harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\calL}_{corr}=-{\frac{i}{4\pi}}\int{\frac{dp_{y}}{2\pi}}\,\sum_{E}{\vert\Psi_{E}(x)\vert}^{2}\,\int_{0}^{\infty}{\frac{ds}{s^{2}}}\,{e^{-i(m^{2}+E-i\epsilon)s}}.
{\calL}_{\mathrm{scalar}}^{(2)}=-e^{2}\intd^{4}x^{\prime}\,{\calD}(x-x^{\prime})\,\Bigl[\langlex|D_{\alpha}G|x^{\prime}\rangle\langlex^{\prime}|D_{\alpha}G|x\rangle+\langlex|G|x^{\prime}\rangle\langlex^{\prime}|D_{\alpha}GD_{\alpha}|x\rangle+4\,\langlex|x^{\prime}\rangle\langlex^{\prime}|G|x\rangle\Bigr].
{\calD}_{2}=\left(\begin{array}{cc}{\begin{array}{c}{D^{2}V(DV)^{\gamma-3}+V(DV)^{\gamma-3}}\\{DV(DV)^{\gamma-3}D}\\\end{array}}&{\begin{array}{c}{D^{2}U+(\gamma-2)UD^{2}}\\{(\gamma-1)DUD}\\\end{array}}\\{\begin{array}{c}{(\gamma-2)D^{2}U+UD^{2}}\\{(\gamma-1)DUD}\\\end{array}}&{D^{2}V+VD^{2}-(\gamma-2)DVD}\\\end{array}\right)
S=-\frac{1}{4\pi\alpha^{\prime}}\intd^{2}\xi\left\{\partial_{a}X^{\mu}\partial^{a}X^{\nu}G_{\mu\nu}(X)+\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}B_{\mu\nu}(X)-\alpha^{\prime}R^{(2)}\phi(X)\right\}
\gamma_{\bar{1}}=\frac{1}{\sqrt{2}}(\Gamma_{1}-i\Gamma_{2}),\gamma_{\bar{2}}=\frac{1}{\sqrt{2}}(\Gamma_{3}-i\Gamma_{3}),\gamma_{\bar{3}}=\frac{1}{\sqrt{2}}(\Gamma_{5}-i\Gamma_{6}),\gamma_{\bar{4}}=\frac{1}{\sqrt{2}}(\Gamma_{7}-i\Gamma_{8}).
\{\psi,\chi\}=\left\{\left(\begin{array}{c}{\psi_{l}}\\{0}\\\end{array}\right),\left(\begin{array}{c}{0}\\{\chi_{r}}\\\end{array}\right)\right\}+\left\{\left(\begin{array}{c}{0}\\{\psi_{r}}\\\end{array}\right),\left(\begin{array}{c}{\chi_{l}}\\{0\}\\\end{array}\right)\right\}
{\calH}^{l_{1}}(X,\wedge^{s_{1}}E)^{(0)}\otimes{\calH}^{l_{2}}(X,\wedge^{s_{2}}E)^{(0)}\longmapsto{\calH}^{l_{1}+l_{2}}(X,\wedge^{s_{1}+s_{2}}E)^{(0)}.
\beta_{g}^{(2)}=-\frac{g^{5}}{(4\pi)^{4}}\left[\frac{34}{3}(C_{2}(G))^{2}-\kappa\left[4C_{2}(F)+\frac{20}{3}C_{2}(G)\right]S_{2}(F)-\left[2C_{2}(S)+\frac{1}{3}C_{2}(G)\right]S_{2}(S)+\frac{2\kappaY_{4}(F)}{g^{2}}\right]
G(x,y;B)=\frac{1}{2\pi}\frac{1}{(x-y)^{2}}\left(\begin{array}{cc}{0}&{e^{-[\chi(x)-\chi(y)]}\;\overline{{(\tilde{x}-\tilde{y})}}}\\{e^{+[\chi(x)-\chi(y)]}\;(\tilde{x}-\tilde{y})}&{0}\\\end{array}\right)\;.
{\calL}_{bulk}^{4d}=-2\frac{3A^{3}BM_{5}^{3}}{N}\int_{0}^{R}dyD^{3}(y,Y)\left[\left(\frac{\dot{A}}{A}\right)^{2}+3\frac{\dot{A}}{A}\frac{\dot{D}}{D}+{\frac{1}{2}}\left(\frac{\dot{D}}{D}\right)^{2}\right],
=\sum_{\gamma\beta^{\prime\prime}\bar{\beta}^{\prime\prime}}w(\bar{\beta},\beta,\beta^{\prime\prime},\bar{\beta}^{\prime\prime})w(\beta^{\prime\prime},\alpha,\beta^{\prime},\gamma)w(\bar{\beta}^{\prime\prime},\gamma,\bar{\beta}^{\prime},\alpha^{\prime})~.
V(z)={\frac{1}{2}}{\frac{A^{\prime\prime}}{A}}+{\frac{n}{2}}{\frac{B^{\prime\prime}}{B}}+{\frac{n}{2}}{\frac{A^{\prime}}{A}}{\frac{B^{\prime}}{B}}-{\frac{1}{4}}\left({\frac{A^{\prime}}{A}}\right)^{2}+{\frac{n(n-2)}{4}}\left({\frac{B^{\prime}}{B}}\right)^{2}+m^{2}.
M={\frac{1}{2}}\left(\begin{matrix}{0}&{ime^{(q_{1}-q_{2})/2}}&{0}&{ime^{-q_{1}}}\\{i\,me^{(q_{1}-q_{2})/2}}&{0}&{i\,me^{q_{2}}}&{0}\\{0}&{i\,me^{q_{2}}}&{0}&{i\,me^{(q_{1}-q_{2})/2}}\\{ime^{-q_{1}}}&{0}&{i\,me^{(q_{1}-q_{2})/2}}&{0}\\\end{matrix}\right).
D_{ij}^{g}(q)=\frac{n^{2}q_{i}q_{j}}{q^{2}+i{\epsilon}}{\delta}^{\prime}(q_{-}){\pi}\mathrm{sgn}(q_{+})-i\frac{n_{+}q_{i}q_{j}}{q^{2}+i{\epsilon}}{\delta}(q_{-}){\int}_{-\infty}^{\infty}dk_{+}\frac{1}{(k_{+}-a)^{2}},
\breve{T}_{(\mu\nu)n}(x)=\Big(\delta_{\mu}^{\rho}\delta_{\nu}^{\sigma}-\mathrm{\large~\frac{1}{2h}~}\;g_{\mu\nu}g^{\rho\sigma}\Big)H_{n}^{(2h)}\!\left(x^{2}|\square\right)T_{(\rho\sigma)n}(x).
a_{1}(x,y)=\sum_{n=1}^{\infty}\Bigl[a_{1}^{(c,2n-1)}(x)Ec_{2}^{2n-1}(\omegay,k)+a_{1}^{(s,2n-1)}(x)Es_{2}^{2n-1}(\omegay,k)\Bigr].
K=\frac{4R_{2}}{\mathrm{Im}\,\tau}\left(\begin{matrix}{1}&{\mathrm{Re}\,\tau+\mathrm{~\frac{R_1}{R_2}~}\,\mathrm{Im}\,\tau}\\{\mathrm{Re}\,\tau-\mathrm{~\frac{R_1}{R_2}~}\,\mathrm{Im}\,\tau}&{|\tau|^{2}}\\\end{matrix}\right)
\big[{\frac{d^{2}}{d\,x^{2}}}+16x^{N-2}(x+B)^{2}+8\lambda_{n+1}+\sum_{i=1}^{n}{\frac{\lambda_{i}}{x+A_{i}}}\big]\Psi_{j}(x;\lambda_{1}\ldots\lambda_{n+1})=0
{\calQ}=\left(\begin{matrix}{P^{\Lambda}}\\{Q_{\Sigma}}\\\end{matrix}\right);\\\\P^{\Lambda}\equiv\oint_{S^{2}}\,\mathrm{Re}\,{\calF}^{-\,\Lambda},\\Q_{\Sigma}\equiv\oint_{S^{2}}\,\mathrm{Re}\,{\calG}_{\Sigma}^{-},
H_{j}\;=\;\frac{i}{2}[\:\eta\tau_{j}^{y}\tau_{j+1}^{x}\:-\:\eta^{-1}\tau_{j}^{x}\tau_{j+1}^{y}\:+\:q\tau_{j}^{x}\tau_{j}^{y}\:+\:q^{-1}\tau_{j+1}^{x}\tau_{j+1}^{y}].
X^{i}=\left(\begin{matrix}{fJ_{i}^{(1)}}&{0}\\{0}&{fJ_{i}^{(2)}}\\\end{matrix}\right)\quad(1\leqi,j,k\leq3),\quad\quadX^{4}=\left(\begin{matrix}{0}&{0}\\{0}&{r}\\\end{matrix}\right),\quad\quadX^{\mu}=0\quad(\mu\geq5)
{\calH}=\intd^{3}x\left[{\frac{1}{4}}F_{ij}^{a}F^{aij}-{\frac{1}{2}}F_{0i}^{a}F^{a0i}+{\frac{1}{2}}D_{i}\phi^{a}D^{i}\phi^{a}-{\frac{1}{2}}D_{0}\phi^{a}D^{0}\phi^{a}\right]
S_{\mathrm{eff}}={\frac{1}{2}}\intd^{4}x\sqrt{-g}e^{-2\phi}\left[\lambda_{s}^{-2}({\calR}+4\partial_{\mu}\phi\partial^{\mu}\phi)-{\frac{1}{12}}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+V\right]\\,
A_{c}(-i\Omega_{m},\stackrel{\rightharpoonup}{p},\stackrel{\rightharpoonup}{l})=T\sum_{n}\frac{1}{[(\omega_{n}+i\mu_{U})^{2}+\omega_{Ul}^{2}][(\omega_{n}+\Omega_{m}+i\mu_{D})^{2}+\omega_{Dl+p}^{2}]},
T_{cl}^{(m+n)}(z)=\sum_{i=1}^{p-1}{\frac{1-q_{i}^{2}}{2h(z-z_{i})^{2}}}-{\frac{1}{2\pih}}\sum_{i=1}^{p-3}\left({\frac{1}{z-z_{i}}}+{\frac{z_{i}-1}{z}}-{\frac{z_{i}}{z-1}}\right){\frac{\partialS_{cl}^{(0,m+n)}}{\partialz_{i}}}.
V_{\mu}\left(x+\pi(\xi)\right)=V_{\mu}+V_{\mu;\lambda}\xi^{\lambda}+{\frac{1}{2!}}\left(V_{\mu;\lambda;\sigma}+{\frac{1}{3}}R_{~\lambda\mu\sigma}^{\kappa}V_{\kappa}\right)\xi^{\lambda}\xi^{\sigma}+...
{\cal{S}}^{[1,1]}\;\;=\;\;-\frac{1}{2}\intd^{D}x\,h^{\mu\nu}E_{\mu\nu}\;\;=\;\;\intd^{D}x\,\left(\,\partial^{[\mu}h^{\nu]\rho}\partial_{[\mu}h_{\nu]\rho}-2\partial^{[\mu}h^{\nu]\nu}\partial_{[\mu}h_{\rho]\rho}\,\right)
I_{0}(u,\mu)={\frac{1}{\Gamma({\frac{1}{2}})}}\mu^{{\frac{1}{2}}}\left\{\Gamma(-{\frac{1}{4}})^{2}F(-{\frac{1}{4}},-{\frac{1}{4}};{\frac{1}{2}};{\frac{u^{2}}{4\mu^{2}}})-{\frac{u}{\mu}}\Gamma({\frac{1}{4}})^{2}F({\frac{1}{4}},{\frac{1}{4}};{\frac{3}{2}};{\frac{u^{2}}{4\mu^{2}}})\right\}.
g_{0}^{-1}\partial_{+}g_{0}=\frac{1+\epsilon}{2+\epsilon}\,(\frac{{\hat{q}}}{\Lambda}+\frac{x^{-}}{2\Lambda^{2}});\;\;\;\;\;\;g_{0}^{-1}\partial_{-}g_{0}=\frac{1+\epsilon}{\epsilon}\,(\frac{{\hat{p}}}{\Lambda}-\frac{x^{+}}{2\Lambda^{2}}).
\gamma^{0}~=~\left(\begin{array}{cccc}{\sigma_{3}}&{0}\\{0}&{-\sigma_{3}}\\\end{array}\right)\,\quad\gamma^{1}~=~\left(\begin{array}{cccc}{i\sigma_{1}}&{0}\\{0}&{-i\sigma_{1}}\\\end{array}\right)\,\quad\gamma^{2}~=~\left(\begin{array}{cccc}{i\sigma_{2}}&{0}\\{0}&{-i\sigma_{2}}\\\end{array}\right)~.
\dot{x}^{A}{}_{D^{\prime}}\acute{x}^{BC^{\prime}}\dot{x}_{AC^{\prime}}\acute{x}_{B}{}^{D^{\prime}}=-\dot{x}^{A}{}_{C^{\prime}}\dot{x}_{AD^{\prime}}\acute{x}^{BC^{\prime}}\acute{x}_{B}{}^{D^{\prime}}
\tilde{\hat{\omega}}(z_{1})-\textstyle{\frac{1}{2}}\hat{\lambda}(z_{1})-4\hat{\rho}^{i}(z_{1})\bar{\theta}_{12i}+\mathrm{b}{\calX}_{12}=\tilde{\hat{\omega}}(z_{2})-\textstyle{\frac{1}{2}}\hat{\lambda}(z_{2})
S={\frac{1}{2\kappa_{6}^{2}}}\intdx^{6}\sqrt{-G}\,e^{-\Phi}\left[{\calR}_{G}+\partial_{\bar{\mu}}\Phi\partial^{\bar{\mu}}\Phi-{\frac{1}{2\cdot3!}}H_{\bar{\mu}\bar{\nu}\bar{\rho}}H^{\bar{\mu}\bar{\nu}\bar{\rho}}\right],
\overline{{\psi}}^{(2)}(x)P_{+}\psi^{(2)}(x)\;\longleftrightarrow\;\frac{1}{2\pi}c^{(2)}:e^{-ia\varphi^{(1)}(x)}:_{M^{(1)}}:e^{+ib\varphi^{(2)}(x)}:_{M^{(2)}}\;e^{i\frac{\theta}{2}}\;,
G_{5}^{(R)}=-\int~\frac{d^{4}k}{(2\pi)^{4}}~e^{ik_{\mu}(x^{\mu}-x^{\mu})}\left[\frac{\ell^{-1}e^{-2(|y|+|y^{\prime}|)/\ell}}{{\bfk}^{2}-(\omega+i\epsilon)^{2}}+\int_{0}^{\infty}~dm\frac{u_{m}(y)u_{m}(y^{\prime})}{m^{2}+{\bfk}^{2}-(\omega+i\epsilon)^{2}}\right],
\begin{array}{c}{(B_{\rho}-(l+\frac{d-3}{2})^{2})R_{l}=-(l+1)(l+2)R_{l+2}}\\{(B_{\rho^{\prime}}-(l+\frac{d-3}{2})^{2})R_{l}=-(l+1)(l+2)R_{l+2},}\\\end{array}
X_{i}\:(\tau,z)={\frac{1}{2}}\left\{NS_{i}\left(\frac{\theta_{3}}{\eta}\right)^{m}-{\widetilde{NS}}_{i}\left(\frac{\theta_{4}}{\eta}\right)^{m}-R_{i}\left(\frac{\theta_{2}}{\eta}\right)^{m}+{\widetildeR}_{i}\left(\frac{\theta_{1}}{\eta}\right)^{m}\right\}(\tau,z),
P(y_{1},y_{2},y_{3})=x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+\zeta_{1}^{1/5}x_{1}x_{2}x_{3}^{3}+\zeta_{2}^{1/5}x_{1}^{2}x_{2}^{2}x_{3},
L^{(-)}=\left(\begin{array}{cc}{q^{\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)}}&{0}\\{(q-q^{-1})X^{-}}&{q^{\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)}}\\\end{array}\right)
R_{abc}^{i_{1}i_{2}i_{3}}R_{i_{1}i_{2}d}^{i_{4}i_{5}i_{6}}R_{i_{4}i_{3}i_{6}}^{ei_{7}i_{8}}R_{i_{5}i_{7}i_{8}}^{fgh}=R_{bcd}^{i_{1}i_{2}i_{3}}R_{ai_{2}i_{3}}^{i_{4}i_{5}i_{6}}R_{i_{4}i_{1}i_{6}}^{i_{7}i_{8}h}R_{i_{7}i_{8}i_{5}}^{efg}
S_{0}[\Phi,\Phi^{*}]=I[\phi]+\omega^{A}\,C_{A}^{r}[\phi]\,\phi_{r}^{*}+\mathrm{\small~\frac{1}{2}~}\omega^{A}\omega^{B}\,C^{C}{}_{AB}[\phi]\,\omega_{C}^{*}-\bar{\omega}_{A}^{*}\,h^{A}\;,
{\frac{dW[\bar{\phi}(\beta)]}{d\beta}}={\frac{\partialW[\bar{\phi},\beta]}{\partial\beta}}+\left.\int_{M^{2}}{\frac{\deltaW[\phi,\beta]}{\delta\phi}}\right|_{\phi=\bar{\phi}(\beta)}{\frac{\partial\bar{\phi}(\beta)}{\partial\beta}}~~~.
Q^{2}\left({\frac{1-\sqrt{1-4G_{o}^{4}}}{2G_{o}^{4}}}\right)\ll\left({\frac{T}{\gamma}}\right)^{3/2}\llQ^{2}\left({\frac{1+\sqrt{1-4G_{o}^{4}}}{2G_{o}^{4}}}\right)\.
\varepsilon_{\psi}\sum_{l=-\infty}^{h-1}(w-z)^{l-h}\langle\varphi,V(\psi,\zeta)V(S_{-l}\chi,z)\;\Omega\rangle+\sum_{m=-\infty}^{h-1}(w-\zeta)^{m-h}\langle\varphi,V(S_{-m}\psi,\zeta)\;V(\chi,z)\;\Omega\rangle.
\{\nabla_{\alpha},\nabla_{\beta}\}=c_{\alpha\beta}^{\enspace\enspace\gamma}\nabla_{\gamma}\qquad\{\bar{\nabla}_{\dot{\alpha}},\bar{\nabla}_{\dot{\beta}}\}=c_{\dot{\alpha}\dot{\beta}}^{\enspace\enspace\dot{\gamma}}\bar{\nabla}_{\dot{\gamma}}
\mathbf{B}(\mathbf{x})=\frac{1}{4\pi}\int\rho_{m}(\mathbf{x}^{\prime})\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}\,+\frac{1}{4\pi}\int\mathbf{J}_{e}(\mathbf{x}^{\prime})\,{\bf\times}\,\frac{\mathbf{r}}{r^{3}}\,d^{3}x^{\prime}.
G^{(a)}:=\bar{C}^{a}\left[(\partial^{\mu}\delta^{ab}-\xig\epsilon^{ab}a^{\mu})A_{\mu}^{b}+\frac\alpha2B^{a}-i\frac\zeta2g\epsilon^{ab}\bar{C}^{b}C^{3}+i\etag\epsilon^{ab}\bar{C}^{3}C^{b}\right].
\mathrm{Re}\,\chi_{T}^{\mathrm{D}}(\omega)=-\frac{\hbarA}{4\pi^{2}}\left(\frac{k_{B}T}{\hbar}\right)^{5}\int_{0}^{\Omega}dv\int_{\Omega}^{\infty}du\,(u-v)\frac{\sqrt{uv(u-\Omega)(\Omega-v)}}{e^{\frac{u+v}{2}}-1},
[{{\omega}^{-1}}]={\frac{1}{ac-b{b^{\prime}}}}\left(\begin{matrix}{0}&{-c}&{0}&{{b^{\prime}}}\\{c}&{0}&{-b}&{0}\\{0}&{b}&{0}&{-a}\\{-{b^{\prime}}}&{0}&{a}&{0}\\\end{matrix}\right)
H^{2}=\frac{\kappa_{(4)}^{2}}{3}\rho+\frac{\kappa_{(4)}^{2}}{6\lambda}\rho^{2}-\frac{k}{a^{2}}+\frac{\sigma^{2}}{6}+\frac{\Lambda_{(4)}}{3}+\frac{2{\calU}}{\lambda\kappa_{(4)}^{2}},
\Sigma_{-}\:\\tilde{x}^{3}-\frac{3}{4}\tilde{u}_{0}^{4}\tilde{x}-\frac{1}{4}(\tilde{u}_{0}^{6}+\tilde{u}_{1})+\frac{1}{4}(\zeta+\frac{1}{\zeta})+{\calO}(\tilde{\epsilon})=0\,
R_{0}(\zeta)\;=\;{\frac{\left(1-{\frac{1}{2\tilde{z}_{0}}}\right)}{8\tilde{z}_{0}^{2}\left(1-{\frac{3}{4\tilde{z}_{0}}}\right)^{2}}}{\frac{\zeta}{2\tilde{z}_{0}+\zeta}}\;.
W_{7_{3}}=-\frac{1}{3}\Bigl(0^{\tilde{u}},1^{\tilde{u}},2^{\tilde{u}},0^{1+\tilde{u}},1^{1+\tilde{u}},2^{1+\tilde{u}},0^{2+\tilde{u}},1^{2+\tilde{u}},2^{2+\tilde{u}}\Bigr)
{\calO}_{i}^{(q)}=\left.\left(\left.{\frac{\partial\Gamma_{\infty}^{(\beta)}}{\partial\beta_{i}}}\right|_{\Phi,K}\right)\right|_{\beta=0}=\left.\left(\left.{\frac{\partialW_{\infty}^{(\beta)}}{\partial\beta_{i}}}\right|_{J,K}\right)\right|_{\beta=0}=\left.<{\frac{\partial\Sigma_{\infty}^{(\beta)}}{\partial\beta_{i}}}>_{J}\right|_{\beta=0}.
\Psi_{R}={\frac{1}{2^{{\frac{1}{4}}}}}\left(\begin{array}{c}{u_{+}}\\{0}\\{0}\\{u_{-}}\\\end{array}\right)\quad\Psi_{L}={\frac{1}{2^{{\frac{1}{4}}}}}\left(\begin{array}{c}{0}\\{v_{+}}\\{v_{-}}\\{0}\\\end{array}\right)
\left(\begin{array}{c}{A}\\{\frac{1}{\omega}B}\\\end{array}\right)\to\left(\begin{array}{cc}{\alpha}&{\beta}\\{\gamma}&{\delta}\\\end{array}\right)\left(\begin{array}{c}{A}\\{\frac{1}{\omega}B}\\\end{array}\right),\qquad\alpha\delta-\beta\gamma=1.
A_{z}^{a}(z,\bar{z})={\frac{i}{2\pi\kappa}}\sum_{\alpha}Q_{\alpha}^{a}{\frac{1}{z-z_{\alpha}}}={\frac{i}{2\pi\kappa}}\partial_{z}\left(\sum_{\alpha}Q_{\alpha}^{a}\mathrm{ln}(z-z_{\alpha})\right).
\partial_{\tau}^{2}X^{\mu}-c^{2}\partial_{\sigma}^{2}X^{\mu}+\Gamma_{\nu\rho}^{\mu}\left[\partial_{\tau}X^{\nu}\partial_{\tau}X^{\rho}-c^{2}\partial_{\sigma}X^{\nu}\partial_{\sigma}X^{\rho}\right]=0,
\begin{array}{l}{e^{+2}\nabla\psi_{+2\dot{q}}^{1,2-}\!+\!e^{-2}\nabla\psi_{-2\dot{q}}^{1,2-}\!+\!e^{+}\nablah_{+\dot{q}}^{1,2-}\!+\!e^{-}\nabla\chi_{\dot{q}}^{1,2}\!+\!T^{+2}\psi_{+2\dot{q}}^{1,2-}\!+\!T^{-2}\psi_{-2\dot{q}}^{1,2-}\!+\!T^{+}h_{+\dot{q}}^{1,2-}\!+\!T^{-}\chi_{\dot{q}}^{1,2}}\\{=-\frac12\tilde{\gamma}_{\dot{q}q}^{i}\Omega^{-2i}\pi_{q}^{1,2+}.}\\\end{array}
H_{B}(ND)=\sum_{j}\left(N_{B}(ND)+\frac{1}{24}\right)=\sum_{j}\left(\sum_{r=1/2}^{\infty}({\alpha_{-r}}^{-j}{\alpha_{r}}_{j}+{\alpha_{-r}}^{j}{\alpha_{r}}_{-j})+\frac{1}{24}\right),
\begin{array}{rlllll}{{[a_{m}^{i},a_{n}^{j}]}}&{=}&{m\delta^{ij}\delta_{m,-n}\,}&{}\\{{[\widetilde{a}_{m}^{i},\widetilde{a}_{n}^{j}]}}&{=}&{m\delta^{ij}\delta_{m,-n}\,}&{\;\;\;m,n\in\mathbb{Z}\setminus\{0\}}\\\end{array}
{\calX}^{2}\,g^{AB}=\overline{{(\gamma_{\mu})_{gh}}}\,\overline{{\epsilon}}^{\,CGH}\,\overline{{\psi_{H}^{h}}}\,\overline{{\psi_{G}^{g}}}\,\overline{{\psi_{C}^{c}}}\,\psi^{Ad}\,V^{Bef}\,\{2\,\beta_{ce}\,(\gamma^{\mu})_{fd}+\beta_{cd}\,(\gamma^{\mu})_{fe}\}.
\hat{\Omega}^{quad}={\frac{1}{2}}\sum_{r,s=-1}^{0}f^{-{\frac{1}{2}}}\hat{\calP}_{a_{r}}f\Omega^{a_{r}b_{s}}\hat{\calP}_{b_{s}}f^{-{\frac{1}{2}}}+\eta^{o}\lambda.
\Pi^{\mu\nu}(k)=ie^{2}\int\frac{d^{2}p}{(2\pi)^{2}}\mathrm{Tr}\left(\gamma^{\mu}\frac{1}{\not\!p+i\varepsilon}\gamma^{\nu}\frac{1}{\not\!p+\not\!k+i\varepsilon}\right)=\frac{e^{2}}{\pi}\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{k^{2}}\right)\;,
\begin{array}{c}{\left[M,b^{\dagger}\right]=b^{\dagger},}\\{\left[M,b\right]=-b,}\\{b^{p+1}=\left(b^{\dagger}\right)^{p+1}=0,}\\{b^{\dagger}b=M\left(p+1-M\right)=\left[M\right],}\\{bb^{\dagger}=\left(M+1\right)\left(p-M\right)=\left[M+1\right],}\\{M=\frac{1}{2}\left(\left[b^{\dagger},b\right]+p\right),}\\\end{array}
\langle\chi_{T}(k)|\chi_{T}(k^{\prime})\rangle=Kl_{n^{\prime}}^{2{k^{\prime}}^{2}}\tilde{l}_{\tilde{n}}^{2k^{2}}\left(\frac{1}{n^{\prime}-1}\right)^{k^{2}+{k^{\prime}}^{2}}\langlee^{ik^{\prime}.X\left((\tilde{n}+1-\varepsilon_{n^{\prime}})\frac{\pi}{4}\right)}e^{ik.X(0)}\rangle_{C_{\tilde{n}+1-\varepsilon_{n^{\prime}}}}.
H_{V}\simeq-V(q_{-})-{\frac{1}{2}}\omega_{-}^{2}\left[q(\tau_{0})-q_{-}\right]^{2}+{\frac{1}{\lambda}}\omega_{-}^{2}F(\tau_{0})\left[q(\tau_{0})-q_{-}\right]-{\frac{1}{2\lambda}}F(\tau_{0})^{2}.
\sum_{{s_{1},s_{2}}={0,1}}(-1)^{s_{1}+s_{2}}\theta(_{s_{2}}^{s_{1}})^{\prime}\theta^{3}(_{s_{2}}^{s_{1}})\frac{d}{d\tau}[\sum_{s_{1},s_{2}}(-1)^{s_{1}+s_{2}}\theta^{4}(_{s_{2}}^{s_{1}})]=0,
\hat{\phi}=\sum_{\tau=\pm}\hat{\phi}^{(\tau)}=\sum_{\tau=\pm}\hat{B}_{L}(f_{B}^{(\tau)})\cdot\hat{\phi}_{RA}^{(\tau)},\;\;\;\hat{\phi}^{\dagger}=\sum_{\tau=\pm}\hat{\phi}^{(\tau)\dagger}=\sum_{\tau=\pm}\hat{\phi}_{RA}^{(\tau)\dagger}\cdot\hat{B}_{R}(f_{B}^{(\tau)}).
\Gamma_{0}(b_{i}\tob_{f}+\gamma)={\frac{2}{3\pi}}\left({\frac{M(N_{i})^{2}-M(N_{f})^{2}}{2\hbarM(N_{i})}}\right)^{3}A_{H}(N_{i})\;{\frac{2J_{f}+1}{2J_{i}+1}}\;{\frac{N_{f}}{N_{i}}}.
Z[T]=\frac{1}{1}{vol~G}\int[d\chi(\vec{x})]\int[da_{i}(\vec{x})]<a\vert\mathrm{~e}^{-H/T}\mathrm{~e}^{i\kappa\int\chi(B_{M}+B/2)+i\sume_{i}\chi(\vec{x}_{i})}\verta+d\chi>
\frac{1}{2}\alphak_{1}B^{\mathrm{T}}-Y^{\mathrm{T}}N=-B^{\mathrm{T}}C_{3}^{-2}-\frac{1}{2}\frac{\alpha}{1+\mu\alphak}(k-\muk_{1})(\frac{1}{2}\alphakB^{\mathrm{T}}+\muY^{\mathrm{T}}N).
\langle:\prod_{a=1}^{r}O_{2}^{I_{a}}(x_{1}^{(a)}):O_{2}^{I_{r+1}}(x_{2}):\prod_{b=1}^{r}O_{2}^{J_{b}}(x_{3}^{(b)}):O_{2}^{J_{r+1}}(x_{4})\rangle
G_{l+\nu}\left(\left.r^{\prime\prime},r^{\prime};E\right|\lambda;a\right)=-\frac{2M}{\hbar^{2}}\,\frac{\sqrt{r^{\prime}r^{\prime\prime}}}{K_{s_{l}}(\kappaa)}\,\left[K_{s_{l}}(\kappaa)I_{s_{l}}(\kappar_{<})-I_{s_{l}}(\kappaa)K_{s_{l}}(\kappar_{<})\right]K_{s_{l}}(\kappar_{>})\;,
{\calL}=\frac{\kappa_{1}}{2}\epsilon^{\mu\nu\lambda}a_{\mu}^{1}\partial_{\nu}a_{\lambda}^{1}+\frac{\kappa_{2}}{2}\epsilon^{\mu\nu\lambda}a_{\mu}^{2}\partial_{\nu}a_{\lambda}^{2}-|D_{\mu}\phi|^{2}-|D_{\mu}\chi|^{2}-U(\phi,\chi)\,,
d{\tilde{s}}_{\mathrm{AdS}_{3}/\mathrm{Z}_{N}}^{2}={\frac{r^{2}}{R^{2}}}\left[-dudv+{\frac{cN}{r^{2}}}du^{2}\right]+{\frac{R^{2}}{r^{2}}}dr^{2}+R^{2}d\Omega_{3}^{2}.
\psi_{b}(w,z)=\psi\left(\frac{w}{z}\right)\psi\left(wz\right)=\frac{1}{(xw/z;x^{4})_{\infty}(xz/w;x^{4})_{\infty}(xwz;x^{4})_{\infty}(x/zw;x^{4})_{\infty}}.
C(p)\,S(p)=\,\frac{F_{-\gamma}(p)\,F_{\gamma}(p)\,\Gamma(1+\alpha-i\frac{\gammap}{2\pi})\,\Gamma(1+\alpha+i\frac{\gammap}{2\pi})}{[p^{2}+(\frac{\hbar\gamma}{2})^{2}]^{\frac{1}{2}}},
i\left<a_{\mu}^{i}a_{\nu}^{j}A_{\rho}^{a}A_{\sigma}^{b}\right>_{\mathrm{bare}}=ig^{2}f^{aic}f^{cjb}\left[2g_{\mu\nu}g_{\rho\sigma}-\left(1-\frac1\alpha\right)(g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\nu\rho})\right].
{\mathcalL}(\phi_{c,B},R)=\frac12\phi_{c,B}\Box_{x}\phi_{c,B}+\frac{m_{B}^{2}}2\phi_{c,B}^{2}+\frac{\lambda_{B}\phi_{c,B}^{4}}{4!}+\varepsilon_{B}R+\frac12\xi_{B}R\phi_{c,B}^{2},
B_{c}(-i\Omega_{m},\stackrel{\rightharpoonup}{p},\stackrel{\rightharpoonup}{l})=T\sum_{n}\frac{\omega_{n}}{[(\omega_{n}+i\mu_{U})^{2}+\omega_{Ul}^{2}][(\omega_{n}+\Omega_{m}+i\mu_{D})^{2}+\omega_{Dl+p}^{2}]}.
\begin{array}{c}{A(k){\calJ}_{l+1/2}(\bar{z}_{2})+B(k){\calY}_{l+1/2}(\bar{z}_{2})={\calJ}_{l+1/2}(\bar{z}_{1}),}\\{A(k){\calJ}_{l+1/2}^{\prime}(\bar{z}_{2})+B(k){\calY}_{l+1/2}^{\prime}(\bar{z}_{2})=\sqrt{\frac{\epsilon_{1}\mu_{2}}{\epsilon_{2}\mu_{1}}}{\calJ}_{l+1/2}^{\prime}(\bar{z}_{1}),}\\\end{array}
\left(\hat{A}_{\mu}^{a}\hat{A}_{\mu}^{b}\right)\left(\hat{A}^{b\mu}\hat{A}^{a\mu}\right)-\left(\left(\hat{A}_{\mu}^{a}\hat{A}_{\mu}^{b}\right)\hat{A}^{b\mu}\right)\hat{A}^{a\mu}=\lambda\hat{A}_{\mu}^{b}\hat{A^{b\mu}}.
\int_{0}^{x}d^{n}x\prod_{i=1}^{p}\int_{x_{k_{i}}}^{x_{k_{i}+1}}dy_{i}f(y_{1},y_{2},\ldots,y_{p})=\frac{x^{p}}{n^{p}}f\left(\frac{k_{1}x}{n},\frac{k_{2}x}{n},\ldots,\frac{k_{p}x}{n}\right)I_{n}(x)
\frac{F(2,-\frac{i}{2}k^{\prime}+2,-\frac{i}{2}k^{\prime}+3,e^{-4x_{0}})}{k^{\prime}+4i}=\sum_{n=1}^{\infty}\frac{ne^{-4(n-1)x_{0}}}{k^{\prime}+i(2+2n)}.
\begin{array}{ccl}{P(i\rightarrowf)}&{=}&{\sum_{n=0}^{\infty}\sum_{a_{1},a_{2},\cdots,a_{n}}\sum_{m=0}^{\infty}\sum_{b_{1},b_{2},\cdots,b_{m}}<\psi_{f}|a_{1},a_{2},\cdots,a_{n}>\times}\\{}&{\times}&{<a_{1},a_{2},\cdots,a_{n}|{\calE}e^{-i\hat{H}_{0}(t_{f}-t_{i})}{\calE}|b_{1},b_{2},\cdots,b_{m}><b_{1},b_{2},\cdots,b_{m}|\psi_{i}>\\\.}\\\end{array}
{\frac{\partial}{\partialX^{+}}}={\frac{\partial}{\partialx_{0}}}+{\frac{\partial}{\partial\phi_{2}}},\quad{\frac{\partial}{\partialX^{-}}}={\frac{(1+e^{2U_{0}})^{-1/2}}{R^{2}}}\left({\frac{\partial}{\partialx_{0}}}-{\frac{\partial}{\partial\phi_{2}}}\right).
m\langle\bar{Q}T^{a}Q\;\mathrm{Tr}\;T^{a}A^{2}\rangle=-\frac{1}{32\pi^{2}}\langle\mathrm{Tr}\;T^{a}W_{\alpha}W^{\alpha}\;\mathrm{Tr}\;T^{a}A^{2}\rangle=\Lambda^{11/3}m^{4/3}=\Lambda_{LE}^{5},
{\frac{\tilde{C}_{i}(t^{\prime})}{22902511064(e^{t^{\prime}}-1)}}\equiv{\bfB}_{i}(t^{\prime})=b_{i}^{0}+{\frac{\sum_{j=1}^{24-i}(b_{j}^{1}+e^{t^{\prime}}b_{j}^{2})t^{\prime}\^{j}}{(e^{t^{\prime}}-1)}}.
\frac{\mathrm{d}^{2}\varphi_{n}(\lambda)}{\mathrm{d}\lambda^{2}}-\mathcal{F}_{n}(\lambda)\frac{\mathrm{d}\varphi_{n}(\lambda)}{\mathrm{d}\lambda}+\mathcal{G}_{n}(\lambda)\varphi_{n}(\lambda)=0
I_{2n}=\int\prod_{i=1}^{2n}{\frac{dt_{i}}{2\pi}}d\theta_{i}~\Theta(t_{1}-t_{2})\cdots\Theta(t_{2n-1}-t_{2n})\prod_{i<j}|e^{it_{i}}-e^{it_{j}}-ie^{\frac{i}{2}(t_{i}+t_{j})}\theta_{i}\theta_{j}|~{\frac{1}{2}}\sum_{k,l}{e^{i(t_{k}-t_{l})}}\theta_{k}\theta_{l}={\frac{1}{2^{n}}}~\delta_{n>0}~.
\begin{array}{c}{(a)\quadF(N)=1,\quada\bar{a}-\bar{a}a=G_{1}(N),}\\{(b)\quadG(N)=1,\quada\bar{a}-F_{1}(N)\bar{a}a=1,}\\{(c)\quadF(N)=q,\quada\bar{a}-q\bar{a}a=G_{q}(N).}\\\end{array}
\int_{0}^{R}\mathrm{d}r\;u_{kl}\left(r\right)u_{k^{\prime}l}\left(r\right)=\frac{u_{kl}\left(R\right)u_{k^{\prime}l}^{\prime}\left(R\right)-u_{kl}^{\prime}\left(R\right)u_{k^{\prime}l}\left(R\right)}{k^{2}-k^{2}}\,,
\langle{\calO}_{1}(x){\calO}_{1}(y)\rangle_{n}^{\mathrm{disc.}}=\sum_{m=0}^{n}C_{nm}\,\langle{\calO}_{\bf20}^{LM}(x){\calO}_{\bf20}^{NP}(y)\rangle_{m}\\langle{\calO}_{\bf20}^{LM}(x){\calO}_{\bf20}^{NP}(y)\rangle_{n-m}\,.
{}^{(D-1)}R_{~\beta\gamma\delta}^{\alpha}={}^{(D)}R_{~\nu\rho\sigma}^{\mu}q_{\mu}^{~\alpha}q_{\beta}^{~\nu}q_{\gamma}^{~\rho}q_{\delta}^{~\sigma}+K_{~\gamma}^{\alpha}K_{\beta\delta}-K_{~\delta}^{\alpha}K_{\beta\gamma}\,.
[\partial^{2}-{\frac{1}{2P}}(R-{\frac{Q^{2}}{4P}})]\Pi_{i}=-2Q\epsilon_{ij}\partial_{j}(\partial_{k}B_{k})+{\frac{Q}{2P}}(2R-{\frac{Q^{2}}{4P}})\epsilon_{ij}B_{j}+2Q\epsilon_{ik}\partial_{j}\partial_{j}B_{k}-2Q\partial_{i}(\epsilon_{jk}\partial_{j}B_{k}).
S_{\mathrm{boson}}^{\mathrm{root}}=\intd^{4}x\left\{\frac1{4g^{2}}F_{\mu\nu}^{2}+\bar{\nabla}_{\mu}\bar{q}_{Af}\nabla_{\mu}q^{fA}+\frac{g^{2}}8[\mathrm{~Tr~}(\bar{q}\tau^{m}q)]^{2}\right\},
\renewcommand{\arraystretch}{2}\begin{array}{lcl}{\displaystyle\Omega_{1}=-\frac{1}{4}(F_{2}+F_{3}+F_{4}^{\prime})+\cdots,}&{~}&{\displaystyle\Omega_{2}=-\frac{1}{4}(-F_{1}+F_{3}+F_{4}^{\prime})+\cdots,}\\{\displaystyle\Omega_{3}=-\frac{1}{4}(-F_{1}-F_{2}+F_{4}^{\prime})+\cdots,}&{~}&{\displaystyle\Omega_{4}=-\frac{1}{4}(-F_{1}-F_{2}-F_{3})+\cdots.}\\\end{array}
{\tilde{Z}}_{m}^{T}~=~\int\prod_{\mu;<{ij}>}dP_{ij}^{\mu}e^{-\sum_{\mu;<{ij}>}(P_{ij}^{\mu})^{2}}Det\left(\sum_{\mu}(\sigma^{\mu}P^{\mu})\right)\prod_{\mu;i}\delta(\sum_{j}P_{ij}^{\mu})~~,
\left\langle{\Phi}\right\vert\bar{\psi}(x)\psi(x)\left\vert{\Phi}\right\rangle=N\int_{-\infty}^{\infty}{\frac{dp_{-}}{2\pi}}\left[{\frac{2\Phi(p_{-})}{1+\Phi^{2}(p_{-})}}{\frac{p_{-}}{\sqrt{c}\omega_{p}}}-{\frac{1-\Phi^{2}(p_{-})}{1+\Phi^{2}(p_{-})}}{\frac{m}{\omega_{p}}}\right].
\left\langle0\right|\alpha_{n}^{l}K^{i-}K^{j-}\alpha_{-m}^{k}\left|0\right\rangle+\left\langle0\right|b_{s}^{l}K^{i-}K^{j-}b_{-s}^{k}\left|0\right\rangle=\sum_{i=1}^{4}D_{i}\tag{A-1}
D\left\langle\mathrm{{\bf~F}}^{n-1}Q_{\alpha}\right\rangle=\left\langleD(\mathrm{{\bf~F}}^{n-1})Q_{\alpha}\right\rangle=\left\langle\mathrm{{\bf~F}}^{n-1}\left\{\Psi,Q_{\alpha}\right\}\right\rangle=0,
\deltax^{+\alpha}=-{\frac{1}{2}}(\epsilon_{a}^{\alpha}\theta^{+a}+\epsilon^{+a}\theta_{a}^{\alpha})\,\quad\deltax^{-\alpha}=-\epsilon^{-a}\theta_{a}^{\alpha}\,\quad\delta\theta^{+a}=u_{\alpha^{\prime}}^{+}\epsilon^{\alpha^{\prime}a}\,\quad\delta\theta_{a}^{\alpha}=\epsilon_{a}^{\alpha}\,\quad\deltau_{\alpha^{\prime}}^{\pm}=0\.
F_{\mu\mathrm{m}}={\frac{1}{2}}\biggl({\frac{\partialu^{\mu}}{\partialX^{\mathrm{m}}}}+{\frac{\partialu^{\mathrm{m}}}{\partialX^{\mu}}}+\sum_{\mathrm{n}=1}^{3}{\frac{\partialu^{\mu}}{\partialX^{\mathrm{n}}}}{\frac{\partialu^{\mathrm{m}}}{\partialX^{\mathrm{n}}}}\biggr).
\begin{array}{lcllcll}{P^{i}}&{=}&{p^{i}\;,}&{M^{ij}}&{=}&{x^{i}p^{j}-x^{j}p^{i}\;,}&{}\\{P^{0}}&{=}&{\omega_{p}\;,}&{M^{i0}}&{=}&{x^{i}\omega_{p}\;,}&{\omega_{p}\equiv(p^{i}p^{i}+m^{2})^{1/2}\;.}\\\end{array}
R_{t}=\left(\begin{array}{cccccc}{0}&{1}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{K_{tts}}&{K_{ttt}}&{0}\\{0}&{0}&{0}&{K_{tss}}&{K_{tst}}&{0}\\{0}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{0}&{0}&{0}\\\end{array}\right)
\begin{array}{rcl}{\nu}&{=}&{{\nu^{\prime}}\pm\left(E_{q-\mathrm{vib}}(v^{\prime}+1)-E_{q-\mathrm{vib}}(v)\right)}\\{}&{=}&{\displaystyle{\nu^{\prime}}\pm\frac{1}{2}\left([v+2+b\gamma]_{q}-[v+b\gamma]_{q}\right)\nu_{vib}~,}\\\end{array}
Z^{\prime}=\intd[B_{\mu},\theta]\delta(\partial_{\mu}B^{\mu})exp\i\int[\frac{\lambda^{2}}{2\pi}B^{\mu}(g_{\mu\nu}-\frac{\partial_{\mu}\partial_{\nu}}{\Box})B^{\nu}-\frac{\theta^{2}}{2}+\epsilon_{\mu\nu}(B^{\mu}+J^{\mu})\partial^{\nu}\theta]d^{2}x
\hat{S}_{H}=-\intd^{D+1}x\sqrt{|\hat{g}_{D+1}|}\left[\frac{1}{2n!}\hat{H}_{M_{1}M_{2}\ldotsM_{n}}\hat{H}_{M_{1}^{\prime}M_{2}^{\prime}\ldotsM_{n}^{\prime}}\hat{g}^{M_{1}M_{1}^{\prime}}\hat{g}^{M_{2}M_{2}^{\prime}}\ldots\hat{g}^{M_{n}M_{n}^{\prime}}\right]
S={\frac{1}{\alpha^{\prime}}}\mathrm{tr}(\intd^{2}xd^{4}\thetae^{gV}\Phie^{-gV}\bar{\Phi}^{\dagger}+{\frac{1}{64g^{2}}}\intd^{2}xd^{4}\thetaW^{2}+{\frac{ig}{3!\sqrt{\alpha^{\prime}}}}\intd^{2}xd^{2}\theta\epsilon_{ijk}\Phi^{i}\Phi^{j}\Phi^{k}+cc)