harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I=\intd\tau\left(\overline{{\eta}}_{a}\partial_{\tau}\eta^{a}-\overline{{\chi}}_{a}\chi^{a}-\overline{{\eta}}_{a}M_{0b}^{a}\chi^{b}-\overline{{\chi}}_{a}M_{0b}^{a}\eta^{b}+\overline{{\eta}}_{a}M_{1b}^{a}\eta^{b}\right)\.
\sqrt2\pib_{\scriptscriptstyle{l-1,L}}\,a^{l-1}=\left(\frac{d}{dr}+(l-\Phi/2\pi)/r\right)\,\psi_{\scriptscriptstyle{l,L}}^{>}(k,r)\,(1+k^{2}\chi_{\scriptscriptstyle{l,L}}(r)+\mathrm{O}(k^{4}))\,,
\begin{array}{ll}{\tilde{f}_{\omega}^{E}(\sigma^{\pm})=[(a/b)_{\omega}+1]exp(-i\omega{\sigma_{\pm}^{E}})/\sqrt{4\pi<\Phi_{\omega},\Phi_{\omega}>}}\\{\tilde{f}_{\omega}^{L}(\sigma^{\pm})=[(c/b)_{\omega}+(1/b)_{-\omega}]exp(-i\omega{\sigma_{\pm}^{L}})/\sqrt{4\pi<\Phi_{\omega},\Phi_{\omega}>}}\\\end{array}
{\calL}=\mathrm{Tr}\left\{\frac{\mu}{2}\,(\dot{X}^{i})^{2}+\frac{\mu}{2R^{2}}\,(X^{i})^{2}+\frac{1}{4g^{2}l^{5}}\,[X^{i},X^{j}]^{2}+\cdots\right\}
V(y_{i},\theta)=-V_{p+1}(i)^{p}\int_{0}^{\infty}\frac{dt}{t}(8\pi^{2}\alpha^{\prime}\,t)^{-\frac{p+1}{2}}e^{-(\frac{y^{2}}{2\pi\alpha^{\prime}}-\pi)t}\eta^{-12}(it)\theta_{11}^{4}(it/2,it)\,,
\begin{array}{ccc}{I_{c}=I(\mu_{c})={\frac{2\sqrt{2}}{3\pi}}\,}&{J_{c}=J(\mu_{c})={\frac{2\sqrt{2}}{15\pi}}\,}&{K_{c}=K(\mu_{c})={\frac{2\sqrt{2}}{35\pi}}\,\nonumber}\\{\I(\mu_{c}-\mu)=I_{c}+\deltaI\,}&{\J(\mu_{c}-\mu)=J_{c}+\deltaJ\,}&{\K(\mu_{c}-\mu)=K_{c}+\deltaK\.}\\\end{array}
\chi=\frac{\hat{X}^{2}}{D}-2c_{1}-4\zeta(2)T^{\prime}\left(\frac{\hat{X}^{2}}{D}-2c_{1}\right)+8\zeta(3)\left(T^{\prime}\left(\frac{\hat{X}^{2}}{D}-2c_{1}\right)\right)^{2}+\ldots
N_{Q}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})=I_{Q}-\frac{1}{2}(\Omega_{m}^{2}+{\stackrel{\rightharpoonup}{p}}^{2}+4m_{Q}^{2})K_{Q}^{T}(-i\Omega_{m},\stackrel{\rightharpoonup}{p}),
f_{k}^{(D)}(\Omega^{({D})})=-\frac{1}{4\pi}\left(\frac{k}{2\pi}\right)^{({D}-3)/2}\intd^{{D}}r^{\prime}e^{-i{\bfk}^{\prime}\cdot{\bfr}^{\prime}}V({\bfr}^{\prime})\Psi^{(+)}({\bfr}^{\prime})\;
G_{ab}^{\mu\nu}(x_{1},x_{2};x_{3},x_{4})=\left.\frac{\delta^{4}}{\deltaJ_{\mu}(x_{1})\deltaJ_{\nu}(x_{2})\delta\overline{{\eta}}_{a}(x_{3})\delta\eta_{b}(x_{4})}Z(\overline{{\eta}},\eta,J)\right|_{\mathrm{curr.}=0}^{\mathrm{connected}}\;,
\begin{array}{l}{L_{f}(u_{f})=J_{\psi}L(u)=}\\{J_{\psi}\left\{S(B_{f})\displaystyle{\frac{i}{2}}\left[\bar{\Psi}_{f}(u_{f})\gamma_{0}D\Psi_{f}(u_{f})-\right.\right.\left.\left.\left(D\bar{\Psi}_{f}(u_{f})\right)\gamma_{0}\Psi_{f}(u_{f})\right]-m\bar{\Psi}_{f}(u_{f})\Psi_{f}(u_{f})\right\}.}\\\end{array}
{\mathcal{F}_{D}^{0}}[\hat{g}]={\frac{Q^{2}}{16\pi}}\int{d^{2}}\xi{d^{2}}\xi^{\prime}\sqrt{\hat{g}(\xi)}\hat{R}(\xi){\hat{G}_{D}}(\xi,\xi^{\prime})\sqrt{\hat{g}(\xi^{\prime})}\hat{R}(\xi^{\prime}).
C_{k}=\frac{1}{n^{2}p^{2}}\sum_{i=1}^{n}\Gamma_{0}^{i}\Gamma^{n-k}\Gamma_{0}^{k-i}=\frac{1}{n^{2}p^{2}}\sum_{i=1}^{n}\Gamma_{0}^{i}\left(\sum_{\lambda=0}^{3}\pi_{\lambda}Q_{\lambda}\right)^{n-k}\Gamma_{0}^{k-i}
V(|\Phi|,|\xi|)=\frac{1}{4}\lambda\left(|\Phi|^{2}-\eta^{2}\right)^{2}+\frac{1}{4}\lambda_{\xi}\left(|\xi|^{2}-\eta_{\xi}^{2}\right)^{2}+\beta|\Phi|^{2}|\xi|^{2},
D_{++}^{(ML)ab}(x)=D_{++}^{(ML)}(x)\delta^{ab}={\frac{i\delta^{ab}}{\pi^{2}}}\intd^{2}k\,e^{ikx}{\frac{k_{+}^{2}}{(k^{2}+i\epsilon)^{2}}}={\frac{\delta^{ab}}{\pi}}{\frac{(x^{-})^{2}}{(-x^{2}+i\epsilon)}}\
S_{\alpha\beta}(x,y)\simeq\phi_{0}^{\dagger}(x_{2})\phi_{0}(y_{2})\langlex_{0},x_{1},\alpha\mid\frac{\widehat{/\!\!\!\partial}+ie\;\gamma_{0}V_{0,0}P_{L}}{(\widehat{\partial}+ie\;\gamma_{0}V_{0,0})^{2}}\midy_{0},y_{1},\beta\rangle\,.
\Big[\begin{array}{c}{v}\\{\emptyset}\\\end{array}\Big],\qquad\Big[\begin{array}{c}{\emptyset}\\{v}\\\end{array}\Big],\qquad\Big[\begin{array}{c}{v}\\{v+F}\\\end{array}\Big]\sim\Big[\begin{array}{c}{v}\\{v}\\\end{array}\Big],
J=\frac{1}{2}\{-f_{2}^{\sharp(NN+DD)}(q)f_{3}^{\sharpDN}(q)\frac{\Theta_{2}(\epsilont,it)}{\Theta_{2}(0,it)}+f_{3}^{\sharp(NN+DD)}(q)f_{2}^{\sharpDN}(q)\frac{\Theta_{3}(\epsilont,it)}{\Theta_{3}(0,it)}\}
X_{m}(a,b,c;q)=\sum_{\begin{array}{c}{\scriptstyle\sigma_{1},\ldots,\sigma_{m-1}=1}\\{\scriptstyle|\sigma_{j+1}-\sigma_{j}|=1}\\\end{array}}^{r-1}q^{\sum_{k=1}^{m}k|\sigma_{k+1}-\sigma_{k-1}|/4},\qquad\sigma_{0}=a,\;\sigma_{m}=b,\;\sigma_{m+1}=c.
p_{i}^{\nu}=\frac{1}{2}\intf_{i}^{2}\left\{3a(u^{\nu}s_{\sigma}-s^{\nu}u_{\sigma})+\Theta\delta_{\sigma}^{\nu}+3(\sigma_{.\sigma}^{\nu}+\omega_{.\sigma}^{\nu})\right\}dx^{\sigma}dV.
T_{\\mu\rho\sigma}^{\mu}=\left(p^{2}T_{1}+2\omegaT_{2}+4T_{3}\right)\Biggl[\left(p_{\rho}p_{\sigma}-p^{2}g_{\rho\sigma}\right)\mp\frac{1}{4}\left(\varepsilon_{\rho\lambda}p^{\lambda}p_{\sigma}+\varepsilon_{\sigma\lambda}p^{\lambda}p_{\rho}\right)\Biggl],
\widetilde{H}=-\sum_{k=1}^{K}\sum_{j=1}^{L^{\prime}}(\tilde{J}_{x}^{k}\sigma_{x,j}\sigma_{x,j+k}+\tilde{J}_{y}^{k}\sigma_{y,j}\sigma_{y,j+k})\sigma_{z,j+1}\ldots\sigma_{z,j+k-1}\,+\,h\sum_{j=1}^{L^{\prime}}\sigma_{z,j}\,\nonumber
{\calL}=\frac{\mu}{2}\,\dot{x}_{i}^{2}+\frac{\mu}{2R^{2}}\,x_{i}^{2}-\frac{\kappa}{2}\,\epsilon_{ij}\left(\dot{x}_{i}\ddot{x}_{j}+\frac{1}{R^{2}}x_{i}\dot{x}_{j}\right)\equiv{\calL}_{0}+{\calL}_{\mathrm{CS}}
\begin{array}{ccc}{\pi}&{:}&{\Omega^{*}({\calA})\longrightarrow{\calL}({\calH})}\\{\pi_{p}((DF_{1})...(DF_{p})F)}&{=}&{\prod_{i=1}^{p}[D,\pi_{0}(F_{i})]\pi_{0}(F),}\\\end{array}
C_{\{k\}}={\frac{1}{k!}}\sum_{a_{1},\dots,a_{k}}Tr_{F}\{t_{F}^{a_{1}}t_{F}^{a_{2}}\cdotst_{F}^{a_{k}}\}t_{R}^{(a_{1}}t_{R}^{a_{2}}\cdotst_{R}^{a_{k})}
f^{(n)}=\left(\begin{array}{c|ccc}{f}&{A^{(0)}}&{\cdots}&{A^{(n)}}\\\hline{-{\tilde{A}}^{(0)}}&{0}&{\cdots}&{0}\\{\vdots}&{\vdots}&{{}}&{\vdots}\\{-{\tilde{A}}^{(n)}}&{0}&{\cdots}&{0}\\\end{array}\right)
V(z)=[1-{\frac{z}{Q_{p}}}{\frac{1+{\frac{(Q_{p}b)^{\frac{n}{2}}}{2(x^{2}+4Q_{p}z)^{\frac{n}{2}}}}}{(1+{\frac{(Q_{p}b)^{\frac{n}{2}}}{(x^{2}+4Q_{p}z)^{\frac{n}{2}}}})^{2}}}]+{\frac{J_{\perp}^{2}}{2m_{f}Q_{p}^{2}}}{\frac{1}{{(1+{\frac{(Q_{p}b)^{\frac{n}{2}}}{(x^{2}+4Q_{p}z)^{\frac{n}{2}}}}})^{2}}},
(\pi({\vec{x}}))^{*}\pi({\vec{x}})=\sum_{\vec{n},\vec{m}}\frac{1}{X^{3}}{\sqrt{\kappa_{\vec{n}}\kappa_{\vec{m}}}}(q_{\vec{n}}^{*}q_{\vec{m}})exp\left(2\pii({\vec{n}}-{\vec{m}})\cdot{\vec{x}}/X\right).
B(z(w)\circ|\mathcal{B}_{p}\rangle)=\ldots-\intd^{25-p}q_{\bot}\,\left(\frac{kW^{2}}{1+W^{2}}\right)^{q_{\bot}^{2}}\frac{2}{W^{2}}\alpha_{-1}\cdot\tilde{\alpha}_{-1}c_{1}\tilde{c}_{1}|q_{\bot}\rangle+\ldots
A_{NS-NS}={\frac{1}{2}}\int_{0}^{\infty}{\frac{dt}{t}}(8\pi^{2}\alpha^{\prime}t)^{-1/2}e^{-{\frac{tY^{2}}{2\pi\alpha^{\prime}}}}f_{4}^{-8}(q)\left(-f_{2}^{8}(q)+f_{3}^{8}(q)\right)={\frac{1}{2}}\int_{0}^{\infty}{\frac{dt}{t}}(8\pi^{2}\alpha^{\prime}t)^{-1/2}e^{-{\frac{tY^{2}}{2\pi\alpha^{\prime}}}}
V(r)=E\left[1-\frac{1+\frac{\rho_{*}^{4}}{2r^{2}(r^{2}+l^{2})}}{\left(1+\frac{\rho_{*}^{4}}{r^{2}(r^{2}+l^{2})}\right)^{2}}\right]+\frac{L^{2}}{2m_{p}r^{2}}\frac{1}{\left(1+\frac{\rho_{*}^{4}}{r^{2}(r^{2}+l^{2})}\right)^{2}}.
ds_{(10)}^{2}=e^{\frac23\phi}\left[dx^{a}\,dx^{b}\,\eta_{ab}+{\frac{1}{\sqrt{2\kappa|u|^{2}+u_{0}}}}\left(du^{i}+\epsilon^{ijk}\tilde{A}^{j}u^{k}\right)^{2}+\sqrt{2\kappa|u|^{2}+u_{0}}\;d\tilde{s}_{3}\right]
p_{*}u_{0}|_{r^{*}=-\infty}=(r-r_{h})(\phi^{(2)^{\prime}}-\phi^{(1)^{\prime}})|_{r=r_{h}}=\left((r-r_{h})\frac{\partial\phi_{h}^{\prime}}{\partial\phi_{h}}u_{0}\right)\left|_{r=r_{h}}\right.
\frac{SO(4,n_{A}n_{B})}{SO(4)\timesSO(n_{A}n_{B})}\to\frac{SO(2,n_{1}n_{3}+n_{2}n_{4})}{SO(2)\timesSO(n_{1}n_{3}+n_{2}n_{4})}\times\frac{SO(2,n_{1}n_{4}+n_{2}n_{3})}{SO(2)\timesSO(n_{1}n_{4}+n_{2}n_{3})}.
\Psi_{E}\left(r,\theta\right)=\sum_{n=-\infty}^{\infty}\left(\begin{array}{l}{A_{n}^{+}J_{n-\kappa}\left(kr\right)e^{in\theta}}\\{-i\frac{k}{E-m}A_{n}^{+}J_{n+1-\kappa}\left(kr\right)e^{i\left(n+1\right)\theta}}\\{A_{n}^{-}J_{n-\kappa}\left(kr\right)e^{in\theta}}\\{-i\frac{k}{E+m}A_{n}^{-}J_{n+1-\kappa}\left(kr\right)e^{i\left(n+1\right)\theta}}\\\end{array}\right)\,.
\begin{array}{ll}{S_{gauged}=}&{\intd^{2}z\,\left(\partial_{a}X\partial^{a}X+\Omega_{a}\Omega^{a}+2\Omega_{a}\partial^{a}X-\epsilon^{ab}\partial_{a}\lambda\Omega_{b}\right)}\\{}&{+\ointds\left(ct_{a}+\kappab_{a}+t_{a}\lambda\right)\Omega^{a}.}\\\end{array}
\begin{array}{ccc}{(kl3)+j\rightarrow(1234),\qquad}&{(1234)+(kl3)\rightarrowj,\qquad}&{j+(1234)\rightarrow(kl3),}\\{(j3)+(kl3)\rightarrow(12334),\,\,\,}&{(kl3)+(12334)\rightarrow(j3),\,\,\,}&{(12334)+(j3)\rightarrow(kl3),}\\{(1234)+3\rightarrow(12334),\,\,\,}&{3+(12334)\rightarrow(1234),\,\,\,}&{(12334)+(1234)\rightarrow3,}\\\end{array}
\tilde{\calA}\vphantom{\calA}_{n,TT}^{0}+\frac{2\alpha}{1+\alpha}(T+\tilde{\Lambda})^{-1}\tilde{\calA}_{n,T}^{0}+\left[n^{2}-\frac{\alpha}{(1+\alpha)^{2}}(T+\tilde{\Lambda})^{-2}\right]\tilde{\calA}_{n}^{0}=0
\mathrm{tr}\,e^{-\beta(\frac{1}{2}K^{ij}v_{i}v_{j}-J^{i}v_{i})}=\left.e^{-\frac{1}{2}\betaK^{ij}u_{i}u_{j}}\mathrm{tr}\,e^{\varphi^{i}v_{i}}\right|_{\varphi=\betaJ}.
f(k,\theta,z_{1})=\sum_{n=-\infty}^{\infty}a_{n}e^{i\frac{n\pi}{l}z_{1}},\;\;\psi_{n}^{(\rho_{-})}=\sum_{n=-\infty}^{\infty}b_{n}e^{i\frac{n\pi}{l}z_{1}},
a_{2}=(Q-R/6)^{2}-\frac{1}{3}\BoxQ-\frac{1}{90}R_{\mu\nu}R^{\mu\nu}+\frac{1}{90}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}+\frac{1}{15}\BoxR+\frac{1}{6}\tilde{R}_{\mu\nu}\tilde{R}^{\mu\nu},
S_{\mu\nu}={\textstyle{\frac{1}{12}}}T_{\alpha}{}^{\alpha}T_{\mu\nu}-{\textstyle{\frac{1}{4}}}T_{\mu\alpha}T^{\alpha}{}_{\nu}+{\textstyle{\frac{1}{24}}}g_{\mu\nu}\left[3T_{\alpha\beta}T^{\alpha\beta}-\left(T_{\alpha}{}^{\alpha}\right)^{2}\right]\,.
Z_{\mathrm{all}}\left(0\right)\equiv\sum_{\alpha=1}^{N+1}Z_{\alpha}=\sum_{\alpha=1}^{N+1}{\frac{e^{-iQc_{\alpha}T}}{\prod_{\beta\ne\alpha}\left\{1-e^{-i\left(c_{\beta}-c_{\alpha}\right)T}\right\}}}\.
\left\langle\alpha_{1},\alpha_{2},...,\alpha_{2N},t\right|U_{+}\left|\alpha_{1}^{\prime},\alpha_{2}^{\prime},...,\alpha_{2N}^{\prime},t\right\rangle=R_{\alpha_{1}\alpha_{2}}^{\alpha_{1}^{\prime}\alpha_{2}^{\prime}}\:R_{\alpha_{3}\alpha_{4}}^{\alpha_{3}^{\prime}\alpha_{4}^{\prime}}\:...\:R_{\alpha_{2N-1}\alpha_{2N}}^{\alpha_{2N-1}^{\prime}\alpha_{2N}^{\prime}}
{\frac{d^{2}x^{i}}{d\tilde{\lambda}^{2}}}=\sqrt{h}\left[{\frac{h^{\prime}}{2}}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-y\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}-{\frac{2}{y}}\right]{\frac{dx^{i}}{d\tilde{\lambda}}}.
\mathrm{~\cal~L~}_{\mathrm{eff}}=\frac{1}{4\pi}\sum_{\varsigma=\pm}\left[\mu_{\varsigma}^{2}m-\frac{(eB)^{2}}{4m}\right]=\frac{\mu^{2}m}{2\pi}+\frac{(eB)^{2}}{8\pim}[(2\sigma)^{2}-1],
\vec{v}(\vec{x})\equiv\epsilon^{ij}\,\partial_{i}\hat{n}\times\partial_{j}\hat{n}\quad,\quadq(\vec{x})\equiv\epsilon^{ij}\,\epsilon^{abc}\,\hat{n}^{a}\,\partial_{i}\hat{n}^{b}\,\partial_{j}\hat{n}^{c}
{\frac{h}{r^{3}}}{\frac{d}{dr}}\left(hr^{3}{\frac{dR}{dr}}\right)~+~\omega^{2}\left[{\frac{(r_{n}r_{1}r_{5})^{2}}{r^{6}}}~+~{\frac{(r_{1}r_{5})^{2}}{r^{4}}}~+~{\frac{(r_{1}^{2}+r_{5}^{2})}{r^{2}}}\right]R~=~0~.
\{f,g\}_{0}=i\left(\frac{\partialf}{\partialz^{n}}\frac{\partialg}{\partial{\bar{z}}^{n}}-\frac{\partialf}{\partial{\bar{z}}^{n}}\frac{\partialg}{\partialz^{n}}\right)+\frac{\partial^{R}f}{\partial\eta^{n}}\frac{\partial^{L}g}{\partial{\bar{\eta}}^{n}}+\frac{\partial^{R}f}{\partial{\bar{\eta}}^{n}}\frac{\partial^{L}g}{\partial\eta^{n}}
\left.\mathrm{Tr}K^{(1)}\right|_{{\calC}_{\beta}\times\Sigma}=\mathrm{Tr}\left.K^{(1)}\right|_{{\calC}_{\beta}}\left.\mathrm{Tr}K^{(0)}\right|_{\Sigma}+\mathrm{Tr}\left.K^{(0)}\right|_{{\calC}_{\beta}}\left.\mathrm{Tr}K^{(1)}\right|_{\Sigma}~~~.
\chi_{2,1}^{4,5}(q)=q^{\frac{49}{120}}\sum_{\vec{k}}\frac{q^{\frac52(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+\vec{k}\cdot\vec{B}_{2,1}^{4,5}}}{(q^{5})_{k_{1}}\ldots(q^{5})_{k_{5}}}\quad\mathrm{with}\quad\vec{B}_{2,1}^{4,5}=\{-\frac32,\frac32,\frac52,2,3\}\.
\Pi({\bf{x}}\sigma)=X_{0\sigma}+\int^{{\bf{x}}}d{\bf{l}}\mathrm{~}(-1/\rho({\bf{y}}\sigma)){\bf{J}}({\bf{y}}\sigma)+\Phi([\rho];{\bf{x}}\sigma)-\int^{{\bf{x}}}d{\bf{l}}\mathrm{~}[-i\Phi,\nabla\Pi]({\bf{y}}\sigma)
\omega_{rp}=\omega_{r}^{(1)}\delta_{rp}+\int_{C_{r}}\eta_{p}(t)J_{r}^{(1)}(t)d\thetadz=\omega_{r}^{(1)}\delta_{rp}+\int_{C_{p}}\eta_{r}(t)J_{p}^{(1)}(t)d\thetadz
\langle\,\partial_{-}f(\sigma)\,\partial_{+}f(\sigma^{\prime})\,\rangle\,_{\mathrm{{in}}}\sim-\,\frac{1}{4\pi}\,\frac{2\lambda/M}{(\sigma^{+}-x_{H})^{2}}\,e^{-\,2\lambda\sigma^{-}}\,.
\epsilon^{ij}\Gamma_{ij}\left(\frac12\left(1+\Gamma_{B}\right)C\Gamma_{\mu_{1}\cdots\mu_{p}}\right)=\left(\frac12\left(1+\Gamma_{B}\right)C\Gamma_{\mu_{1}\cdots\mu_{p}}\right)\Gamma_{A}\epsilon^{ij}\Gamma_{ij}\Gamma_{A}
<\theta^{a}|\Phi>=\sum_{i=0,1,..,3,5,..,d}\quad\sum_{\{a_{1}<a_{2}<...<a_{i}\}\in\{0,1,..,3,5,..,d\}}\alpha_{a_{1},a_{2},...,a_{i}}\theta^{a_{1}}\theta^{a_{2}}\cdots\theta^{a_{i}},
P^{n}=|[n]_{q^{-1}}!T(P^{n})=|[n]_{q^{-1}}!\int_{\infty}^{z}dt_{1}\int_{\infty}^{t_{1}}dt_{2}\cdots\int_{\infty}^{t_{n-1}}dt_{n}J(t_{n})\cdotsJ(t_{1})~.
\langle\mathrm{out}|f(\phi)|\mathrm{in}\rangle={\frac{e^{-i\pi/4}}{\sqrt{2\pip}}}\left({\frac{1}{1+e^{i\phi}}}\right)\langle\mathrm{out}|{\calR}^{\phi/\pi}\left({\calR}^{-1}-{\calR}\right)|\mathrm{in}\rangle\;,
\begin{aligned}{-H^{*}=\sum_{i}\Big(}&{{}c_{++}(J_{i}^{+})^{2}+c_{00}(J_{i}^{0})^{2}+c_{--}(J_{i}^{-})^{2}+\frac{c_{0+}}2\{J_{i}^{0},J_{i}^{+}\}+\frac{c_{0-}}2\{J_{i}^{0},J_{i}^{-}\}}\\{}&{{}+c_{+}J_{i}^{+}+c_{0}J_{i}^{0}+c_{-}J_{i}^{-}\Big),}\\\end{aligned}
\left[N_{i},p_{j}\right]=i\,\delta_{ij}\left({\frac{\kappa}{2}}\left(1-e^{-2{p_{0}/\kappa}}\right)+{\frac{1}{2\kappa}}\vec{p}\,{}^{2}\,\right)-i\,{\frac{1}{\kappa}}p_{i}p_{j}
\left(\hat{\Lambda}_{n}^{\mu}(Q_{1},Q_{2})\right)_{ji}^{\ell}\simeq-(-)^{i+j+\ell}\gamma^{0}\,\hat{Q}_{1\tau}^{\mu}\sum_{\rho=\pm}\left[{\calN}_{j\ell}^{(\rho)}{\calN}_{\elli}^{(-\rho)}\left\{\frac{-2i\rho\tau\gamma_{q}}{E-\tau\hat{{\bfq}}_{1}\cdot{\bfp}-2i\rho\tau\gamma_{q}}\right\}^{n}\right]\,,
\hat{\Pi}^{\underline{{m}}}=d\hat{X}^{\underline{{m}}}-id\hat{\Theta}^{1}{\sigma}^{\underline{{m}}}\hat{\Theta}^{1}-id\hat{\Theta}^{2}\tilde{\sigma}^{\underline{{m}}}\hat{\Theta}^{2}=d\xi^{m}\hat{\Pi}_{m}^{~\underline{{m}}}(\xi)
\theta_{d-1}=\frac{J}{l\sqrt{2}}\left[\sqrt{1+E}\,EllipticE\left(\frac{T}{l},\frac{1}{E+1}\right)-\frac{E}{\sqrt{1+E}}\,EllipticF\left(\frac{T}{l},\frac{1}{E+1}\right)\right]\;,
\omega|_{G}=\omega_{M}d\,x^{M}|_{g}=\overline{{L}}_{M}^{i}L_{i}^{N}d\,x^{M}|_{g}\otimes\left.\frac{\partial}{\partialx^{N}}\right|_{g}=\omega_{M}^{\mu}d\,x^{M}|_{g}\otimes\left.\frac{\partial}{\partialx^{\mu}}\right|_{g}\,.
H^{(0)}=\sum_{r=\pm1}\sum_{\vec{p},p^{02}=\vec{p}^{2}}|p^{0}|\;(b_{\vec{p},r}^{(0)+}b_{\vec{p},r}^{(0)}-d^{(0)+}{}_{\vec{p},r}d_{\vec{p},r}^{(0)}).
W_{m}=\left[\begin{array}{cccccccc}{{\calR}_{2}(\theta)}&{}&{}&{}&{}&{}&{}&{}\\{}&{.}&{}&{}&{}&{}&{}&{}\\{}&{}&{\pm1}&{}&{}&{}&{}&{}\\{}&{}&{}&{.}&{}&{}&{}&{}\\{}&{}&{}&{}&{{\calR}_{2}^{-1}(\theta)}&{}&{}&{}\\{}&{}&{}&{}&{}&{.}&{}&{}\\{}&{}&{}&{}&{}&{}&{\pm1}&{}\\{}&{}&{}&{}&{}&{}&{}&{.}\\\end{array}\right]
L^{\prime}[C]=-\frac{1}{2\pi}\oint_{C}du^{\mu}\,\partial_{\mu}\theta^{\prime}=-\frac{1}{2\pi}\oint_{C}du^{\mu}\,\partial_{\mu}\left[\theta+\chi\right]=L[C]-\frac{1}{2\pi}\oint_{C}du^{\mu}\,\partial_{\mu}\chi,
S=\frac{1}{2}\int\left((e^{-\Phi}Q_{B}e^{\Phi})(e^{-\Phi}\eta_{0}e^{\Phi})-\int_{0}^{1}dt(e^{-t\Phi}\partial_{t}e^{t\Phi})\left\{(e^{-t\Phi}Q_{B}e^{t\Phi}),(e^{-t\Phi}\eta_{0}e^{t\Phi})\right\}\right)\,
\eta_{R}^{\prime}\equiv{\frac{1}{2}}\phi_{1}^{\prime}-{\frac{1}{2}}\pi_{1}^{\phi},~~\eta_{L}^{\prime}\equiv{\frac{1}{2}}\phi_{2}^{\prime}+{\frac{1}{2}}\pi_{2}^{\phi};~~~\chi_{R}^{\prime}\equiv{\frac{1}{2}}\rho_{1}^{\prime}-{\frac{1}{2}}\pi_{1}^{\rho},~~\chi_{L}^{\prime}\equiv{\frac{1}{2}}\rho_{2}^{\prime}+{\frac{1}{2}}\pi_{2}^{\rho}.
\biggl\{\eta_{\mu\nu}\biggl[1-\Bigl(1+\frac{\partial_{t}^{2}}{\nabla^{2}}\Bigr)\,\frac{m^{2}}{\nabla^{2}}\biggr]\Box+\biggl[\frac{1}{\alpha}-1+\Bigl(1+\frac{\partial_{t}^{2}}{\nabla^{2}}\Bigr)\,\frac{m^{2}}{\nabla^{2}}\biggr]\,\partial_{\mu}\partial_{\nu}\biggr\}\,A^{\nu}=0\,.
\frac{{e^{4}}}{(2\pi)^{3}}\int_{0}^{\infty}\frac{{q}^{2}}{{2Q^{0}}}\mathrm{d}qN({Q^{0}})\intd\Omega\sum_{ijkl}{B}_{(ijkl)}^{\mu\nu\lambda\sigma}({k_{1},k_{2},k_{3}};Q).
e_{\mathrm{M}}=\left(\begin{matrix}{u^{-t}}&{}&{}\\{}&{1_{16}}&{}\\{}&{}&{u}\\\end{matrix}\right)\cdot\left(\begin{matrix}{1_{3}}&{\zeta}&{\beta-\frac{\zeta\zeta^{t}}{2}}\\{}&{1_{16}}&{-\zeta^{t}}\\{}&{}&{1_{3}}\\\end{matrix}\right)\,
\frac{(|[\alpha]+1|-[\alpha])M_{\kappa-\frac{1}{2},|\kappa+\frac{1}{2}|}(\alpha)-M_{\kappa+\frac{1}{2},|\kappa+\frac{1}{2}|}(\alpha)}{(|[\alpha]+1|+[\alpha]+2)M_{\kappa+\frac{3}{2},|\kappa+\frac{1}{2}|}(\alpha)-M_{\kappa+\frac{1}{2},|\kappa+\frac{1}{2}|}(\alpha)},
R_{22}=R_{33}=-\frac{1}{g^{2}}\left(\frac{h^{\prime\prime}}{h}-\frac{h^{\prime}}{h}\frac{g^{\prime}}{g}+\frac{f^{\prime}}{f}\frac{h^{\prime}}{h}\right)+\frac{1}{h^{2}}\left(1-\left(\frac{h^{\prime}}{g}\right)^{2}\right)
\int_{{\calM}_{g,N+1}}\!\!\!\![dm][dz]\;\langle\;\oint_{C_{w}}\!\!G^{-}\bar{G}^{-}{\calO}_{\bar{i}}(w)\prod_{i=1}^{N}\:\oint_{C_{z_{i}}}\!\!G^{-}\bar{G}^{-}\:\phi_{i}(z_{i})\;\prod_{a,{\bar{a}}=1}^{3g-3}\:G^{-}(\chi_{a})\bar{G}^{-}(\bar{\chi}_{\bar{a}})\;\rangle_{\Sigma_{g,N+1}}
\mathcal{F}_{\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}}=\mathcal{F}_{\lambda_{1}^{*},\lambda_{2}^{*},\lambda_{3}^{*},\lambda_{4}^{*}}=\sqrt{\frac{\Gamma(\lambda_{1})\Gamma(\lambda_{2})\Gamma(\lambda_{3})\Gamma(\lambda_{4})}{\Gamma(\lambda_{1}^{*})\Gamma(\lambda_{2}^{*})\Gamma(\lambda_{3}^{*})\Gamma(\lambda_{4}^{*})}}\,,
C_{i_{p},\beta_{p}\;[j_{q},\gamma_{q}}^{k_{s},\alpha_{s}}C_{l_{t},\rho_{t}\;m_{u},\sigma_{u}]}^{i_{p},\beta_{p}}=0\;,\;\;i_{p,q,s}=1,2,\ldots,\mathrm{dim}V_{p,q,s}
{\partial^{2}x^{i}}/{\partialz_{+}{\partialz_{-}}}-\sum_{j}G^{ij}(x){\frac{\deltaT(x)}{\deltax^{j}}}+\sum_{j,l}\Gamma_{jl}^{i}(x){\frac{\partialx^{j}}{\partialz_{+}}}{\frac{\partialx^{l}}{\partialz_{-}}}=0.
c_{2}(V^{1})-\frac{1}{2}c_{2}(TX)=(\eta^{(1)}-6c_{1}(B))\sigma-\frac{1}{2}\left(c_{2}(B)+21c_{1}(B)^{2}-\frac{15}{\lambda^{(1)}}\left(\lambda^{(1)2}-\frac{1}{4}\right)\right)F
\tilde{E}_{2}=-q^{a+b-1}\tilde{J}_{0}^{0}\tilde{J}_{0}^{0}+q^{a}\tilde{J}_{0}^{0}\tilde{J}_{0}^{-}+[q^{a+b-1}-\{a\}q^{b}-\{b\}]\tilde{J}_{0}^{0}+\{a\}\tilde{J}_{0}^{-}\,
{\calC}_{k}^{\prime\prime}+\biggl\{\frac{k^{2}}{1+r^{2}}\biggl[1+r^{2}\biggl(-\frac{\eta}{\eta_{1}}\biggr)^{-1}\biggr]-\frac{3}{4\eta^{2}}\biggr\}{\calC}_{k}=0.
x\equiv\frac{\sqrt{3}\alpha^{\prime}}{\varphi^{\prime}}\,\y\equiv\frac{\sqrt{6}\beta^{\prime}}{\varphi^{\prime}}\,\z\equiv\frac{Q^{2}}{2(\varphi^{\prime})^{2}}\,\u\equiv-\frac{6k\mathrm{e}^{-(5\alpha+\varphi-6\beta)}}{(\varphi^{\prime})^{2}}\;\\Omega\equiv\frac{\mathrm{e}^{-(\varphi+3\alpha-6\beta)}\rho}{2\left(\varphi^{\prime}\right)^{2}}\.
\begin{array}{c}{\left\{l_{t}\left(x\right)_{1},l_{t}\left(y\right)_{2}\right\}=\frac12\left[t,l_{t}\left(x\right)_{1}-l_{t}\left(y\right)_{2}\right]\delta\left(x-y\right),}\\{\left\{g_{1}\left(x\right),l_{t}\left(y\right)_{2}\right\}=-g_{1}\left(x\right)t\delta\left(x-y\right),}\\{\left\{g_{1}\left(x\right),g_{2}\left(y\right)\right\}=0,}\\\end{array}
\sum_{N=1}^{\infty}\frac{[C_{N}^{\frac{3}{2}}(cos{\theta_{4}^{0}})]^{2}}{N^{2}(N+3)^{2}}\sum_{N^{\prime}=0}^{N_{0}^{\prime}}(2N^{\prime}+3)K^{2}\frac{1}{\epsilon}\int_{C}d^{n}\etaD_{c}(\eta_{1}-\eta)D_{c}(\eta-\eta_{2})+F.T.
\Theta^{\mu}[X_{a}]=-\tilde{T}_{\cdot\,\nu}^{\mu\,\cdot}k_{a}^{\nu}+\frac{1}{4}\,\overline{{\psi}}\{\gamma^{\hat{\alpha}},S^{\hat{\beta}\hat{\gamma}}\}\psi\,e_{\hat{\alpha}}^{\mu}\,\Omega_{a\,\hat{\beta}\hat{\gamma}}
\tilde{W}^{\mu\nu\alpha\beta}\left(p\right)=2\pi\theta\left(p_{0}\right)\left[\delta\left(p^{2}\right)M^{(0)\mu\nu\alpha\beta}\left(p\right)+\delta\left(p^{2}-\mu^{2}\right)M^{(\mu)\mu\nu\alpha\beta}\left(p\right)\right]\;\,,
0=\ddot{\psi}_{i}+3H\dot{\psi}_{i}+{\frac{\partialV_{mf}}{\partial\psi_{i}}}=[\ddot{b}+3H\dot{b}+(V_{mf}^{\prime\prime}/2)b]x_{i}+[b\ddot{x}_{i}+\dot{x}_{i}(2\dot{b}+3Hb)].
I_{N}(p;\{\nu\})=\frac{(-1)^{N}}{(-4\pi)^{(N-1)\ell}}\left(\prod_{i=1}^{N}\!\int_{0}^{1}\!\frac{d\alpha_{i}}{\alpha_{i}^{\nu_{i}-\ell+1}}\right)\frac{\Gamma(\sum_{i}\nu_{i}+\ell-N\ell)\delta(1-\sum_{i}\alpha_{i})}{[p^{2}\!-\!\sum_{i}(m_{i}^{2}/\alpha_{i})]^{\sum_{i}\nu_{i}+\ell-N\ell}},
Z=\sum_{\{{\sigma}_{0}\}}exp\left\{J_{3}\sum_{\triangle}{\sigma}_{0}^{(1)}{\sigma}_{0}^{(2)}{\sigma}_{0}^{(3)}+h\sum_{j}\sigma_{0}^{(j)}\right\}{[g_{n}({\sigma}_{0}^{(1)})]}^{\gamma-1}{[g_{n}({\sigma}_{0}^{(2)})]}^{\gamma-1}{[g_{n}({\sigma}_{0}^{(3)})]}^{\gamma-1},
{\prod_{\bfk}}^{1/2}\delta\left({\calE}_{a,\bfk}^{M\mathrm{R}}-{\calE}_{a,-\bfk}^{M\mathrm{R}}\right)\delta\left({\calE}_{a,\bfk}^{M\mathrm{I}}+{\calE}_{a,-\bfk}^{M\mathrm{I}}\right).
a(z)\!\!=\!\!\left(\!\!q^{-2j+1}x_{+}z\!-\!\frac{2x_{0}}{sk}q^{-j}\!-\!x_{-}k^{-2}\!z^{-1}\!\!\right)\!\!\left(\!\!-q^{-2j}y_{+}z\!+\!2y_{0}tkq^{-j}\!+\!q^{-1}y_{-}k^{2}z^{-1}\phantom{\frac{q_{0}}{s}}\!\!\!\!\!\!\!\!\right),
L:=\frac12\int_{s_{0}}^{s_{1}}[\frac12Tr(D_{1}A_{2}-D_{2}A_{1})^{*}(D_{1}A_{2}-D_{2}A_{1})+(D_{1}^{*}Z^{*})(D_{1}Z)+(D_{2}^{*}Z^{*})(D_{2}Z)]ds
q=\left.\int_{0}^{\pi}d\chi\,\int_{0}^{\pi}d\theta\,\int_{0}^{2\pi}d\phi\,K^{r}\right|_{r=0}^{r=\infty}+\left.\int_{0}^{\infty}dr\,\int_{0}^{\pi}d\chi\,\,\int_{0}^{2\pi}d\phi\,K^{\theta}\right|_{\theta=0}^{\theta=\pi}.
\frac{1}{2}\int_{\varepsilon^{2}}^{\infty}\frac{dt}{t^{3}}\sum_{k=-\infty}^{\infty}e^{-k^{2}\pit/R^{2}}={R^{-4}}{\calI}(\varepsilon^{2}/R^{2},\alpha=-2)=\frac{1}{R^{4}}\frac{3}{4\pi^{2}}\zeta(5)+\frac{1}{5}\varepsilon^{-5}R
C_{\alpha\beta}\left(z\right)=\left(\begin{array}{cccc}{1}&{-g}&{0}&{-h}\\{-g}&{-1}&{-h}&{0}\\{0}&{-h}&{1}&{-g}\\{-h}&{0}&{-g}&{-1}\\\end{array}\right)\varepsilon\left(z\right),
\phi_{\mathrm{\bf~\tiny~k}}\left(r\right)=-\frac{\Omega_{n}G_{n+4}\,\mu\,\psi_{\mathrm{\bf\tiny~k}}^{\star}\left(0\right)}{2}\frac{1}{r}e^{-\frac{\left|\mathrm{\tiny\bf~k}\right|}{R}}.
Tr_{V}e^{-\frac{i}{\hbar}HT}|_{\psi}=\underset{antiperiodic}{\int{{\mathcal{D}}}\bar{\psi}{{\mathcal{D}}}\psi}e^{\frac{i}{\hbar}\underset{LxT}{\intdt}dx\frac{1}{2}\bar{\psi}[i\not\partial-m]\psi}+O(\hbar^{2})
E_{2}=\frac{v^{2}}{2}\int\frac{d^{3}q}{(2\pi)^{2}}\left\{G_{h}^{-1}(-\vec{q}\,^{2})h(\vec{q})h(-\vec{q})+G_{p}^{-1}(-\vec{q}\,^{2})\vec{p}(\vec{q})\cdot\vec{p}(-\vec{q})\right\}
[{\calD}\mu]\equiv{\calD}A^{\mua}{\calD}\phi^{\dagger}{\calD}\phi{\calD}\Phi^{ia}{\calD}\xi^{a}\prod_{\beta=0}^{3}\delta(G_{\alpha}^{a}[A_{0}^{a}+\xi^{a},A_{i}^{a},\Phi^{ia}])\middet\{\Gamma_{\alpha}^{a},G_{\beta}^{b}\}\mid
\begin{array}{ccccccc}{}&{}&{3+1}&{}&{}&{}&{}\\{}&{\diagup}&{}&{\diagdown}&{}&{}&{}\\{4}&{}&{}&{}&{2+1+1}&{\;\;-\;\;}&{1+1+1+1,}\\{}&{\diagdown}&{}&{\diagup}&{}&{}&{}\\{}&{}&{2+2}&{}&{}&{}&{}\\\end{array}
(-1)^{F}=(-1)^{\sum_{j=2}^{n}\sum_{n>0}(d_{-n}^{-j}d_{n,j}+d_{-n}^{j}d_{n,-j})+\sum_{j=n+1}^{5}\sum_{r>0}(d_{-r}^{-j}b_{r,j}+b_{-r}^{j}b_{r,-j})}\cdot\prod_{k=2}^{n}(d_{0}^{k}d_{0}^{-k}-d_{0}^{-k}d_{0}^{k})
\left.\begin{array}{ll}{\textrm{D3_{(m,0,\pm1)}}:}&{U(1)_{0,(m,0,\pm1)}}\\{\widetilde{\textrm{D7}}_{3,(m)}:}&{U(1)_{1,(m)}}\\{\textrm{D7_{4,(0)}}:}&{U(2)=U(1)_{2,(0)}\timesSU(2)}\\\end{array}\right.
\gamma\cdot{\calF}(a)={\calF}(a)+{\frac{a_{11}a_{21}}{2}}a_{D}^{2}+{\frac{a_{12}a_{22}}{2}}a^{2}+a_{12}a_{21}aa_{D},\qquad\left(\begin{array}{c}{a_{11}}\\{a_{21}}\\\end{array}\begin{array}{cc}{a_{12}}\\{a_{22}}\\\end{array}\right)\in\Gamma.
{{\calA}^{\prime}}_{(2)}^{I}=\frac{(2\pi\sqrt{\alpha^{\prime}})^{2}}{\sqrt{V}}\;{\calA}_{(2)}^{I},\qquad{b^{\prime}}^{I}=\frac{(2\pi\sqrt{\alpha^{\prime}})^{2}}{\sqrt{2\;V}}\;{b}^{I}
{\calJ}_{j}^{\mu}=-N_{jk}^{\mu\nu}\varepsilon_{\nu\lambda}\partial^{\lambda}\Phi^{k}=\frac{-1}{1+4\Phi^{2}}\Bigl((\delta^{jl}+4\Phi^{j}\Phi^{l})\varepsilon^{\mu\nu}\partial_{\nu}\Phi^{l}+2\varepsilon^{jlk}\Phi^{l}\partial^{\mu}\Phi^{k}\Bigr),
\Gamma_{PV}(A)=(-1\pm\mathrm{sign}(m))\frac{i\pi}{2}\eta(0)+i\sum_{n=0}^{\infty}\frac{P_{2n+1}}{(m^{2})^{n-(d-1)/2}}\frac{\Gamma\left(n-\displaystyle{\frac{d}{2}}+1\right)}{2n+1-d}.
\frac{f^{2}}{6}\,E\int_{-\infty}^{\infty}dx\,\psi_{n}^{(0)*}(x)\bigl\{1-4\mux^{2}[V(x)-E]-{\textstyle\frac{2}{3}}x^{3}\muV^{\prime}(x)\bigr\}\psi_{n}^{(0)}(x)=0.
m^{2}tr\\gamma^{5}G_{L}(x,y)=-m^{2}\int\frac{d^{4}k}{(2\pi)^{4}}e^{ik\cdot(x-y)}\epsilon^{\lambda\mu\rho\sigma}\frac{1}{(m^{2}-k^{2})^{3}}G_{\lambda\mu}(y)G_{\rho\sigma}(y)\.
V(\beta,\varphi_{0})=\frac{1}{2}\varphi_{0}^{2}+N\sum_{s=1}^{\infty}p(D,s)(g\varphi_{0})^{2s}\beta^{2s-D}\sum_{n=-\infty}^{\infty}\frac{1}{(n+\frac{1}{2})^{2s-d}}.
{\calZ}_{(0)}^{(1)}=\sum_{\{\nu_{i}\}}(-1)^{\sum_{i=even}^{}\nu_{i}}(\frac{2\pi}{g^{2}A})^{\frac{\nu}{2}}\frac{\left[1^{\nu_{1}}2^{\nu_{2}}3^{\nu_{3}}...N^{\nu_{N}}\right]^{-\frac{3}{2}}}{\nu_{1}!\nu_{2}!\nu_{3}!...\nu_{N}!}.