harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
c_{n}=\frac{(-1)^{n(n+1)/2}x^{n^{2}/2}}{(x^{2};x^{2})_{\infty}^{n(n-3)/2}(x^{2r};x^{2r})_{\infty}^{3n(n-1)/2}}\frac{\{x^{2}\}_{\infty}^{n}\{x^{6}\}_{\infty}^{n}\{x^{2r+2}\}_{\infty}^{n}\{x^{2r+6}\}_{\infty}^{n}}{\{x^{4}\}_{\infty}^{n}\{x^{8}\}_{\infty}^{n}\{x^{2r}\}_{\infty}^{n}\{x^{2r+4}\}_{\infty}^{n}},
{\nabla_{0}^{2}f^{(0,0)}(\tau,\bar{\tau})\equiv4\tau_{2}^{2}{\frac{\partial}{\partial\tau}}{\frac{\partial}{\partial\tau^{*}}}f^{(0,0)}(\tau,\bar{\tau})={\frac{3}{4}}f^{(0,0)}(\tau,\bar{\tau}).}
\beta_{1}\beta_{2}K^{T}(s_{1},s_{1})\,\equiv\!\intdp_{0}^{\prime}dp_{0}\delta(p_{0}^{\prime}\!-\!s_{1})\beta_{1}\beta_{2}K^{T}(p_{0}^{\prime},p_{0})\delta(p_{0}\!-\!s_{1}).
\underbrace{\phi^{i_{2(k+r)}^{\prime}}\phi^{i_{2(k+r)-1}^{\prime}}}\ldots\underbrace{\phi^{i_{2}^{\prime}}\phi^{i_{1}^{\prime}}}F_{a_{p-2r+1}a_{p-2r+2}}\ldotsF_{a_{p-1}a_{p}}).
w_{i\bar{\imath}}^{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}|RR}(\theta_{1,}\theta_{2})=\frac{f_{1}(\theta_{1})f_{1}(\theta_{2})}{(f_{2}(\theta_{1})-f_{3}(\theta_{1}))(f_{2}(\theta_{2})-f_{3}(\theta_{2}))},
\gamma_{ij}dx^{i}dx^{j}=\left(1+\textstyle{\frac{k}{4}}\delta_{mn}x^{m}x^{n}\right)^{-2}\delta_{ij}dx^{i}dx^{j}={\frac{dr^{2}}{1-kr^{2}}}+r^{2}d\Omega_{n-1}^{2}.
\intdx\psi^{\dagger}(j,x)\psi(j^{\prime},x)=\delta_{j,j^{\prime}}=\delta_{s,s^{\prime}}\,\delta_{l,l^{\prime}}\,\delta(p_{3}-p_{3^{\prime}})\,{\frac{\delta(p_{\perp}-p_{\perp}^{\prime})}{\sqrt{p_{\perp}p_{\perp}^{\prime}}}}\,,
g^{-1}=\left(\begin{array}{ll}{-2\mu_{2}-2\mu_{2}^{2}+\frac23\mu_{3},}&{-\mu_{3}(6+\frac{16}{3}\mu_{2})}\\{-\mu_{3}(6+\frac{16}{3}\mu_{2}),}&{\frac83\mu_{2}^{2}\mu_{3}-8\mu_{3}^{2}}\\\end{array}\right)
2B_{IJ}=\left(\begin{array}{cccc}{0}&{d}&{a}&{c}\\{-d}&{0}&{-a-c}&{a}\\{-a}&{a+c}&{0}&{b}\\{-c}&{-a}&{-b}&{0}\\\end{array}\right)~,~~~G_{IJ}=\left(\begin{array}{cccc}{2u}&{-u}&{f}&{h}\\{-u}&{2u}&{-f-h}&{f}\\{f}&{-f-h}&{2v}&{-v}\\{h}&{f}&{-v}&{2v}\\\end{array}\right)~.
V_{rad}=V_{\|}\:\left[\bfr\cdot\left(v_{\mathrm{1}}+v_{\mathrm{2}}\right)\right]^{\mathrm{2}}/4\mathrm{r^{2}\;+\;V_{L}\:\left[\bfr\cdot\left(v_{\mathrm{1}}-v_{\mathrm{2}}\right)\right]^{\mathrm{2}}/4\mathrm{r^{2}.}}
W(\gamma,A)=2+\sum_{i=1}^{\infty}\mathrm{Tr}[\ointds_{1}\int_{0}^{s_{1}}ds_{2}...\int_{0}^{s_{n-1}}ds_{n}\dot{\gamma}^{a_{1}}(s_{1})...\dot{\gamma}^{a_{n}}(s_{n})A_{a_{1}}(\gamma(s_{1}))...A_{a_{n}}(\gamma(s_{n}))]
u_{-1}=-5u_{1}+v_{-1}+2w_{1}\,a^{-\gamma}+(\mathrm{~\frac{9}{2}~}u_{1}^{2}-\mathrm{~\frac{1}{2}~}v_{-1}^{2}-3u_{1}v_{-1}+w_{2})a^{-2\gamma}
\Sigma(z)=\frac{\int_{-\infty}^{z}\sqrt{-g^{(4)}}dz^{\prime}\left({T^{t}}_{t}-{T^{z}}_{z}-{T^{r}}_{r}-{T^{\phi}}_{\phi}\right)\intdxdy}{\sqrt{g^{(2)}(z)}\intdxdy}
\left<\hat{H}_{\alpha}(t)\right>_{n_{\alpha}}=\frac{\piH_{0}}{8}\Biggl[\Bigl|\bigl(\frac{3}{2}+\nu\bigr)H_{\nu}^{(1)}(z)+zH_{\nu+1}^{(1)}(z)\Bigr|^{2}+\bigl(\frac{9}{4}-\nu^{2}+z^{2}\bigr)|H_{\nu}^{(1)}(z)|^{2}\Biggr]\hbar(2n_{\alpha}+1).
A_{\mu}^{a(\theta)}(\boldsymbol{x},\tau)=\sum_{n=-\infty}^{+\infty}\mathrm{e}^{-\theta|n|}\left[A_{\mu}^{a(I)}(\boldsymbol{x},\tau+T_{1}+nT)+A_{\mu}^{a(A)}(\boldsymbol{x},\tau-T_{1}+nT)\right]\;.
\beta_{00}(t)=\frac{\pig\mathrm{e}^{-\pigt/4}}{16\tilde{\omega}\tilde{\tilde{\omega}}}\left[\left(2\pig-i\tilde{\tilde{\omega}}\right)\mathrm{e}^{i\tilde{\tilde{\omega}}t}-\left(2\pig-i\tilde{\tilde{\omega}}\right)\mathrm{e}^{-i\tilde{\tilde{\omega}}t}\right]\;.
E_{bind}=T_{0}+E_{D4}-E_{D4/D0}=\frac{(2\pi)^{2}T_{4}}{(\mathrm{Pf}B+(2\pi)^{2})}\Bigl[\frac{|\mathrm{tr}\,\tilde{B}g\tilde{B}g|}{4\mathrm{Pf}B}-\sqrt{\mathrm{det}g}\Bigr]\,.
H=R\,\mathrm{Tr}\left(\frac{1}{2\hbar}\Pi_{i}^{2}-\frac{1}{16\alpha^{3}\hbar^{2}}\left[Y^{i},Y^{j}\right]^{2}-\frac{i}{2\pi(\alpha^{\prime}\hbar)^{3/2}}\pi\gamma^{j}\left[Y_{j},\theta\right]\right),
{\frac{\deltaW}{\deltaJ_{n}}}\vert_{J_{1},J_{2}}\equiv{\frac{1}{n!}}\phi^{n}+{\frac{1}{2~(n-2)!}}G\phi^{n-2}+\sum_{m=3}^{n}{\frac{\delta\chi_{m}}{\deltaJ_{n}}}{\frac{\deltaW_{2}}{\delta\chi_{m}}}
{\calN}_{F}(t,\star;\rho,w,\rho^{\prime},w^{\prime},\bullet,\bullet^{\prime};\alpha)=\sum_{p=0}^{\infty}N^{(p)}(t,\star;\rho,w,\rho^{\prime},w^{\prime},\bullet,\bullet^{\prime}){\frac{\alpha^{p}}{p!}},\\\(\mathrm{A}.4)\notag
\begin{array}{rcl}{\tau}&{\in}&{{\calW}_{3}\oplus{\calW}_{4}\oplus{\calW}_{5}\,,}\\{\mathrm{~with~}}&{}&{2\,{\calW}_{4}+{\calW}_{5}=0}\\{\mathrm{~and~}}&{}&{{\calW}_{4}\,,\,{\calW}_{5}\quad\mathrm{~exact~and~real.}}\\\end{array}
\langle[A_{2}]_{N}({\bfx})\rangle=\frac{1}{\sqrt{2k\omega}}\left\{\frac{1}{2}(1-\frac{m}{\omega})J_{N+1}(kr)e^{i(N+1)\varphi}-\frac{1}{2}(1+\frac{m}{\omega})J_{N-1}(kr)e^{i(N-1)\varphi}\right\}.
\left(\begin{array}{cccc}{\left(E-m\right)}&{0}&{-P_{3}}&{-\sqrt{2eBn}}\\{0}&{\left(E-m\right)}&{-\sqrt{2eBn}}&{P_{3}}\\{-P_{3}}&{-\sqrt{2eBn}}&{\left(E+m\right)}&{0}\\{-\sqrt{2eBn}}&{P_{3}}&{0}&{\left(E+m\right)}\\\end{array}\right)\left(\begin{array}{c}{a_{1}}\\{a_{0}}\\{a_{\widetilde{0}}}\\{a_{2}}\\\end{array}\right)=0
E_{k}^{(j)}=-i\rho\,t_{k}^{(j)},\qquad\overline{{E}}_{k}^{(j)}=i\rho\,\overline{{t}}_{k}^{(j)},\qquadA_{j}(t_{k}^{(j)})=\overline{{A}}_{j}(\overline{{t}}_{k}^{(j)})=0,\qquadk=1,2,\ldots,\infty
\begin{array}{c}{\mathrm{curl}\overrightarrow{H}-\partial_{0}\epsilon\overrightarrow{E}=\overrightarrow{j}_{e},\quad\mathrm{curl}\overrightarrow{E}+\partial_{0}\mu\overrightarrow{H}=\overrightarrow{j}_{mag},}\\{\mathrm{div}\epsilon\overrightarrow{E}=\rho_{e},\quad\mathrm{div}\mu\overrightarrow{H}=\rho_{mag,}}\\\end{array}
\Gamma_{2}=\frac{1}{C_{2}}=\frac{3}{a^{2}(4-a^{2})},\\\\Gamma_{4}=-\frac{C_{4}}{(C_{2})^{4}}=\frac{9(1-a^{2})(5-2a^{2})}{a^{4}(4-a^{2})^{4}}
D_{\mu\nu}^{0}(k)_{=}-{\frac{1}{k^{2}+i\epsilon}}\bigl(g_{\mu\nu}-{\frac{k_{\mu}\eta_{\nu}+k_{\nu}\eta_{\mu}}{\eta\cdotk}}+k_{\mu}k_{\nu}{\frac{\lambdak^{2}+\eta^{2}}{(\eta\cdotk)^{2}}}\biggr),
S_{0}=\frac{ik}{4\pi}\int_{D\timesR}dxdtd\tau\mathrm{Tr}\partial_{\tau}[h^{-1}*\partial_{x}h*h^{-1}*(\partial_{x}+\partial_{t})h]+\frac{ik}{12\pi}\int_{D\timesR}\mathrm{Tr}(h^{-1}dh)^{3}.
\sum_{j=0}^{i}q^{j}\chi_{1,2k-i+2j+1}^{(3,3k+2)}(q)=\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k}^{2}+n_{i+1}+\cdots+n_{k}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+1}},
(T_{\mu\nu}^{-}T_{\rho\sigma}^{+})(T_{-}^{\mu\nu}T_{+}^{\rho\sigma})=4(T_{\mu\rho}^{-}T_{+}^{\mu\sigma})(T_{-}^{\nu\rho}T_{\nu\sigma}^{+})=4(T_{\mu\rho}^{-}T_{+}^{\mu\sigma})(T_{\nu\sigma}^{-}T_{+}^{\nu\rho}),
A_{\mu}^{\prime}=\mp\,{\frac{i}{g}}\,\lambda_{3}\,{\frac{v_{\mu}}{\rho}}+\kappa\,E_{\pm}^{12}\,R_{\mu},\quadA_{\mu}^{\prime\prime}=\mp\,{\frac{i}{g}}\,{\frac{\lambda_{8}}{\sqrt3}}\,{\frac{v_{\mu}}{\rho}}.
F_{g}={\frac{1}{2}}\epsilon_{ij}^{1}\epsilon_{ij}^{2}\int_{0}^{1}dx\left[{\frac{(1{-}k_{0}^{2}{-}\vec{k}\cdot\vec{p})^{2}}{x^{2}}}+2{\frac{k_{0}^{4}{-}(\vec{k}\cdot\vec{p})^{2}}{x}}+(1{-}k_{0}^{2}{+}\vec{k}\cdot\vec{p})^{2}\right]x^{\vec{k}\cdot\vec{p}+k_{0}^{2}}(1-x)^{-2k_{0}^{2}}\,.
{\delta_{\rho}}{\mathcal{F}_{B}^{00}}=-{\frac{Q_{B}^{2}}{8\pi}}\oint{d}\hat{s}(\xi)\int{d^{2}}\xi^{\prime}\rho(\xi){\partial_{\hat{n}}}{\hat{G}_{D}}(\xi,\xi^{\prime})\sqrt{\hat{g}(\xi)}\hat{R}(\xi^{\prime})+{\frac{Q_{B}^{2}}{4\pi}}\oint{d}\hat{s}{k_{\hat{g}}}\rho.
\Phi=i\sum_{i=1}^{N}\left({\frac{\partialH}{\partialq_{i}}}{\frac{\partial}{\partialp_{i}}}-{\frac{\partialH}{\partialp_{i}}}{\frac{\partial}{\partialq_{i}}}\right)+\sum_{i,j=1}^{N}{\frac{\partialH}{\partialp_{i}}}G_{ij}{\frac{\partial}{\partialp_{j}}}
\displaystyle{\langle\bar{\psi}^{a}(z_{1})\psi_{b}(z_{2})\rangle=C_{\psi}\frac{\hat{v}{}^{a}{}_{b}(z_{12})}{(x_{\bar{1}2}^{2})^{\frac{1}{2}(\eta-(\frac{4}{{\calN}}-1)\kappa)}(x_{\bar{2}1}^{2})^{\frac{1}{2}(\eta+(\frac{4}{{\calN}}-1)\kappa)}}}\,.
I_{on-shell}=\frac{1}{16\pi}\intd^{4}x\\partial_{\mu}\Biggl[\sqrt{-g}\,\left(\omega^{\mu}+A_{\nu}^{n}({\mathrm{e}}^{-2\phi}F^{n\,\mu\nu}-ia*F^{n\,\mu\nu})+\tilde{B}_{\nu}^{n}({\mathrm{e}}^{+2\phi}\tilde{G}^{n\,\mu\nu}-ia*\tilde{G}^{n\,\mu\nu})\right)\Biggr]\.
\Gamma_{{\mu}_{1}{\mu}_{2}}^{(12)}=\Gamma_{{\mu}_{1}{\mu}_{2}}^{(11)}\equiv\left[\Gamma_{{\mu}_{1}}^{(11)}\,,\Gamma_{{\mu}_{2}}^{(12)}\right]\,,\qquad\Gamma_{11{\mu}}^{(12)}=\Gamma_{{\mu}}^{(11)}\,,
n_{1}=\frac{2}{\zeta}z_{1}\bar{z}_{1},\quadn_{2}=\frac{2}{\zeta}z_{2}\bar{z}_{2},\quadn_{3}=\frac{2}{\zeta^{\prime}}z_{3}\bar{z}_{3},\quadn_{4}=\frac{2}{\zeta^{\prime}}z_{4}\bar{z}_{4}\,.
\mathbf{x}=\sum_{\mu=1}^{8}\tilde{\sigma}_{\mu}x^{\mu}=\left(\begin{array}{cc}{z_{2}}&{z_{1}}\\{-\bar{z}_{1}}&{\bar{z}_{2}}\\\end{array}\right)\,,\quad\mathbf{x}^{\prime}=\sum_{\mu=1}^{8}\tilde{\sigma}_{\mu}^{\prime}x^{\mu}=\left(\begin{array}{cc}{z_{4}}&{z_{3}}\\{-\bar{z}_{3}}&{\bar{z}_{4}}\\\end{array}\right)\,,
\langleI\rangle=\int_{G^{4)}}d^{4}x\,\mu\left(u(x)\right)\,Z(x)\,L^{*}\left({\dot{x}}^{\mu\nu}(x)\right)=\int_{G^{4)}}d^{4}x\,L^{*}\left(\rho(x){\dot{x}}^{\mu\nu}(x)\right)\,
X_{1}=\int_{0}^{\sqrt{2(e-d)+e^{2}}}{dx{\frac{2}{\left(2e+e^{2}-x^{2}\right)}}\left[{\frac{(x^{4}-3\delta^{2}x^{2})}{(x^{2}+\delta^{2})^{3}}}-{\frac{2}{\left(2e+e^{2}-x^{2}\right)}}{\frac{x^{2}}{(x^{2}+\delta^{2})}}\right]},
V_{D4}(x,\tilde{b},c)=2\pi\sqrt{\alpha^{\prime}}\left(\frac{m^{4}n^{2}}{16\pi}\right)\frac{n}{g_{s}}x^{2}\Biggl[x^{2}-2x\mathrm{Re}\left(e^{-i\varphi}+e^{3i\varphi}\frac{\tilde{b}}{3}\right)+1+\frac{|\tilde{b}|^{2}}{3}-\mathrm{Re}\left(e^{2i\varphi}c\right)\Biggr],
\left(\bar{a}_{\alpha}^{n^{\prime}}\bar{b}^{m^{\prime}\beta}-\frac{1}{C}\delta_{\alpha}^{\,\beta}\bar{a}^{n^{\prime}}\cdot\bar{b}^{m^{\prime}}\right)|0>,\quad\bar{a}_{\alpha_{1}}^{n_{1}^{\prime}}\cdots\bar{a}_{\alpha_{k}}^{n_{k}^{\prime}}|0>,\quadetc.
\sum_{A=2}^{m}\phi_{A}^{a}f_{1t}^{A}=-\sum_{A=2}^{m}[\phi_{At}^{a}f_{1}^{A}+\phi_{A1}^{a}f_{t}^{A}+\sum_{B=2}^{m}(\phi_{AB}^{a}f_{t}^{B})f_{1}^{A}]-\phi_{1t}^{a},\;\;\;a=1,2,...,m-1.
\Omega_{0;0}^{0}={\frac{e^{-ar}}{1+r}}\,\left[a+(a+{\frac{1}{1+r}})(\omega^{0}\wedge\omega^{3}+\omega^{1}\wedge\omega^{2})+a\,\omega^{0}\wedge\omega^{1}\wedge\omega^{2}\wedge\omega^{3})\right].
\int_{-T/2}^{T/2}ds_{i}\simeq\int_{-T/2}^{T/2}dt_{i}\,\sqrt{1+h_{11}[x(t_{i})]}\simeqT+\frac{1}{2}\int_{-T/2}^{T/2}dt_{i}\,h_{11}[x(t_{i})]
\bar{\Theta}_{1}^{\mu}\equivS^{\mu\nu}P_{\nu}+k_{1}P^{\mu}\pi~~~;~~~\bar{\Theta}_{2}^{\mu}\equiv(\Lambda^{0\mu}-\frac{P^{\mu}}{M})+k_{2}(\Lambda^{0\mu}+\frac{P^{\mu}}{M})\phi~,
f\mapsto\left(\begin{array}{cccccc}{0}&{0}&{0}&{\ldots}&{0}&{0}\\{1}&{0}&{0}&{\ldots}&{0}&{0}\\{0}&{1}&{0}&{\ldots}&{0}&{0}\\{0}&{0}&{1}&{\ldots}&{0}&{0}\\{.}&{.}&{.}&{.}&{.}&{.}\\{0}&{0}&{0}&{\ldots}&{1}&{0}\\\end{array}\right)
{\calL}={\frac{1}{2}}m_{2}(v_{\parallel}^{2}-2)+{\frac{1}{2}}V_{2}\left({\frac{\mu_{2}Z_{6}}{g}}v_{\perp}^{2}+{\frac{g_{s}}{\mu_{2}Z_{6}}}\left({\dot{s}}+\mu_{2}C_{\phi}{\dot{\phi}}\right)^{2}\right)\,
<\psi_{i}(z_{1})\psi^{j\dagger}(z_{2})\psi^{k\dagger}(z_{3}^{*})\psi_{l}(z_{4}^{*})>\to{\frac{\delta_{l}^{j}\delta_{i}^{k}}{(2\pi)^{2}[(z_{1}-z_{3}^{*})(z_{4}^{*}-z_{2})]^{1-2/N}|\tau_{1}-\tau_{2}|^{4/N}}}
=2\sum_{s=odd}^{\infty}e^{-\pi\tau_{2}\frac{s^{2}}{2}}(\frac{s^{2}}{4}+\frac{1}{2\pi}\frac{d}{d\tau_{2}})\left(\frac{1}{\sqrt{2\tau_{2}}}e^{-\frac{\pi\tau_{1}^{2}s^{2}}{2\tau_{2}}}\sum_{n}e^{-\frac{\pin^{2}}{2\tau_{2}}-\frac{n\pi\tau_{1}s}{\tau_{2}}}\right)
\begin{aligned}{\mu(T_{X})}&{{}=\frac{c_{1}(T_{X})\cdoth}{2}=0\,}\\{\mu(T_{X}\otimesh^{2a})}&{{}=\frac{(c_{1}(T_{X}\otimesh^{2a})\cdoth}{2}=2ah^{2}=8a\,}\\\end{aligned}
\varepsilon_{0}^{\beta}=\frac{\verte\vertH}{(2\pi)^{2}}\sum_{n\sigma}\int_{-\infty}^{\infty}dp\;\epsilon_{n\sigma}\;\left\{\frac{1}{1+\mathrm{e}^{\beta\epsilon_{n\sigma}}}-\frac{1}{1+\mathrm{e}^{-\beta\epsilon_{n\sigma}}}\right\}
\partial_{y}\phi^{(2)}=-{\frac{l^{2}}{4}}\left(ye^{4y/l}+{\frac{l}{2}}e^{2y/l}\right)\Box^{2}\phi_{0}-{\frac{l^{2}}{8}}(e^{6y/l}+e^{2y/l})\BoxC(x)+e^{4y/l}D(x)\.
G(E;\xi_{b},\eta_{b},\varphi_{b};\xi_{a},\eta_{a},\varphi_{a})=\sum_{n}\phi_{n}(E_{n};\xi_{b},\eta_{b},\varphi_{b})\frac{\hbar}{(E_{n}-E)(-\frac{\partial\lambda_{n}(E_{n})}{\partialE_{n}})}\phi_{n}^{\star}(E_{n};\xi_{a},\eta_{a},\varphi_{a})
\rho_{N}(p)\!=\!\frac{\pi^{1\!-\!\ell(N\!-\!1)}[\Gamma(\ell\!-\!\!1)]^{N\!-\!1}(p^{2}\!\!-\!\!m^{2})^{2N\ell\!-\!2\ell\!-\!2N\!+\!1}\theta(p^{2}\!\!-\!\!m^{2})}{2^{2\ell(N-1)-1}(m^{2})^{N\ell-N-\ell+1}\Gamma(2(N-1)(\ell-1))}F(\!N(\ell\!-\!1),(N\!-\!1)(\ell\!-\!1);2(\!N\!-\!1)(\ell\!-\!1);1\!-\!\frac{p^{2}}{m^{2}}\!)
2\partial_{\sigma}\left(Ng_{s}e^{\Phi_{D}}\rho^{\prime}\right)-\frac{\partiale^{\Phi_{D}}}{\partial\rho}\left(e^{2}-r^{2}e^{2}\omega^{2}+(r^{\prime})^{2}+Ng_{s}(\rho^{\prime})^{2}\right)=0\,,
S=\int\!\sqrt{-g}d^{4}x\left[\sum_{k=1}^{N}\bar{\psi_{k}}i\gamma^{\mu}\nabla_{\mu}\psi_{k}+\frac{G}{2N}\left((\sum_{k=1}^{N}\bar{\psi_{k}}\psi_{k})^{2}-(\sum_{k=1}^{N}\bar{\psi_{k}}\gamma_{5}\psi_{k})^{2}\right)\right],
D_{\tau}X_{i}^{M}\rightarrow\left(\Lambda^{-1}\right)_{\,\,\,N}^{M}\left(\kappa,\Theta\right)\left(\partial_{\tau}X_{i}^{N}-\varepsilon_{ij}A_{\kappa}^{jk}X_{k}^{N}-\Omega_{g}^{NK}X_{iK}\right)\,.
S_{\mathrm{YM/\Phi}}^{(\mathrm{q})}={\frac{NV}{4g_{0}^{2}}}\mathrm{Tr}\left(\left[\mathbf{A}_{\mu}^{(\mathrm{q})},\mathbf{A}_{\nu}^{(\mathrm{q})}\right]^{2}\right)\.
\phi_{c}(r)=\int_{r}^{\infty}\frac{dx}{\sqrt{1+\frac{x^{4}}{e^{2}}}}=\phi_{c}(0)-\int_{0}^{r}\frac{dx}{\sqrt{1+\frac{x^{4}}{e^{2}}}}\stackrel{r\rightarrow0}{\simeq}\left[\phi_{c}(0)-r+\frac{r^{5}}{10e^{2}}\right]
B_{IJ}=\left(\begin{array}{cccccc}{0}&{0}&{}&{}&{}&{}\\{0}&{0}&{}&{}&{}&{}\\{}&{}&{0}&{0}&{}&{}\\{}&{}&{0}&{0}&{}&{}\\{}&{}&{}&{}&{0}&{1}\\{}&{}&{}&{}&{-1}&{0}\\\end{array}\right)\nonumber
\left(\begin{array}{c}{\Psi_{1}(r,x^{+})}\\{\Psi_{2}(r,x^{+})}\\\end{array}\right)=P(r,x^{+};r_{0},x_{0}^{+})\left(\begin{array}{c}{\Psi_{1}(r_{0},x_{0}^{+})}\\{\Psi_{2}(r_{0},x_{0}^{+})}\\\end{array}\right)\qquad,
ds_{\mathrm{rel}}^{2}=\left(\mu+\frac{2\pi}{e^{2}r}\right)d{\vec{r}}^{2}+\frac{4\pi^{2}/e^{4}}{\mu+\frac{2\pi}{e^{2}r}}\left(d{\psi}+{\vec{w}}({\vec{r}})\cdotd{\vec{r}}\right)^{2},
\tau^{\prime}=\frac{a\tau+b}{c\tau+d},\quad\quad\quad\left(\begin{matrix}{H_{(3)}^{\prime}}\\{F\,_{(3)}^{\prime}}\\\end{matrix}\right)=\left(\begin{matrix}{d}&{c}\\{b}&{a}\\\end{matrix}\right)\left(\begin{matrix}{H_{(3)}}\\{F_{(3)}}\\\end{matrix}\right)\
\langle\bar{\sigma}_{k-1}\bar{\sigma}_{S}\rangle_{g}=\sum_{j\inS}(2j+1)\langle\bar{\sigma}_{j-1}\bar{\sigma}_{S-j}\rangle_{g}+{\frac{1}{4}}\langle\bar{P}\bar{P}\bar{\sigma}_{S}\rangle_{g-1}+{\frac{1}{8}}\sum_{X\cupY}\sum_{g_{1}+g_{2}=g}\langle\bar{\sigma}_{X}\bar{P}\rangle_{g_{1}}\langle\bar{P}\bar{\sigma}_{Y}\rangle_{g_{2}}\.
\frac{1}{e^{2}}\sum_{\vec{x}}\sum_{\mu<\nu}\left[1-\frac{1}{2}(U_{\mu\nu}(\vec{x})+U_{\mu\nu}^{\dagger}(\vec{x}))\right]\simeq\frac{1}{4}\sum_{\vec{x},\mu,\nu}a^{4}F_{\mu\nu}(\vec{x})F^{\mu\nu}(\vec{x})
P=(P^{+},P^{-}=\frac{M^{2}}{P^{+}},\bbox{0}_{\perp})\,,\;\;p_{\gamma}=(0,p_{\gamma}^{-}=2p_{\gamma}\cdotP/P^{+},\bbox{p}_{\gamma\perp})\,,\;\;Q^{2}\equivp_{\gamma\perp}^{2}\,,
\begin{array}{|c|c|}\hline{\mathrm{closed~string}}&{\mathrm{open~string}}\\\hline{\mathrm{NS-NS}}&{\mathrm{NS}}\\\hline{\mathrm{NS-NS~(-1)^F~}}&{\mathrm{R}}\\\hline{\mathrm{R-R}}&{\mathrm{NS~(-1)^F~}}\\\hline{}\\\end{array}
\pi(x)~=~\frac{\delta\,{\calL}}{\delta\,\dot{\varphi}\,(x)}~=~\dot{\varphi}+\lambda\,\frac{\delta\,{\calL}_{int}}{\delta\,\dot{\varphi}}\,\left(\varphi,\vec{\bigtriangledown}\,\varphi,\dot{\varphi}\right)~=~\dot{\varphi}+\lambda\,\dot{\varphi}\,\left[\left(\vec{\bigtriangledown}\,\varphi\right)^{2}-\dot{\varphi}^{2}\right],
\begin{array}{l}{S({\bfX},\sigma,\hat{g})=S_{M}({\bfX},\hat{g})+\frac{1}{4\pi}\intd^{2}z\sqrt{\hat{g}}\left(\hat{g}^{ab}\partial_{a}\sigma\partial_{b}\sigma+Q\hat{R}\sigma+\mue^{2\alpha\sigma}\right)}\\{S_{M}({\bfX},\hat{g})=\frac{1}{4\pi\alpha^{\prime}}\intd^{2}z\sqrt{\hat{g}}\,\hat{g}^{ab}\partial_{a}{\bfX}\cdot\partial_{b}{\bfX}\.}\\\end{array}
{\calI}_{\calK}^{2}={\calI}_{\calK}^{3}=\left(\begin{array}{cc}{0}&{-1}\\{1}&{0}\\\end{array}\right)\;,\qquad{\calI}_{\calK}^{4}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right).
\left\langleU_{\lambda}\left(w\right)\right\rangle_{w_{n_{i^{\prime}}^{*}=2}^{\prime}}^{\left({\itG}_{1}\right)_{X_{i^{\prime}}^{\prime}}lc}=1-\left(L^{(2\beta-d)}\lambda-\beta_{1}\lambda^{2}+O(\lambda^{3})\right)\times
r_{crit}=\left[{\frac{(p-3)^{2}}{(p-3)^{2}+16}}\right]^{\frac{1}{p-1}}L_{p}^{\frac{p+1}{p-3}}\bar{\calE}^{\frac{2}{p-1}}\bar{\calP}_{\phi}^{-{\frac{2}{p-1}}},
\frac{1}{2}{\calE}_{3}^{2}=\frac{1}{2L^{2}}\int_{0}^{L}dz_{3}\int_{0}^{L}dy_{3}\sum_{n}\left(1-\delta_{n,0}\right)\frac{G_{\perp}(\vec{x}_{\bot},z_{3})G_{\perp}(\vec{x}_{\bot},y_{3})}{\left[\frac{2\pin}{L}\right]^{2}}e^{i2\pin(z_{3}-y_{3})/L}\\nonumber
{\calO}_{2}^{i}{}_{k}{\calO}_{1}^{k}{}_{j}=\left[\begin{matrix}{\mathrm{e}^{i{\frac{\pi}{4}}}\Omega_{-}^{\frac{1}{2}}}&{0}\\{0}&{\mathrm{e}^{-i{\frac{\pi}{4}}}\Omega_{-}^{-{\frac{1}{2}}}}\\\end{matrix}\right]{\calO}_{B}^{i}{}_{j}\left[\begin{matrix}{\mathrm{e}^{-i{\frac{\pi}{4}}}\Omega_{-}^{-{\frac{1}{2}}}}&{0}\\{0}&{\mathrm{e}^{i{\frac{\pi}{4}}}\Omega_{-}^{\frac{1}{2}}}\\\end{matrix}\right].
S_{\mathrm{anomalous}}={\frac{1}{T_{H}}}\int_{\Sigma}\{\varrho_{L}-{\calL}_{E}\}K^{\mu}d\Sigma_{\mu}={\frac{k}{\hbar}}\int_{\Omega}\{\varrho_{L}-{\calL}_{E}\}\sqrt{g_{E}}d^{4}x.
\{\tilde{H}(f_{1})\,,\tilde{H}(f_{2})\}=\tilde{H}([f_{1},f_{2}])\,,\quad[f_{1},f_{2}]^{i}=f_{1}^{j}\frac{\partial}{\partialx^{j}}f_{2}^{i}-f_{2}^{j}\frac{\partial}{\partialx^{j}}f_{1}^{i}\,.
e_{a}^{\;\;\mu}=\left(\begin{array}{cc}{e_{\hat{0}}^{\;\;0}}&{e_{\hat{0}}^{\;\;i}}\\{e_{\alpha}^{\;\;0}}&{e_{\alpha}^{\;\;i}}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{1}{\sqrt{a}}}&{0}\\{-v_{\alpha}}&{{\bar{e}}_{\alpha}^{\;\;i}}\\\end{array}\right).
{P}=\left(\begin{array}{ccc}{f-e^{-2\phi}v^{2}}&{-e^{-2\phi}v}\\{-e^{-2\phi}v}&{-e^{-2\phi}}\\\end{array}\right),\quad{\Omega}=\left(\begin{array}{ccc}{\omega}&{-q}\\{-q}&{qv-b}\\\end{array}\right)
E_{n(s)}^{(r)}=\sum_{\alpha,\beta=0}^{1}\sum_{\sigma=\alpha}^{\alphar}\sum_{\tau=\beta}^{\betar}\,E_{n(s)\sigma\tau}\left(\frac{\Lambda^{2}}{p_{1}^{2}+u_{r,2\sigma}^{\prime\prime}\Lambda^{2}}\right)^{\alpha}\left(\frac{\Lambda^{2}}{p_{2}^{2}+u_{r,2\tau}^{\prime\prime}\Lambda^{2}}\right)^{\beta}\,.
L(x,\dot{x},\{q_{n},\dot{q}_{n}\},t)=\frac{M}{2}\dot{x}^{2}-V(x,t)+\sum_{n}\left\{\frac{1}{2}m\dot{q}_{n}^{2}-\frac{1}{2}m\omega_{n}^{2}\left[q_{n}-Q_{n}(x,t)\right]^{2}\right\}.
\frac{\delta^{2}S[\bar{\phi}]}{\delta\bar{\phi_{1}}(x)\delta\bar{\phi_{2}}(x^{\prime})}=-\frac{\delta^{2}S[\bar{\phi}]}{\delta\bar{\phi_{2}}(x)\delta\bar{\phi_{1}}(x^{\prime})}=2ie\mu\frac{\partial}{\partialt}\delta(x,x^{\prime})\;.
\left\langle{^0t},\left(w-w_{\Lambda}\right)x^{\alpha_{1}}x^{\alpha_{2}}\right\rangle=-\left({D(\Lambda)^{\alpha_{1}}}_{\beta_{1}}{D(\Lambda)^{\alpha_{2}}}_{\beta_{2}}-\delta_{\beta_{1}}^{\alpha_{1}}\delta_{\beta_{2}}^{\alpha_{2}}\right)c^{\beta_{1}\beta_{2}}.
M_{F4}^{grav-med.}=4\left(\frac{\kappa}{12}\right)^{2}\left(T_{F}^{\alpha\beta}(k_{1},k_{2})\frac{1}{s}\:G_{\alpha\beta;\mu\nu}T_{F}^{\mu\nu}(k_{3},k_{4})+\mathrm{perm.}\right).
{\frac{b-a}{n}}\,\sum_{k=0}^{n-1}f\Bigl(a+k{\frac{b-a}{n}}\Bigr)=\int_{a}^{b}\!f-{\frac{\bigl(b-a\bigr)\bigl(f(b)-f(a)\bigr)}{2n}}+{\calO}(1/n^{2})
{^{(k)}}S_{\nuo}^{j}dx^{\nu}dx_{j}\:=\:\Big(\Big\{_{\nuo}^{o}\Big\}+{^{(k)}}S_{\nuo}^{o}\Big)dx^{\nu}dx^{o}-\Big\{_{\nuo}^{j}\Big\}dx^{\nu}dx_{j}\:,
W_{\mu}^{T,I_{1}I_{2}...I_{r}I_{r+1}}(x_{2})=\frac{1}{r}\sum_{c=1}^{r}:\prod_{\substack{a=1\,(a\neqc)}}^{r}O_{2}^{I_{a}}(x_{2})[O_{2}^{I_{c}}(x_{2})\partial_{\mu}O_{2}^{I_{r+1}}(x_{2})-O_{2}^{I_{r+1}}(x_{2})\partial_{\mu}O_{2}^{I_{c}}(x_{2})]:
{\mathrm{\Large~v~}}_{D}(m)={\frac{m^{2}}{2g^{2}}}+{\frac{{\mathrm{tr}}{\bf1}}{2(4\pi)^{D/2}}}\int_{1/\Lambda^{2}}^{\infty}\!d\tau\,\tau^{-{\frac{D}{2}}-1}\left\{{\mathrme}^{-\tau(m^{2}+\epsilon)}G_{D}(\tauF)-1\right\}\.
a_{\beta\dot{\beta}}^{\{\alpha\gamma\}}=\frac{1}{4g}\,G_{\beta\sigma}^{\{\alpha\gamma\}}\,x^{\sigma\dot{\sigma}}\,\bar{\omega}_{\dot{\sigma}\dot{\beta}}\,,\qquad\left\|\frac{\partiala_{\nu}^{a}}{\partial\bar{\omega}^{b}}\right\|=\frac{2\pi\rho}{g}\,.
\mathrm{d}\equiv\mathrm{d}z^{M}\frac{\partial~}{\partialz^{M}}=e^{M}D_{M}=e^{\mu}\partial_{\mu}+\mathrm{d}\theta^{a\alpha}D_{a\alpha}-\mathrm{d}\bar{\theta}_{a}^{\dot{\alpha}}\bar{D}{}_{\dot{\alpha}}^{a}\,,
\Lambda_{k}=\left(\begin{array}{ccc}{n_{k}}&{{\frac{1}{2}}q_{k}^{\prime\prime}+Tq_{k}+{\frac{1}{2}}(\zeta\delta_{k}+\eta\beta_{k})}&{q_{k}\zeta+\beta_{k}\Phi+\beta_{k}^{\prime}}\\{q_{k}}&{n_{k}+q_{k}^{\prime}}&{\beta_{k}}\\{\delta_{k}}&{q_{k}\eta+\delta_{k}\Phi-\delta_{k}^{\prime}}&{2n_{k}+q_{k}^{\prime}}\\\end{array}\right)
\sum_{\alpha,\nu,\rho}A(\beta,\mu,\rho|\alpha,\nu,\sigma)v_{1}(\alpha)v_{2}(\nu)v_{3}(\sigma)=\sum_{\alpha,\nu,\rho}B(\beta,\mu,\rho|\alpha,\nu,\sigma)v_{1}(\alpha)v_{2}(\nu)v_{3}(\sigma).
\int_{0}^{2\pi}d\theta\int_{0}^{\infty}dr\,r\Bigl(\mathrm{e}^{-{\frac{\phi({\bfr}_{ij})}{kT}}}-1\Bigr)={\frac{2}{N}}\Bigl\{b-{\frac{a}{(RT)^{2}}}\Bigr\}\.
e^{iW}=\intdg_{\mu\nu}~d\bar{\Gamma}_{~\mu\nu}^{\sigma}e^{i\left(S_{clas}+S_{gf}\right)}\left(det\omega_{\mu\nu}\right)^{\frac{1}{2}}\left(det\omega_{\mu\alpha}\triangle_{~\nu}^{\alpha}\right)\left(det\zeta_{\mu\nu}\right)^{\frac{3}{2}}\left(det\varsigma_{\mu\nu}\right)^{\frac{3}{2}}
\begin{array}{c}{\overline{{A}}^{\prime1}{}_{\nu}\longrightarrowW_{\nu}^{-},}\\{\overline{{A}}^{\prime2}{}_{\nu}\longrightarrowW_{\nu}^{+},}\\{\overline{{A}}^{\prime3}{}_{\nu}\longrightarrowZ_{\nu},}\\{\overline{{A}}^{\prime4}{}_{\nu}\longrightarrowA_{\nu}.}\\\end{array}
A_{i}=(2\piR)^{p-3}\sqrt{\left(\left(n_{1}^{(i)}\right)^{2}+\left(m_{1}^{(i)}\right)^{2}\right)\left(\left(n_{2}^{(i)}\right)^{2}+\left(m_{2}^{(i)}\right)^{2}\right)}.
\varphi^{ij}\left(x\right)=\frac{1}{\sqrt{L}}\sum_{k=-\infty}^{\infty}{\binom{a_{k}^{ij}}{b_{k}^{ij}}}e^{ip_{k}x}\quad,\quadp_{k}=\frac{2\pi}{L}\left(k+\frac{1}{2}\right)\.
S_{\mathrm{CS}}=-\mathrm{i}\int\mathrm{d}^{3}x\epsilon^{\mu\rho\nu}\left(\frac{1}{2}A_{\mu}^{a}\partial_{\rho}A_{\nu}^{a}+\frac{1}{3!}gf^{abc}A_{\mu}^{a}A_{\rho}^{b}A_{\nu}^{c}\right),
=:\int_{q^{\prime},t^{\prime}}^{q^{\prime\prime},t^{\prime\prime}}{\cal{D}}q(t)\;e^{\frac{i}{\hbar}{\underset{T}{\int}}dt[\frac{1}{2}\dot{q}^{2}-V(q)]}=\int_{q^{\prime},t^{\prime}}^{q^{\prime\prime},t^{\prime\prime}}{\cal{D}}q(t)\;e^{\frac{i}{\hbar}S[q(t),T]}.
e^{-1}{\calL}=-{\frac{1}{2}}R-{\frac{1}{4}}e^{{\frac{2}{3}}\phi}F_{\mu\nu}F^{\mu\nu}-{\frac{1}{4}}e^{-{\frac{4}{3}}\phi}G_{\mu\nu}G^{\mu\nu}-{\frac{1}{6}}(\partial_{\mu}\phi)^{2}+{\frac{e^{-1}}{4\sqrt2}}\epsilon^{\mu\nu\rho\sigma\lambda}F_{\mu\nu}F_{\rho\sigma}B_{\lambda}\.
{\cal{S}}=\frac{1}{16\piG}\int_{M}d^{5}x\sqrt{-g}\left({\calR}_{5}+\frac{12}{\ell^{2}}-F_{MN}F^{MN}\right)+\frac{1}{8\piG}\int_{\partialM}d^{4}x\sqrt{-\gamma}\,{\cal{K}}\,,
G_{0}(\tau):=\left(\frac{\theta_{E_{8}}^{2}(\tau)}{\eta(\tau)^{9}}\right)^{24}=\left(\frac{\eta(\frac{\tau}{6})\eta(\frac{\tau}{10})\eta(\frac{\tau}{15})}{\eta(\frac{\tau}{2})\eta(\frac{\tau}{3})\eta(\frac{\tau}{5})\eta(\frac{\tau}{30})}\right)^{24}.
\frac{1}{2}\left(\frac{\partialg_{\beta\gamma}}{\partialx^{\alpha}}+\frac{\partialg_{\alpha\gamma}}{\partialx^{\beta}}-\frac{\partialg_{\alpha\beta}}{\partialx^{\gamma}}\right)=\sum_{\sigma=0}^{3}g_{\gamma\sigma}\Gamma_{\alpha\beta}^{\sigma}.
\lambda^{^{S,A}}=\left(\begin{matrix}{\zeta_{\lambda}\Theta_{[j]}\phi_{_L}^{\ast}(p^{\mu})}\\{\phi_{_L}(p^{\mu})}\\\end{matrix}\right)\,\,,\quad\rho^{^{S,A}}=\left(\begin{matrix}{\phi_{_R}(p^{\mu})}\\{(\zeta_{\rho}\Theta_{[j]})^{\ast}\phi_{_R}^{\ast}(p^{\mu})}\\\end{matrix}\right)\,\,.
\nabla^{2}\Phi-2\left(\nabla\Phi\right)^{2}-{\frac{m^{2}}{8}}T^{2}-{\frac{1}{12}}e^{2\Phi}\left\{f_{+}(T)F_{+3}^{2}+f_{-}(T)F_{-3}^{2}\right\}=0.
H={\frac{1}{2}}\intd^{3}x\,\left(E^{2}+B^{2}\right)+\intd^{3}x\,\left\{\bar{\psi}\left(i{\vec{\gamma}}\cdot{\vec{\partial}}+m\right)\psi+e\,\bar{\psi}A\cdot\gamma\psi\right\}.
\int_{0}^{2\pi}d\theta\int_{0}^{\infty}dr\,r\Bigl(\mathrm{e}^{-{\frac{\phi({\bfr}_{ij})}{kT}}}-1\Bigr)=2\pi\int_{0}^{\infty}dr\,r\Bigl\{I_{0}\Bigl({\frac{C_{l}}{kT}}r^{2l+2}\Bigr)-1\Bigr\}\.
I^{(2)}\simeq\frac{2\pi}{m_{H}^{2}}\left[\frac12\left(c^{2}-1+\left(\frac{2}{\mathrm{e}^{\gamma}}\right)^{\frac{8\piT}{g^{2}}}\frac{1-c^{2-\frac{8\piT}{g^{2}}}}{1-\frac{4\piT}{g^{2}}}\right)+\mathrm{e}^{\frac{a}{\mathrm{e}}}-1+\frac{a}{\mathrm{e}}\right].
{\calL}={\frac{1}{16\piG_{10}}}\sqrt{-\hat{G}}\,e^{-\hat{\Phi}}[{\calR}_{\hat{G}}+\hat{G}^{MN}\partial_{M}\hat{\Phi}\partial_{N}\hat{\Phi}-{\textstyle{\frac{1}{12}}}\hat{H}_{MNP}\hat{H}^{MNP}-{\textstyle{\frac{1}{4}}}\hat{F}_{MN}^{I}\hat{F}^{I\,MN}],