harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E={\frac{1}{2}}m_{f}v_{\parallel}^{2}+{\frac{3m_{f}Q_{F}v_{\parallel}^{4}}{8(x^{2}+4Q_{p}z)^{\frac{n}{2}}}}+{\frac{5m_{f}Q_{F}^{2}v_{\parallel}^{6}}{16(x^{2}+4Q_{p}z)^{n}}}+...,
h\sum_{\ell=1}^{\infty}f\left(h\ell\right)=-\frac{h}{2}f\left(0\right)+\int_{0}^{\infty}dx\,f\left(x\right)+i\int_{0}^{\infty}\frac{f\left(iy\right)-f\left(-iy\right)}{e^{2\piy/h}-1},
V_{eff}=\left[e^{2}a^{i}a_{i}-\frac{1}{2\alpha^{\prime}}+\left(e\phi_{a}-\frac{\delta^{a}}{2\pi\alpha^{\prime}}\right)^{2}\right]T\overline{{T}}+\frac{e^{2}}{2}\left(T\overline{{T}}\right)^{2}
{\calZ}_{a}={\frac{1}{(8\pi^{2}\alpha^{\prime}t)^{2}}}{\frac{1}{\left[\eta(e^{-2\pit})\right]^{2+d_{a}}}}\left[{\calX}_{a}(e^{-2\pit})-{\calY}_{a}(e^{-2\pit})\right]~,
\left.\left(S+S_{\mathrm{gf}}+S_{\mathrm{gh}}\right)\right|_{\mathrm{bilinear}}=\intd^{4}x\sqrt{-g}\;\left({\Phi}\hat{H}{\Phi}^{T}+c_{\mu}\hat{H}_{\mathrm{gh}c^{\mu}}\right)\;,
U(g_{1})|\bbox{w}\pm\rangle\simeq(\alpha_{\bbox{v}}^{\pm}(g_{\bbox{w}^{\prime}}^{-1}g_{1}g_{\bbox{w}}))^{-1/2}|_{\bbox{v}=\bbox{0}}|{\bbox{w}}_{g_{1}}\pm\rangle\qquadg_{1}\in{\calG}.
\begin{array}{rcl}{I_{3}(0,m)}&{=}&{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}D^{2}(k_{1})\overline{{D_{m}(k_{2}+p)}}K_{12}.}\\\end{array}
{}^{(4)}G_{\mu}^{\nu}=-E_{\mu}^{\nu}=\frac{4\rho_{*}^{2}r^{4}}{(r^{2}+\rho_{*}^{2})^{4}}(\delta_{\mu}^{\tau}\delta_{\tau}^{\nu}-3\delta_{\mu}^{r}\delta_{r}^{\nu}+\delta_{\mu}^{\vartheta}\delta_{\vartheta}^{\nu}+\delta_{\mu}^{\varphi}\delta_{\varphi}^{\nu}).
\int_{-1}^{1}du\,\frac{(1+u)^{n}(1-u)^{n-1}}{(1+\alphau)^{n}}=\int_{-1}^{1}du\,(1-u)^{n-1}+\epsilon\left[n\int_{-1}^{1}du\,\frac{u(1-u)^{n-1}}{(1+u)}\right].
R_{\mu\nu}=\sum_{\kappa=0}^{3}\frac{\partialA_{\mu\nu}^{\kappa}}{\partialx^{\kappa}}-\frac{\partial}{\partialx^{\mu}}\left(\sum_{\kappa=0}^{3}A_{\kappa\nu}^{\kappa}\right)+\sum_{\kappa,\sigma=0}^{3}(A_{\mu\nu}^{\sigma}A_{\kappa\sigma}^{\kappa}-A_{\kappa\nu}^{\sigma}A_{\sigma\mu}^{\kappa}),
{\tilde{\Upsilon}}=\frac{a^{3/2}[m_{1}^{2}+m_{2}^{2}+\frac{1}{2}\,\dot{\phi}_{1}^{2}+\frac{1}{2}\,\dot{\phi}_{2}^{2}]^{1/2}}{m_{1}\,\dot{\phi}_{2}^{2}-m_{2}\,\dot{\phi}_{1}^{2}}\left(\dot{\phi}_{1}\,\chi_{1}+\dot{\phi}_{2}\,\chi_{2}\right),
\begin{array}{ccccc}{L_{1}}&{L_{1/2}}&{L_{0}}&{L_{-1/2}}&{L_{-1}}\\{P_{\mu}}&{Q_{A},\overline{{Q}}_{\dot{A}}}&{M_{\mu\nu},D,A}&{S_{A},\overline{{S}}_{\dot{A}}}&{K_{\mu}}\\\end{array}\,,
M=\sqrt{(P_{1}^{(1)})^{2}+\beta^{2}}+\sqrt{(P_{1}^{(2)})^{2}+\beta^{2}}+\sqrt{(Q_{2}^{(1)})^{2}+\beta^{2}}+\sqrt{(Q_{2}^{(2)})^{2}+\beta^{2}},
\frac{SU(1,1)}{U(1)}\times\frac{SO(2,\frac{n_{A}(n_{A}-1)}{2}+\frac{n_{B}(n_{B}-1)}{2})}{SO(2)\timesSO(\frac{n_{A}(n_{A}-1)}{2}+\frac{n_{B}(n_{B}-1)}{2})}\to\frac{SU(1,1)}{U(1)}\times\frac{SO(2,n_{1}n_{2}+n_{3}n_{4})}{SO(2)\timesSO(n_{1}n_{2}+n_{3}n_{4})}.
|\deltae|^{2}=\intd\lambda({\hat{e}}(\lambda;l))^{-1}(\deltae(\lambda;l))^{2}=\intd\lambda[-{\hat{e}}(\lambda)(\deltaf){\frac{d^{2}}{d\lambda^{2}}}(\deltaf)+{\frac{(\delta\rho[f(\lambda)])^{2}}{{\hat{e}}}}],
\hat{\Gamma}_{\mu}(\partial,\stackrel{\leftarrow}{\partial}^{\dagger})=\stackrel{\leftarrow}{\partial}^{\dagger\;\;a}g_{aj}\stackrel{\leftarrow}{\zeta}_{\mu}^{-\;\;j}+\stackrel{\leftarrow}{\zeta}_{\mu}^{-\;\;j}g_{ja}\partial^{a}
g_{\mu\nu}^{(6)}=\left(\begin{array}{cc}{g_{\alpha\beta}^{(5)}+e^{\psi_{1}}A_{\alpha}A_{\beta}}&{e^{\psi_{1}}A_{\alpha}}\\{e^{\psi_{1}}A_{\beta}}&{e^{\psi_{1}}}\\\end{array}\right),
\left<0|\hat{H}|0\right>=-c(D)\prod_{i=1}^{d-1}\left(\frac{L_{i}}{2\pi}\right)\int_{0}^{\infty}dp_{1}\int_{0}^{\infty}dp_{2}\,...\int_{0}^{\infty}dp_{d-1}\sum_{n=0}^{\infty}\left(p_{1}^{2}+...+p_{d}^{2}(n,L)\right)^{\frac{1}{2}}.
\hat{Q}={\frac{1}{2}}\sum_{h}\int\frac{d^{3}{\bfp}}{(2\pi)^{3}}\,{\frac{1}{2p_{0}}}\left[a_{h}^{\dagger}(p^{\mu})c_{h}(p^{\mu})+c_{h}^{\dagger}(p^{\mu})a_{h}(p^{\mu})-b_{h}(p^{\mu})d_{h}^{\dagger}(p^{\mu})-d_{h}(p^{\mu})b_{h}^{\dagger}(p^{\mu})\right]\quad,
iF_{ab}^{l}(x)e^{{\frac{ik}{4\pi}}S_{CS}}=i\epsilon_{abc}B^{c\,l}e^{{\frac{ik}{4\pi}}S_{CS}}={\frac{4\pi}{k}}\epsilon_{abc}{\frac{\delta}{\deltaA_{c}^{l}}}e^{{\frac{ik}{4\pi}}S_{CS}}.
S=\intdt\left[n^{-1}e^{-\varphi}\left(3\dot{\alpha}^{2}-\dot{\varphi}^{2}+6kn^{2}e^{-2\alpha}+6\dot{\beta}^{2}\right)+\frac{1}{2n}e^{\varphi+6\alpha}\dot{\sigma}^{2}-\frac{n}{2}Q^{2}e^{-6\beta+3\alpha}\right],
\frac{1}{l_{p}^{2}}ds^{2}=g^{-\frac{1}{3}}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+g^{-\frac{1}{3}}g_{m\overline{{n}}}dz^{m}dz^{\overline{{n}}}+g^{\frac{2}{3}}(4t^{2}dt^{2}+t^{4}d\Omega_{2}^{2})
\begin{array}{c}{\{p_{i}(x),p_{j}(y)\}=-\delta_{ij}\delta^{\prime}(x-y)\1\leqi,j\leql}\\{\{p_{i}(x),p_{j}(y)\}=\delta_{ij}\delta^{\prime}(x-y)\l+1\leqi,j\leqk+1+2l}\\{\{p_{i}(x),p_{j}(y)\}=0\1\leqi\leql,\l+1\leqj\leqk+1+2l}\\\end{array}
\Lambda={\frac{12(16-9a^{2})}{(4-9a^{2})^{2}}}e^{2aC}\bar{K}^{2},\\\\\\\\\\sigma_{\mathrm{DW}}={\frac{1}{\kappa_{5}^{2}}}{\frac{24}{4-9a^{2}}}e^{aC}\bar{K},
\Psi_{\vec{k},n}\longrightarrow\;\;\;e^{i\vec{k}.\vec{x}}e^{in\phi}+\sum_{m=-[KR]}^{+[KR]}T(\vec{k^{\prime}},m;\vec{k},n)\frac{e^{ik_{mn}^{\prime}|\vec{x}|}}{|\vec{x}|}e^{im\phi};
H=\dot{X}P+\left\langle\dot{\varphi}\,\,\pi\right\rangle-L=\frac12\left\{\left\langle\pi^{2}+\varphi^{\,\,\prime\,2}+2U(\varphi)\right\rangle+\frac{\left(\left\langle\varphi^{\,\prime}\pi\right\rangle+P\right)^{2}}{\left\langle\varphi^{\,\prime}{\calP}\varphi^{\,\prime}\right\rangle}\right\}\,.
\gamma_{i}=\left[\begin{array}{ll}{0}&{i\sigma_{i}}\\{-i\sigma_{i}}&{0}\\\end{array}\right],\gamma_{4}=\left[\begin{array}{ll}{0}&{1}\\{1}&{0}\\\end{array}\right],\gamma_{5}=\left[\begin{array}{ll}{1}&{0}\\{0}&{-1}\\\end{array}\right].
\begin{array}{c}{trM\left(\lambda\right)=A\left(\lambda\right)+D\left(\lambda\right),}\\{\mathrm{where~}M\left(\lambda\right)=\left(\begin{array}{cc}{A\left(\lambda\right)}&{B\left(\lambda\right)}\\{C\left(\lambda\right)}&{D\left(\lambda\right)}\\\end{array}\right)}\\\end{array}
\hat{\calO}_{\alpha}(t)=f_{\alpha}^{(-)}(t)\hat{b}_{\alpha}+f_{\alpha}^{(+)}(t)\hat{b}_{\alpha}^{\dagger}+g_{\alpha}^{(-)}(t)\hat{d}_{\alpha}+g_{\alpha}^{(+)}(t)\hat{d}_{\alpha}^{\dagger},
\hat{T}_{\mu\nu}=-\frac{1}{2}\left(\hat{F}_{\mu\lambda}\star\hat{F}_{\nu}^{\,\lambda}+\hat{F}_{\nu\lambda}\star\hat{F}_{\mu}^{\,\lambda}-\frac{1}{2}\,\eta_{\mu\nu}\,\hat{F}_{\lambda\rho}\star\hat{F}^{\lambda\rho}\right)
\left({\frac{d}{d\rho}}\left(F_{,\phi\dot{\phi}}+F_{,\dot{\phi}\dot{\phi}}{\frac{\partial\dot{\phi}}{\partial\phi}}\right)\right)_{\dot{\phi}}-F_{,\phi\phi}+F_{,\dot{\phi}\dot{\phi}}\left({\frac{\partial\dot{\phi}}{\partial\phi}}\right)^{2}~=~0.
{\calL}={\calL}_{gauge}+{\calL}_{mat}=-\,\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}-\,\mathrm{Im}\left\{\overline{{\psi_{K}}}\,{(\gamma^{\mu})^{K}}_{L}\,\left(D_{\mu}\psi^{L}\right)\right\}.
{\calD}^{-1}=\left(\begin{array}{cc}{2\betap^{2}\theta_{\mu\nu}+2\left(1+\frac{m\xi^{2}}{2\pi}\right)p^{2}\omega_{\mu\nu}}&{-\frac{im\xi\alpha}{2\pi}p_{\mu}}\\{\frac{im\xi\alpha}{2\pi}p_{\nu}}&{p^{2}-\mu^{2}-\frac{m\alpha^{2}}{\pi}}\\\end{array}\right).
g\left(q^{i},p_{i},z^{a},\overline{{p}}_{a}\right)=\sum_{k=1}^{\infty}\stackrel{(k)}{g}_{a_{1}\ldotsa_{k}}\left(q^{i},p_{i},\overline{{p}}_{a}\right)z^{a_{1}}\ldotsz^{a_{k}}.
\begin{aligned}{x^{-}(s)}&{{}=x_{0}^{-}}\\{x^{+}(s)}&{{}=x_{0}^{+}+2x_{0}^{1}\,s+x_{0}^{-}\,s^{2}}\\{x^{1}(s)}&{{}=x_{0}^{1}+x_{0}^{-}\,s}\\\end{aligned}
V=-\frac{\Gamma(-1/2)}{\sqrt{8\pi^{2}\alpha^{\prime}}}[(\frac{b^{2}}{2\pi\alpha^{\prime}}-\pi(\epsilon_{1}-\epsilon_{2}))^{1/2}+(\frac{b^{2}}{2\pi\alpha^{\prime}}+\pi(\epsilon_{1}-\epsilon_{2}))^{1/2}-2(\frac{b^{2}}{2\pi\alpha^{\prime}})^{1/2}]
{\calL}={\frac{m}{2}}\,\epsilon^{\mu\nu\lambda}A_{\mu}^{a}(\partial_{\nu}A_{\lambda}^{a}+{\frac{g}{3}}f^{abc}A_{\nu}^{b}A_{\lambda}^{c})-F^{a}(n^{\mu}A_{\mu}^{a})-\overline{{c}}^{a}n^{\mu}(D_{\mu}c^{a})
Z=\int{\calD}\bar{\Psi}{\calD}\Psi{\calD}A_{\mu}e^{\intd^{2}x[\bar{\Psi}\gamma^{\mu}(\partial_{\mu}-{\bfA}_{\mu})\Psi-m\bar{\Psi}\Psi-{\frac{1}{4{\tilde{g}}^{2}}}\mathrm{Tr}F^{\mu\nu}F_{\mu\nu}]}.
=\;2^{N}\prod_{a=1}^{N}\sum_{n_{a}=0}^{\infty}\frac{(\beta^{(a)})^{2n_{a}}}{(2n_{a})!}\prod_{b=1}^{N}\sum_{q_{b}=0}^{2n_{b}}{\binom{2n_{b}}{q_{b}}}\left\langle\prod_{c=1}^{N}\left[\chi_{+}^{(c)}[Q^{M}]\right]^{q_{c}}\left[\chi_{-}^{(c)}[Q^{M}]\right]^{2n_{c}-q_{c}}\right\rangle_{Q^{M}}.
\begin{array}{c}{D(\beta,R)\k(t,\vec{x};t^{\prime},\vec{x^{\prime}})=\delta(\vec{x}-\vec{x^{\prime}})\delta(t-t^{\prime})}\\{Be^{\mu\betat}k(t,\vec{x};t^{\prime},\vec{x^{\prime}})=0,{\\mathrm{for\}}|\vec{x}|=1,}\\{k(1,\vec{x};t^{\prime},\vec{x^{\prime}})+e^{-\mu\beta}k(0,\vec{x};t^{\prime},\vec{x^{\prime}})=0.}\\\end{array}
H_{\mathrm{R}}^{(\sigma)}=\int_{0}^{\infty}\!d\Omega\,\Omega\!\sum_{\{T,\lambda\}}\sum_{A,B}b_{\Omega\{T,\lambda\}}^{(\sigma)A\dagger}\eta_{AB}b_{\Omega\{T,-\lambda\}}^{(\sigma)B}\.
E={\frac{1}{2}}m_{f}\dot{z}^{2}+V(z);\\\\V(z)=E[1-{\frac{1}{H_{p}}}{\frac{1+{\frac{E}{2m_{f}}}H_{F}}{(1+{\frac{E}{m_{f}}}H_{F})^{2}}}]+{\frac{J_{\perp}^{2}}{2m_{f}H_{p}^{2}z^{2}}}{\frac{1}{(1+{\frac{E}{m_{f}}}H_{F})^{2}}}.
D^{\mu\nu}(q;\phi)\equiv{\frac{-i}{q^{2}}}\Bigl\{g^{\mu\nu}+iq^{\mu}\widetilde\phi^{\nu}(q)-i{\widetilde\phi^{\mu*}}(q)q^{\nu}+q^{\mu}q^{\nu}\vert\widetilde\phi(q)\vert^{2}\Bigr\}\,
\begin{array}{l}{T(x^{\prime\prime})=L(l^{\prime})T(x)L^{+-1}(l^{\prime})+T(x^{\prime})+}\\{\phantom{T(x^{\prime\prime})=}1/2i\left[\Xi(\xi^{\prime})U(u^{\prime})\Xi(\xi)L^{+-1}(l^{\prime})-\right.}\\{\phantom{T(x^{\prime\prime})=asdf}\left.L(l^{\prime})\Xi\left(\xi\right)U^{-1}(u^{\prime})\Xi^{+}(\xi^{\prime})\right]}\\\end{array}
\tilde{A}_{\tilde{\mathcal{N}}_{i},\tilde{M_{i}},\tilde{\lambda}_{i}}\equiv\tilde{\mathcal{N}}_{i}e^{-\bar{\eta}\tilde{M}_{i}\eta-\bar{\eta}\tilde{\lambda}_{i}},\quad\tilde{\mathcal{N}}_{i}=g(\tau_{i},p_{i})e^{\bar{\epsilon}_{i}H(\tau_{i},p_{i})-\bar{\epsilon}_{i}F(\tau_{i})\epsilon_{i}},\,\tilde{M}
\begin{array}{c}{\bar{W}_{k}(\sigma^{+},\sigma^{-})=\left[\frac1{U_{k}(\sigma^{+})+v_{k}}+\frac1{U_{k}(\sigma^{-})+v_{k}}-\frac1{u_{k-1}+v_{k}}\right]^{-1}-v_{k}}\\{U_{k+1}(z)=\left[\frac1{U_{k}(z)+v_{k}}+\frac1{u_{k}+v_{k}}-\frac1{u_{k-1}+v_{k}}\right]^{-1}-v_{k}}\\{u_{k+1}=\left[\frac2{u_{k}+v_{k}}-\frac1{u_{k}+v_{k-1}}\right]^{-1}-v_{k}}\\\end{array}
\phi^{\alpha_{1},\dots,\alpha_{2j}}=v\,U_{\dot{\alpha}_{1}}^{\alpha_{1}}\,\cdots\,U_{\dot{\alpha}_{2j}}^{\alpha_{2j}}\,c^{\dot{\alpha}_{1}\cdots\dot{\alpha}_{2j}}\,,\quad\bar{\phi}^{\alpha_{1},\dots,\alpha_{2j}}=\bar{v}\,U_{\dot{\alpha}_{1}}^{\alpha_{1}}\,\cdots\,U_{\dot{\alpha}_{2j}}^{\alpha_{2j}}\,\bar{c}^{\dot{\alpha}_{1}\cdots\dot{\alpha}_{2j}}\,,
\hat{H}_{0}=\hbarg\omega\left[{\alpha_{i}^{+}}^{\dagger}\alpha_{i}^{+}+{\alpha_{i}^{-}}^{\dagger}\alpha_{i}^{-}+d\right]=\hbarg\omega\left[\hat{N}+d\right]\\,\\\hat{\phi}=-\hbar\left[{\alpha_{i}^{+}}^{\dagger}\alpha_{i}^{+}-{\alpha_{i}^{-}}^{\dagger}\alpha_{i}^{-}\right]\\.
X_{0}=\frac{i}{\kappa}\,\left(\begin{array}{ccc}{0}&{\mathbf{0}}&{1}\\{\mathbf{0}}&{\mathbf{0}}&{\mathbf{0}}\\{1}&{\mathbf{0}}&{0}\\\end{array}\right)\quad\vec{X}=\frac{i}{\kappa}\,\left(\begin{array}{ccc}{0}&{{\vec{\epsilon}\,{}^{T}}}&{0}\\{\vec{\epsilon}}&{\mathbf{0}}&{\vec{\epsilon}}\\{0}&{-\vec{\epsilon}\,{}^{T}}&{0}\\\end{array}\right),
S_{TM}^{4}=\intd^{4}x\Bigl[-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}-{\frac{1}{12\mu^{2}}}H_{\mu\nu\lambda}H^{\mu\nu\lambda}-{\frac{1}{4}}\varepsilon^{\mu\nu\lambda\rho}B_{\mu\nu}F_{\lambda\rho}\Bigl]\;,
S=\frac{1}{16\pi}\int_{B}d^{D}x\,\sqrt{|g|}\left(R+\alpha^{2}\left(d(d-1)\right)\right)-\frac{1}{8\pi}\int_{\partial{B}}d^{d}x\sqrt{|\gamma|}K+\frac{1}{8\pi}S_{(ct)}(\gamma).
\begin{array}{rrrll}{{\calB}_{0}}&{:}&{f_{0}-1}&{=}&{0\;,}\\{{\calB}_{1}}&{:}&{f_{1}-f_{2}}&{=}&{0}\\{\mathrm{~and~}\;\;\;\tilde{\calB}_{12}}&{:}&{F(f_{0}^{-1/2}f_{1},f_{0}^{-1/2}f_{2})}&{=}&{0}\\\end{array}
\begin{array}{rcl}{w_{\alpha}(z_{1},\cdots,z_{N})}&{=}&{\displaystyle\prod_{l=1}^{m}~{\calT}_{1n}(z_{1},\cdots,z_{N}|z_{l}^{(n)})\cdots\prod_{l=1}^{m}~{\calT}_{12}(z_{1},\cdots,z_{N}|z_{l}^{(2)})\Omega,}\\{\Omega}&{=}&{\displaystylev_{1}\otimes\cdots\otimesv_{1}\quad\inV^{\otimesN}.}\\\end{array}
S_{DBI}=\intd^{4}\xi-\sqrt{-g}f\,\f=\sqrt{1+{\tilde{H}}_{i,r}(\epsilon^{T}M\epsilon)^{rs}{{\tilde{H}}^{i}}_{,s}+\frac{1}{2}({\tilde{H}}_{i,r}\epsilon^{rs}{\tilde{H}}_{j,s})({{\tilde{H}}^{i}}_{,t}\epsilon^{tu}{{\tilde{H}}^{j}}_{,u})}.
\eta_{1}(t)=\frac{1}{W_{0}}\left\lbrackf_{2}(t-t_{0})\int_{t_{0}}^{t}f_{1}(t^{\prime}-t_{0})h(t^{\prime})dt^{\prime}-f_{1}(t-t_{0})\int_{t_{0}}^{t}f_{2}(t^{\prime}-t_{0})h(t^{\prime})dt^{\prime}\right\rbrack\;.
I=-\intd^{4}x\left[\frac{1}{2}B_{i}^{\alpha}{\cal{L}}_{\alpha\beta}E_{i}^{\beta}+\sqrt{1+B_{i}^{\alpha}B_{i}^{\alpha}+\frac{1}{4}(\epsilon^{ijk}B_{j}^{\alpha}{\cal{L}}_{\alpha\beta}B_{k}^{\beta})^{2}}-1\right],
\begin{array}{c}{\nablax^{\pm2}=dx^{\pm2}\pm\Omega^{(0)}x^{\pm2},\nablax^{i}=dx^{i}-\Omega^{ij}x^{j},}\\{\nabla\theta_{q}^{\pm}=d\theta_{q}^{\pm}\pm\frac12\Omega^{(0)}\theta_{q}^{\pm}-\frac14\Omega_{qp}\theta_{p}^{\pm},\nabla\theta_{\dot{q}}^{\pm}=d\theta_{\dot{q}}^{\pm}\pm\frac12\Omega^{(0)}\theta_{\dot{q}}^{\pm}-\frac14\Omega_{\dot{q}\dot{p}}\theta_{\dot{p}}^{\pm},}\\\end{array}
\tilde{T}^{a}=d\tilde{e}^{a}+\tilde{\omega}_{\cdotb}^{a}\wedge\tilde{e}^{b},\;\;\;\;\;\tilde{R}^{ab}=d\tilde{\omega}^{ab}+\tilde{\omega}_{\cdotc}^{a}\wedge\tilde{\omega}^{cb}-\frac{1}{4}\tilde{e}^{a}\wedge\tilde{e}^{b}+\frac{1}{2}C_{\;\cdotc}^{ab}\tilde{T}^{c},
{\calL}_{0}\left(P_{\alpha}f\right)(\lambda)=\frac{1}{\Gamma\left(\alpha\right)}\int_{\gamma_{0}}\mathrm{e}^{\mathrm{i}\lambda\theta}d\theta\int_{\gamma_{\left(0,\theta\right)}}\left(\theta-\theta^{\prime}\right)^{\alpha-1}f\left(\theta^{\prime}\right)d\theta^{\prime}\tag{B.13}
\underline{{\underline{R}}}_{ab}=-\frac1{\ell^{2}}\underline{{\Theta}}_{a}\wedge\underline{{\Theta}}_{b}+\stackrel{(0)}{\underline{{\underline{{\calR}}}}}_{ab}+\frac1{\ell^{2}}(\underline{{\theta}}_{a}\wedge\underline{{\gamma}}_{b}+\underline{{\gamma}}_{a}\wedge\underline{{\theta}}_{b})+O(r^{-2})\qquad,
S_{\mathrm{ct}}=-\frac{1}{8\piG}\intd^{3}x\sqrt{-h}\frac{c_{0}}{l_{\mathrm{eff}}},\\\c_{0}=2\sqrt{\frac{3N}{4N-1}},\\\\frac{1}{l_{\mathrm{eff}}}=e^{b\phi}\sqrt{\frac{\Lambda}{3}}.
ds^{2}=(Ff_{1}\cdotsf_{7}f_{8})^{1/2}(a_{1}(r)\cdotsa_{7}(r))^{1/2}\times\left({\frac{(\sigma_{1})^{2}}{f_{1}^{2}a_{1}^{2}}}+\cdots{\frac{(\sigma_{7})^{2}}{f_{7}^{2}a_{7}^{2}}}+{\frac{dr^{2}}{f_{8}^{2}}}\right)
f\astg:=f\cdotg+\sum_{n=1}^{\infty}{\frac{(\lambda/2)^{n}}{n!}}\epsilon^{r_{1}s_{1}}\dots\epsilon^{r_{n}s_{n}}{\frac{\partial^{n}f}{\partial\phi^{r_{1}}\dots\partial\phi^{r_{n}}}}{\frac{\partial^{n}g}{\partial\phi^{s_{1}}\dots\partial\phi^{s_{n}}}},
w=\left(\begin{array}{c}{n^{1}}\\{n^{2}}\\{0}\\{0}\\{n^{1}}\\{n^{2}}\\\end{array}\right)\,\\n^{1},n^{2}\inZ\\\\mathrm{and}\\\p=\left(\begin{array}{c}{m_{1}}\\{m_{2}}\\{-m_{1}}\\{-m_{2}}\\{m_{1}}\\{m_{2}}\\\end{array}\right)\,\;m_{1},m_{2}\inZ.
{\calV}=-{\frac{f_{\pi}^{2}}{2}}{\calR}_{\mathrm{i}}\cdot{\calR}_{\mathrm{i}}-{\frac{1}{4e^{2}}}\biggl[{\calR}_{\mathrm{i}}\cdot{\calR}_{\mathrm{i}}\;{\calR}_{\mathrm{j}}\cdot{\calR}_{\mathrm{j}}-{\calR}_{\mathrm{i}}\cdot{\calR}_{\mathrm{j}}\;{\calR}_{\mathrm{j}}\cdot{\calR}_{\mathrm{i}}\biggr].
ds_{E}^{2}=\breve{g}_{\mu\nu}dx^{\mu}dx^{\nu}=(1+r^{2}\breve{\Omega}^{2})d\tau^{2}-2\breve{\Omega}r^{2}d\taud\tilde{\varphi}+r^{2}d\tilde{\varphi}^{2}+dr^{2}+dz^{2}~.
{\frac{\partial}{\partialt}}V(\varphi,t)+{\frac{1}{2}}(D-2)\varphi_{a}{\frac{\partialV}{\partial\varphi_{a}}}-DV={\frac{\partial^{2}V}{\partial\varphi_{a}\partial\varphi_{a}}}-\left({\frac{\partialV}{\partial\varphi_{a}}}\right)^{2}
\hat{b}_{n}=i\sqrt{\frac{2\pi^{2}}{|k|n}}\,\hat{\pi}_{n},\qquad\qquad\hat{b}_{n}^{\dagger}=-i\sqrt{\frac{2\pi^{2}}{|k|n}}\,\hat{\pi}_{-n},\;\;\;(n>0)
\begin{array}{l}{\S_{k}f_{(+)}(\theta_{+k})f_{(-)}(\theta_{-k})=\S_{k}f_{(+)}^{\prime}(\theta_{+k}^{\prime})f_{(-)}^{\prime}(\theta_{-k}^{\prime})=\cdots=inv,}\\{\S_{k}f_{(-)}(\theta_{-k})f_{(+)}(\theta_{+k})=\S_{k}f_{(-)}^{\prime}(\theta_{-k}^{\prime})f_{(+)}^{\prime}(\theta_{+k}^{\prime})=\cdots=inv.}\\\end{array}
m_{F,0}^{2}-m_{B,0}^{2}\simeq\frac{4N(g_{1}^{2}\phi^{2})\cdots(g_{N}^{2}\phi^{2})\theta^{2}}{\sum_{k=1}^{N}(g_{1}^{2}\phi^{2})\cdots\widehat{(g_{k}^{2}\phi^{2})}\cdots(g_{N}^{2}\phi^{2})}
Q_{BRST}=\intd^{3}x\{C\Phi_{A}+C_{i}\Phi_{H}^{i}+iC_{(1)}\partial_{i}\overline{{P}}^{i}+PB+P_{i}B^{i}+P_{(1)}B_{(1)}+P_{1}^{1}B_{1}^{1}\}.
G_{o}(K;\vec{x},\phi;\vec{x^{\prime}},\phi^{\prime})=-\frac{1}{(2\pi)^{4}}\sum_{n=-\infty}^{n=+\infty}\intd^{3}p\frac{e^{i\vec{p}.(\vec{x}-\vec{x^{\prime}})}e^{in(\phi-\phi^{\prime})}}{[p^{2}+n^{2}/R^{2}-K^{2}-i\epsilon]},
R=\left[\begin{array}{llll}{q}&{\,}&{\,}&{\,}\\{\,}&{1}&{0}&{\,}\\{\,}&{\lambda}&{1}&{\,}\\{\,}&{\,}&{\,}&{q}\\\end{array}\right]\;,\quad\hat{R}=\left[\begin{array}{llll}{q}&{\,}&{\,}&{\,}\\{\,}&{\lambda}&{1}&{\,}\\{\,}&{1}&{0}&{\,}\\{\,}&{\,}&{\,}&{q}\\\end{array}\right]\;,\quad{\calP}R{\calP}=R^{t}\;,
Q_{\mathrm{BRST}}^{\mathrm{L}}\,V_{\zeta}^{(-1,0)}=\Big(\oint\!\frac{\mathrm{d}w}{2\pii}\,e^{\phi}\partialX^{\mu}\Psi_{\mu}(w)+\cdots\Big)\Big(\int\!\mathrm{d}^{2}z\,\zeta_{\mu\nu}\Psi^{\mu}\bar{\partial}X^{\nu}e^{-\phi}(z)+\cdots\Big)\,,
{\calL}_{\mathrm{FP}}=\frac{1}{2\xi}\bar{c}\,\partial^{2}\,c+\frac{i}{4\xi}\left(\bar{c}\astA_{\mu}\ast\partial^{\mu}c-A_{\mu}\ast\partial^{\mu}c\ast\bar{c}\right)+i\xiv\bar{c}*\chi*c,
\phi(z,\bar{z})={\mathrme}^{-i\pi/4}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\left[d_{kl}z^{1+4k}(z\bar{z})^{l}+e_{kl}{\bar{z}}^{3+4k}(z\bar{z})^{l}\right].
\hat{E}_{\mathrm{gh}}^{\mu}:=\hat{K}_{\mathrm{gh}}^{-1}\hat{L}_{\mathrm{gh}}^{\mu}\;\;,\;\;\hat{D}_{\mathrm{gh}}:=\hat{K}_{\mathrm{gh}}^{-1}\hat{M}_{\mathrm{gh}}\;\;.
Q_{\alpha}^{-}=\int_{0}^{\infty}[...]b_{\alpha}^{\dagger}(k_{3})a_{I}(k_{1})a_{I}(k_{2})+...+(\beta_{I}\beta_{J}^{T}-\beta_{J}\beta_{I}^{T})_{\alpha\beta}[...]b_{\beta}^{\dagger}(k_{3})a_{I}(k_{1})a_{J}(k_{2})+...
\begin{array}{c}{a_{i}a_{j}=e^{c_{ji}-c_{ij}}a_{j}a_{i}\quadi\neqj}\\{a_{i}a_{j}^{+}=e^{c_{ij}+c_{ji}^{*}}a_{j}^{+}a_{i},\quadi\neqj}\\{a_{i}a_{i}^{+}=|\varphi_{i}(N_{i}+1)|e^{\sum_{j}(c_{ij}+c_{ij}^{*})N_{j}}e^{(c_{ii}+c_{ii}^{*})}}\\{a_{i}^{+}a_{i}=|\varphi_{i}(N_{i})|e^{\sum_{j}(c_{ij}+c_{ij}^{*})N_{j}}.}\\\end{array}
R^{\alpha}{}_{\beta\gamma}{}^{\delta}=2\delta_{(\beta}^{\alpha}\delta_{\gamma)}^{\delta}-C_{\beta\gamma\epsilon}\bar{C}^{\alpha\delta\epsilon}\;\qquad\bar{C}^{\alpha\delta\epsilon}\partial_{\bar{\alpha}}\left(e^{-K}C_{\alpha\beta\gamma}\right)=0\;\qquad{\calD}_{[\alpha}e^{K}C_{\beta]\gamma\delta}=0\.
T_{1}=\frac{1}{2}\left(\begin{array}{cccc}{-1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{-1}\\\end{array}\right)\;\;\;T_{2}=\frac{1}{2}\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{-1}\\\end{array}\right)
{\bfX}_{a}=\left[\begin{array}{cc}{\hat{r}_{a}+\hat{X}_{a}+\hat{Z}_{a}}&{Y_{a}}\\{\Y_{a}^{\dagger}}&{\tilde{r}_{a}+\tilde{X}_{a}+\tilde{Z}_{a}}\\\end{array}\right],\;\;\;\;\;\;\;\;\;\;\Theta_{\alpha}=\left[\begin{array}{cc}{\hat{\theta}_{\alpha}+\hat{\eta}_{\alpha}}&{\chi_{\alpha}}\\{\chi_{\alpha}^{\dagger}}&{\tilde{\theta}_{\alpha}+\tilde{\eta}_{\alpha}}\\\end{array}\right]\;.
S_{\mathrm{eff}}(T=0)=\intdt\left(\frac{1}{2g_{s}\sqrt{\alpha^{\prime}}}\sum_{\alpha=1}^{2}({\dot{\vec{q}}}\,{}^{\alpha})^{2}-\frac{15}{16}\left(\alpha^{\prime}\right)^{3}\frac{\vert{\dot{\vec{q}}}\,{}^{1}-{\dot{\vec{q}}}\,{}^{2}\vert^{4}}{\vert{\vec{q}}\,{}^{1}-{\vec{q}}\,{}^{2}\vert^{7}}+\ldots\right)
\Omega_{1}=\left(\begin{array}{cc}{I_{d}}&{\gamma}\\{0}&{I_{d}}\\\end{array}\right)\quad\Omega_{2}=\left(\begin{array}{cc}{I_{d}}&{0}\\{\alpha}&{I_{d}}\\\end{array}\right)\quad\Omega_{3}=\left(\begin{array}{cc}{{A^{-1}}^{T}}&{0}\\{0}&{A}\\\end{array}\right),
\frac{\partial\zeta^{(\lambda\mu\alpha)}}{\partial\zeta^{(\tau,\nu,\beta)}}=\psi_{(\tau,\nu,\beta)}^{(\lambda\mu\alpha)}=\frac{1}{2}\left(D_{(\tau,\nu,\beta)}^{(\lambda\mu\alpha)}+\omega_{(\tau,\nu,\beta)}^{(\rho,\omega,\gamma)}D_{(\rho,\omega,\gamma)}^{(\lambda\mu\alpha)}\right).
H_{D}=\lambda(\tau)H(\tau)-\vec{\lambda}(\tau)\cdot\vec{H}_{p}(\tau)+\int{d^{3}\sigma\Big[-A_{\tau}(\tau,\vec{\sigma})\Gamma(\tau,\vec{\sigma})+\mu_{\tau}(\tau,\vec{\sigma})\pi^{\tau}(\tau,\vec{\sigma})\Big]},
Z_{66^{\prime}}(\theta^{k})\rightarrow(1-1)\frac{1}{16\pi^{4}t}\frac{L_{8}}{L_{9}}\prod_{i=1,2}(-1)^{p_{i}+1}\sum_{I}(\mathrm{Tr}\gamma_{k,6,I})\sum_{I^{\prime}}(\mathrm{Tr}\gamma_{k,6^{\prime},I^{\prime}}).
\tilde{R}=\frac{\tilde{a}M^{2}(\gamma-1)^{2}e^{\phi_{0}}}{\rho^{4}}\;\left(1-\frac{\rho_{1}}{\rho}\right)\;\left(1-\frac{\rho_{0}}{\rho}\right)^{-\frac{1+3k^{2}-2l}{1+k^{2}}}\;.
\rho_{(1)}^{\;I}=\sqrt{\frac{\pi}{2\kappa_{(10)}^{2}}}\;(4\pi^{2}\alpha^{\prime})\\\\mathrm{and}\\\\rho_{(5)}^{\;I}=2\times\sqrt{\frac{\pi}{2\kappa_{(10)}^{2}}}\;(4\pi^{2}\alpha^{\prime})^{-1}\.
g(z)=\frac{(x^{6}z;x^{4},x^{4},x^{2r})_{\infty}(x^{2}z^{-1};x^{4},x^{4},x^{2r})_{\infty}(x^{2r+6}z;x^{4},x^{4},x^{2r})_{\infty}(x^{2r+2}z^{-1};x^{4},x^{4},x^{2r})_{\infty}}{(x^{8}z;x^{4},x^{4},x^{2r})_{\infty}(x^{4}z^{-1};x^{4},x^{4},x^{2r})_{\infty}(x^{2r+4}z;x^{4},x^{4},x^{2r})_{\infty}(x^{2r}z^{-1};x^{4},x^{4},x^{2r})_{\infty}}.
\frac{k\mu}{2\pi}Q_{\alpha}\equivJ^{-}F_{+\alpha}-\frac{J^{3}}{2}F_{-\alpha}+\frac{d-1}{2C_{\rho}}B^{a}\eta^{\beta\gamma}\left(\lambda^{a}\right)_{\alpha\beta}F_{-\gamma}+\frac{k}{2\pi}\partial_{\theta}F_{-\alpha}(\theta)
C_{R}^{MN}(p,x^{i})={\delta^{M}}_{\mu}{\delta^{N}}_{\nu}\left[2\epsilon^{\mu\nu}{}_{\sigma\rho}F_{-}^{\sigma\rho}(p)-L^{2}p^{2}C_{R}^{\mu\nu}(p,0)\right]\int{\frac{d^{2}k}{(2\pi)^{2}}}~{\frac{e^{-ik\cdotx}}{k^{2}+p^{2}}}~.
\tilde{J}_{+}=\frac{\sqrt{2}}{\zeta}\,\bar{\tilde{z}}_{2}\bar{\tilde{z}}_{1}\,,\quad\tilde{J}_{-}=\frac{\sqrt{2}}{\zeta}\,\tilde{z}_{1}\tilde{z}_{2}\,,\quad\tilde{J}_{3}=\frac{1}{\zeta}\left(\bar{\tilde{z}}_{2}\tilde{z}_{2}-\tilde{z}_{1}\bar{\tilde{z}}_{1}\right)\,.
\vec{W}=C\left(\frac{\alpha\Omegay}{r^{3}}-\frac{\betaxz}{r^{5}},\quad-\frac{\alpha\Omegax}{r^{3}}-\frac{\betayz}{r^{5}},\quad\frac{\beta(x^{2}+y^{2})}{r^{5}}-\frac{2\alpha}{r^{3}}\right).
\zeta_{A}^{\prime}(s/2)=\frac{1}{\sqrt{\pi}}\,\frac{\Gamma(s-1/2)}{\Gamma(s)}\,\sum_{n=1}^{\infty}\frac{y_{-}}{(y_{+}y_{-})^{s}}\,F(s/2,1/2;s;1-\eta),\\\\,Re(s)>1
\frac{\psi_{1}{}_{(\alpha_{1},\alpha_{2},\beta_{3}+s_{3})}\prod_{j=3}^{p-2}\psi_{1}{}_{(\beta_{j}+s_{j},\alpha_{j},\beta_{j+1}+s_{j+1})}\psi_{1}{}_{(\alpha_{p-1},\beta_{p-1}+s_{p-1},\alpha_{p})}}{\psi_{1}{}_{(\alpha_{1},\alpha_{2},\beta_{3})}\prod_{j=3}^{p-2}\psi_{1}{}_{(\beta_{j},\alpha_{j},\beta_{j+1})}\psi_{1}{}_{(\alpha_{p-1},\beta_{p-1},\alpha_{p})}}=\psi_{0,p}[\alpha,\beta,s]
S=\frac{N}{4g^{2}}\sum_{\mu,\nu=1}^{10}\sum_{\bfm,n,k}{\calS}({\bfm-n},{\bfn}){\calS}({\bf-m-k},{\bfk})\,A_{\bfm-n}^{\mu}A_{\bf-m-k}^{\mu}A_{\bfn}^{\nu}A_{\bfk}^{\nu},
ds^{2}=\left({\frac{R}{\hat{x}^{3}}}\right)^{2}[dx^{i}\eta_{ij}dx^{j}+(d\hat{x}^{3})^{2}]=\left({\frac{R}{\hat{x}^{3}}}\right)^{2}d\hat{x}^{m}d\hat{x}^{n}\eta_{mn}.
\hat{\xi}^{\alpha}=\hat{\xi}_{\mathrm{triv}}^{\alpha}=\sum_{k,l}\left(\frac{d^{l}}{dt^{l}}f_{\alpha^{\prime}}\right)u^{l\alpha^{\prime}|k\alpha}\frac{d^{k}}{dt^{k}}\,,\;\;u^{l\alpha^{\prime}|k\alpha}=-u^{k\alpha|l\alpha^{\prime}}\,,
|v|^{2}\left({\frac{n}{2}}\right)+\bar{v}v_{-}^{n/2-1}\sqrt{{\frac{n}{2}}}\left({\frac{n}{2}}-c\right)+c.c.+|v_{-}^{n/2-1}|^{2}\left({\frac{n}{2}}-c\right)^{2}.
M^{05}(n_{1}^{\prime}n_{2}^{\prime}n^{\prime}k^{\prime}n_{1}n_{2}nk)=(2+s+2n+2n_{1}+2n_{2})\delta_{n_{1}n_{1}^{\prime}}\delta_{n_{2}n_{2}^{\prime}}\delta_{nn^{\prime}}\delta_{kk^{\prime}}
\frac{(m-1)k_{\mu}k^{\lambda}l_{\,\,\,\,\nu\lambda}^{\mu\nu}}{m^{2}(k^{2})^{2-1/m}((k^{2})^{1/m}-\lambda)^{3}}-\frac{(m-1)k^{\mu}k^{\lambda}l_{\nu}^{\,\,\nu\mu\lambda}}{m^{2}(k^{2})^{2-1/m}((k^{2})^{1/m}-\lambda)^{3}}
2^{-N}\sum_{k=-\frac{N}{2}}^{\frac{N}{2}}\binom{N}{\frac{N}{2}+k}\phi_{k}\longrightarrow2^{-N}\sum_{k=-\frac{N}{2}}^{\frac{N}{2}}\binom{N}{\frac{N}{2}+k}\phi_{k}\mathrm{e}^{ik\pi}
{\calL}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}(\partial_{\mu}\phi)^{2}+e(g^{\mu\nu}-\epsilon^{\mu\nu})(\partial_{\mu}\phi)A_{\nu}+\frac{1}{2}e^{2}\alphaA_{\mu}^{2}
H_{\mu\nu}\equiv2g^{-1/2}\frac{\delta}{\deltag^{\mu\nu}}\int{\calE}\sqrt{g}d^{4}x=2g^{-1/2}\frac{\deltag^{\mu\nu}}{\deltae_{n}^{\rho}}\frac{\delta}{\deltag^{\mu\nu}}\int{\calE}\sqrt{g}d^{4}x=g^{-1/2}g_{\mu\nu}e_{n}^{\rho}\frac{\delta}{\deltae_{n}^{\rho}}\int{\calE}\sqrt{g}d^{4}x
H_{B}=1+{\frac{r_{B}^{7-B}}{V_{a}}}{\frac{1}{r^{7-B-a}}}=1+{\frac{g_{s}N_{B}}{V_{a}/\ell_{s}^{a}}}\left({\frac{\ell_{s}}{|x_{\perp}|}}\right)^{7-A-b}\,.