\alpha}^{pq}{\xi}^{q\alpha},
\!\!\!\!\!\!\!\!\!\!\!R_{x}(\theta)=\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{c\theta}&{-s\theta}\\{0}&{s\theta}&{c\theta}\\\end{array}\right),R_{y}(\theta)=\left(\begin{array}{ccc}{c\theta}&{0}&{s\theta}\\{0}&{1}&{0}\\{-s\theta}&{0}&{c\theta}\\\end{array}\right),R_{z}(\theta)=\left(\begin{array}{ccc}{c\theta}&{-s\theta}&{0}\\{s\theta}&{c\theta}&{0}\\{0}&{0}&{1}\\\end{array}\right),
\left(\begin{array}{c}{\pi}\\{\phi^{\prime}}\\\end{array}\right)\rightarrow\,\left(\begin{array}{cc}{cosh\,\varphi}&{sinh\,\varphi}\\{sinh\,\varphi}&{cosh\,\varphi}\\\end{array}\right)\,\left(\begin{array}{c}{\tilde{\pi}}\\{\tilde{\phi^{\prime}}}\\\end{array}\right)
\langleA\rangle=v\left(\begin{array}{ccccc}{}&{{\mathbf1}}&{}&{\ldots}&{}\\{}&{}&{{\mathbf1}}&{\ldots}&{}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{}&{}&{}&{\ldots}&{{\mathbf1}}\\{{\mathbf1}}&{}&{}&{\ldots}&{}\\\end{array}\right),
\frac{(-1)^{j}l^{+}!}{(l^{+}-j)!}(1-x)^{l-m-j}=\frac{\frac{d^{j}}{dx^{j}}(1-x)^{l^{+}}}{(1-x)^{m^{+}}}\quad\mathrm{and}\quad\frac{l^{-}!}{(m^{-}+j)!}(1+x)^{j}=\frac{\frac{d^{l-m-j}}{dx^{l-m-j}}(1+x)^{l^{-}}}{(1+x)^{m^{-}}},
\begin{array}{c}{A_{1}=\left(\begin{array}{cc}{m}&{m}\\{-m}&{3m}\\\end{array}\right)}\\{A_{2}=\left(\begin{array}{cc}{3m}&{m}\\{-m}&{m}\\\end{array}\right)}\\{A_{3}=\left(\begin{array}{cc}{1}&{2}\\{0}&{1}\\\end{array}\right)}\\\end{array}
\langleN\rangle_{\mathrm{T}}^{(2l-1)}\sim\delta_{l,1}\,\frac{1}{2\pi}\intdx\,\theta^{\prime}-\sqrt{\frac{2mT}{\pi}}\,e^{-m/T}\,\frac{1}{(2l-1)!}\,\intdx\,\left(\frac{\theta^{\prime}}{2T}\right)^{2l-1}+\dots
4\piL_{M}=-\frac{1}{4}e^{2\kappa\Psi}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-\frac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-e^{-2\kappa\Psi}V(\Phi^{a})\,
\begin{array}{c}{s\lambda_{2n}^{B}=\rho_{2n+1}^{F},\\\s\rho_{2n+1}^{F}=0,\\s\bar{C}_{2n-1}^{F}=-b_{2n}^{B},\\\sb_{2n}^{B}=0,}\\{s\lambda_{2n-1}^{F}=\rho_{2n}^{B},\\\s\rho_{2n}^{B}=0,\\s\bar{C}_{2n}^{B}=-b_{2n+1}^{F},\\\sb_{2n+1}^{F}=0,}\\\end{array}
S=\int\!\int_{W.S.}d\taud\sigma\left\{\frac{\alpha}{U^{2}}(X^{2}-Y^{2})\,\partial_{a}U\partial^{a}U-\partial_{a}U\partial^{a}V+\sum_{j=2}^{D-1}\partial_{a}X^{j}\partial^{a}X^{j}\right\}
\int_{-(n-1)\pi/2+3\epsilon_{0}/2}^{(n-1)\pi/2-\epsilon_{0}/2}d\theta_{2}\int_{-(n-1)\pi/2+\epsilon_{0}/2}^{\theta_{2}-\epsilon_{0}}d\theta_{1}~{\calF}_{0}(\theta_{1})^{k^{2}-1}e^{ikX(e^{i\theta_{1}})}{\calF}_{0}(\theta_{2})^{k^{2}-1}e^{ikX(e^{i\theta_{2}})}.
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[{B}_{i}^{w^{\prime}(2)}\right]=-\frac{1}{2}{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[\frac{\partial}{\partialw^{i}}[\Gamma^{(2)}]\right]+{K}_{\hat{0}\alpha}^{w^{\prime}},
\begin{array}{ll}{D_{\alpha}=\partial_{\alpha}+i\bar{\theta}^{\dot{\alpha}}\partial_{\alpha\dot{\alpha}}=\partial_{\alpha}^{-}~~}&{~\overline{{D}}_{\dot{\alpha}}=-\bar{\partial}_{\dot{\alpha}}-i\theta^{\alpha}\partial_{\alpha\dot{\alpha}}=-\bar{\partial}_{\dot{\alpha}}^{+}}\\\end{array}
\alpha:=\frac{3m^{2}}{p_{0}^{4}}-\frac{m^{4}}{p_{0}^{6}}-\frac{2}{p_{0}^{2}}\,,\quad\beta:=\frac{3m^{2}}{p_{0}^{2}}-\frac{m^{4}}{p_{0}^{4}}\,,\quad\gamma:=-\frac{m^{4}}{p_{0}^{2}}\,.
\begin{array}{rcl}{ds^{2}}&{=}&{\displaystyle\frac{dr^{2}}{V(r)}+V(r)\left[d\tau+4\,n\,\frac{x\,dy-y\,dx}{\left[1+\kappa(x^{2}+y^{2})\right]}\right]^{2}+}\\{}&{+}&{\displaystyle4\frac{r^{2}-n^{2}}{\left[1+\kappa(x^{2}+y^{2})\right]^{2}}\left(dx^{2}+dy^{2}\right)\.}\\\end{array}
(t\partial_{z}+z^{-1}(D_{\jmath_{1}}^{+}D_{\jmath_{1}}^{-}-2\jmath_{0}D_{\jmath_{1}}^{0}))-(\tilde{J}^{+}(z)D_{\jmath_{1}}^{-}+2\tilde{J}^{0}(z)D_{\jmath_{1}}^{0}+\tilde{J}^{-}(z)D_{\jmath_{1}}^{+})
\times\left[\Sigma_{\mu\nu}^{1}(x)\Sigma_{\mu\nu}^{1}(y)+\Sigma_{\mu\nu}^{1}(x)\Sigma_{\mu\nu}^{2}(y)+\Sigma_{\mu\nu}^{2}(x)\Sigma_{\mu\nu}^{2}(y)\right]\frac{K_{1}\left(m_{B}|x-y|\right)}{|x-y|},
\bigg(\frac{\partial^{2}}{\partialr^{2}}+\frac{g^{2}}{4}e^{\mp2i\theta}(\lambda_{2}-\lambda_{1})\bigg)\psi_{\pm}^{(1)}=0,\;\;\psi_{\pm}^{(2)}=\frac{2e^{\pmi\theta}}{g(\sqrt{\lambda_{1}}+\sqrt{\lambda_{2}})}\frac{\partial}{\partialr}\psi_{\pm}^{(1)}.
\frac{ds^{2}}{\sqrt{g_{s}}l_{s}}=\sqrt{\frac{u}{\sqrt{N}l_{s}}}\left[-\left(1-\frac{u_{0}^{2}}{u^{2}}\right)dt^{2}+\sum_{i=1}^{5}(dy^{i})^{2}+Nl_{s}^{2}\left(\frac{du^{2}}{u^{2}-u_{0}^{2}}+d\Omega_{3}^{2}\right)\right]\,
\psi(p)=-\frac{\lambda}{\pi^{2}i}\intd^{4}q\frac{\psi(q)}{\left[\left(p-q\right)^{2}+io\right]\left[\left(\frac{P}{2}-q\right)^{2}-m^{2}\right]\left[\left(\frac{P}{2}+q\right)^{2}-m^{2}\right]}.
\epsilon_{T}(\omega,{\bfk})=1+{\frac{m^{2}}{\omega|{\bfk}|}}[-{\frac{\omega}{2{\bfk}}}+{\frac{1}{4}}[1-{\frac{\omega^{2}}{{\bfk}^{2}}}]\mathrm{ln}|{\frac{\omega-|{\bfk}|}{\omega+|{\bfk}|}}|+{\frac{i\pi}{4}}[1-{\frac{\omega^{2}}{{\bfk}^{2}}}]\Theta(|{\bfk}|-\omega)],
\begin{array}{ll}{a_{0}=-3\displaystyle\int_{-\infty}^{\infty}(3\phi_{K}+\eta)\eta^{2}\eta_{D}\mathrm{d}x;}&{a_{1}=-9\displaystyle\int_{-\infty}^{\infty}(2\phi_{K}+\eta)\eta\eta_{D}^{2}\mathrm{d}x}\\{a_{2}=-9\displaystyle\int_{-\infty}^{\infty}\eta\eta_{D}^{3}\mathrm{d}x-6\alpha;}&{a_{3}=-3\displaystyle\int_{-\infty}^{\infty}\eta_{D}^{4}\mathrm{d}x=-{\frac{12}{35}}\.}\\\end{array}
\Gamma_{\mathrm{{1-loop}}}\propto\frac{1}{r^{4}}\intd^{4}y\bigl[F^{\mu}{_\nu}F^{\nu}{_\rho}F^{\rho}{_\kappa}F^{\kappa}{_\mu}-\frac{1}{4}F^{\mu}{_\nu}F^{\nu}{_\mu}F^{\rho}{_\kappa}F^{\kappa}{_\rho}\bigr].
(\mathcal{D}_{\mu}\,e)_{a}^{\nu}\;=\;\left[(\mathcal{D}_{\mu})^{\nu}{}_{\rho}\,\right]_{a}{}^{c}\;e_{c}^{\rho}\;:=\;(\nabla_{\mu})^{\nu}{}_{\rho}\,e_{a}^{\rho}+(\omega_{\mu})_{a}{}^{c}\,e_{c}^{\nu}=0
K(\alpha_{i},{\bar{\alpha}}_{i})\approx\sum_{i}^{n}\alpha_{i}\bar{\alpha}_{i}+{\frac{1}{4}}\sum_{i}^{n}\alpha_{i}^{2}\bar{\alpha}_{i}^{2}+\sum_{i<j}^{n}(\alpha_{i}\bar{\alpha}_{i}\alpha_{j}\bar{\alpha}_{j}-{\frac{1}{4}}\alpha_{i}^{2}\bar{\alpha}_{j}^{2}-{\frac{1}{4}}\alpha_{j}^{2}\bar{\alpha}_{i}^{2}),
\sum_{\nu\in\Sigma(G,n,i)}1/|S_{\nu}|=\sum_{p_{1},\ldots,p_{i}\inP_{n}}\sum_{s_{1},t_{1},\ldots,s_{G},t_{G}\inS_{n}}\left[\frac{1}{n!}\delta(p_{1}\ldotsp_{i}\prod_{j=1}^{g}s_{j}t_{j}s_{j}^{-1}t_{j}^{-1})\right].
w_{(m)}^{a}=\frac{w_{(n)}^{a}}{w_{(n)}^{m}},\;\;\quad\theta_{(m)}^{k}=\frac{\theta_{(n)}^{k}}{w_{(n)}^{m}},\quad\mathrm{where}\quadw_{(n)}^{m}=(w_{(n)}^{a},w_{(n)}^{n}=1).
\begin{array}{lllll}{\mathrm{for~any~\psi~,}}&{\mathrm{at~}\phi=+1}&{~:~}&{(i^{k},1,\pm\sqrt{\psi},0,0,1)}&{,(k=0,...,3)~,}\\{}&{\mathrm{or~}\phi=-1}&{~:~}&{(i^{k}e^{i\pi/4},1,\pm\sqrt{\psi},0,0,1)}&{,(k=0,...,3)~,}\\{\mathrm{and~for~any~\phi~,}}&{\mathrm{at~}\psi=\pm\sqrt{\phi^{2}-1}}&{~:~}&{(\phi^{1/4}i^{k},1,0,0,0,1)}&{,(k=0,...,3)~.}\\\end{array}
B\,=\,m+\frac{e^{2}}{2\pi}\frac{\lambda+1}{\lambda}\int_{0}^{\infty}dk\frac{kB}{A^{2}k^{2}+B^{2}}\frac{1}{\sqrt{(p^{2}+k^{2}+\mu^{2})^{2}-4p^{2}k^{2}}}
P^{-}=\int_{-\infty}^{\infty}dx^{-}\mathrm{Tr}\biggl[m_{b}^{2}\phi_{+}\phi_{-}-{\frac{g^{2}}{2}}{J}^{+}{\frac{1}{\partial_{-}^{2}}}{J}^{+}+{\frac{tg^{2}}{2}}\left[\phi_{+},\phi_{-}\right]^{2}+\sum_{h}{F}_{h}^{\dagger}{\frac{1}{\mathrm{i}\partial_{-}}}{F}_{h}^{\dagger}\biggr]
A_{\mu}^{I}(p^{I})=i{\frac{f(p)}{\sqrt{f^{2}(p)+g^{2}(p)}}}Q_{\mu\rho}A_{\rho}(p)-{\frac{g(p)}{\sqrt{f^{2}(p)+g^{2}(p)}}}\delta_{\mu\rho}^{\perp}A_{\rho}(p)
W_{-2}^{a}W_{-2}^{b}\mu=g^{ab}\left(L_{-2}^{2}+\frac52L_{-3}L_{-1}-\frac78L_{-4}\right)\mu+f_{c}^{ab}\left(2L_{-2}W_{-2}^{c}+\frac14W_{-4}^{c}\right)\mu\,.
V(x_{1},x_{2},x_{3}\cdotsx_{N})=\sum_{i<j}{\frac{g}{(x_{i}-x_{j})^{2}}}-\frac{g^{\prime}}{\sum_{i<j}(x_{i}-x_{j})^{2}}+U(\sqrt{\sum_{i<j}(x_{i}-x_{j})^{2}})\,\,\,.
\begin{array}{c}{\Theta_{1}^{+}=\left(\pi_{a}+iG_{,a}(z,\bar{z})\right)\eta_{1}^{a}+i{\bar{U}}_{,\bar{a}}({\bar{z}}){\bar{\eta}}_{2}^{\bar{a}},}\\{\Theta_{2}^{+}=\left(\pi_{a}-iG_{,a}(z,\bar{z})\right)\eta_{2}^{a}-i{\bar{U}}_{,{\bar{a}}}({\bar{z}}){\bar{\eta}}_{1}^{\bar{a}},}\\\end{array}
{\calN}^{-1}<\dot{x}(\tau_{1})\dot{x}(\tau_{2})>_{o}=D\ddot{G}_{B}(\tau_{1},\tau_{2})-{\frac{D}{2}}\frac{\dot{G}_{B}^{2}(\tau_{1},\tau_{2})}{L+G_{B}(\tau_{1},\tau_{2})}
S_{\mathrm{INT}}=\intd^{3}x\,d^{2}\theta\;\left(-\frac{1}{2}C^{\alpha\beta}ie\,\Gamma_{\alpha}\PhiD_{\beta}\Phi^{*}+\frac{1}{2}C^{\alpha\beta}ie\,\Gamma_{\alpha}\Phi^{*}D_{\beta}\Phi-\frac{1}{2}e^{2}C^{\alpha\beta}\Gamma_{\beta}\Gamma_{\alpha}\Phi^{*}\Phi\right).
G_{\lambda}(w,w^{\prime})=\frac{1}{2\pii}\,\int_{-i\infty}^{+i\infty}\,ds\,\Gamma(-s)\,e^{i\pis}\,\frac{\Gamma(\lambda+2s)}{\Gamma(\tilde{\nu}+s+1)}\,\frac{1}{2\pi^{\frac{d}{2}}}\,\left[\frac{w_{0}w_{0}^{\prime}}{w_{0}^{2}+{w^{\prime}}_{0}^{2}+(\vec{w}-\vec{w^{\prime}})^{2}}\right]^{\lambda+2s}
\sigma_{(2n+2)}^{1}=\Gamma_{(2n)}^{D+1},\quad\sigma_{(2n+2)}^{k}=-i\Gamma_{(2n)}^{k-1},\quadk=2,\dots,2n,\quad\sigma_{(2n+2)}^{2n+1}=\Gamma_{(2n)}^{0},
\zeta=\frac{a\tilde{\zeta}+b}{-\overline{{b}}\tilde{\zeta}+\overline{{a}}},\qquad\eta=\frac{\tilde{\eta}}{(-\overline{{b}}\tilde{\zeta}+\overline{{a}})^{2}},\qquad|a|^{2}+|b|^{2}=1,
S=\frac{1}{16\pi}\int\left\{-R+2\partial_{\mu}\phi\partial^{\mu}\phi+\frac{1}{2}e^{4\phi}{\partial_{\mu}}\kappa\partial^{\mu}\kappa-e^{-2\phi}F_{\mu\nu}F^{\mu\nu}-\kappaF_{\mu\nu}{\tilde{F}}^{\mu\nu}\right\}\sqrt{-g}\,d^{4}x,
{\frac{1}{-(\gamma\cdotD)^{2}+M^{2}}}={\frac{1}{-{\bfD}^{2}+\Phi^{2}+M^{2}}}+{\frac{1}{-{\bfD}^{2}+\Phi^{2}+M^{2}}}\left({\frac{i}{2}}\gamma_{\mu}\gamma_{\nu}G_{\mu\nu}\right){\frac{1}{-{\bfD}^{2}+\Phi^{2}+M^{2}}}+\cdots
D={\frac{\partial}{\partial\overline{{\theta}}}}-i\partial\!\!\!/\theta,\qquad\qquad\overline{{D}}=-{\frac{\partial}{\partial\theta}}+i\bar{\theta}\partial\!\!\!/,\qquad\qquadQ={\frac{\partial}{\partial\overline{{\theta}}}}+i\partial\!\!\!/\theta~.
{\calK}(\alpha^{1},\alpha^{2})=\sum_{\beta^{1},\beta^{2}=0,1}{\widetilde{\calK}}(\beta^{1},\beta^{2};\beta^{1}+\alpha^{1}~(\mathrm{mod}~2),\beta^{2}+\alpha^{2}~(\mathrm{mod}~2))~.
M(A,A^{\prime},T)=\left(\begin{array}{cc}{-iA_{\mu}({\bfX})D{\bfX}^{\mu}}&{\overline{{T}}({\bfX})}\\{T({\bfX})}&{-iA_{\mu}^{\prime}({\bfX})D{\bfX}^{\mu}}\\\end{array}\right),\{\bfX}^{\mu}=X^{\mu}+\sqrt{\alpha^{\prime}}\theta\psi^{\mu}\,
\begin{array}{c}{{\bfk}=\left\{\left(x,x\right),x\in{\bfa}\right\},}\\{{\bfp}=\left\{\left(x,-x\right),x\in{\bfa}\right\},}\\{\mathrm{so~that~correspondingautomorphism~}\sigma\mathrm{~is~}\sigma\left(x,y\right)=\left(y,x\right).}\\\end{array}
{\mathcalM}=\left(\begin{array}{c}{\frac12(M+M^{*})}\\{\frac1{2i}(M-M^{*})}\\\end{array}\right),\quad{\mathcalZ}=\left(\begin{array}{c}{\frac12(Z+Z^{*})}\\{\frac1{2i}(Z-Z^{*})}\\\end{array}\right).
A_{2}\astA_{1}({\mit\Gamma},\bar{\mit\Gamma})=A_{2}({\mit\Gamma},\bar{\mit\Gamma})\cdotA_{1}({\mit\Gamma},\bar{\mit\Gamma})+iA_{2}({\mit\Gamma},\bar{\mit\Gamma})\frac{\stackrel{\leftarrow}{\partial}\phantom{0}}{\partial{\mit\Gamma}^{A}}\Omega^{A\bar{B}}\frac{\stackrel{\rightarrow}{\partial}\phantom{0}}{\partial{\mit\Gamma}^{\bar{B}}}A_{1}({\mit\Gamma},\bar{\mit\Gamma})+{\calO}(s^{-2})\,,
{\mathcal{L}}_{I}=i{\overline{{\Psi_{I}}}}\gamma^{\mu}\partial_{\mu}\Psi_{I}-m_{I}{\overline{{\Psi_{I}}}}\Psi_{I}-\frac{1}{4}F_{I}^{\mu\nu}F_{\mu\nu}^{I}+ie_{I}\overline{{\Psi_{I}}}{\gamma^{\mu}}{A_{\mu}^{I}}{\Psi_{I}}~.
det\left|\begin{array}{ccc}{Q(\lambda-3i/2)}&{Q(\lambda\pmi/2)}&{Q(\lambda+3i/2)}\\{P(\lambda-3i/2)}&{P(\lambda\pmi/2)}&{P(\lambda+3i/2)}\\{R(\lambda-3i/2)}&{R(\lambda\pmi/2)}&{R(\lambda+3i/2)}\\\end{array}\right|=T^{\pm}(\lambda).
\left(\begin{array}{c}{\frac{1}{2}\astd\phi}\\{e^{\phi}\astd\chi}\\\end{array}\right)=\left(\begin{array}{cc}{1}&{\frac{1}{2}\chi}\\{0}&{e^{-\phi}}\\\end{array}\right)\left(\begin{array}{c}{\frac{1}{2}d\widetilde{\phi}}\\{d\widetilde{\chi}}\\\end{array}\right)
\Gamma[A]=\intd^{5}x\left\{A_{M}^{B}\Pi_{MN}^{BC}[-\partial_{5}^{2}]A_{N}^{C}+\frac{1}{2}\bigl(\delta(x_{5})+\delta(x_{5}-\piR)\bigr)A_{M}^{B}\widetilde\Pi_{MN}^{BC}[i\!\stackrel{\leftarrow}{\partial_{5}},i\!\stackrel{\rightarrow}{\partial_{5}}]A_{N}^{C}\right\}.
H=\left(\frac{P_{Q}^{2}}{2M}+\frac{P_{\varphi}^{2}}{2MQ^{2}}+\frac{1}{2}M\Omega^{2}{\bfQ}^{2}\right)\cdot{\bf1}+k\cdot\left(\begin{array}{cc}{0}&{Q\,\mathrm{e}^{-i\varphi}}\\{Q\,\mathrm{e}^{i\varphi}}&{0}\\\end{array}\right).
V=m_{3/2}^{2}\biggl\lbrace|1-{\frac{24\pi^{2}}{b_{0}}}Y|^{2}+\sum_{i=1}^{3}{\frac{Y}{8\pi^{2}Y-\delta_{GS}^{i}}}(1-3{\frac{\delta_{GS}^{i}}{b_{0}}})(T_{i}+\bar{T}_{i})^{2}|\hat{G}_{2}(T_{i})|^{2}-3\biggr\rbrace{}.
\hat{D}_{\mu\nu}^{-1}=\Bigl(1+\sum_{n=1}^{\infty}A_{n}\square^{n}\Bigr)\squareT_{\mu\nu}+\Bigl(1+\sum_{n=1}^{\infty}\frac{A_{n}}{\xi^{n}}\square^{n}\Bigr)\frac{1}{\xi}\squareL_{\mu\nu},
\intd^{4}xdy\sqrt{-G}\,{\frac{1}{2}}\bar{\Psi}_{M}\Gamma^{MNP}D_{N}\Psi_{P}=\int_{-1/K}^{1/K}dy\,e^{A}f_{0}^{2}\intdx^{4}\,{\frac{1}{2}}\bar{\psi}_{\alpha}^{(0)}\gamma^{\alpha\beta\delta}\partial_{\beta}\psi_{\delta}^{(0)}.
{\calF}=\left\{\mathrm{cos}\,\left(2\mathrm{Im}g_{n,m}(z)\right),\,\mathrm{sin}\,\left(2\mathrm{Im}g_{n,m}(z)\right)\bigg|({n,m})\in{\bfZ}^{2h}\right\},\quadg_{n,m}(z)\equiv\int_{P_{0}}^{z}\omega_{n,m},
M^{2}=\left({\frac{n_{p}}{R}}+{\frac{n_{w}R}{\alpha^{\prime}}}\right)^{2}+{\frac{4}{\alpha^{\prime}}}N_{R}=\left({\frac{n_{p}}{R}}-{\frac{n_{w}R}{\alpha^{\prime}}}\right)^{2}+{\frac{4}{\alpha^{\prime}}}N_{L}.
\sum_{m=0}^{\hat{l}}x^{-m}\frac{1}{\Gamma(\bar{l}-m+1)}\frac{1}{\Gamma(\hat{l}-m+1)}\frac{1}{\Gamma(m+1)}=\frac{x^{-\hat{l}}}{\Gamma(\bar{l}+1)}L_{\hat{l}}^{\bar{l}-\hat{l}}(-x),