harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{(z-t)^{2}}=\frac{2\pii}{z-\bar{z}}e^{-2\pii\tau(t,z)}\frac{\partial\tau(t,z)}{\partialt},\qquad\frac{1}{(\bar{z}-t)^{2}}=\frac{2\pii}{z-\bar{z}}e^{2\pii\tau(t,z)}\frac{\partial\tau(t,z)}{\partialt}.
\left(\begin{matrix}{{\bf1}}&{0}\\{{\tilde{\bfc}}}&{{\bf1}}\\\end{matrix}\right)\qquad\qquad{\tilde{\bfc}}=\sum_{i}N_{i}g_{i}^{t}{\tilde{\bfc}}^{v}g_{i}+\sum_{j}M_{j}g_{j}^{t}{\tilde{\bfc}}^{h}g_{j}
K_{n}=-\sum_{n^{\prime}=1}^{n}(-1)^{n+n^{\prime}}\left(e^{-igh{\calA}_{n^{\prime}}}-1\right)\left(K_{n^{\prime}}^{\vphantom{()}}+K_{n^{\prime}}^{(0)}\right)+(-1)^{n}\left(e^{igh{\calA}_{0}/2}-1\right)K_{0}^{(0)},
{\calP}_{0}^{(1)}\oplus{\calP}_{0}^{(2)}={\calP}_{0}^{(1)}+{\calP}_{0}^{(2)},\quad{\calP}_{i}^{(1)}\oplus{\calP}_{i}^{(2)}=e^{-{\calP}_{0}^{(2)}/\kappa}\,{\calP}_{i}^{(1)}+{\calP}_{i}^{(2)}
\frac{-2}{X(t)r^{2}}\frac{\mathrm{d}M(t)}{\mathrm{d}t}\left(\begin{matrix}{0}&{1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{matrix}\right)
\begin{array}{l}{D_{1}U=-\sqrt{\frac{2}{\lambda}}e^{2\Phi}Y;\qquadD_{2}U=-2iD_{2}\Phi\cdotZ}\\{\sqrt{2\lambda}Y=2iD_{2}\Phi-D_{2}Z+2iD_{2}\Phi\cdotZ\cdotZ}\\{\sqrt{2\lambda}Z=D_{2}Y-2iD_{2}\Phi\cdotZ\cdotY}\\\end{array}
\psi_{n,s;z_{1},z_{2}}^{f,g;\alpha,\beta}(u^{0},x,y)=\frac{\left(A_{1}^{+\;f,g}-z_{1}^{\ast}\right)^{n}\left(A_{2}^{+\;\alpha,\beta}-z_{2}^{\ast}\right)^{s}}{\sqrt{\Gamma(n+1)\Gamma(s+1)}}\psi_{z_{1},z_{2}}^{f,g;\alpha,\beta}(u^{0},x,y),\,\,\psi_{z_{1},z_{2}}^{f,g;\alpha,\beta}=\psi_{0,0;z_{1},z_{2}}^{f,g;\alpha,\beta}\,.
S(p+k)\Gamma^{\nu}(p+k,k)S(p)=\frac{k^{\nu}}{k^{2}}\left[S(p)-S(p+k)\right]-\frac{\varepsilon^{\nu\alpha}k_{\alpha}}{k^{2}}\left[\gamma^{5}S(p)+S(p+k)\gamma^{5}\right]\;.
g_{\nu_{1}\nu_{2}}F_{r,n}^{\nu_{1}...\nu_{r}}=\frac{(-1)^{n}}{N_{r,n-1}}G_{r,n-1}^{\nu_{3}...\nu_{r}}+\frac{(-1)^{n}}{N_{r,n}}G_{r,n}^{\nu_{3}...\nu_{r}},\quadn=1,...,\Bigl[\frac{r}{2}\Bigr],
\begin{array}{l}{\langleK_{\tau}(x,y)\rangle=\langleE[\delta\left(y-x-\tau^{\frac{1}{2}-\sigma\gamma}\sqrt{\xi}\tilde{q}\left(1\right)\right)]\rangle}\\{=\xi^{-2}\tau^{-\frac{2}{1+\gamma}}F\left(\tau^{-\frac{1}{2(1+\gamma)}}\xi^{-\frac{1}{2}}\left(y-x\right)\right)}\\\end{array}
T_{\mu\nu}={\frac{\hbar}{\ell_{P}^{2}R^{2}}}\left[\begin{array}{cccc}{\xi\quad}&{0\quad}&{0\quad}&{0}\\{0}&{\chi}&{0}&{0}\\{0}&{0}&{\chi}&{0}\\{0}&{0}&{0}&{-1}\\\end{array}\right]
{\calK}{\cdot{\calP}}=a^{\mu}P_{\mu}+\bar{\varepsilon}_{a}Q^{a}+\lambdaD+\textstyle{\frac{1}{2}}w^{\mu\nu}M_{\mu\nu}+b^{\mu}K_{\mu}+\bar{\rho}_{a}S^{a}+\textstyle{\frac{1}{2}}t^{ab}A_{ab}\,,
\left.\begin{array}{lc}{\alpha_{n}^{\mu}|\Omega\rangle=0}&{n>0}\\{b_{n}|\Omega\rangle=0}&{n\ge0}\\{c_{n}|\Omega\rangle=0}&{n>0}\\{p^{\mu}|\Omega\rangle\propto\alpha_{0}^{\mu}|\Omega\rangle=0.}&{}\\\end{array}\right.
\sum_{a}\left(K_{a}+\sum_{\mu}M_{a}^{\mu}{\mathrm{Tr}}(\gamma_{a}^{\mu})+\sum_{\mu,\nu}A_{a}^{\mu\nu}{\mathrm{Tr}}(\gamma_{a}^{\mu}){\mathrm{Tr}}(\gamma_{a}^{\nu})\right)~.
S_{E}={\frac{4\,\pi}{\lambda}}\intd\tau\intdr\;r^{2}\left[{\frac{1}{2}}{({\frac{\partial\varphi}{\partial\tau}})^{2}+(\nabla\varphi)^{2}}+{\frac{1}{2}}(\varphi^{2}-1)^{2}-f\varphi^{3}\right].
f\starg=\Omega^{-1}(\Omega(f)\Omega(g))=\sum_{l=0}^{b}\frac{1}{l!}\left(\frac{\partial^{l}}{\partialp^{l}}x^{a}p^{b}\right)\left((\theta(x)\frac{\partial}{\partialx})^{l}x^{r}p^{s}\right).
\ddot{\xi^{r}}-\xi^{r}=-\frac{Kr_{0}}{1-Kr_{0}^{2}}[(\dot{\eta}^{r})^{2}-(\eta^{r})^{2}]+r_{0}(1-Kr_{0}^{2})\sum_{i}[(\dot{\eta}^{i})^{2}-(\eta^{i})^{2}],
\varepsilon_{0}^{\beta}=\langle0|{\calH}|0\rangle_{\beta}=\sum_{{\bf{p}}}\epsilon_{{\bf{p}}}\left\{\frac{-2+\mathrm{e}^{\beta\epsilon_{{\bf{p}}}}}{1-\mathrm{e}^{\beta\epsilon_{{\bf{p}}}}}\right\}
\Gamma_{\nu}(p,P)=-e^{2}\int\mathrm{~\,~\frac{d^3q}{(2\pi)^3}~\,~}D(p-q)\gamma_{\mu}S(\mathrm{~\,~\frac{1}{2}~\,~}P+q)\Gamma_{\nu}(q,P)S(-\mathrm{~\,~\frac{1}{2}~\,~}P+q)\gamma_{\mu},
\nu={\nu_{0}^{\prime\prime}}-{\frac{1}{2}}\left([v^{\prime\prime}+1+b^{\prime\prime}\gamma^{\prime\prime}]_{q^{\prime\prime}}+[v^{\prime\prime}+b^{\prime\prime}\gamma^{\prime\prime}]_{q^{\prime\prime}}\right){\nu_{vib}^{\prime\prime}}~,
H=\frac{t}{2a}\sum_{x}\left(e^{iA}S^{+}(x+1)S^{-}(x)+e^{-iA}S^{-}(x+1)S^{+}(x)\right)+\frac{e_{L}^{2}a}{2}\sum_{x,y}S^{3}(x)V(x-y)S^{3}(y)+e_{L}^{2}E^{2}/2N
\sum_{c}\sum_{Q"}\Gamma_{P}^{Q^{\prime}bQ"c}(p)\left[\delta_{Q"Q}\delta^{ca}-N_{Q"5}^{ca}(p)g_{Q"Q}^{\prime}{(-1)}^{a+1}\right]=\frac{i}{2}g_{Q^{\prime}Q}^{\prime}\delta^{ba}{(-1)}^{a+1},\\Q^{\prime},Q=U,D,\\b,a=1,2.
\tilde{g}_{\mu\nu}=e^{\alpha\phi/M_{p}}(\zeta+b)g_{\mu\nu},\quad\tilde{e}_{a\mu}=e^{\frac{1}{2}\alpha\phi/M_{p}}(\zeta+b)^{1/2}e_{a\mu},\quad\Psi^{\prime}=e^{-\frac{1}{4}\alpha\phi/M_{p}}\frac{(\zeta+k)^{1/2}}{(\zeta+b)^{3/4}}\Psi.
\begin{array}{llccl}{J_{L}}&{:}&{T_{L}^{*}({\calM}_{n})}&{\rightarrow}&{T_{L}({\calM}_{n})}\\{}&{}&{U}&{\mapsto}&{\(LU)_{+}\,L\,-\,L\,(UL)_{+}\,+\,L\,\Phi_{U}(L)\,+\,\bar{\Phi}_{U}(L)\,L}\\\end{array}
\sigma_{3}^{2}=\frac{1}{4}+\frac{1}{4\sigma_{2}^{4}}-\frac{1}{2\sigma_{2}^{2}}-\frac{{\sqrt{1-4\sigma_{2}^{2}+6\sigma_{2}^{4}+4\sigma_{2}^{6}+\sigma_{2}^{8}}}}{4\sigma_{2}^{4}}\quad.
\phi^{\prime}(x)\;\mathrm{exp}\left[-{\frac{i}{\hbar}}\intdx\;\Pi(x)\phi^{\prime}(x)\right]=i\hbar{\frac{\delta}{\delta\Pi(x)}}\;\mathrm{exp}\left[-{\frac{i}{\hbar}}\intdx\;\Pi(x)\phi^{\prime}(x)\right]\;,
\gammaA_{0}^{a}=\eta_{1}^{a},\;\gammaA_{i}^{a}=\partial_{i}\eta_{2}^{a}+f_{\;\;bc}^{a}\eta_{2}^{b}A_{i}^{c},\;\gammaB^{0i}=\eta_{1}^{i},\;\gammaB^{ij}=\partial^{\left[i\right.}\eta_{2}^{\left.j\right]},
\phi_{m_{k}}^{2}(0)=\frac{1}{{\calA}_{n}R_{0}}\frac{\pic}{8}\frac{(n+1)^{2}}{\Gamma^{2}[(n+3)/2]}\left(\frac{m_{k}}{c}\right)^{n}e^{-\frac{c}{2}\rho_{\mathrm{max}}}~,
2\;\int_{0}^{\infty}dp\;p\int_{0}^{\infty}\frac{dq}{q}\int_{0}^{4\pi}d\varphi\left(\biggm|\psi,_{p}\;+\;\frac{i\psi,_{\varphi}}{p}\;\biggm|^{2}\;+\;\bigm|\psi,_{q}\bigm|^{2}\right)\left(q,p,\varphi\right)\;<\;\widetilde{K},
R_{\pm}\tilde{\Theta}_{1}R_{\mp}^{-1}\tilde{\Theta}_{2}+\tilde{\Theta}_{2}R_{\pm}\tilde{\Theta}_{1}R_{\pm}^{-1}=k_{q^{\pm1}}\left(R_{\pm}\tilde{\Theta}_{1}^{2}R_{\mp}^{-1}+\tilde{\Theta}_{2}^{2}\right).
{\hat{F}[L,~f]}~=~\alpha^{-\sigma[L,~f]}~\sum_{\{j_{i}\}}~~\mu_{j_{1}}~\mu_{j_{2}}~....~\mu_{j_{r}}~V[L;~n_{1},~n_{2},~...~n_{r};~j_{1},~j_{2},~....~j_{r}]
R_{z\bar{z}z\bar{z}}^{\prime}\equivR_{z\bar{z}z}^{\prime}{}^{z}g_{z\bar{z}}^{\prime}=-2g_{z\bar{z}}g_{z\bar{z}}^{\prime}+C_{zzz}g^{z\bar{z}}\bar{C}_{\bar{z}\bar{z}\bar{z}}\,.
I_{i}^{E}[g,\varphi_{i}]=\int_{{\calM}_{\beta}}d^{4}x\,\sqrt{g}\,\varphi_{i}^{+}L_{i}\varphi_{i}~~~,~~~I_{i}^{C}[\bar{g},\bar{\varphi}_{i}]=\int_{\bar{{\calM}}_{\beta}}d^{4}x\,\sqrt{\bar{g}}\,\bar{\varphi}_{i}^{+}\bar{L}_{i}\bar{\varphi}_{i}~~~,
\frac{f(-s\triangle)-1-\frac16s\triangle}{s\triangle}=\int_{0}^{1}\!\!{\mathrm{d}}\alpha_{1}\,{\alpha_{1}}^{2}(1-\alpha_{1})^{2}\int_{0}^{1}\!\!{\mathrm{d}}\alpha_{2}\,\alpha_{2}\int_{0}^{1}\!\!{\mathrm{d}}\alpha_{3}\,{\mathrme}^{\alpha_{1}(1-\alpha_{1})\alpha_{2}\alpha_{3}s\triangle},
\begin{array}{ccccccc}{{\calL}_{\mathbf{I}}K^{x}}&{=}&{\epsilon^{xyz}K^{y}W_{\mathbf{I}}^{z}}&{;}&{{\calL}_{\mathbf{I}}\omega^{x}}&{=}&{\nablaW_{\mathbf{I}}^{x}}\\\end{array}
\left(\begin{array}{l}{f_{-}^{q}}\\{f_{-}^{-q}}\\\end{array}\right)=e^{-2iq\sigma_{3}\phi}\left(\begin{array}{l}{f_{+}^{q}}\\{f_{+}^{-q}}\\\end{array}\right)\qquad\mathrm{in}\:U_{+}\capU_{-}.
\frac{\partial\Gamma_{\Lambda}[\phi]}{\partial\Lambda}=-\frac{\Omega}{2}\mathrm{tr}\int_{0}^{\infty}\!\!\!\!\!dq\,\frac{q^{D-1}}{C(q,\Lambda)}\frac{\partialC(q,\Lambda)}{\partial\Lambda}\left\langle\left[1+C.\frac{\delta^{2}\Gamma_{\Lambda}}{\delta\phi\delta\phi}\right]^{-1}\!\!\!(\mathbf{q},\mathbf{-q})\right\rangle
p_{i}=0~,\qquadq_{i}=\frac{{\calC}}{\cal{A}}+\sigma_{i}^{2}\frac{{\calE}}{\cal{A}}-\frac{{\cal{B}}^{2}}{{\cal{A}}^{2}}-p_{*}\left(\frac{{\calB}}{{\calA}}\right),~~~~~i=1,2
{}^{L,R}\langlex|2k\rangle_{\Omega}^{L,R}\rightarrow\left({\frac{\Omega}{\pi}}\right)^{1/4}[2^{2k}(2k)!]^{-1/2}H_{2k}(0)\sqrt{\pi}\left({}^{L,R}\langlex|p\rangle_{+}^{L,R}\right)
G_{MN}^{(I)}dx^{M}dx^{N}={\frac{r^{2}}{\sqrt{Q_{e}Q_{p}}}}(-dt^{2}+dx_{1}^{2})+{\frac{\sqrt{Q_{e}Q_{p}}}{r^{2}}}dr^{2}+\sqrt{\frac{Q_{e}}{Q_{p}}}\sum_{i=2}^{5}dx_{i}^{2}+\sqrt{Q_{e}Q_{p}}d\Omega_{3}^{2}.
Q=\int_{x^{0}=const}\!\!d^{3}x\,\Big(H^{\alpha\beta}(x)_{,\beta}+\frac{1}{4}\Phi(x)^{,\alpha}\Big){\stackrel{\longleftrightarrow}{\partial_{x}^{0}}}u^{\gamma}(x)\,\eta_{\alpha\gamma}\,,
\psi_{1}=\intd^{D}x\left(-\mu_{3a}^{(\varphi)}\partial^{\alpha}\mu_{2\alpha}^{a}+\left(\partial^{\alpha}\mu_{3\alpha}^{a}\right)\mu_{2a}^{(\varphi)}+\left(\partial^{\alpha}A_{\alpha}^{a}\right)\nu_{1a}^{(\varphi)}-\left(\partial^{\alpha}\varphi_{a}\right)\nu_{1\alpha}^{a}\right)\,,
\times\;\Big\langle\prod_{b=1}^{N}\prod_{i=1}^{n_{b}^{\prime}}\overline{{\psi}}^{(b)}({x^{\prime}}_{i}^{(b)})P_{+}\psi^{(b)}({x^{\prime}}_{i}^{(b)})\prod_{i=1}^{m_{a}^{\prime}}\overline{{\psi}}^{(b)}({y^{\prime}}_{i}^{(b)})P_{-}\psi^{(b)}({y^{\prime}}_{i}^{(b)})\Big\rangle_{0}\;,
M^{2}=-g^{4}C_{2}(R)C(R^{\prime})\int_{q}\int_{k55}{\frac{N(k,k^{5},{\hat{k}}^{5},q)}{(k^{2})(k^{2}-(k^{5})^{2})(k^{2}-({\hat{k}}^{5})^{2})(q^{2}-m_{1}^{2})((q-k)^{2}-m_{2}^{2})}}
\left.{\frac{\delta\langleJ_{L}^{\mu}(x)\rangle}{\delta{\tilde{A}}_{\nu}(x^{\prime})}}\right|_{{\tilde{A}}=0}\propto\int{\frac{d^{4}k}{(2\pi)^{4}}}e^{ik.(x-x^{\prime})}\Pi^{\mu\nu}.
\dot{x}^{AA^{\prime}}=o^{A}\bar{o}^{A^{\prime}}-\frac{\rho^{\sigma}}{\rho^{\tau}}\acute{x}^{AA^{\prime}},\,\,\,\,\acute{x}^{AA^{\prime}}=|q|(o^{A}\bar{\iota}^{A^{\prime}}+\iota^{A}\bar{o}^{A^{\prime}}).
:e^{i(\alpha_{\mu}\widehat{v}^{\mu}+b_{\mu}^{\nu}\widehat{\Lambda}_{\\nu}^{\mu})}:\,=e^{-i\alpha_{0}\widehat{v}_{0}}e^{i\vec{\alpha}\vec{v}}e^{ib_{\nu}^{\mu}\widehat{\Lambda}_{\mu}^{\\nu}}
D_{\mu}^{AC}J^{\mu\,C}=\partial_{\mu}J^{\mu\,A}-{\frac{ig}{2}}d^{ABC}[A_{\mu}^{B},J^{\mu\,C}]_{\mathrm{M}}+{\frac{g}{2}}\,f^{ABC}\{A_{\mu}^{B},J^{\mu\,C}\}_{\mathrm{M}}=0
S_{boundary}=-\frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(1)}}{\sqrt{-g}V^{-1}\alpha^{(1)}}-\frac{\sqrt{2}}{\kappa_{5}^{2}}\int_{M_{4}^{(2)}}{\sqrt{-g}V^{-1}\alpha^{(2)}}
S_{f}^{(0)}\left(g_{0},m_{e0}\right)=1+\sum_{n=1}^{\infty}\frac{\left(-i\right)^{n}}{n!}\int_{-\infty}^{\infty}d^{4}x_{1}...\int_{-\infty}^{\infty}d^{4}x_{n}T:\mathcal{H}_{FI}^{(0)}...\mathcal{H}_{FI}^{(0)}\tag{2.5.3}
\sum_{ij}\left\langle\frac{1}{(x_{i}-\phi_{a})(y_{j}-\chi_{b})}\right\rangle_{N}=N(N-1)\left\langle\frac{1}{(x_{1}-\phi_{a})(y_{2}-\chi_{b})}\right\rangle_{N}+N\left\langle\frac{1}{(x_{1}-\phi_{a})(y_{1}-\chi_{b})}\right\rangle_{N}
\begin{aligned}{J^{1}}&{{}=e^{12}+e^{34}+e^{56}+e^{78},}\\{J^{2}}&{{}=e^{14}+e^{23}+e^{58}+e^{67},}\\{J^{3}}&{{}=e^{13}+e^{42}+e^{57}+e^{86}.}\\\end{aligned}
\frac{V_{i,k,l}V_{k,l,i}}{4\pi(m_{l}^{2}-m_{k}^{2})}\int_{0}^{\infty}\frac{ds}{s}e^{-m_{l}^{2}s}+\frac{V_{i,k,l}V_{k,l,i}}{4\pi(m_{k}^{2}-m_{l}^{2})}\int_{0}^{\infty}\frac{ds}{s}e^{-m_{k}^{2}s}.
\frac{1}{4}\left[k^{2}-G^{-2}(k)\right]+\frac{\pi^{2}}{g^{2}}\left[2zk^{-2}G^{-3}(k)-4\pi^{2}\frac{\deltaz}{\deltaG(k)}\int\frac{d^{2}p}{(2\pi)^{2}}p^{-2}G^{-2}(p)\right]=0
\left(e^{s\bar{D}^{2}}\right)_{Ax,By}=\intd^{4}z_{1}\dots\intd^{4}z_{n}\left(e^{\frac{s\bar{D}^{2}}{n+1}}\right)_{Ax,C_{1}z_{1}}\left(e^{\frac{s\bar{D}^{2}}{n+1}}\right)_{C_{1}z_{1},C_{2}z_{2}}\dots\left(e^{\frac{s\bar{D}^{2}}{n+1}}\right)_{C_{n}z_{n},By}
\begin{array}{l}{R_{12}e_{1}e_{2}=ce_{2}e_{1}\;,\;\;R_{12}A_{1}R_{21}A_{2}+A_{2}R_{12}A_{1}R_{12}^{-1}=0\;,}\\{e_{1}A_{2}=(\pm)R_{21}A_{2}R_{12}e_{1}\;,\;\;R_{12}F_{1}R_{21}F_{2}=F_{2}R_{12}F_{1}R_{21}\;,}\\{e_{1}F_{2}=R_{21}F_{2}R_{12}e_{1}\;,\;\;R_{21}A_{2}R_{12}F_{1}=F_{1}R_{21}A_{2}R_{12}\;.}\\\end{array}
S_{lc}^{F}=\frac{i}{4\pil_{s}^{2}p^{+}}\intd\tau\int_{0}^{2\pil_{s}^{2}p^{+}}d\sigma\left[S^{1}\left(\partial_{+}-\frac{\mu}{2}\gamma^{12}\right)S^{1}+S^{2}\left(\partial_{-}-\frac{\mu}{2}\gamma^{12}\right)S^{2}+2\muS^{1}\gamma^{234}S^{2}\right],
\deltax^{\alpha\dot{\alpha}}\=\2(\theta^{\alpha}\tilde{\epsilon}^{\dot{\alpha}}-\epsilon^{\alpha}\tilde{\theta}^{\dot{\alpha}})\quad,\qquad\delta\theta^{\alpha}\=\\epsilon^{\alpha}\quad,\qquad\delta\tilde{\theta}^{\dot{\alpha}}\=\\tilde{\epsilon}^{\dot{\alpha}}\quad.
\pi\left(\sum_{i}a_{0}^{i}da_{1}^{i}\cdotsda_{n}^{i}\right)=\sum_{i}\pi\left(a_{0}^{i}\right)\left[D,\pi\left(a_{1}^{i}\right)\right]\cdots\left[D,\pi\left(a_{n}^{i}\right)\right]
\begin{array}{ccl}{J_{n}^{\pm}}&{\rightarrow}&{J_{n\pmw}^{\pm}}\\{J_{n}^{3}}&{\rightarrow}&{J_{n}^{3}-{\frac{k}{2}}w\delta_{n}\qquad\qquadw\in{\bfZ}}\\{L_{n}}&{\rightarrow}&{L_{n}-wJ_{n}^{3}-{\frac{k}{4}}w^{2}\delta_{n}.}\\\end{array}
\left.\frac{e^{2}}{\pi}\Big(A,T\sum_{b=1}^{N}a^{(b)}\Big)-\frac{e\sqrt{g}}{\pi}\Big(h,T\sum_{b=1}^{N}a^{(b)}\Big)-\frac{e^{2}}{2\pi}\sum_{b=1}^{N}\Big(a^{(b)},Ta^{(b)}\Big)\right)\;.
T_{\mu\nu}^{(f,canonical)}=\frac{i}{2}[\overline{{\Psi}}^{\prime}\gamma^{a}e_{a(\mu}^{\prime}\nabla_{\nu)}\Psi^{\prime}-(\nabla_{(\mu}\overline{{\Psi}}^{\prime})\gamma^{a}e_{\nu)a}^{\prime}\Psi^{\prime}]
\begin{array}{c}{\{\quad,\quad\}=i{\tilde{g}}^{A\bar{B}}\nabla_{A}\wedge{\bar{\nabla}}_{\bar{B}}+g^{A\bar{B}}\frac{\partial}{\partial\theta^{A}}\wedge\frac{\partial}{\partial{\bar{\theta}}^{\bar{B}}}\;,\;}\\\end{array}
S_{s}={\frac{1}{2\kappa_{10}^{2}}}\intd^{10}x\sqrt{-G^{s}}\left[e^{-2\Phi}\left\{{\calR}_{G^{s}}+4(\partial\Phi)^{2}\right\}-{\frac{1}{2\cdot(p+2)!}}F_{p+2}^{2}\right],
J^{\mu}=\frac{1}{A(S^{n-1})(n-1)!}\epsilon^{\mu\mu_{1}...\mu_{n}}\epsilon_{\hat{\phi}_{1}...\hat{\phi}_{n}}\partial_{\mu_{1}}\hat{\phi}^{a_{1}}...\partial_{\mu_{n}}\hat{\phi}^{a_{n}}
T_{\underline{{A}}\\\underline{{B}}\dot{A}}^{\\\dot{A}\\\\dot{D}}H_{\dot{D}\underline{{C}}}=-\frac{i}{12}\left(\nabla_{\underline{{A}}}G_{\underline{{B}}}^{\\\dot{D}}\right)\overline{{R}}\chi_{\underline{{C}}\dot{D}}
W_{gr}=\int{}\sqrt{g}d^{4}x\left(-{\frac{1}{16\piG_{B}}}R+c_{1,B}R^{2}+c_{2,B}R_{\mu\nu}R^{\mu\nu}+c_{3,B}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}\right)
z_{i}^{a}=\sqrt{\frac{\omega}{2\hbarg}}\left[q_{i}^{a}+i\frac{g}{\omega}p_{i}^{a}\right]\\,\\\bar{z}_{i}^{a}=\sqrt{\frac{\omega}{2\hbarg}}\left[q_{i}^{a}-i\frac{g}{\omega}p_{i}^{a}\right]\\.
\tilde{C}_{\alpha}^{\,\alpha_{0}}(p):=\int_{\alpha_{0}}^{\alpha}d\alpha^{\prime}\,e^{\,-\alpha^{\prime}(\,p\etap+(\varepsilon+i)m^{2})}\quad,\qquad\varepsilon>0\,\0<\alpha_{0}\leq\alpha<\infty\quad,
R_{\alpha{\bar{m}}\nu{\bar{n}}}=\nabla_{\bar{m}}\partial_{\bar{n}}H_{\alpha\nu}=\partial_{\bar{m}}\partial_{\bar{n}}H_{\alpha\nu}-\Gamma_{\bar{m}\bar{n}}^{\bar{k}}\partial_{\bar{k}}H_{\alpha\nu}\,,
D_{3}=i\left(\begin{matrix}{{\frac{M+1}{2}}}&{0}\\{0}&{{\frac{M-1}{2}}}\\\end{matrix}\right)\qquad\Rightarrow\qquadD_{3}D_{3}=-\left(\begin{matrix}{\left({\frac{M+1}{2}}\right)^{2}}&{0}\\{0}&{\left({\frac{M-1}{2}}\right)^{2}}\\\end{matrix}\right).
Z_{x}(q)=\sum_{i,j}{\tilde{V}}_{ij^{*};1}{}^{x}\chi_{i}(q){\chi_{j}(q)}^{*},\qquad\qquad{\tilde{V}}_{ij^{*};1}^{(G)}{}^{x}=\sum_{c\inT_{\gamma}}n_{ic}{}^{a}n_{jc}{}^{b}
(J_{-}^{\mathrm{L}},J_{+}^{\mathrm{L}})=(h_{a\overline{{b}}}\lambda^{a}\lambda^{\overline{{b}}},0),\quad(J_{-}^{\mathrm{R}},J_{+}^{\mathrm{R}})=(0,g_{i\overline{{j}}}\psi^{i}\psi^{\overline{{j}}}).
\frac{\partialP_{p}}{\partialt}=\frac{1}{2}\frac{\partial}{\partial\phi}\left(\frac{H^{3/2}(\phi)}{2\pi}\,\frac{\partial}{\partial\phi}\Bigl(\frac{H^{3/2}(\phi)}{2\pi}P_{p}\Bigr)+\frac{V^{\prime}(\phi)}{3H(\phi)}\,P_{p}\right)+3H(\phi)P_{p}\.
4\pi\rho^{2}\Theta^{\mu\nu}K_{\nu}=\frac{K^{\mu}}{2\rho^{2}}(1-K^{2}W^{2})=\frac{K^{\mu}}{2}{\big\{}\frac{1+K^{2}}{\rho^{2}}-2\frac{K^{2}\mathrm{\Large~a}.K}{\rho}-K^{2}(\mathrm{\Large~a}^{2}-{\mathrm{\Large~a}_{K}}^{2}){\big\}},
\Omega^{\prime\prime}-2\rho^{\prime}\Omega^{\prime}+\frac{1}{2}\Omegaf^{\prime2}+\frac{1}{2}\Omegaf_{1}^{\prime2}+\frac{1}{2}\Omegaf_{2}^{\prime2}+\frac{1}{2}\Omegaf_{1}^{\prime}f_{2}^{\prime}=0.
w_{i}=\frac{1}{\varepsilon}\left[\left(2\pi\alpha^{\prime}\right)^{2}B_{ij}P_{1}^{j}-\left(2\pi\alpha^{\prime}\right)^{2}B_{ij}B^{jk}\left(X_{2k}-X_{1k}\right)+g_{ij}\left(X_{2}^{j}-X_{1}^{j}\right)\right]\approx0,
\left.-\frac{1}{2g}(\theta_{\mu}\cdot{\tilde{f}}_{\mu}\theta_{\nu}\cdot{\tilde{f}}_{\nu}\,-\,\theta_{\mu}\cdot{\tilde{f}}_{\nu}\,\theta_{\mu}\cdot{\tilde{f}}_{\nu}\,+\,{\tilde{f}}_{\mu}\cdot{\tilde{f}}_{\mu}\,\theta_{\nu}\cdot\theta_{\nu}-\,{\tilde{f}}_{\mu}\cdot{\tilde{f}}_{\nu}\,\theta_{\mu}\cdot\theta_{\nu})\right]\;,
\left\langleU_{\lambda}\left(w\right)\right\rangle_{w_{n_{i^{\prime}}^{*}=1}^{\prime}}^{\left({\itG}_{1}\right)_{X_{i^{\prime}}^{\prime}},l.c.}=1-\left(\gamma_{1}\lambda-\gamma_{2}\lambda^{2}+O(\lambda^{3})\right)\sum_{i_{\alpha_{1}}^{\prime}\inJ_{X_{i^{\prime}}^{\prime}}}t_{i_{\alpha_{1}}^{\prime}}^{\prime}
U_{g}^{a\prime\prime}=V^{\prime}(y^{a}+L^{a})U_{g}^{\prime}V^{\prime\dagger}(y^{a})=\left(\begin{array}{cccc}{\mathrm{e}^{-i\theta_{a,1}}}&{}&{}&{}\\{}&{\ddots}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{\mathrm{e}^{-i\theta_{a,N}}}\\\end{array}\right)\equivU_{g}^{a:inv}.
<\phi_{m}^{*}(t_{1})\phi_{n}^{*}(t_{1})\phi_{m}(t_{2})\phi_{n}(t_{2})>=\frac{1}{Z}\left[{\calG}_{0}^{(2)}(t_{1},t_{2})+{\calG}_{1}^{(2)}(t_{1},t_{2})+{\calG}_{2}^{(2)}(t_{1},t_{2})\right]\.
\begin{array}{rcl}{{\sfs}G_{i\mu}}&{=}&{\partial_{\mu}c_{i}+ig_{i}[c_{i},G_{i\mu}],}\\{{\sfs}\phi}&{=}&{ig_{1}c_{1}\Phi-ig_{2}\Phic_{2},}\\{{\sfs}\phi^{\dagger}}&{=}&{ig_{2}c_{2}\Phi^{\dagger}-ig_{1}\Phi^{\dagger}c_{1},}\\{{\sfs}c_{i}}&{=}&{ig_{i}c_{i}c_{i},}\\{{\sfs}\bar{c}_{i}}&{=}&{\displaystyle-\frac{1}{\xi_{i}}f_{i},}\\\end{array}
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\lambda_{s}^{2}\,(\gamma_{2}+\frac{\gamma_{1}^{2}}{2})^{2}+\frac{2\gamma_{1}^{2}\lambda_{s}^{2}}{r_{s}^{2}}+\frac{4\,(1-\lambda_{s})^{2}}{r_{s}^{4}}+{\cal{O}}(r-r_{s})
S_{\mathrm{DBI}}^{(\mathrm{M10})}=-\int_{R^{9+1}}|{\hat{k}}|^{3}\sqrt{|\mathrm{det}\left({\hat{\Pi}}+|{\hat{k}}|^{-1}{\hat{\calF}}\right)|}\,{\hat{R}}({\hat{T}},\partial{\hat{T}},\dots)\,.
S=\intdx^{+}dx^{-}\,\mathrm{tr}\Bigg[\partial_{+}\phi\partial_{-}\phi+i\psi\partial_{+}\psi+\frac{g^{2}}{2}J^{+}\frac{1}{\partial_{-}^{2}}J^{+}-\frac{1}{2}ig^{2}[\phi,\psi]\frac{1}{\partial_{-}}[\phi,\psi]\Bigg].
\thetaV_{0}(a_{-1}^{i}a_{-1}^{j}|0\rangle)c_{-1/2}^{i}\chi={\frac{1}{2}}\thetac_{-1/2}^{j}\chi={\frac{1}{2}}\sum_{k}c_{-1/2}^{k}{S^{j}}_{k}\chi\,,
D(\vec{V},\vec{\sigma})=\left(\begin{array}{c|c}{\gamma}&{0}\\\hline{\displaystyle-\frac{\vec{V}}{c}\gamma^{-1}}&{I+\displaystyle\frac{\vec{V}\otimes\vec{V}^{\mathrm{T}}}{c^{2}\gamma\left[\gamma+\sqrt{\gamma^{2}+\displaystyle\frac{\vec{V}}{c}^{2}}\right]}-\frac{\vec{V}\otimes\vec{\sigma}^{\mathrm{T}}}{c^{2}\gamma\gamma_{0}}}\\\end{array}\right).
S=\int\frac{1}{2}\epsilon^{\mu\nu\lambda\rho}\left(B_{\mu\nu}^{a}(x)R_{\lambda\rho}^{a}(x)+\underbrace{\phi^{ab}(x)B_{\mu\nu}^{a}(x)B_{\lambda\rho}^{b}(x)}_{constraint\:term}\right).
D=\gamma\left(\mu(\infty)-\frac{1}{2}x_{\mathrm{H}}^{2}AN^{\prime}|_{x_{\mathrm{H}}}\right)=\gamma\mu(\infty)\left(1-\frac{\mu_{\mathrm{S}}}{\mu(\infty)}\frac{T}{T_{\mathrm{S}}}\right)\,
\hat{A}_{[1]}=e^{i\Omega_{G}t}\Biggl[\hat{a}_{0}+\frac{\lambda}{(2\Omega_{G})^{2}}\Biggl\{\frac{1}{4\Omega_{G}}\hat{a}_{0}^{\dagger3}+\frac{3}{2\Omega_{G}}\hat{a}_{0}^{\dagger2}\hat{a}_{0}+(3it+c_{0})\hat{a}_{0}^{\dagger}\hat{a}_{0}^{2}-\frac{1}{2\Omega_{G}}\hat{a}_{0}^{3}\Biggr\}\Biggr],
i\gamma_{ab}^{\mu}\partial_{x}^{\mu}\frac{\deltaW}{\delta\overline{{\eta}}_{b}(x)}-e\frac{\deltaW}{\deltaJ^{\mu}(x)}\gamma_{ab}^{\mu}\frac{\deltaW}{\delta\overline{{\eta}}_{b}(x)}+ie\gamma_{ab}^{\mu}\frac{\delta^{2}W}{\deltaJ^{\mu}(x)\delta\overline{{\eta}}_{b}(x)}+\eta_{a}(x)=0\;.
\sum_{\{T\}}q^{c(T)}=\sum_{\{\nu\}}q^{{\frac{1}{2}}L(L-1-\sum_{j}s_{j}^{(1)})}\prod_{a=1}^{n-1}\prod_{j\geq1}q^{-{\frac{1}{2}}s_{j}^{(a)}h_{j}^{(a)}}\biggl[\begin{array}{c}{s_{j}^{(a)}+h_{j}^{(a)}}\\{s_{j}^{(a)}}\\\end{array}\biggr],
K_{R}=t_{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}\mu_{4}\nu_{4}}\epsilon_{1}^{\mu_{1}}k_{1}^{\nu_{1}}\epsilon_{2}^{\mu_{2}}k_{2}^{\nu_{2}}\epsilon_{3}^{\mu_{3}}k_{3}^{\nu_{3}}\epsilon_{4}^{\mu_{4}}k_{4}^{\nu_{4}},
\psi_{12}(p_{30})=\,{\frac{\Lambda_{12}^{++}}{(P_{0}-S)-(p_{30}-h_{3})+i\epsilon}}\,\left[\,V_{12}^{0}\psi_{12}(p_{30})+V_{23}^{0}\psi_{23}(E_{1})+V_{31}^{0}\psi_{31}(E_{2})\,\right]
\left(\bar{\lambda}\chi\right)^{*}=\left(\bar{\lambda}^{i}\chi_{i}\right)^{*}=\chi_{i}^{\dagger}\left(\bar{\lambda}^{i}\right)^{\dagger}=\bar{\chi}^{i}\lambda_{i}=\bar{\lambda}^{i}\chi_{i}=\bar{\chi}\lambda=\bar{\lambda}\chi\.
\begin{array}{c}{A_{~\mu}^{\lambda}\eta_{\lambda\rho}A_{~\nu}^{\rho}\propto\eta_{\mu\nu}}\\{{}}\\{B_{\alpha}^{\mu}B_{\mu\beta}=\bar{B}_{\dot{\alpha}}^{\mu}\bar{B}_{\mu\dot{\beta}}=A_{~\nu}^{\mu}B_{\mu\alpha}=A_{~\nu}^{\mu}\bar{B}_{\mu\dot{\alpha}}=B_{\alpha}^{\mu}\bar{B}_{\mu\dot{\alpha}}=0}\\\end{array}
\frac{\tilde{M}_{b}\pi}{\lambda^{2}}=\frac{M_{b}\pi}{\lambda^{2}}+\frac{(M_{b}\pi/\lambda^{2})(M_{a}\pi/\lambda^{2})}{\left(e^{\lambdar_{c}}+\sqrt{M_{1}\pi}/\lambda\right)\left(e^{\lambdar_{c}}-\sqrt{M_{1}\pi}/\lambda\right)}
\begin{aligned}{\mathcal{A}_{1}=}&{{}\mathcal{A}_{1}^{(a_{1},0)}T_{(a_{1},0)},}\\{\mathcal{A}_{2}=}&{{}\mathcal{A}_{2}^{(a_{1},a_{2})}T_{(a_{1},a_{2})}\quad\textrm{with}\quada_{2}\neq0.}\\\end{aligned}
\lambda_{\alpha}^{\mathrm{LD}}(x)=2\gamma_{\alpha}^{\mu\\beta}\\xi_{\beta}\{\frac{x_{\mu}}{x^{3}}}\,\quad\psi_{\alpha}^{\mathrm{LD}}(x)=\2\gamma_{\alpha}^{\mu\\beta}\{\xi^{\prime}}_{\beta}\{\frac{x_{\mu}}{x^{3}}}\,
S_{CS}=\frac{1}{4\kappa^{2}\mu}\intd^{3}x\epsilon^{\lambda\mu\nu}\Gamma_{\lambda\rho}^{\sigma}(\partial_{\mu}\Gamma_{\sigma\nu}^{\rho}+\frac{2}{3}\Gamma_{\mu\tau}^{\rho}\Gamma_{\nu\sigma}^{\tau}).
\alpha^{a}=C_{\;\;bc}^{a}\left(\eta^{*b\mu}A_{\mu}^{c}+\frac12\varepsilon^{\mu\nu\rho\sigma}B_{\mu\nu}^{*b}B_{\rho\sigma}^{*c}+g^{cd}\varphi^{*b}\partial_{\mu}A_{d}^{*\mu}\right),
\chi_{c}\:C\:\gamma^{I}\:q_{k}^{I_{k}}\cdotsq_{1}^{I_{1}}\;\tilde{V}_{J_{k},c_{k}}^{(k)}(q_{k})\cdots\tilde{V}_{J_{1},c_{1}}^{(1)}(q_{1})\:\partial_{p}^{K}\;S^{\vee(h)}(p)\;\;\;,
p_{l}=\frac{\partialL}{\partial\dot{q}_{l}}=\sqrt{2m[E(V_{1}(q_{1})+V_{2}(q_{2}))-(U_{1}(q_{1})+U_{2}(q_{2}))]}\times\frac{\dot{q}_{l}}{\sqrt{(\dot{q}_{1}^{2}+\dot{q}_{2}^{2})}}
dR_{2}=\delta^{4}(p_{1}+p_{2}-p_{\mathrm{T}}-p_{\mathrm{B}})\delta(p_{1}^{2}+m_{1}^{2})\delta(p_{2}^{2}+m_{2}^{2})\frac{d^{4}p_{1}}{(2\pi)^{4}}\frac{d^{4}p_{2}}{(2\pi)^{4}}
E[\sigma]=\sum_{\ell}N_{\ell}\left[\int_{0+i\epsilon}^{\infty+i\epsilon}\frac{dk}{\pi}\sqrt{k^{2}+m^{2}}\int_{0}^{\infty}dr\left[\psi_{\ell}^{*}(k,r)\psi_{\ell}(k,r)\right]_{N}+\sum_{j}\frac{\omega_{j}}{2}\right]+\sum_{i=1}^{N}E_{\mathrm{FD}}^{(i)}+E_{\mathrm{CT}}\,,
\alpha=\sqrt{\frac{N_{k}}{\hbar\tau_{2}}}\left[-i\tau\hat{A}_{1}+i\hat{A}_{2}\right]\\,\\\alpha^{\dagger}=\sqrt{\frac{N_{k}}{\hbar\tau_{2}}}\left[i\bar{\tau}\hat{A}_{1}-i\hat{A}_{2}\right]\\.
{}^{(i)}\langle\zeta_{1},\ldots,\zeta_{N}|T(\zeta_{1},\ldots,\zeta_{N})={}^{(i)}\langle\zeta_{1},\ldots,\zeta_{N}|\Lambda_{W}^{(i)}(\zeta;r)\Lambda_{V}^{(i)}(\zeta;\bar{r}).
\Biggl[\psi_{1}({x}_{1},\ldots,{x}_{n})\prod_{i=1}^{m}\frac{1}{{1-{y}_{i}-{b}}}(1-{b})^{-s}\psi_{2}(\frac{{y}_{1}}{1-{y}_{1}-{b}},\ldots,\frac{{y}_{m}}{1-{y}_{m}-{b}})\Biggr].
\begin{array}{rcl}{{\bf{Q}}_{-{\frac{1}{k}}}^{+}\vert\Omega_{-s}^{-}\rangle}&{=}&{0}\\{\{{\frac{({\bf{Q}}_{-{\frac{1}{k}}}^{-})^{r}}{[r]!}}\vert\Omega_{-s}^{-}\rangle}&{;}&{o\ler\lek-1\}}\\\end{array}
S=\intd^{5}x\sqrt{-g}\left({\frac{R}{2}}-\Lambda_{b}+{\calL}^{(M)}\right)+\sum_{j=1,2\\mathrm{branes}}\intd^{4}x\sqrt{-g^{(j)}}\left({\calL}_{j}^{(M)}-\Lambda_{j}\right)~,
\Gamma_{div}=-{\frac{1}{2\epsilon}}\intd^{2}x\,\sqrt{g}\,\left[{\frac{n}{12}}R+\mathrm{Tr}\,(\hat{E}^{\lambda}\hat{E}_{\lambda})+\mathrm{Tr}\,(\nabla^{\lambda}\hat{E}_{\lambda})-\mathrm{Tr}\,\left(\hat{\Pi}+{\frac{R}{6}}\hat{1}\right)\right],\nonumber
\begin{aligned}{M_{4(a)}(q)}&{{}={\frac{1}{2!}}\int{\frac{d^{4}l}{(2\pi)^{4}}}\tau_{2}^{\alpha\beta\gamma\delta}(k_{1},k_{2},m_{1})\tau_{2}^{\sigma\rho\mu\nu}(k_{3},k_{4},m_{2})\times\Big[\frac{i{\calP}_{\alpha\beta\mu\nu}}{(l+q)^{2}}\Big]\Big[\frac{i{\calP}_{\gamma\delta\sigma\rho}}{l^{2}}\Big]}\\\end{aligned}
S_{t}(x)=\frac{e^{iet_{\mu}x_{\mu}}}{2\piL_{1}}\left(\begin{array}{cc}{0}&{\frac{\vartheta_{1}^{\prime}(0)\vartheta_{1}(z+\bar{t})}{\vartheta_{1}(\bar{t})\vartheta_{1}(z)}e^{-ieL_{1}t_{1}z}}\\{-\frac{\vartheta_{1}^{\prime}(0)\vartheta_{1}(\bar{z}-t)}{\vartheta_{1}(t)\vartheta_{1}(\bar{z})}e^{-ieL_{1}t_{1}\bar{z}}}&{0}\\\end{array}\right).
N_{1}={\frac{1}{\sqrt{2}}}\left(\begin{matrix}{0}&{0}&{0}&{i}\\{0}&{0}&{0}&{0}\\{i}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{matrix}\right),\qquadN_{2}={\frac{1}{\sqrt{2}}}\left(\begin{matrix}{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{i}\\{0}&{i}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{matrix}\right),