harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}{\hat{\partial}_{i}\hat{\partial}_{j}=q^{-1}\hat{\calR}_{ji}^{lk}\hat{\partial}_{k}\hat{\partial}_{l}~,}\\{\hat{\partial}_{i}\partial_{j}=q\hat{\calR}_{ji}^{lk}\partial_{k}\hat{\partial}_{l}~.}\\\end{array}
\renewcommand{\arraystretch}{1.4}T_{i}^{(2)}(s)=\left(\begin{array}{cc}{\rho_{i}/s+O(1)}&{O\left(s^{(k-n-1)/2}\right)}\\{O\left(s^{(k-n-1)/2}\right)}&{T_{i}^{(1)}(1)+O(s)}\\\end{array}\right),\quadi=1,2,3.
\deltaL=\frac{\partialL}{\partialq^{k}}dq^{k}+\frac{\partialL}{\partial\dot{q}^{k}}d\dot{q}^{k}=\left[-\frac{d}{d\tau}\frac{\partialL}{\partial\dot{q}^{k}}+\frac{\partialL}{\partialq^{k}}\right]dq^{k}+\frac{d}{d\tau}\left(\frac{\partialL}{\partial\dot{q}^{k}}dq^{k}\right)=F_{k}dq^{k}+\frac{d}{d\tau}\Omega_{1}
m_{i}\beta^{i}=_{def}\left(\begin{array}{rrrrrrrr}{}&{m_{z}}&{m_{y}}&{}&{m_{x}}&{}&{}&{}\\{m_{z}}&{}&{}&{m_{y}}&{}&{m_{x}}&{}&{}\\{m_{y}}&{}&{}&{-m_{z}}&{}&{}&{m_{x}}&{}\\{}&{m_{y}}&{-m_{z}}&{}&{}&{}&{}&{m_{x}}\\{m_{x}}&{}&{}&{}&{}&{-m_{z}}&{-m_{y}}&{}\\{}&{m_{x}}&{}&{}&{-m_{z}}&{}&{}&{-m_{y}}\\{}&{}&{m_{x}}&{}&{-m_{y}}&{}&{}&{m_{z}}\\{}&{}&{}&{m_{x}}&{}&{-m_{y}}&{m_{z}}&{}\\\end{array}\right)
\begin{array}{l}{\nabla_{\rho}\nabla^{\rho}h_{\mu_{1}\ldots\mu_{s}}-s\nabla_{\rho}\nabla_{\mu_{1}}h_{\mu_{2}\ldots\mu_{s}}^{\rho}+{\frac{1}{2}}s(s-1)\nabla_{\mu_{1}}\nabla_{\mu_{2}}h_{\rho\mu_{3}\ldots\mu_{s}}^{\rho}+}\\{+2(s-1)(s+d-3)h_{\mu_{1}\ldots\mu_{s}}=0}\\\end{array}
\begin{array}{rcl}{\partial_{\mu}F^{\mu\nu}(x)+\partial^{\nu}(\partial_{\rho}A^{\rho}(x))+m^{2}A^{\nu}(x)\equiv\left[\partial_{\mu}\partial^{\mu}+m^{2}\right]A^{\nu}(x)}&{=}&{0}\\{\left[\partial_{\mu}\partial^{\mu}+m^{2}\right]\varphi(x)}&{=}&{0}\\\end{array}
\left(\begin{matrix}{a_{1}^{1}}&{\;a_{1}^{2}}&{\;a_{1}^{3}}\\{a_{2}^{1}}&{\;a_{2}^{2}}&{\;a_{2}^{3}}\\\end{matrix}\right)=\left(\begin{matrix}{\tilde{a}_{1}^{1}}&{\;\tilde{a}_{1}^{2}}&{\;0}\\{\tilde{a}_{2}^{1}}&{\;\tilde{a}_{2}^{2}}&{\;0}\\\end{matrix}\right)\:U_{2}^{\dagger}(\varphi^{2})\;U_{1}^{\dagger}(\varphi^{1})
{\calL}_{GN}^{\prime}=\frac{1}{2}\left[\bar{\Psi},(i\gamma^{\mu}\partial_{\mu})\Psi\right]-\bar{\Psi}(\sigma+i\gamma^{5}\pi)\Psi-\frac{1}{2\tilde{G}}\left(\sigma^{2}+\pi^{2}\right).
q^{+}=f^{i}(x)u_{i}^{+}+\theta^{+\alpha}\psi_{\alpha}(x)+\bar{\theta}_{\dot{\alpha}}^{+}\bar{\xi}^{\dot{\alpha}}(x)+2i\theta^{+\alpha}\bar{\theta}^{+\dot{\alpha}}\partial_{\alpha\dot{\alpha}}f^{i}(x)u_{i}^{-}
{\calH}(Q,R)={\calH}(q,r)+\frac{1}{2}\left[-3r^{\prime}q-i\kappar^{2}q^{2}-\frac{r^{\prime\prime}rq}{r^{\prime}}+\frac{i}{\kappa}\left(\frac{r^{\prime\prime}}{r}-\frac{1}{2}\frac{r^{2}}{r^{2}}+\frac{1}{2}\frac{r^{2}}{r^{2}}\right)\right]^{\prime}.
\mathrm{\hat{H}=\frac{\kappa^{2}}{12}R^{p-3}\frac{\partial}{\partialR}R^{-p+2}\frac{\partial}{\partialR}-\frac{1}{2R^{3}}\frac{\partial^{2}}{\partial\varphi^{2}}+\frac{\kappa^{12}}{R^{9}}V(\varphi),}
R_{{\calN}=2}^{2222}=\frac{[12]^{2}[34]^{2}}{x_{12}^{4}\,x_{34}^{4}}\s+\frac{[14]^{2}[23]^{2}}{x_{14}^{4}\,x_{23}^{4}}\t+\frac{[12][34][14][23]}{x_{12}^{2}\,x_{34}^{2}\,x_{14}^{2}\,x_{23}^{2}}\(s+t-1)\,,
{\calR}_{m}=\left(\begin{array}{ccccc}{\{u_{1{\phantom{-1}}},\partial_{t}^{-1}~~\}}&{1}&{0}&{\ldots}&{0}\\{\{u_{2{\phantom{-1}}},\partial_{t}^{-1}~~\}}&{0}&{1}&{\ldots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{\{u_{m-1},\partial_{t}^{-1}~~\}}&{0}&{0}&{\ldots}&{1}\\{\{u_{m{\phantom{-1}}},\partial_{t}^{-1}~~\}}&{0}&{0}&{\ldots}&{0}\\\end{array}\right)
y^{2}=x^{2}\left(y-P(z^{2})\right)+P^{2}(z^{2})-\Lambda^{2\beta_{0}}z^{4}\prod_{i=1}^{N_{s}}(x_{1}-m_{i}^{4})\prod_{j=1}^{N_{f}}(z^{2}-m_{j}^{2}).
\deltax^{\alpha\alpha^{\prime}}=-{\frac{1}{2}}(\epsilon_{a}^{\alpha}\theta^{\alpha^{\prime}a}+\epsilon^{\alpha^{\prime}a}\theta_{a}^{\alpha})\,\\\delta\theta_{a}^{\alpha}=\epsilon_{a}^{\alpha}\,\\\delta\theta^{\alpha^{\prime}a}=\epsilon^{\alpha^{\prime}a}\.
E_{c}={\frac{Q^{2}}{2\pi\epsilon_{0}}}({\frac{1}{2}}+ln[\sqrt{{\frac{\epsilon_{0}}{2\pim}}}{\frac{J_{\theta}}{Qr_{0}}}])~={\frac{Q^{2}}{2\pi\epsilon_{0}}}({\frac{1}{2}}+ln{\frac{r}{r_{0}}})~,
\frac{1}{m!n!}\left<\prod_{a}T_{j_{a}m_{a}\bar{m}_{a}}(w_{a})\left(\intd^{2}zS_{\mathrm{int}}^{+}(z)\right)^{m}\left(\intd^{2}zS_{\mathrm{int}}^{-}(z)\right)^{n}\right>
e^{\textstyle-J\alpha_{-}\Phi(\sigma,\tau)}={\frac{1}{\sqrt{\varpi}}}\sum_{M=-J}^{J}(-1)^{J-M}e^{ih(J-M)}\xi_{M}^{(J)}(x_{+})\,{\overline{\xi}_{-M}^{(J)}}(x_{-})\sqrt{\varpi}
\,-\,\frac{1}{3!}\intd^{3}x_{1}d^{3}x_{2}d^{3}x_{3}{\mathcalW}_{\mu_{1}\mu_{2}\mu_{3}}^{(3)}(x_{1},x_{2},x_{3})b_{\mu_{1}}(x_{1})b_{\mu_{2}}(x_{2})b_{\mu_{3}}(x_{3})\,+\,\ldots
L^{+}=\left(\begin{array}{ll}{q^{J_{3}^{q}}}&{-\omegaJ_{+}^{q}}\\{0}&{q^{-J_{3}^{q}}}\\\end{array}\right)\;,\;L^{-}=\left(\begin{array}{ll}{q^{-J_{3}^{q}}}&{0}\\{-\omegaJ_{+}^{q}}&{q^{J_{3}^{q}}}\\\end{array}\right)
G_{i}\equiv\int{\calD}\phi^{\prime}\;\mathrm{exp}\left[-{\frac{i}{\hbar}}\intdx\;\Pi(x)\phi^{\prime}(x)\right]\;\langle\Phi+{\frac{1}{2}}\phi^{\prime}\vert\;[\hat{H}_{\alpha},\hat{\rho}]\;\vert\Phi-{\frac{1}{2}}\phi^{\prime}\rangle\;,
h_{\mu\nu}^{ab}(x;x^{\prime})=\delta^{4}(x-x^{\prime})\delta^{ab}\partial_{\mu}^{\prime}\partial_{\nu}^{\prime}+\frac{1}{4}\partial_{\mu}^{\prime}(\partial\cdotA^{b})_{x^{\prime}}\frac{\delta}{\deltaA_{\nu}^{a}(x)}.
\tilde{Z}^{\Lambda,\Lambda_{0}}(\varrho\,;J)=\intd\mu_{\Lambda,\Lambda_{0}}(\phi)e^{-\frac{1}{\hbar}\bigl(\tilde{L}^{\Lambda_{0},\Lambda_{0}}(\varrho\,;\phi)+\tilde{I}^{\Lambda_{0},\Lambda_{0}}(\varrho)\bigr)+\frac{1}{\hbar}\langle\phi,J\rangle}.
\mathrm{Re}\,\chi_{T}^{\mathrm{(TM)}}(\omega)=\frac{\hbarA}{4\pi^{2}}\left(\frac{k_{B}T}{\hbar}\right)^{5}\int_{0}^{\Omega}dv\int_{\Omega}^{\infty}du\,\frac{u-v}{e^{\frac{u+v}{2}}-1}\frac{\left[uv-\frac{\Omega}{2}\left(u+v\right)\right]^{2}}{\sqrt{uv(u-\Omega)(\Omega-v)}},
H_{T}^{l.c.}=\int_{-L}^{L}dx^{-}\left[\frac{1}{2}p^{2}+m\Psi_{R}^{\dagger}\Psi_{L}+\frac{g}{\sqrt{2}}(\Psi_{R}^{\dagger}T^{a}\Psi_{R})_{\sim}\widetilde{A}_{+}^{a}\right],
\begin{array}{lcl}{\hat{C}_{(2n)\,\mu_{1}\cdots\mu_{2n}}^{\pm}}&{=}&{C_{(2n)\,\mu_{1}\cdots\mu_{2n}}^{\pm}-2nA^{(2)}{}_{[\mu_{1}}C_{(2n-1)\,\mu_{2}\cdots\mu_{2n}]}^{\pm}\,,}\\{}&{}&{}\\{\hat{C}_{(2n)\,\mu_{1}\cdots\mu_{2n-1}\underline{{y}}}^{\pm}}&{=}&{-C_{(2n-1)\,\mu_{1}\cdots\mu_{2n-1}}^{\pm}\,.}\\\end{array}
a_{1}=k[\tilde{H}]_{1}^{n-2}(B_{0}^{2}A_{1}^{0}+B_{1}^{1}A_{0}^{1})+\frac{1}{\alpha}(-)^{n-3}k{\overline{A}}_{1}^{*}B_{1}^{1}+m_{1}A_{1}^{0},
(\pi({\vec{x}}))^{*}\pi({\vec{x}})-\pi2({\vec{x}})=-\sum_{\vec{n}}\frac{1}{X^{3}}\kappa_{\vec{n}}+2\sum_{\vec{n},\vec{m}}\frac{1}{X^{3}}{\sqrt{\kappa_{\vec{n}}\kappa_{\vec{m}}}}(p_{\vec{n}}^{*}p_{\vec{m}})exp\left(2\pii({\vec{n}}-{\vec{m}})\cdot{\vec{x}}/X\right),
h^{-1}(\Psi_{\vertA}^{(\pm)},\Phi_{\vertA}^{(\pm)})={\frac{1}{2}}\,\Bigl((\psi_{\pm}^{A,(0)})^{c}\phi_{\pm}^{A,(0)}\,+(\psi_{\pm}^{A,(1)})^{c}\phi_{\pm}^{A,(1)}\Bigr)\;.
\deltaS=\int_{\tau_{i}}^{\tau_{f}}d\tau[\frac{1}{2}(\frac{dX}{d\tau})^{2}+X^{2}(3\frac{\mu^{2}}{a^{2}}{\phi_{c}}^{2}-\mu^{2})]\equiv\int_{\tau_{i}}^{\tau_{f}}(X\hat{M}X)d\tau
+\sqrt{|g_{E}|}\,\textstyle{\frac{1}{2}}e^{\varphi}a^{2}\left\{\left[\mathrm{Tr}\,\left(Q\mathcal{M}\right)\right]^{2}-2\mathrm{Tr}\,\left(Q\mathcal{M}Q\mathcal{M}\right)\right\}\;,
\begin{array}{c|c}{2d-\mathrm{surface}}&{3d-\mathrm{solution}}\\\hline{\mathrm{sphere~}(g=0)}&{\mathrm{adS}_{3}}\\{\mathrm{torus~}(g=1)}&{\mathrm{black~holes~/~TadS}}\\{\mathrm{Riemann~surface~}(g>1)}&{\mathrm{?}}\\\end{array}
\left\langleT_{\ell}^{ij}(p)\,T_{\ell^{\prime}}^{km}(-p)\right\rangle=-{\frac{3}{4\kappa_{10}^{2}{\alpha^{\prime}}^{3}}}\,F_{\ell}^{ij;km}(p)\,\delta_{\ell\ell^{\prime}},
J_{\alpha}^{\mu{\mathbfi}}=192\left[i\sigma_{\alpha\dot{\alpha}}^{\mu}{\bar{D}}_{\mathbfj}^{\dot{\alpha}}D^{{\mathbfi}{\mathbfj}}T-3i{\bar{\sigma}}^{\mu\dot{\alpha}\beta}{\bar{D}}_{\dot{\alpha}}^{\mathbfi}D_{\alpha\beta}T-12\partial^{\mu}D_{\alpha}^{\mathbfi}T+a\sigma_{\alpha}^{\mu\nu\,\beta}\partial_{\nu}D_{\beta}^{\mathbfi}T\right].
{\bar{\partial}}^{-1}\lambda_{n}=(-1)^{{\frac{n(n-1)}{2}}}\int_{ST_{2}}\prod_{j=2}^{n+1}d\tau_{j}\prod_{i=1}^{n-1}({\hat{\zeta}_{i,i+1}}B_{i+1}D_{i+1}){\hat{\zeta}_{n,n+1}}\lambda_{1}(a_{n+1}).
\frac{-S_{D4}}{V}=\frac{2\mu_{4}}{g_{s}n\alpha^{\prime}}\Biggl[|z|^{4}-2\pin\alpha^{\prime}\mathrm{Im}\left(mzz\bar{z}\right)+(\pin\alpha^{\prime})^{2}m^{2}|z|^{2}\Biggr],
W^{[a_{1},a_{2}]}(\theta=0)=C^{i_{1}\cdotsi_{a_{1}+a_{2}}\;j_{1}\cdotsj_{a_{2}}}\;u_{i_{1}}^{1}\cdotsu_{i_{a_{1}+a_{2}}}^{1}u_{j_{1}}^{2}\cdotsu_{j_{a_{2}}}^{2}\,,
F[a]=\intd^{9}x~\left\{\frac{1}{(2\pi)^{7}(\alpha^{\prime})^{5}g_{s}}\sum_{b}\sqrt{1+\left(2\pi\alpha^{\prime}e_{b}\vec{\nabla}a(\vec{x})\right)^{2}}+\sum_{b,c}V_{\mathrm{eff}}\Bigl[(e_{b}-e_{c})a(\vec{x})\Bigr]+\dots\right\}
\begin{array}{}{]{c|cccc}\mathrm{Field}}&{SU(n_{c})_{0}}&{SU(n_{c})_{N+1}}&{U(1)}&{U(1)_{R}}\\\hline{Q_{0}}&{n_{c}}&{1}&{1}&{-1}\\{Q_{i}}&{1}&{1}&{(-1)^{i}}&{-1}\\{Q_{N}}&{1}&{\bar{n}_{c}}&{(-1)^{N}}&{-1}\\{\lambda_{j}}&{1}&{1}&{0}&{1}\\\end{array}
\begin{array}{l}{\displaystyle{-Q(z)\frac{d^{2}\Psi(z)}{dz^{2}}+\left((j-\frac{1}{2})Q^{\prime}(z)+2P(z)\right)\frac{d\Psi(z)}{dz}}\,-}\\{-\displaystyle{\left(\frac{1}{3}j(j-\frac{1}{2})Q^{\prime\prime}(z)+2jP^{\prime}(z)\right)\Psi(z)=E\Psi(z)}\,,}\\\end{array}
Z=\sum_{G}\rho_{1}^{m_{1}}\ldots\rho_{V}^{m_{V}}\sum_{\mathrm{\scriptsize~heights}}\prod_{i}\prod_{a_{i},b_{i}=1}^{L_{i}}\left(\frac{S_{b_{i}}^{i}}{S_{a_{i}}^{i}}\right)^{m_{b_{i}a_{i}}},
U^{\prime}=\left(\begin{array}{cc}{U_{oo}^{\prime}}&{U_{oe}^{\prime}}\\{U_{eo}^{\prime}}&{U_{ee}^{\prime}}\\\end{array}\right),\;\;\;\;\;\bar{U}^{\prime}=\left(\begin{array}{cc}{U_{oo}^{\prime}}&{-U_{oe}^{\prime}}\\{-U_{eo}^{\prime}}&{U_{ee}^{\prime}}\\\end{array}\right).
\begin{array}{lcr}{\sum_{\stackrel{m_{i}n_{i}}{i=1,2,3}}}&{\rightarrow}&{\sum_{pq}\sum_{\stackrel{m_{i},n_{i}}{i=1,2}}}\\{\omega(\{m_{i}\},\{n_{i}\})_{i=1,2,3}}&{\rightarrow}&{\omega(\{m_{i}\},\{n_{i}\},p,q)_{i=1,2}=\widetilde{\omega}}\\{\epsilon_{123}}&{\rightarrow}&{\epsilon_{12\bar{3}}\equiv\widetilde{\epsilon}}\\{F_{(123)}}&{\rightarrow}&{F_{(12\bar{3})}={\widetildeF}_{sym},}\\\end{array}
\begin{array}{l}{\tilde{N}_{0,0}^{r,s}=N_{0,0}^{r,s}-\frac{1}{2}N_{0,0}^{r,r}-\frac{1}{2}N_{0,0}^{s,s},}\\{\tilde{N}_{0,n}^{r,s}=N_{0,n}^{r,s}-\frac{1}{2}K_{n}^{s},}\\{\tilde{N}_{n,m}^{r,s}=N_{n,m}^{r,s},}\\\end{array}
\begin{array}{lll}{d_{v}}&{=}&{2kH+Nk+N(k+1)=2kH+(2k+1)N}\\{d_{H}}&{=}&{[\frac{2k(2k+1)}{2}2H-2Hk(2k+1)]+[N]+[2\frac{2k}{2}]}\\{}&{+}&{[\frac{2k(2k+2)}{2}2N-Nk(2k+1)-N(k+1)(2k+1)]}\\{}&{=}&{2k.}\\\end{array}
S_{\mathrm{eff}}^{d=2}|_{\mu=0}={\frac{4}{\left(\beta\sqrt{\pi}\right)^{3}}}\sum_{k=1}^{\infty}{\frac{{\calB}_{2k}(\beta^{2}B/2)^{2k}}{(2k)!}}\sum_{j=0}^{\infty}{\frac{\left(-{\frac{\beta^{2}m^{2}}{4}}\right)^{j}\Gamma\left(\frac{3}{2}-j-2k\right)\eta(3-2j-4k)}{j!}}
\begin{array}{ll}{\bar{\bfs}X^{i*}=\psi^{i*}}&{{\bfs}X^{i*}=\xi^{i*}}\\{\bar{\bfs}\psi^{i}=-\frac{i}{2}\partial_{+}X^{i}}&{{\bfs}\psi^{i*}=-\frac{1}{2}F^{i*}}\\{\bar{\bfs}\xi^{*}=\frac{1}{2}F^{i*}}&{{\bfs}\xi^{i}=-\frac{i}{2}\partial_{-}X^{i}}\\{\bar{\bfs}F^{i}=i\partial_{+}\xi^{i}}&{{\bfs}F^{i}=-i\partial_{-}\psi^{i}\,}\\\end{array}
\mathrm{eigenv}(H)=\sqrt{{\mathsfs}_{R}}=(M_{R}-i\frac{\Gamma_{R}}{2})=\sqrt{\bar{M}_{Z}^{2}-i\bar{M}_{Z}^{2}\bar{\Gamma}_{Z}}=\sqrt{\frac{(m_{1}^{2}-im_{1}\Gamma_{1})}{(1+\frac{\Gamma_{1}}{m_{1}})}}\,.
{\calA}_{3}={\frac{1}{2i}}\left(-{\frac{\overline{{w}}_{1}d{w}_{1}}{1+|w_{1}|^{2}}}+{\frac{\overline{{w}}_{1}dw_{1}+\overline{{w}}_{2}dw_{2}}{2(1+|w|^{2})}}-c.c.\right).
X^{i}(\sigma,\tau)=X_{u}^{i}+p^{i}\tau+\frac{w^{i}\sigma}{2\pi}+\frac{i}{2\pi}\sum_{n\neq0}\left(\frac{\alpha_{n}^{i}}{n}e^{-in(\tau-\sigma)}+\frac{{\tilde{\alpha}}_{n}^{i}}{n}e^{-in(\tau+\sigma)}\right),
ds_{E}^{2}=-\frac{dt^{2}}{\sqrt{\left(-k+\frac{t_{+}^{2}}{t^{2}}\right)\left(1-\frac{t_{-}^{2}}{t^{2}}\right)}}+t^{2}\sqrt{\left(-k+\frac{t_{+}^{2}}{t^{2}}\right)\left(1-\frac{t_{-}^{2}}{t^{2}}\right)}d\Omega_{k}^{2}.
{\calL}_{m}[h,\Lambda]=-{\frac{1}{4}}(h_{\mu\nu}^{i}+\partial_{\mu}\Lambda_{\nu}^{i}-\partial_{\nu}\Lambda_{\mu}^{i})^{2}-{\frac{1}{4\eta^{2}}}h_{\mu\nu}^{i}\partial_{\lambda}\partial^{\lambda}h^{\mu\nu}{}^{i}+{\frac{1}{4\gamma^{4}}}h_{\mu\nu}^{i}(\partial_{\lambda}\partial^{\lambda})^{2}h^{\mu\nu}{}^{i}.
\left|\alpha_{k}\right|=\left|\beta_{k}\right|=\frac{2^{3n_{1}-n_{s}-1}}{|n_{1}(1-2n_{1})|}\left|\frac{\Gamma\left(-n_{s}^{\pi}\right)}{\Gamma\left(n_{1}^{\pi}\right)}\right|z_{S}^{-1/2-m-n_{1}}\left(\frac{g_{S}}{g_{1}}\right)^{l}\left(\frac{k}{k_{s}}\right)^{n_{1}+n_{s}}\mathrm{~for~}n_{1}<0.
\Lambda_{\vec{k}\\vec{k}^{\prime}}=\delta_{\vec{k}\\vec{k}^{\prime}}\\left(\begin{array}{clcr}{{\calH}^{(1)}(\vec{k},\,t)}&{-\dot{\calH}^{(1)}(\vec{k},\,t)}\\{{\calH}^{(2)}(\vec{k},\,t)}&{-\dot{\calH}^{(2)}(\vec{k},\,t)}\\\end{array}\right)\.
\mathcal{U}_{\bar{\Lambda}}=-16\sum_{i=0}^{\infty}c_{i}m_{i}^{2}\sum_{n=1}^{\infty}\int\,dz_{0}\dotsdz_{n}\bar{\Lambda}(z_{0})\starG(z_{0},z_{1})\starv(z_{1})\star\bar{D}^{2}D^{2}G(z_{1},z_{2})\star\dots\starv(z_{n})\star\bar{D}^{2}D^{2}G(z_{n},z_{0}),
P^{\mu}\equiv{\frac{\partial{\calL}_{DJ}}{\partial\dot{f}^{\mu}}}\approx0~~,~~\Pi^{0}\equiv{\frac{\partial{\calL}_{DJ}}{\partial\dot{A}^{0}}}\approx0~~,\Pi^{j}\equiv{\frac{\partial{\calL}_{DJ}}{\partial\dot{A}^{j}}}=\epsilon^{ij}(f^{i}-{\frac{\mu}{2}}A^{i}).
\Psi_{\kappa}(x)=\frac1{2^{3/2}\pi}\,e^{\pi\kappa/2}\left\{\Gamma(i\kappa)\left(\frac{mx_{+}}2-i0\right)^{-i\kappa}+\Gamma(-i\kappa)\left(-\frac{mx_{-}}2+i0\right)^{i\kappa}+...\right\}.
(\tilde{Z}_{E}[\tilde{J}])_{b}=-\sum_{n\geq1}\sum_{k\geq0}\frac{1}{(n)!}\langle(\tilde{G}_{k}^{(n)})_{b}(p_{1},\dots,p_{n};\lambda_{b},m_{b},\epsilon)Z_{\varphi}^{-\frac{n}{2}}\tilde{J}_{1}\dots\tilde{J}_{n}\rangle,
\hat{a}_{n}=\mathcal{U}\tilde{a}_{n}^{in}\mathcal{U}^{-1}\,\mathcal{U}=e^{K}\,K=\sum_{n}(c(\tilde{b}_{n}^{in})^{2}-\overline{{c}}(\tilde{a}_{n}^{in})^{2})\
S=\sum_{n=0}^{\infty}\intd^{6}x\[\dot{A}_{ab}^{(n)}E_{ab}^{(n)}-\frac{(-1)^{n}}{4}(E_{ab}^{(n)}E_{ab}^{(n)}+B_{ab}^{(n)}B_{ab}^{(n)})+{\Lambda}_{ab}^{(n+1)}{\Psi}_{ab}^{(n+1)}-2A_{ta}^{(n)}{\calG}_{a}^{(n)}],
f_{\epsilon}(y)=\int_{0}^{1}dxx^{-\frac{1}{2}+\frac{\epsilon}{2}}y^{\frac{1}{2}+\frac{\epsilon}{2}}\left\{\frac{q^{2}}{p^{2}}[y^{2}(1-x)^{2}-y(1-x)]+2\frac{p.q}{p^{2}}y^{2}x(1-x)+yx(yx-1)\right\}^{-2\epsilon}\;.
\gamma_{s}=i\sum_{m\neqs}\int_{\Sigma}\int_{\partial\Sigma={\bfc}}\langle\hat{\varphi}_{s}|\nabla_{R}\hat{\varphi}_{m}\rangle\times\langle\hat{\varphi}_{m}|\nabla_{R}\hat{\varphi}_{s}\rangle\cdotd\vec{\Sigma}
w\equivG_{B_{1}+B_{2}}^{-1}(y)=R_{B_{1}+B_{2}}+\frac{1}{y}=R_{B_{1}}+R_{B_{2}}+\frac{1}{y}=G_{B_{1}}^{-1}(y)+G_{B_{2}}^{-1}(y)-\frac{1}{y}=2z-\frac{1}{y}
g\left(q^{i},p_{i},z^{a},\overline{{p}}_{a}\right)=\sum_{k=1}^{\infty}\frac1{k!\lambda^{k}}\left[C_{a_{1}}^{0},\left[C_{a_{2}}^{0},\ldots,\left[C_{a_{k}}^{0},\tilde{H}\right]\ldots\right]\right]z^{a_{1}}\ldotsz^{a_{k}}.
\begin{array}{l}{\hat{ds}^{2}=-dt^{2}+{\calV}^{-2}(dx^{i}dx^{i}+dx^{4}dx^{4})\,\qquadF=0}\\{\nonumber}\\{e^{2\hat{\phi}}=2\,\chi^{(2)}={\calV}^{-2}\quad,\quad\partial_{i}\hat{B}_{4j}-\partial_{j}\hat{B}_{4i}=\epsilon_{ijm}\partial_{m}e^{2\hat{\phi}}}\\\end{array}
a_{i}(x)={\frac{1}{\sqrt{2\piL_{1}}}}\sum_{n_{1}}\intdp_{2}\,{\frac{1}{p^{0}}}[({\frac{p_{i}}{\mu}}g(\vec{p})+i{\frac{\epsilon_{ij}p_{j}}{2}}h(\vec{p}))e^{-ip\cdotx}+c.c]_{p^{0}=|\vec{p}|}\,,
\psi_{+}=\left[\begin{array}{l}{\xi}\\{0}\\\end{array}\right]~~,~~\psi_{-}=\left[\begin{array}{c}{0}\\{\left(\frac{1}{i\partial^{+}}\right)[\sigma^{i}(i\partial^{i}+gA^{i})+im_{F}]\xi}\\\end{array}\right].
\hat{\Gamma}_{\mu}(\partial,\stackrel{\leftarrow}{\partial}^{\dagger})=\stackrel{\leftarrow}{\zeta}_{\mu}^{-\;\;j}\Lambda_{j}+\sum_{l=1}^{N-1}\sum_{k=0}^{l}\stackrel{\leftarrow}{\partial}^{\dagger\;\;\mu_{1}}...\stackrel{\leftarrow}{\partial}^{\dagger\;\;\mu_{k}}\stackrel{\leftarrow}{\zeta}_{\mu}^{-\;\;j}\Lambda_{\mu_{1}...\mu_{k}j\mu_{k+1}...\mu_{l}}\partial^{\mu_{k+1}}...\partial^{\mu_{l}}
W_{i_{1}j}^{(1)}(x,y)=\left(\begin{array}{ccccc}{\widetilde{W}_{ij}^{(0)}}\\\hline{}\\{\begin{array}{ccccccc}{0}&{0}&{0}&{-iz_{1}^{*}}&{-iz_{2}^{*}}&{iz_{1}}&{iz_{2}}\\{0}&{0}&{\frac{\kappa}{\pi}}&{0}&{0}&{0}&{0}\\{0}&{-\frac{\kappa}{\pi}}&{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{z_{1}^{*}}&{z_{2}^{*}}&{z_{1}}&{z_{2}}\\\end{array}}\\\end{array}\right)\delta^{2}(x-y),
=\epsilon^{ij}({\frac{\mu}{2}}A^{j}-f^{j})\dot{A}^{i}-{\frac{1}{2}}f^{i}f^{i}-{\frac{1}{2}}(\epsilon^{ij}\partial^{i}A^{j})^{2}+A^{0}(\mu\epsilon^{ij}\partial^{i}A^{j}-\epsilon^{ij}\partial^{i}f^{j}),
\mathcalL_{\mathrm{mass}}=\frac{1}{2}\left(\lambda_{L}^{1}\\lambda_{L}^{2}\right)\left(\begin{array}{cc}{\omega/R}&{i\,p_{5}}\\{-i\,p_{5}}&{\omega/R}\\\end{array}\right)\left(\begin{array}{c}{\lambda_{L}^{1}}\\{\lambda_{L}^{2}}\\\end{array}\right)+h.c.
\int_{1}^{\rho}\frac{(s^{2}-1)^{n}}{s^{2}}ds=\int_{0}^{\rho-1}\frac{(y^{2}+2y)^{n}}{(y+1)^{2}}dy=\frac{2^{n}(\rho-1)^{n+1}}{n+1}\,\mathrm{{Appell}}F_{1}[n+1,2,-n,n+2,1-\rho,\frac{1-\rho}{2}]
G_{\alpha\beta}=\epsilon\left(\begin{array}{ccccc}{-(1-\tilde{v}^{2})}&{0}&{0}&{0}&{-\tilde{v}}\\{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{0}\\{-\tilde{v}}&{0}&{0}&{0}&{1}\\\end{array}\right),
\bar{\lambda}_{1,2}=\frac{1}{2}\left(\lambda_{1}+\lambda_{2}+{\calP}_{11}+{\calP}_{22}\right)\pm\sqrt{\frac{1}{4}\left(\lambda_{1}-\lambda_{2}+{\calP}_{11}-{\calP}_{22}\right)^{2}+{\calP}_{12}{\calP}_{21}}\,\,.
\chi_{\ell}^{(k)}(q)=\frac{\theta_{\ell+1,k+2}-\theta_{-\ell-1,k+2}}{\theta_{1,2}-\theta_{-1,2}}(q):=\sum_{m\in{\bfZ}_{2k}}c_{m}^{\ell}(q)\theta_{m,k}(q)~,
\mathrm{d}\mu=\mathrm{d}\mu_{G}\,\frac{\mathrm{d}\Omega_{G/H}}{\Omega_{G/H}}\,(M_{\mathrm{PV}})^{-T(R)}\prod_{R(j)}\left\{e^{-4\pi^{2}\,j\,|v_{\,\mathrm{SU}(2)}|^{2}\,\rho_{\mathrm{inv}}^{2}}\,\prod_{i=1}^{2T(j)}\mathrm{d}\,\bar{\eta}_{i}\right\}\,.
\begin{aligned}{|\Phi_{1}|^{2}+|\Phi_{3}|^{2}}&{{}=s_{1}\,,}\\{|\Phi_{2}|^{2}+|\Phi_{4}|^{2}}&{{}=s_{2}\,,}\\{|\Phi_{1}|^{2}+|\Phi_{2}|^{2}+|\Phi_{5}|^{2}}&{{}=s_{3}\,,}\\\end{aligned}
\frac{1}{4\pi}L_{CM}=\frac{vg}{2k}\left(\sum_{j=1}^{k}\vec{V}_{j}\right)^{2}-\frac{v}{2gk}\left(\sum_{j=1}^{k}q_{j}\right)^{2}+\frac{b}{k}\left(\sum_{j=1}^{k}q_{j}\right)\left(\sum_{l=1}^{k}V_{l\chi}\right),
A_{ji}^{-1}\frac{|\alpha_{i}|^{2}}{2}=\left\langle\lambda_{j},\lambda_{i}\right\rangle=\sum_{x=1}^{h}\frac{x}{h}\,\left\langle\lambda_{j},\sigma^{x+\frac{c_{i}-1}{2}}\gamma_{i}\right\rangle\;,
\mathrm{E}_{5(5)}=\frac{\mathrm{O}(5,5)}{\mathrm{O}(5)\otimes\mathrm{O}(5)}\rightarrow\frac{\mathrm{GL}(4,\mathbb{R})}{\mathrm{O}(4)}\otimes\frac{\mathrm{SL}(2,\mathbb{R})}{\mathrm{O}(2)}
\delta(\sigma)=\prod_{j=1}^{2}\prod_{p,q}\delta(\mathrm{Re}\,\sigma_{pq}^{jj})\delta(\mathrm{Im}\,\sigma_{pq}^{jj})\\prod_{j\neql}\prod_{p,q}\delta(\sigma_{pq}^{jl*})\delta(\sigma_{pq}^{jl})\.
\{F(\bar{\psi},\psi),G(\bar{\psi},\psi)\}=(2iN)^{-1}\intdx~g^{ij}\left(\frac{\deltaF}{\delta\bar{\psi}_{i}(x)}\frac{\deltaG}{\delta\psi_{j}(x)}-\frac{\deltaG}{\delta\bar{\psi}_{i}(x)}\frac{\deltaF}{\delta\psi_{j}(x)}\right)
\begin{array}{rcl}{}&{}&{\hat{\hat{C}\,}_{abc}=e^{-\varphi/2}C_{abc}\,,}\\{}&{}&{}\\{}&{}&{\hat{\hat{C}\,}_{ab\,m}=e^{-\varphi/3}B_{m\,ab}\,,}\\{}&{}&{}\\{}&{}&{\hat{\hat{C}\,}_{a\,mn}=\epsilon_{mnp}e^{-\varphi/6}A^{2\,p}{}_{a}\,,}\\\end{array}
z(1-z)d^{2}\psi_{1}^{\prime}+\frac12\left(1-3z\right)d\psi_{1}^{\prime}+\frac14\left(\frac{-\imathlk^{+}+l^{2}k^{+2}}{z}+\imathlk^{-}-l^{2}k^{-2}-\frac{1}{1-z}\right)\psi_{1}^{\prime}=0
K_{lm}(p_{1j}^{2};M_{l}^{2},\epsilon)=\frac{-i}{32\pi^{4}}\frac{1}{M_{l}^{2\epsilon}}\left[I(\frac{p_{12}^{2}}{M_{l}^{2}},\epsilon)+I(\frac{p_{13}^{2}}{M_{l}^{2}},\epsilon)+I(\frac{p_{14}^{2}}{M_{l}^{2}},\epsilon)\right].
g_{\tilde{r}\tilde{r}}(G_{\tilde{r}}^{\tilde{r}}+G_{\varphi}^{\varphi})=16\piG\frac{m}{f}(L_{M}-g^{\tilde{r}\tilde{r}}\frac{\partialL_{M}}{\partialg^{\tilde{r}\tilde{r}}}-g^{\varphi\varphi}\frac{\partialL_{M}}{\partialg^{\varphi\varphi}})
\deltaL_{0}=i\Lambda^{a}D_{-}\chi_{a}(\varphi,\tilde{\varphi})+\Lambda^{a}A_{-}^{b}\left(\partial_{\alpha}\chi_{a}\xi_{b}^{\alpha}(\varphi)+\partial_{\overline{{\alpha}}}\chi_{a}\tilde{\xi}_{b}^{\overline{{\alpha}}}\right)(\varphi,\tilde{\varphi}).
{\calF}_{a}^{\mu\nu}=\frac{1}{a}\phi_{a}{\calF}^{\mu\nu}=\frac{1}{a}\phi_{a}\,\frac{1}{a^{3}e}\phi\cdot(\partial^{\mu}\phi\wedge\partial^{\nu}\phi)+\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}.
T_{\phantom{a}a}^{a}=-{\frac{1}{2\alpha^{\prime}}}\beta_{\mu\nu}^{G}g^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}-{\frac{i}{2\alpha^{\prime}}}\beta_{\mu\nu}^{B}\epsilon^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}-{\frac{1}{2}}\beta^{\Phi}R\.
\hat{\Psi}(x)=\sum_{\sigma=\pm}\intd^{3}{\bfk}\left[\hat{b}_{{\bfk}\sigma}\psi_{{\bfk}\sigma}^{(+\omega)}(x)+\hat{d}_{{\bfk}\sigma}^{\dagger}\psi_{-{\bfk}-\sigma}^{(-\omega)}(x)\right]\;,
\int_{C}A=\int\left(\left(f_{x}\mathbf{g}+\frac{\partialh(x,y)}{\partialx}\mathbf{J}_{1}\right)dx+\left(f_{y}\mathbf{g}+\frac{\partialh(x,y)}{\partialy}\mathbf{J}_{1}\right)dy\right)~,
L_{B}=-{\frac{a}{4}}(B+\bar{A})_{\mu\nu}(B+\bar{A})^{\mu\nu}-{\frac{1}{2g}}B_{\mu}B^{\mu}+{\frac{\alpha}{2}}\epsilon^{\mu\nu\lambda}(B+\bar{A})_{\mu}(B+\bar{A})_{\nu\lambda}.
Z_{0}=Tre^{-\beta(H_{D}+\frac{VB^{2}}{2})}=e^{-\beta\frac{B^{2}}{2}V}\,(1-e^{-\betaE_{0}})^{d}\;e^{\beta\frac{d}{4}E_{0}}\;\prod_{n=1}^{\infty}(1-e^{-\betaE_{n}})^{4d}\;e^{\beta\frac{d}{2}E_{n}},
{\mathcalG}_{l+\nu}^{(0)}(r^{\prime\prime},r^{\prime};E)=\int_{0}^{\infty}dz\,\eta_{l+\nu}(z;\kappa)\,\Psi_{z,l+\nu}(r^{\prime\prime};\kappa)\Psi_{z,l+\nu}(r^{\prime};\kappa)\;.
S_{E}^{p}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-G^{E}}\left[{\calR}_{G^{E}}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(p+2)!}}e^{2b\phi}F_{p+2}^{2}\right],
\Phi_{i}^{(1)a}\equiv\epsilon_{0ijk}\pi^{jka}\approx0,\;\Phi_{i}^{(2)a}\equiv\frac12\epsilon_{0ijk}\left(F^{jka}-\left(D^{\left[j\right.}\right)_{\;\;b}^{a}\pi^{\left.k\right]0b}\right)\approx0,
\gamma_{b}w^{b}f^{a_{1}...a_{l}}+lNf^{a_{1}...a_{l}}-\sum_{i=1}^{l}w^{a_{i}}\left(\gamma_{b}f^{a_{1}...{\nota_{i}}...a_{l}b}\right)=d\left(\gamma_{b}j^{a_{1}...a_{l}b}\right)
R_{2}^{1\;KBq}=\left(\begin{array}{cc}{\frac{1}{2}u+\frac{1}{2}u_{x}D^{-1}}&{1}\\{\varepsilon^{2}D^{2}+\rho+\frac{1}{2}\rho_{x}D^{-1}}&{\frac{1}{2}u}\\\end{array}\right)
{\itS}={\frac{1}{4\pi}}\intd\sigmad\tau\,\Bigl({\frac{1}{2}}({\frac{\partial\Phi}{\partial\sigma}})^{2}+{\frac{1}{2}}({\frac{\partial\Phi}{\partial\tau}})^{2}+e^{\displaystyle2\sqrt{\gamma}\Phi}\Bigr)
\overline{{F}}^{\primea}{}_{\mu\nu}=\partial_{\mu}\overline{{A}}^{\primea}{}_{\nu}-\partial_{\nu}\overline{{A}}^{\primea}{}_{\mu}+\overline{{f}}^{a}{}_{bc}\overline{{A}}^{\primeb}{}_{\mu}\overline{{A}}^{\primec}{}_{\nu}-\overline{{C}}^{a}{}_{c\mu}\\overline{{A}}^{\primec}{}_{\nu}+\overline{{C}}^{a}{}_{c\nu}\overline{{A}}^{\primec}{}_{\mu},
\begin{array}{rl}{}&{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}D_{m}(k_{1}+k_{2})D_{m}(k_{2})}\\{}&{\times[D(k_{1}+k_{2}+p)-D(k_{1}+k_{2}-p)]D(k_{2}+p)K_{1}}\\{=}&{\displaystyle\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}D_{m}(k_{1}+k_{2})D_{m}(k_{2})}\\{}&{\times[D(k_{2}-p)-D(k_{2}+p)]D(k_{1}+k_{2}-p)K_{1}.}\\\end{array}
\begin{array}{ccc}{A_{\bar{\mu}}=\left(\begin{array}{cc}{0}&{c_{\bar{\mu}}^{*}}\\{c_{\bar{\mu}}}&{0}\\\end{array}\right),}&{A_{p}=\left(\begin{array}{cc}{0}&{c^{*}}\\{c}&{0}\\\end{array}\right),}&{X_{p+1}=\left(\begin{array}{cc}{0}&{d^{*}}\\{d}&{0}\\\end{array}\right)}\\\end{array}
+\frac58F^{\lambda\nu}F^{\mu\rho}F_{\lambda\mu}F_{\nu\rho}+2F_{\mu\rho}~^{;\rho}F^{\mu\lambda}~_{;\lambda}-2(F_{\rho}~^{\nu}F^{\rho\mu})_{;\nu\mu}+(F_{\nu\rho}F^{\nu\rho})^{;\mu}~_{\mu}~~.
\intdU\Upsilon_{\bar{\tau}\sigma}(U,U^{\dagger})\Upsilon_{\bar{\tau}^{\prime}\sigma^{\prime}}(U,U^{\dagger})=\prod_{l}\intdU_{l}\Upsilon_{\overline{{\tau^{(l)}}}\sigma^{(l)}}(U_{l},U_{l}^{\dagger})\Upsilon_{\overline{{\tau^{(l)}}}^{\prime}{\sigma^{(l)}}^{\prime}}(U_{l},U_{l}^{\dagger}).
<0|e^{\kappab\tilde{b}}e^{\lambdaa\tilde{a}}e^{\mua\tilde{b}}e^{\nub\tilde{a}}e^{\rhoa^{\dagger}\tilde{a}^{\dagger}}e^{\sigmab^{\dagger}\tilde{b}^{\dagger}}|0>=\frac{1}{(1-\kappa\sigma)(1-\lambda\rho)-\mu\nu\rho\sigma}
\begin{array}{lll}{M_{1}\\equivmax(k_{1},k_{2}),}&{m_{1}\\equivmin(k_{1},k_{2}),}\\{M_{2}\\equivmax(k_{1}+k_{4},k_{3}),}&{m_{2}\\equivmin(k_{1}+k_{4},k_{3}),}\\{M_{3}\\equivmax(k_{2}+k_{4},k_{3}+k_{5}),\qquad}&{m_{3}\\equivmin(k_{2}+k_{4},k_{3}+k_{5}).}\\\end{array}
\bar{Z}_{\;\;j_{1}\ldotsj_{p-k}}^{i_{1}\ldotsi_{p-k+1}}=\frac1{\left(p-k\right)!}\partial_{}^{\left[i_{1}\right.}\delta_{\;\;j_{1}}^{i_{2}}\ldots\delta_{\;\;j_{p-k}}^{\left.i_{p-k+1}\right]},\;k=1,\ldots,p.