harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{-\infty}^{\infty}\frac{dp_{0}}{2\pi}\int_{-\infty}^{\infty}\frac{dp_{1}}{2\pi}~\frac{e^{-|k_{1}\theta^{12}\sigma|\sqrt{p_{0}^{2}+p_{1}^{2}}}}{2\sqrt{p_{0}^{2}+p_{1}^{2}}}=\frac{1}{4\pi|k_{1}\theta^{12}\sigma|}.
G_{m}(p_{2},...,p_{n})=\underset{\{q_{i}=0\}}{\mathrm{Res}}\left(q_{m}\mathrm{det}\left(\frac{\partialF_{i}}{\partialq_{j}}\right)\prod_{i=2}^{n}\frac{dq_{i}}{(F_{i}(q)-p_{i})}\right).
A_{0}^{qu(0)}(y)~~=~~c_{a}^{(0)}~\frac{\partialA_{0}^{cl}(y)}{\partialn^{b}}\left[\intdy\left(\frac{\partialA_{0}^{cl}(y)}{\partialn^{c}}\right)^{2}/2\right]^{-1/2}{}~(1-n^{a}n^{b})
{\calH}=\pm\frac{1}{2e_{5}k}\\epsilon_{ijk}\Tr(F_{ij}\D_{k}\Phi)=\pm\frac{1}{2e_{5}^{2}k}\epsilon_{ijk}\Tr(F_{ij}\F_{k5})=\pm\frac{1}{8e_{5}^{2}k}\epsilon_{0NPQR}\Tr(F^{NP}\F^{QR}).
Z={\mathrm{const}}\times\int[d\psi][d{\widetilde\psi}]({\mathrm{det}}\;\gammaD)^{-2}\delta\Bigr(\gamma^{\rho}\psi_{\rho}-b\Bigr)\delta\Bigr({\widetilde\psi}_{\lambda}\gamma^{\lambda}-{\tilde{b}}\Bigr){\mathrm{e}}^{-S_{cl}[\psi,{\tilde{\psi}}]}\int[dg][d{\overline{g}}]\;.
\vertn_{1},n_{2},n_{3},\cdots;\bar{n}_{1},\bar{n}_{2},\bar{n}_{3},\cdots\rangle=\prod_{j=1}^{\infty}{\frac{(b_{j}^{\dagger})^{n_{j}}}{\sqrt{n_{j}!}}}{\frac{(a_{j}^{\dagger})^{\bar{n}_{j}}}{\sqrt{\bar{n}_{j}!}}}\vert0\rangle.
{\frac{1}{2}}\\langle\bar{\lambda}^{a}\lambda^{a}\rangle\=\{\frac{32\pi^{2}}{g^{2}(M^{2})}}\,\Lambda_{h}^{3}\=\{\frac{32\pi^{2}}{g^{4}(M^{2})}}\,\Lambda_{1}^{3}
{\frac{\partialP}{\partialn^{i}}}=-{\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}\mu_{i}\,\Rightarrow\widetilde{n^{i}}\,{\frac{\partialP}{\partialn^{i}}}=\widetilde{n^{i}}\,{\frac{\partial{\mit\Pi}}{\partialn^{i}}}=\Big({\frac{c_{_\mathrm{S}}^{2}}{c^{2}}}-1\Big)(\rhoc^{2}+P)\.
f\vertj_{k-1},-j_{k-1}\rangle=\left[\begin{array}{c}{j_{k}+j_{k-1}-1}\\{2j_{k-1}}\\\end{array}\right]\frac{[j_{k}+j_{k-1}]}{[j_{k}-j_{k-1}]}\vertj_{k},-j_{k-1}-1\rangle~.
\Phi^{\prime\prime}=\frac{c(2\varepsilon(\hat{\bfp}))^{-1/2}}{(\varepsilon(\hat{\bfp})+mc^{2})^{\frac{1}{2}}}\left(\vec{\sigma}{\bf\hat{p}}\Phi(\varepsilon(\hat{\bfp})+mc^{2})+(\varepsilon(\hat{\bfp})+mc^{2})\Phi\vec{\sigma}{\bf\hat{p}}\right)\frac{(2\varepsilon(\hat{\bfp}))^{-1/2}}{(\varepsilon(\hat{\bfp})+mc^{2})^{\frac{1}{2}}}\,.
\pi\mu\gamma(1-\beta^{2})=\left[\frac{M\pik(G)\Gamma\left(\frac1{(1-\beta^{2})h}\right)}{\Gamma\left(\frac1h\right)\Gamma\left(\frac{\beta^{2}}{(1-\beta^{2})h}\right)}\right]^{2(1-\beta^{2})}.
\psi=\left(\begin{array}{c}{{\psi_{1}}}\\{{\psi_{2}}}\\{{\psi_{3}}}\\{{\psi_{4}}}\\\end{array}\right)~\leftrightarrow~\psi_{1}+e_{2}\psi_{2}+e_{4}\psi_{3}+e_{6}\psi_{4}\quad\quad[~\psi_{1,...,4}\in\mathbb{C}(1,\;e_{1})~]\quad.
{\frac{d(d-1)}{2}}U_{d}\;=\;{\frac{1}{2}}\left({\frac{z^{2}}{\beta^{2}}}-d-{\frac{1}{\beta}}\right)\;=\;{\frac{d}{2}}\left[{\frac{\betad-1}{2\beta}}+{\frac{1}{2\beta}}\sqrt{(1-\betad)^{2}+4\beta}-1\right]\;.
g_{0}^{2}J(p,nx)\equiv-ig_{0}^{2}\int\frac{[1+p.k/p^{2}]d^{D}k/(2\pi)^{D}}{k^{2}((k+p)^{2(1+nx)}}=\frac{\alphaz^{x}}{4\pi}\frac{j_{n+1}}{(p^{2})^{nx}},
F_{8}^{{\mathcal{O}}_{0,0}^{0,0}|++++++--}=\frac{\langle{\mathcal{O}}_{0,0}^{0,0}\rangle(\sigma_{6}^{+}+\sigma_{2}^{+}(\sigma_{2}^{-})^{2}+\sigma_{2}^{-}\sigma_{4}^{+}+(\sigma_{2}^{-})^{3})/\sigma_{6}^{+}}{2^{6}\,\mathrm{cosh}(\theta_{78}/2)\,\prod_{1\leqi<j\leq6}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
\widehat{K}=\left(\begin{array}{ccccc}{Z-{C^{\prime}}^{2}/C}&{C^{\prime}/2}&{0}&{q^{\prime}\bar{\psi}}&{0}\\{C^{\prime}/2}&{0}&{0}&{(q/4)\bar{\psi}}&{0}\\{0}&{0}&{-(C/2)P^{\mu\nu,\alpha\beta}}&{0}&{0}\\{0}&{0}&{0}&{q\hat{1}}&{0}\\{0}&{0}&{0}&{0}&{-f\hat{1}}\\\end{array}\right).
\Gamma_{R(\tau)}^{(0,2)})=(\kappa_{\tau}^{-2\tau})^{\frac{\epsilon_{L}}{2}}(C_{1\tau}(\frac{t}{\kappa_{\tau}^{2\tau}})^{-\alpha_{\tau}}+\frac{\nu_{\tau}}{\tau\nu_{\tau}(d-\frac{m}{2})-2}B_{\tau}(u_{\tau}^{*})).
i^{r}\mathrm{exp}(-i\frac{\pi}{6}\sum_{j=1}^{j=r}a_{j})\prod_{j=r}^{j=1}\left(\begin{array}{cc}{a_{j}}&{2i\epsilon}\\{i/2\epsilon}&{0}\\\end{array}\right)\left(\begin{array}{c}{V}\\{0}\\\end{array}\right)
<\psi_{i}(z_{1})\psi^{j\dagger}(z_{2})\psi^{k\dagger}(z_{3}^{*})\psi_{l}(z_{4}^{*})>={\frac{\delta_{i}^{j}\delta_{l}^{k}}{(2\pi)^{2}(z_{1}-z_{2})(z_{3}^{*}-z_{4}^{*})}}+{\frac{\delta_{l}^{j}\delta_{i}^{k}}{(2\pi)^{2}(z_{1}-z_{3}^{*})(z_{4}^{*}-z_{2})}}
d=\frac{p^{3}}{27}+\frac{q^{2}}{4}=4L^{2}(M^{2}-3N/4)^{2}+(4N^{3}L+2NL^{2})(M^{2}-3N/4)-\frac{N^{2}L^{2}}{12}-\frac{L^{3}}{27}
(\Pi({\vec{x}}))^{\dagger}\Pi({\vec{x}})=\sum_{\vec{n},\vec{m}}\frac{1}{X^{3}}{\sqrt{\kappa_{\vec{n}}\kappa_{\vec{m}}}}(P_{\vec{n}}^{\dagger}P_{\vec{m}})exp\left(2\pii({\vec{n}}-{\vec{m}})\cdot{\vec{x}}/X\right),
\check{R}=\sum_{I\neqJ}(-1)^{p_{I}p_{J}}e_{JI}\otimese_{IJ}+\sum_{I}(-1)^{p_{I}}q^{1-2p_{I}}e_{II}\otimese_{II}+(q-q^{-1})\sum_{I<J}e_{II}\otimese_{JJ}
\hat{A}(x)=\frac{1}{2}\iota\,l_{P}^{2}\sqrt{(J_{e_{+}}-J_{e_{-}})^{2}}=\frac{1}{2}\iotal_{P}^{2}\sqrt{2J_{e_{+}}^{2}+2J_{e_{-}}^{2}-(J_{e_{+}}+J_{e_{-}})^{2}}~,
\gamma_{J}^{I}s^{L}=\delta_{J}^{L}s^{I}+\sum_{L_{1}L_{2}=L}\delta_{J}^{L_{1}}s^{IL_{2}}+\sum_{L_{1}L_{2}=L}\delta_{J}^{L_{2}}s^{L_{1}I}+\sum_{L_{1}L_{2}L_{3}=L}\delta_{J}^{L_{2}}s^{L_{1}IL_{3}}.
Q_{\mathrm{BRST}}^{\#}\=\\sum_{n\in{\mathbbZ}}c_{n}^{\#}\,L_{-n}^{\#}\,-\,\frac12\;\sum_{n,m\in{\mathbbZ}}(n-m)\,\mathrm{\bf:}\,c_{-n}^{\#}\,c_{-m}^{\#}\,b_{n+m}^{\#}\,\mathrm{\bf:}\,-a^{\#}\,c_{0}^{\#}\,
\int_{0}^{\infty}dw~\frac{{\calH}^{2}}{a^{d}{\varphi^{\prime}}^{2}}=6\int_{0}^{\infty}\frac{dw}{a^{d}q}~\biggl(\frac{{\calH}^{2}}{{{\calH}^{2}-{\calH}^{\prime}}}\biggr).
\left|i^{2}\int_{l^{\prime4}}(\int_{l^{4}}[e[(\overline{{\Psi}}\Psi)]A_{\mu}][e^{\prime}[(\overline{{\Psi^{\prime}}}\Psi^{\prime})]A_{\mu}^{\prime}]d^{4}x)d^{4}x^{\prime}\right|=1~,
H^{\mu}{}_{\nu\rho}-H^{\mu}{}_{\lambda\tau}J^{\lambda}{}_{\tau}J^{\tau}{}_{\rho}+H^{\tau}{}_{\nu\lambda}J^{\mu}{}_{\tau}J^{\lambda}{}_{\rho}+H^{\tau}{}_{\lambda\rho}J^{\mu}{}_{\tau}J^{\lambda}{}_{\nu}=0\.
\begin{array}{l}{D_{\mu}^{ij}[A](x)\langleT[j^{\mu,j}(x)j^{\nu,k}(y)]\rangle+i\delta^{4}(x-y)gf^{ilj}A_{\mu}^{l}(x)\langle\frac{\deltaj^{\nu,k}}{\deltaA_{\mu}^{j}}(x)\rangle}\\{+ig\delta^{4}(x-y)f^{ikl}\langlej^{\nu,l}(x)\rangle+i\partial_{\mu}[\delta^{4}(x-y)\langle\frac{\deltaj^{\nu,k}}{\deltaA_{\mu}^{i}}(x)\rangle]=0.}\\\end{array}
\f_{2}=-e^{2}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{d^{4}q}{(2\pi)^{4}}\frac{(k+p+q)^{2}+q^{2}-k^{2}-p^{2}}{k^{2}\Big(1+(-1)^{n}k^{2n}/\Lambda^{2n}\Big)(k+q)^{2}(k+p+q)^{2}q^{2}(q+p)^{2}}
\ddot{\Phi}-\frac{2}{a}\,\dot{\Phi}_{,r}+\frac{1}{a^{2}}\,\Phi_{,rr}+H\dot{\Phi}+\frac{2}{a^{2}}\left(\frac{1}{r}-\dot{a}\right)\Phi_{,r}=8\piG\,\dot{\varphi}\left(\delta\dot{\varphi}-\frac{1}{a}\,\delta\varphi_{,r}\right)\.
\Biggl[\frac{\partial^{2}}{\partialt^{2}}+\frac{1}{4M}\frac{\partial}{\partialt}-\left(\frac{e^{-t/4M}}{4M}\right)^{2}\frac{\partial^{2}}{\partial\sigma^{2}}\Biggr]X^{i}(\sigma,t)=0.
{\dot{X}}^{a}=\{H,\X^{a}\}={\calP}^{a}(1+{\frac{1}{m^{2}}}{\frac{\partialg}{\partialm^{(i)(k)}}}m_{(i)(k)}+{\frac{1}{m^{2}}}{\frac{\partialg}{\partial{\bar{m}}^{(i)(k)}}}{\bar{m}}_{(i)(k)}),
V_{np}=e^{K/m_{pl}^{2}}\left[K^{S\overline{{S}}}D_{S}W\overline{{D_{S}W}}-\frac{3}{m_{pl}^{2}}W\overline{{W}}\right]=\frac{1}{\Phi}\sum_{j=1}^{5}h_{j}(\Phi,A)e^{-(j+1)\alpha\Phi}
V=e^{K/2}\int\Omega^{(3,0)}\;\U_{\alpha}=e^{K/2}\int\Omega_{\alpha}^{(2,1)}\;\,barU_{\bar{\alpha}}=e^{K/2}\int\Omega_{\bar{\alpha}}^{(1,2)}\;\,barV=e^{K/2}\int\Omega^{(0,3)}\.
\Psi_{G}^{(-)}(\vec{x})=\left(\begin{matrix}{ig_{1}(r){\calY}_{G+\frac{1}{2},G+1}(\hat{x})}\\{-f_{1}(r){\calY}_{G+\frac{1}{2},G}(\hat{x})}\\\end{matrix}\right)+\left(\begin{matrix}{ig_{2}(r){\calY}_{G-\frac{1}{2},G-1}(\hat{x})}\\{f_{2}(r){\calY}_{G-\frac{1}{2},G}(\hat{x})}\\\end{matrix}\right)\.
i\frac{\partial}{\partialt}K=\left[\sum_{j=1}^{2}\left(m_{j}+\frac{{\bfp}_{j}^{2}}{2m_{j}}-\frac{{\bfp}_{j}^{4}}{8m_{j}^{3}}\right)+V^{q\overline{{q}}}\right]K,
\partial_{x}\psi_{BA}(t,\lambda)+\sum_{i=1}^{m}\Phi_{i}\Gamma_{i}(t,\lambda)=\lambda\psi_{BA}(t,\lambda)\;\;;\;\;\partial_{x}\Gamma_{i}(t,\lambda)=\Psi_{i}(t)\psi_{BA}(t,\lambda)\,.
C_{1}\alpha_{n}^{\mui}C_{1}^{-1}=\frac{1}{\sqrt{n}}\oint_{\xi_{0i}=0}d\xi_{0i}\(\xi_{0i})^{n}\left(\frac{d}{d\xi_{0i}}V_{i}^{-1}V_{0i}\xi_{0i}\right)P^{\mui}(V_{i}^{-1}V_{0i}\xi_{0i})\.
{\alpha_{1}}^{2}=\frac{4\sqrt\lambda}{a_{+}},\;\;{\alpha_{2}}^{2}=\frac{4\sqrt\lambda}{a_{-}},\;\;{\alpha_{3}}^{2}=\frac{2(a_{-}-2\sqrt\lambda)}{(1-\alpha)a_{+}},\;\;{\alpha_{4}}^{2}=\frac{2(a_{+}-2\sqrt\lambda)}{(1+\alpha)a_{-}}
\mathcal{S}=\int{d^{4}x\left[\frac{1}{4}h^{\mu\nu}\left(\nabla^{2}\left\{P_{\mu\nu\rho\sigma}^{(2)}-2P_{\mu\nu\rho\sigma}^{(0)}\right\}+2m^{-2}\left(\nabla^{2}\right)^{2}P_{\mu\nu\rho\sigma}^{(0)}\right)h^{\rho\sigma}\right]},
S=S_{0}^{L}\left[A_{\mu}^{a},B_{a}^{\mu\nu}\right]+\intd^{4}x\left(\varepsilon^{\mu\nu\lambda\rho}B_{\mu\nu}^{*a}\partial_{\lambda}\eta_{\rhoa}+\varphi^{*a}\partial^{\mu}\eta_{a\mu}\right),
\begin{array}{lcll}{\mathrm{floor}}&{\mathrm{SU(2/1)}}&{\mathrm{field}}&{\mathrm{helicity}}\\{\mid\mathrm{gnd}>}&{(0~-\frac{1}{2})}&{\lambda_{+}}&{+\frac{1}{2}}\\{\mid\mathrm{1st}>}&{(1~-\frac{1}{2})}&{2\Phi}&{~0}\\{\mid\mathrm{2nd}>}&{(0~+\frac{1}{2})}&{\lambda_{-}}&{-\frac{1}{2}.}\\\end{array}
{\calL}_{gK}=\frac{1}{2}\left(i\overline{{\Psi}}_{L}[\partial\!\!\!/+\frac{1}{2}(i{\calW}\!\!\!\!/\,\hat{C}+{\calW}\!\!\!\!/\,\hat{C}i)]\Psi_{L}+\overline{{\Psi}}_{L}[\partial\!\!\!/+\frac{1}{2}(i{\calW}\!\!\!\!/\,\hat{C}+{\calW}\!\!\!\!/\,\hat{C}i)]\Psi_{L}i\right).
\langlen_{1}n_{2}|\vec{\gamma},t\rangle={\frac{1}{g_{1}\sqrt{n_{1}!N_{2}!}}}\mathrm{exp}\\biggl(-{\frac{|\vec{\gamma}|^{2}}{2}}+{\frac{1}{2}}g_{4}\vec{\gamma}\cdot\vec{\gamma}\biggr){\bfH}_{n_{1},n_{2}}^{\{g_{2}{\bfI_{2}}\}}\biggl({\frac{1}{g_{1}g_{2}}}\tilde{\bfR}\vec{\gamma}\biggr)\\.
\lambda\intdt\,d^{d}r\,{\frac{1}{\rho(t,\bbox{r})}}\rightarrow\lambda\intdt\,d^{d}r\,{\frac{|J|}{\rho(T,\bbox{R})}}=\lambda\intdT\,d^{d}\,R{\frac{1}{\rho(T,\bbox{R})}}\.
{\calS}_{D0}\sim\mp\frac{T_{0}}{\kappa}\intd^{3}\xi\sqrt{-\mathrm{det}\,G}\;e^{-\,\frac{1}{4}\,\Phi}\;\int_{{\calC}_{1}}B_{(2)}\,\pm\,\frac{T_{0}}{\kappa}\left[\intC_{(1)}\int_{{\calC}_{1}}B_{(2)}+\intA_{(1)}\right]
\mathrm{Res}_{s=s_{k}}\zeta_{A}(s)=\frac{1}{m}\,\mathrm{res}\A^{-s_{k}}=\frac{1}{m}\int_{S^{*}M}\mathrm{tr}\a_{-n}^{-s_{k}}(x,\xi)\,d^{n-1}\xi.
\omega_{L}(\xi,gg^{\prime})=\sum_{l+m\leL}\tilde{c}_{l,m}^{(L)}\,(\omega\omega^{\prime})^{L-l-m}\bigl[(\partial\omega)^{l}K_{[l]}(\partial\omega^{\prime})^{l})\bigr]\left[(\partial\partial\omega)K_{[0,1]}(\partial\partial\omega^{\prime})\right]^{m}
r^{S}=\left(\begin{array}{cccc}{r_{+}^{S}}&{0}&{0}&{0}\\{0}&{r_{+}^{S}}&{0}&{0}\\{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{r_{\chi}^{S}}\\\end{array}\right)\qquadr^{\delta_{i}}=\left(\begin{array}{cc}{1}&{0}\\{0}&{r_{\delta_{i}}}\\\end{array}\right)
{\widetildej}_{\mu}^{(5)}=\begin{array}{}{t]{l}\displaystyle{\Biggl(}{\widetildeT}_{\mu}+\frac{1}{16}\sigma_{\mu}^{\alpha{\dot{\alpha}}}[D_{\alpha},{\bar{D}}_{\dot{\alpha}}](a_{1}(X+{\bar{X}})+a_{2}(X-{\bar{X}}))}\\{+\frac{i}{8}\partial_{\mu}((b_{1}+2a_{1})(X+{\bar{X}})+(ib_{2}+2a_{2})(X-{\bar{X}}))\displaystyle{\Biggr)}\displaystyle{\biggl\vert}\.}\\\end{array}
\Psi=e^{-{\frac{\sigma}{2}}}\left(\begin{array}{c}{e^{-{\frac{\theta}{2}}}e^{\imathv(x^{-})}p(x^{-})}\\{e^{{\frac{\theta}{2}}}e^{\imathu(x^{+})}f(x^{+})}\\\end{array}\right)
\begin{array}{ll}{\displaystyle{f=}}&{\displaystyle{2n(1-n)\tau_{nn}-n\tau_{n}-2nw(\chi_{n}+n\chi_{nn})+\left(\alphaw^{2}/\,2\right)n\tau_{nn}\,,}}\\{\mathrm{}}\\{\displaystyle{g=}}&{\displaystyle{2n(1-n)\chi_{nn}+(2-3n)\chi_{n}+\alphaw\left(2n\tau_{nn}+\tau_{n}\right)+\left(\alphaw^{2}/\,2\right)\left(n\chi_{nn}+\chi_{n}\right)\,.}}\\\end{array}
z=V\int_{0}^{\infty}dp_{n}e^{-\frac12\betac\left(p_{n}+\frac{m^{2}c^{2}}{p_{n}}\right)}\prod_{k=1}^{n-1}\int_{-\infty}^{\infty}dp_{k}e^{-\frac{\betac}{2p_{n}}p_{k}^{2}}.
\hat{\calS}_{ij}^{\alpha\beta}\equiv\epsilon_{~~\mu\nu}^{\alpha\beta}\bigg(\delta_{ij}\hat{\theta}^{\mu}\hat{P}^{\nu}+{\itg_{2}}\big(\tilde{F}_{(ij)}^{\mu\nu}-{\sqrt2}i\hat{\theta}_{5}F_{(ij)}^{\mu\nu}\big)\bigg)+{\frac{i\delta_{ij}}{2\kappa}}\big(e_{A}+{\sqrt2}ie_{B}\hat{\theta}_{5}\big)\sigma^{\alpha\beta}\,,
\begin{array}{lcl}{{\calC}_{con}}&{\equiv}&{{\calQ}\bigcap\{(864)^{2}\hat{\xi}+\hat{\zeta}+1728\hat{\eta}-\hat{\tau}=0\},}\\{{\calC}_{1}}&{\equiv}&{{\calQ}\bigcap\{\hat{\eta}-\hat{\tau}=0\},}\\{{\calC}_{\infty}}&{\equiv}&{{\calQ}\bigcap\{\hat{\tau}=0\},}\\{{\calC}_{0}}&{\equiv}&{{\calQ}\bigcap\{\hat{\xi}=\hat{\eta}=0\}.}\\\end{array}
U^{\Lambda\Sigma}=\mathrm{e}^{\mathcalK}{\mathcalD}_{i}X^{\Lambda}{\mathcalD}_{\bar{\jmath}}\bar{X}^{\Sigma}g^{i\bar{\jmath}}=-\frac12(\Im{\mathcalN}_{\Lambda\Sigma})^{-1}-\mathrm{e}^{\mathcalK}\bar{X}^{\Lambda}X^{\Sigma}
\begin{array}{l}{{\calX}^{0}\vertn,z\rangle_{0}=q^{n}z\vertn,z\rangle_{0}~,}\\{{\calD}_{0}\vertn,z\rangle_{0}=\displaystyle\frac{q^{1-n}}{q-1}z^{-1}\vertn-1,z\rangle_{0}~,}\\{\mu_{0}\vertn,z\rangle_{0}=\vertn-1,z\rangle_{0}~.}\\\end{array}
Z_{1\mathrm{inst}}=\frac{(-)^{N}}{\sqrt{2\piN\frac{16\pi^{4}m^{4}}{A^{2}}\sqrt{\frac{A}{\pi^{2}m^{2}}-1}}}(\mathrm{e}^{i\frac{3}{4}\pi-\frac{2\pi^{2}m^{2}N}{A}\gamma(\frac{A}{\pi^{2}m^{2}})}+\mathrm{e}^{-i\frac{3}{4}\pi+\frac{2\pi^{2}m^{2}N}{A}\gamma(\frac{A}{\pi^{2}m^{2}})}).
<{\calJ}_{\mu_{1}\cdots\mu_{s}}^{(s)}(x)~{\calJ}_{\nu_{1}\cdots\nu_{s}}^{(s)}(0)>=c_{s}{\prod}_{\mu_{1}\cdots\mu_{s},\nu_{1}\cdots\nu_{s}}^{(s)}\left({\frac{1}{|x|^{4+2h}}}\right),
\langle{\phi}^{k_{1}m_{1}p_{1}n_{1}}{\phi}^{k_{2}m_{2}p_{2}n_{2}}\rangle=\frac{(-1)^{m_{2}+n_{2}}}{R^{2}}\frac{{\delta}^{k_{1}k_{2}}{\delta}^{m_{1},-m_{2}}{\delta}^{p_{1}p_{2}}{\delta}^{n_{1},-n_{2}}}{k_{1}(k_{1}+1)+p_{1}(p_{1}+1)+R^{2}{\mu}_{l}^{2}}.
H_{+}(x)=\left(\begin{matrix}{h_{G}^{(1)}(x)}&{0}\\{0}&{h_{G}^{(1)}(x)}\\\end{matrix}\right)\quad\mathrm{and}\quadH_{-}(x)=\left(\begin{matrix}{h_{G+1}^{(1)}(x)}&{0}\\{0}&{h_{G-1}^{(1)}(x)}\\\end{matrix}\right)
a_{0}^{\mu}=\frac{1}{2}\sqrt{b}\widehat{p}^{\mu}-\frac{i}{\sqrt{b}}\widehat{x}^{\mu},\quada_{0}^{\mu\dagger}=\frac{1}{2}\sqrt{b}\widehat{p}^{\mu}+\frac{i}{\sqrt{b}}\widehat{x}^{\mu}.
\tilde{H}(\bar{z},w,x_{i})=\frac{\langle\partial_{\bar{z}}X(z,\bar{z})\partial_{w}\bar{X}(w,\bar{w})\sigma_{+}(x_{1})\sigma_{-}(x_{2})\sigma_{+}(x_{3})\sigma_{-}(x_{4})\rangle}{\langle\sigma_{+}(x_{1})\sigma_{-}(x_{2})\sigma_{+}(x_{3})\sigma_{-}(x_{4})\rangle}
\langleV({\bfx})V({\bfy})\rangle=gM^{2}\int\frac{d^{2}p}{(2\pi)^{2}}\,\frac{e^{i{\bfp}\cdot({\bfx}-{\bfy})}}{p^{2}+M^{2}}=\frac{gM^{2}}{4\pi}\int_{0}^{\infty}\frac{ds}{s}\,e^{-M^{2}s-({\bfx}-{\bfy})^{2}/4s}\;.
\left\{S_{,ij}^{G}-\left\{\begin{array}{c}{k}\\{i\j}\\\end{array}\right\}{\calE}_{k}-\frac{\delta\chi^{\alpha}}{\delta\Phi^{i}}c_{\alpha\beta}\frac{\delta\chi^{\beta}}{\delta\Phi^{j}}\right\}{\calG}^{jl}=-\delta_{i}^{l}.
S(a,\chi)=\intd^{4}x[{\frac{1}{2}}\eta^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi-{\frac{1}{2}}m^{2}a^{2}\chi^{2}-{\frac{1}{2}}(\xi-{\frac{1}{6}})Ra^{2}\chi^{2}-{\frac{\lambda}{4!}}\chi^{4}]
g=\frac{\sum_{l=1}^{n}\lambda_{l}^{2}\Bigl(e^{k_{l}(r+it-ic_{l})}-1\Bigr)^{-1}}{\sum_{l=1}^{n}\lambda_{l}^{2}\Bigl(1-e^{-k_{l}(r+it-ic_{l})}\Bigr)^{-1}}
\Phi=U_{1}^{-1}({\bfr})\left(\begin{matrix}{t_{4}-{\frac{1}{2y_{R}}}}&{0}&{0}&{0}\\{0}&{t_{2}-{\frac{1}{2y_{L}}}+{\frac{1}{2y_{R}}}}&{0}&{0}\\{0}&{0}&{t_{2}}&{0}\\{0}&{0}&{0}&{t_{1}+{\frac{1}{2y_{L}}}}\\\end{matrix}\right)U_{1}({\bfr})
{\frac{1}{8}}{(m_{L}^{(k)})}^{2}=\frac{1}{2}\sum_{i=3}^{8}{(p_{L}^{i})}^{2}{\delta}_{k,0}+\frac{1}{2}{\sum}_{i=1}^{16}{(P^{I}+kV^{I})}^{2}+N_{L}^{(k)}-1+c_{k},\;i=1,\dots,6
\alpha\Omega^{1/2}e^{-\psi_{1}}\alpha^{\prime}\Omega^{\prime}-\frac{3}{4}\alpha^{2}\Omega^{3/2}e^{-\psi_{1}}\psi_{1}^{\prime2}+\frac{\mul^{2}}{2}\Omega^{-1}e^{\psi_{1}}=\frac{3}{2}
I_{R^{4}}^{\mathrm{tree}}=-\frac{\gamma}{16\piG_{10}}\intd^{10}x\sqrt{g}e^{-3\phi/2}\left[C^{hmnk}C_{pmnq}C_{h}^{\rsp}C_{\rsk}^{q}+\frac{1}{2}C^{hkmn}C_{pqmn}C_{h}^{\rsp}C_{\rsk}^{q}\right],
V\equiv\sum_{\mu\in{\mathcalR}}g_{|\rho|}(\rho\cdot\mu)(\rho^{\vee}\!\cdot\nu)\frac{\ell_{\mu}^{j}-\ell_{s_{\rho}(\mu)}^{j}}{\ell_{\mu}-\ell_{s_{\rho}(\mu)}}\,\ell_{s_{\rho}(\nu)}^{k}\check{s}_{\rho}=0.
Z_{L}^{(n)}(\{q_{N_{s}}\})=\tilde{Z}^{(n)}(\{q_{N_{s}}\},L)H(\{q_{N_{s}}\})\prod_{s=1}^{n}\frac{(-1)^{2l_{1s}+2l_{2s}-1}16^{2l_{1s}}Z^{(1)}(k_{s};l_{1s},l_{2s})}{k_{s}^{(3-2l_{1s})/2}}
\dot{x}^{AA^{\prime}}=r\bar{\pi}^{A}\pi^{A^{\prime}},\,\,\,\,\acute{x}^{AA^{\prime}}=\bar{\zeta}\bar{\pi}^{A}\eta^{A^{\prime}}+\zeta\bar{\eta}^{A}\pi^{A^{\prime}}+g\bar{\pi}{}^{A}\pi^{A^{\prime}}.
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}-\frac{{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=-\left(\frac{1}{{g^{\prime}}_{tt}}E^{2}+\mu^{2}\right),
R_{1/2,1/2}^{\mathrm{p}}(u)=\left(\begin{array}{cccc}{1}&{{}}&{{}}&{{}}\\{{}}&{\frac{q^{-1}(1-u^{2})}{1-q^{-2}u^{2}}}&{\frac{u(1-q^{-2})}{1-q^{-2}u^{2}}}&{{}}\\{{}}&{\frac{u(1-q^{-2})}{1-q^{-2}u^{2}}}&{\frac{q^{-1}(1-u^{2})}{1-q^{-2}u^{2}}}&{{}}\\{{}}&{{}}&{{}}&{1}\\\end{array}\right)\,.
G_{1}\left(m_{eff}\beta\right)=\left(\frac{\beta}{2\pir}\right)^{3}\,m_{eff}\beta\sum_{n=1}^{\infty}\frac{1}{n^{3}}\frac{d^{2}}{d\lambda^{2}}\frac{1}{\lambda}K_{1}\left(2\pin\mu_{eff}\lambda\right).
g_{\mathrm{w}}(x)\;=\;{\frac{{\frac{\sqrt{x}}{8\beta}}-1}{\sqrt{(x-b^{2})(x-a^{2})}}}-{\frac{1}{8\pi\beta}}\int_{0}^{\infty}dy\,{\frac{\sqrt{y}}{x+y}}{\frac{1}{\sqrt{(y+b^{2})(y+a^{2})}}}\;.
\gamma^{\mu}\langle\partial_{\mu}\psi\gamma_{21}\tilde{\psi}\rangle_{2}=\partial\psi\gamma_{21}\tilde{\psi}-\gamma^{\mu}\langle\partial_{\mu}\psi\gamma_{21}\tilde{\psi}\rangle_{0}-\gamma^{\mu}\langle\partial_{\mu}\psi\gamma_{21}\tilde{\psi}\rangle_{4},
S[A_{i}^{a},\Phi^{a}]=\intd^{3}x\left({\frac{1}{4}}F_{ij}^{a}F_{ij}^{a}+{\frac{1}{2}}(D_{i}\Phi)^{a}(D_{i}\Phi)^{a}+{\frac{\lambda}{8}}\left(\Phi^{a2}-v^{a2}\right)^{2}\right)\;,
\psi_{\gamma}^{\alpha_{1},\dots,\alpha_{2j}}=\frac{i}{4\pi\rho\sqrt{2j}}\,{\calD}_{\gamma}^{\dot{\alpha}_{1}}\left[U_{\{\dot{\alpha}_{1}}^{\alpha_{1}}\,\cdots\,U_{\dot{\alpha}_{2j}\}}^{\alpha_{2j}}\right]\bar{\eta}^{\dot{\alpha}_{2}\cdots\dot{\alpha}_{2j}}
{\bfB}^{\mathrm{circ.}}={\frac{B^{(0)}}{\sqrt{2}}}\left[\left(\begin{matrix}{i}\\{1}\\{0}\\\end{matrix}\right)e^{+i\phi}+\left(\begin{matrix}{-i}\\{1}\\{0}\\\end{matrix}\right)e^{-i\phi}\right]\,,
E_{8}\equiv\left(\begin{array}{ccccccccc}{-2}&{}&{}&{1}&{}&{}&{}&{}\\{}&{-2}&{1}&{}&{}&{}&{}&{}\\{}&{1}&{-2}&{1}&{}&{}&{}&{}\\{1}&{}&{1}&{-2}&{1}&{}&{}&{}\\{}&{}&{}&{1}&{-2}&{1}&{}&{}\\{}&{}&{}&{}&{1}&{-2}&{1}&{}\\{}&{}&{}&{}&{}&{1}&{-2}&{1}\\{}&{}&{}&{}&{}&{}&{1}&{-2}\\\end{array}\right).
S_{\mathrm{Liouville}}\equiv\frac{1}{4\pi\gamma}\intd\sigmad\tau\left\{{\textstyle\frac{1}{2}}[\left(\partial_{\tau}\varphi\right)^{2}-\left(\partial_{\sigma}\varphi\right)^{2}]-\mu^{2}e^{2\varphi}\right\}\,.
E_{f}=T_{p}(2\piR)^{p-3}\sqrt{\left(2n_{f1}^{(1)}n_{f2}^{(1)}\right)^{2}\!+\!\left(2n_{f1}^{(1)}m_{f2}^{(1)}\right)^{2}\!+\!\left(2m_{f1}^{(1)}n_{f2}^{(1)}\right)^{2}\!+\!\left(2m_{f1}^{(1)}m_{f2}^{(1)}\right)^{2}}\,.\quad
H^{(1)}=\intd^{2}x\left[m\Phi^{0}\partial_{i}B^{i}+m\sqrt{m}\epsilon_{ij}\Phi^{i}B^{j}+\sqrt{m}\Phi^{i}\partial_{i}B^{0}-\Phi^{3}(mB^{0}-\epsilon_{ij}\partial^{i}B^{j})\right].
\begin{array}{cc}{\left[L_{3},D^{\pm}\right]=}&{\pmD^{\pm}}\\{\left[\frac{\Sigma_{3}}{2},D^{\pm}\right]=}&{\mpD^{\pm}}\\{\left[D^{+},D^{-}\right]=}&{2\left(\frac{\Sigma_{3}}{2}\right)\.}\\\end{array}
\begin{array}{c}{\left(\sum_{i}r_{i}^{(l)}\right)_{L}\times\left(\sum_{i}r_{i}^{(l)}\right)_{R}}\\{\times\left[\begin{array}{l}{\{\left(44+84\right)_{B}+128_{F}\}_{L}}\\{\times\{\left(44+84\right)_{B}+128_{F}\}_{R}}\\\end{array}\right]}\\\end{array}
\frac{1}{(q)_{\infty}}\sum_{r=-\infty}^{\infty}(-1)^{r}q^{r(kr+k-i)}=\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k-1}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k-1}^{2}+n_{i}+\cdots+n_{k-1}}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-2}-n_{k-1}}(q^{2};q^{2})_{n_{k-1}}},
\begin{array}{l}{{\Pi^{\pm}}_{\alpha}{}^{\beta}=\frac12\Big(1\pm\frac1{2b}\Gamma^{\mu\nu}D_{\mu}\Lambda_{\nu}\Big)_{\alpha}{}^{\beta},\qquadb=\sqrt{D^{2}\Lambda^{2}-(D\Lambda)^{2}},}\\{1=\Pi^{+}+\Pi^{-},\qquad(\Pi^{\pm})^{2}=\Pi^{\pm},\qquad\Pi^{+}\Pi^{-}=0,}\\\end{array}
Q^{(1)}={\frac{l}{a^{2}}}\left[{\frac{1}{6}}R(h)+{\frac{ye^{\phi}}{l}}\left(\phi_{\|\alpha}^{|\alpha}+\phi^{|\alpha}\phi_{|\alpha}\right)-{\frac{y^{2}e^{2\phi}}{l^{2}}}\phi^{|\alpha}\phi_{|\alpha}\right]\.
\frac{1}{R^{2}}\frac{d}{dr}(R^{2}A^{2}\frac{d\phi}{dr})=\sqrt{2}e^{2\sqrt{2}(\phi+\chi)}\frac{Q_{m}^{2}}{R^{4}}-\sqrt{2}e^{-2\sqrt{2}(\phi-\chi)}\frac{Q_{e}^{2}}{R^{4}},
\begin{array}{ll}{\Pi^{\mu\nu,ab}(p)=\delta^{ab}}&{\left[\right.(p^{\mu}p^{\nu}-\eta^{\mu\nu}p^{2})(1+\Pi_{1}(p))}\\{}&{}\\{}&{+i\,m\epsilon^{\mu\nu\lambda}p_{\lambda}(1+\Pi_{2}(p))-\displaystyle{\frac{1}{\xi}}p^{\mu}p^{\nu}\left.\right]}\\\end{array}
\tilde{\phi}_{m}=\frac{1}{\sqrt{2\omega_{m}}}(e^{i\omega_{m}t}\tilde{a}_{m}^{\dagger}+e^{-i\omega_{m}t}\tilde{a}_{m}),\;\;\;\;\;\;\tilde{\pi}_{m}=i\sqrt\frac{\omega_{m}}{2}(e^{i\omega_{m}t}\tilde{a}_{m}^{\dagger}-e^{-i\omega_{m}t}\tilde{a}_{m})
I_{\mathrm{D}}=\frac{1}{16\pi}\intd^{D}x\sqrt{-g}\left(R-\frac{1}{2}(\nabla\phi)^{2}-\frac{e^{-\alpha(d)\phi}}{2(d+1)!}F_{d+1}^{2}\right)-\frac{1}{8\pi}\intd^{D-1}x\sqrt{-h}K,
(e)=-{\frac{\lambda^{2}}{32(2\pi)}}{\tilde{G}}_{\beta}^{2}(0)\int_{0}^{\frac{\beta}{2}}d\tau\,\,\int_{-\infty}^{+\infty}{\frac{dk}{g^{2}(k)}}\left[e^{-g(k)\tau}+{\frac{e^{g(k)\tau}+e^{-g(k)\tau}}{e^{\betag(k)}-1}}\right]^{2}\;,
\begin{array}{l}{X.(v_{1}\otimesv_{2}\otimes\cdots\otimesv_{n})=}\\{=Xv_{1}\otimesv_{2}\otimes\cdots\otimesv_{n}+v_{1}\otimesXv_{2}\otimes\cdots\otimesv_{n}+\ldots+v_{1}\otimesv_{2}\otimes\cdots\otimesXv_{n}}\\\end{array}
\begin{array}{c}{[s_{\mu\nu},s_{\lambda\rho}]=-\eta_{\mu\lambda}s_{\nu\rho}+\eta_{\mu\rho}s_{\nu\lambda}+\eta_{\nu\lambda}s_{\mu\rho}-\eta_{\nu\rho}s_{\mu\lambda}\,,}\\{{}}\\{{[}{s_{~b}^{a}},{}{s_{~d}^{c}}{]}=2(\delta_{~d}^{a}s_{~b}^{c}-\delta_{~b}^{c}s_{~d}^{a})\,,}\\\end{array}
\left({\frac{1}{r}}{\frac{d^{2}}{dr^{2}}}r-{\frac{l(l+1)}{r^{2}}}+\omega^{2}\right)\left\{\begin{array}{c}{F_{l}(r,r^{\prime})}\\{G_{l}(r,r^{\prime})}\\\end{array}\right\}=-{\frac{1}{r^{2}}}\delta(r-r^{\prime}),
B=3D-{\frac{\sqrt{-2\nabla^{2}}}{2}}(Q_{1}+Q_{2});~~E_{i}=3D-{\frac{1}{\sqrt{-2\nabla^{2}}}}[\epsilon_{ij}\partial=_{j}(P_{1}+P_{2})+(\mu+m)\partial_{i}Q_{1}+(\mu-m)\partial_{i}Q_{2}],
Q_{3}^{2,m}=\intdzP_{4}^{2,m}+\lambda\intd\bar{z}[\phi_{2}^{3/2,m+1/2}\otimes\overline{{\phi}}^{1/2,-1/2}+\phi_{2}^{3/2,m-1/2}\otimes\overline{{\phi}}^{1/2,1/2}]
\intd^{4}y\frac{\delta}{\delta{\calO}(y)}\intd^{4}x\,\underbrace{{\calO}(x)\star{\calO}(x)\star\ldots\star{\calO}(x)}_{n\mathrm{\;factors}}=n\intd^{4}x\,\underbrace{{\calO}(x)\star{\calO}(x)\star\ldots\star{\calO}(x)}_{n-1\mathrm{\;factors}}.
\tilde{a}_{3}^{(n)}\\equiv\a_{3}^{(n)}\,-{\frac{1}{2}}\,Da_{2}^{(n)}\=\,iota_{n}\,{\calR}\qquad\quad\mathrm{with}\quad\iota_{n}\,\equiv\,\frac{1}{2}\,n(n+1)\\,
I=\left(\begin{array}{cc}{1}&{0}\\{0}&{1}\\\end{array}\right),\quad\sigma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\quad\sigma_{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right),\quad\epsilon=\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right).
V(\Phi)={\frac{1}{2}}\Omega^{2}\Phi^{2}+{\frac{1}{128\pi^{2}}}[2\Omega^{4}ln{\frac{\Omega^{2}}{m^{2}}}-(3-2k)(\Omega^{2}-m^{2})^{2}-2m^{2}(\Omega^{2}-m^{2})]+{\calD}_{1}
I_{r}=\frac{9}{\Lambda^{2}r_{h}}\int_{1+{\hat{\epsilon}}}^{{\hat{L}}}dx\,x^{6}\left(x-1\right)^{-2}\left(x-x_{-}\right)^{-2}\left(x^{2}+(1+x_{-})x+\zeta\right)^{-2},
I=\frac{1}{4}\int_{\Omega}d^{10}x\sqrt{-g}\left(-R+\frac{1}{2}(\partial\Phi)^{2}+\frac{1}{12}e^{-\Phi}H^{2}\right)-\frac{1}{2}\int_{\Sigma}{\calK}d\Sigma\equivI_{vol}+I_{surf}\,.
\Upsilon=64\left[\frac{\left(H^{I}-i\eta^{IJ}H_{J}\right)\nabla_{p}H_{I}-\left(H_{I}+i\eta_{IJ}H^{J}\right)\nabla_{p}H^{I}}{H^{I}\eta_{IJ}H^{J}+H_{I}\eta^{IJ}H_{J}}\right]^{2}\,.
\mathrm{~e}^{-S_{\mathrm{eff}}[A_{0}]}\equiv\sum_{B_{M}}\int[d\vec{a}]<\vec{a}\vert\mathrm{~e}^{-H/T}\mathrm{~e}^{i\frac{\kappa}{T}\intA_{0}(B_{M}+B/2)}\vert\vec{a}+\vec{\nabla}A_{0}/T>
C_{n}^{\parallel}(\tau\rightarrow0_{-})\rightarrow-\frac{(\alpha^{\prime}\sqrt{E^{2}-m^{2}}-inl)B_{n}^{\parallel}+(\alpha^{\prime}\sqrt{E^{2}-m^{2}}+inl)A_{n}^{\parallel}}{\alpha^{2}(E^{2}-m^{2})+n^{2}l^{2}}\;\frac{l}{\tau},