harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\phi}=\left(\psi^{{}^{\parallel}}\;,\;P_{k}+i\psi_{k}\right)\;,\;\;T_{0}=\epsilon_{jkl}\pi_{j}\psi^{k\perp}\psi^{l\perp}-\frac{i\alpha}{2}\zeta\omega\;,\;\;T_{k}=-\zeta\omega\epsilon_{kjl}\psi^{j\perp}\psi^{l\perp}+\frac{i\alpha}{2}\pi_{k}\;.
{\Gamma^{\prime}}_{\mathrm{eff}}^{{(2)}}=\int\frac{d^{4}\!p}{(2\pi)^{4}}\,\phi(p)\phi(-p)\left(p^{2}-\frac{\hbarg^{2}}{96\pi^{2}}\left(\frac{1}{\tilde{p}^{2}}-\frac{1}{\tilde{p}^{2}+\frac{1}{\Lambda^{2}}}\right)+O(g^{4})\right)
B=-\frac{1}{8}\bar{D}(1-\gamma_{5})D(W_{a}^{*})^{2}=(D^{a})^{2}-\frac{1}{2}(F_{\mu\nu}^{a})^{2}-\frac{i}{2}F_{\mu\nu}^{a}\tilde{F}_{\mu\nu}^{a}+O(\theta)
{\calL}_{B}=-\frac{1}{4l^{2}}\partial_{\mu}l\partial^{\mu}l+\frac{1}{4l^{2}}h_{\mu}h^{\mu}-G_{i\bar{\jmath}}\partial_{\mu}{z^{i}}\partial^{\mu}{\bar{z}^{\bar{\jmath}}}-\frac{i}{2}(G_{j}^{(1)}\partial_{\mu}{z^{j}}-G_{\bar{\jmath}}^{(1)}\partial_{\mu}{\bar{z}^{\bar{\jmath}}})h^{\mu}.
{\mathrm{\Large~v~}}_{1}^{(D)}=\frac{1}{2VT}\int_{1/\Lambda^{2}}^{\infty}\!d\tau\\tau^{-1}{e}^{-\taum^{2}}\!\mathrm{tr}\!\int\!d^{D}xK(x,x;\tau)={\frac{4}{(4\pi)^{D/2}}}\int_{1/\Lambda^{2}}^{\infty}\!d\tau\,\tau^{-D/2-1}{e}^{-\taum^{2}}G_{D}(\tauF)\,
M=\left(\begin{array}{ccccc}{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{\partialX^{-}}&{\partialX^{2}}&{-\partialX^{3}}\\{0}&{-\partialX^{-}}&{0}&{\partialX^{3}}&{\partialX^{2}}\\{0}&{-\partialX^{2}}&{-\partialX^{3}}&{0}&{\partialX^{+}}\\{0}&{\partialX^{3}}&{-\partialX^{2}}&{-\partialX^{+}}&{0}\\\end{array}\right).
h=\intd^{2}x_{\bot}\;\left[-\frac{1}{2L}\frac{1}{J(a_{-}({x}_{\bot}))}\frac{\delta}{\deltaa_{-}({x}_{\bot})}J(a_{-}({x}_{\bot}))\frac{\delta}{\deltaa_{-}({x}_{\bot})}+\frac{L}{2\eta^{2}}(\nabla_{\bot}a_{-}({x}_{\bot}))^{2}\right]\;,
\langle\Psi^{(p)}|\Xi^{(q)}\rangle={\frac{1}{2^{p}\pi^{D}}}\int_{{\bfC}^{D}}\langle\Psi^{(p)}|\bar{z}^{i},d\bar{z}^{j}\rangle*\langle\bar{z}^{k},d\bar{z}^{l}|\Xi^{(q)}\rangle.
[a_{k}^{(\sigma)},a_{k^{\prime}}^{(\sigma^{\prime})}]=0\,,\quad[a_{k}^{(\sigma)},a_{k^{\prime}}^{(\sigma^{\prime})\,\dagger}]=\delta_{\sigma\sigma^{\prime}}\delta_{kk^{\prime}}\,,\quad\,\sigma\,,\sigma^{\prime}=\pm\,,\,\,\forall\,k\,,\,k^{\prime}\,,
z_{1}~=~-\,\frac{\Gamma(2\mu-1)}{4\pi^{2\mu}\Gamma(\mu-1)\Gamma(1-\mu)}{}~,~~y_{1}~=~-\,\frac{(2\mu-3+b)\Gamma(2\mu)}{16\pi^{2\mu}\Gamma(2-\mu)\Gamma(\mu)}
H=(a,a,1/2,-x)={\frac{\pi^{{\frac{-1}{2}}}}{2}}\Gamma(a+{\frac{1}{2}})\Gamma(1-a)(1+x)^{-a}\left(P_{2a-1}[{\frac{x^{\frac{1}{2}}}{\sqrt{(1+x)}}}]+P_{2a-1}[{\frac{-x^{\frac{1}{2}}}{\sqrt{(1+x)}}}]\right)
{\frac{1}{2}}\mathrm{e}^{\alpha\varphi(r)}[R(r)E_{i}(r)E_{j}(r)-R^{-1}(r)P_{i}P_{j}]+\chiR(r)\rho^{ij}(r)={\frac{1}{2}}{\frac{\mathrm{d}}{\mathrm{d}r}}[\lambda(r)R(r){\frac{\mathrm{d}\rho^{ij}(r)}{\mathrm{d}r}}]\\,
\left[x,p\right]=i,\quad\left[a_{0},a_{0}^{\dagger}\right]=1,\quad\left[a_{n}^{I},{a_{m}^{I}}^{\dagger}\right]=\delta_{nm},\quad\left[a_{n},{a_{m}}^{\dagger}\right]=\delta_{nm},\quad\left[\tilde{a}_{n},{\tilde{a}_{m}}^{\dagger}\right]=\delta_{nm}
S^{\mu}=R^{\mu}+\frac{1}{3}\left(1-\frac{3T(G^{\prime})}{2N_{f}T(R^{\prime})}-\gamma_{q}\right)(K_{i}^{\mu}+\widetilde{K}_{i}^{\mu}-2K_{M}^{\mu})-\frac{1}{3}(2\gamma_{q}+\gamma_{M})K_{M}^{\mu},
\beta_{\lambda}=\left(\frac{\omega_{\lambda}^{\mathrm{out}}}{\omega_{\lambda}^{\mathrm{in}}}\right)^{1/2}\frac{\Gamma(1-\imath\omega_{\lambda}^{\mathrm{in}}\eta_{0})\Gamma(\imath\omega_{\lambda}^{\mathrm{out}}\eta_{0})}{\Gamma(\imath\omega_{\lambda}^{-}\eta_{0})\Gamma(1+\imath\omega_{\lambda}^{-}\eta_{0})}.
\sum_{M_{1}M_{2}}(J_{1},J_{2})_{M_{1}\,M_{2}}^{P_{2}\,P_{1}}\bigl((J_{1},J_{2})_{-M_{1}\,-M_{2}}^{-N_{2}\,-N_{1}}\bigr)^{*}=\delta_{P_{1},N_{1}}\,\delta_{P_{2},N_{2}}
\left.\Gamma^{(\infty)}(p_{1j}^{2};m_{0},L_{1},L_{2},\lambda_{1B},\lambda_{2B},\epsilon)\right|_{s.p.}=\left.\Gamma^{(0)}(p_{1j}^{2};m_{0},\lambda_{1B}^{\prime},\epsilon)\right|_{s.p.}=g\kappa^{2\epsilon},
\left[\begin{array}{l}{n}\\{j}\\\end{array}\right]=\frac{\left(1-q^{n}\right)\left(1-q^{n-1}\right)\cdot\cdot\cdot\left(1-q^{n-j+1}\right)}{\left(1-q\right)\left(1-q^{2}\right)\cdot\cdot\cdot\left(1-q^{n}\right)},
\chi(q)=\sum_{\vec{k}=0}^{\infty}\frac{q^{\frac{b}{2}({k_{1}}^{2}+\ldots+k_{N}^{2})+\vec{k}\cdot\vec{B}}}{(q^{b})_{k_{1}}\ldots(q^{b})_{k_{N+N^{\prime}}}}\quad.
\overline{{\varphi}}_{nm}\left(x_{j},\sigma\right)=\,\overline{{\varphi}}_{nm}\left(x_{j-1},\sigma\right)+\frac{x}{2}\left(\,\overline{{\phi}}_{nm}\left(x_{j-1},\sigma\right)+\,\overline{{\phi}}_{nm}\left(x_{j},\sigma\right)\right)\,,
{\mathbbC}\,{\bfB}_{\pm}^{(1-j,j)}\,{\mathbbC}=\mp\,(-1)^{j}\{\bfB}_{\pm}^{(1-j,j)}\,,\qquad\quad{\mathbbU}_{\pi}\,{\bfB}_{\pm}^{(1-j,j)}\,{\mathbbU}_{\pi}=\pm\,(-1)^{j}\{\bfB}_{\mp}^{(j,1-j)}\.
g^{(4)}\simeq\Lambda^{2}\,(\zeta^{2}\,dT^{2}+d\zeta^{2}+d\rho^{2})+\Lambda^{-2}\,\rho^{2}\,d\phi^{2},\qquad\Lambda\simeq1+\frac{B^{2}\rho^{2}}{4},
2V^{-1}<H>=\frac{1}{2}\int\frac{d^{2}p}{(2\pi)^{2}}\left[G^{-1}(p)+\kappa^{2}G(p)+(p^{2}+4\kappa^{2})[G^{-1}(p)+\kappa^{2}G(p)]^{-1}\right]
\sum_{\vec{k}}\left\{\delta\Pi\\dot{\varphi}(\vec{k},\,t)+\delta\dot{\varphi}\\Pi(\vec{k},\,t)-\left[\delta\varphi\{\frac{\partialH}{\partial\varphi(\vec{k},\,t)}}+\delta\Pi\{\frac{\partialH}{\partial\Pi(\vec{k},\,t)}}\right]\right\}=\dot{\Omega}\.
2{^{(k)}}\Gamma_{oj}^{i}dx^{j}\:+\:Q_{o\omega}^{i}dx^{\omega}\:=\:Q_{j\omega}^{i}dx^{\omega}\:=\:2{^{(k)}}\Gamma_{\muj}^{o}dx^{j}\:+\:Q_{\mu\omega}^{o}dx^{\omega}\:=\:0\:,
W_{\lambda+\frac{d-1}{2}}(z_{1},z_{2})=W_{\nu}(z_{1},z_{2})=w_{\nu}(\zeta)=\frac{e^{-i\pi\frac{d-2}2}}{(2\pi)^{\frac{d}2}}(\zeta^{2}-1)^{-\frac{d-2}4}Q_{\nu-\frac12}^{\frac{d-2}2}(\zeta),
\begin{array}{ll}{H=}&{\sum_{i=2}^{M}(\mathrm{\cal~A}c_{i}^{+}c_{i-1}+1/2B_{i}c_{i}^{+}c_{i}+\gammac_{i}^{+}c_{i-1}^{+})+1/2B_{1}c_{1}^{+}c_{1}}\\{}&{-\mathrm{\cal~A}(-1)^{\hat{N}}c_{1}^{+}c_{M}-\gamma(-1)^{\hat{N}}c_{1}^{+}c_{M}^{+}+H.c.}\\\end{array}
T^{+2}=-2ie_{q}^{+}e_{q}^{+}-2i\pi_{\dot{q}}^{2+}\pi_{\dot{q}}^{2+},\quadT^{-2}=-2i\pi_{\dot{q}}^{1-}\pi_{\dot{q}}^{1-}-2ie_{q}^{-}e_{q}^{-}.
\begin{array}{ccc}{c}&{}&{\mu_{\mathrm{phys}}^{2}}\\{0}&{}&{1}\\{0.05}&{}&{0.95}\\{0.10}&{}&{0.8}\\{0.15}&{}&{0.65}\\{0.20}&{}&{0.5}\\{0.25}&{}&{0.35}\\{0.30}&{}&{0.2}\\{0.35}&{}&{0.1}\\{0.40}&{}&{0.0}\\\end{array}
\renewcommand{\arraystretch}{1.5}\begin{array}{rcl}{f_{1}^{(0)}}&{=}&{{}_{2}F_{1}(\lambda_{3},1-\lambda_{1};\lambda_{3}+\lambda_{4};x)\,,}\\{*f_{2}^{(0)}}&{=}&{x^{1-\lambda_{3}-\lambda_{4}}(1-x)^{1-\lambda_{2}-\lambda_{3}}{}_{2}F_{1}(\lambda_{1},1-\lambda_{3};\lambda_{1}+\lambda_{2};x)\,.}\\\end{array}
H_{0}=2\pi\left\{\frac{1}{4r}(f_{A}^{\prime2}+f_{B}^{\prime2})+\frac{\eta^{2}}{2}\left[r(H^{\prime2}+K^{\prime2})+\frac{1}{r}(Kf_{A}-Hf_{B})^{2}\right]+\frac{\lambda\eta^{4}}{4}r(1-H^{2}-K^{2})^{2}\right\}
\vec{X}_{,\sigma^{1}}^{rest}=\vec{e}_{1}+\frac{\sigma^{1}}{R_{1}^{rest}}\vec{m},\;\;\;\vec{X}_{,\sigma^{2}}^{rest}=\vec{e}_{2}+\frac{\sigma^{2}}{R_{2}^{rest}}\vec{m}.\nonumber
T_{2/1}^{\sigma}(x,y)=:\!u^{\rho}(x)h^{\mu\nu}(y)\!:\,t_{uh}^{\sigma}(x,y)_{\rho\mu\nu}+:\!u^{\sigma}(x)h^{\mu\nu}(y)\!:\,t_{uh}(x,y)_{\mu\nu}\,.
\hat{\epsilon}_{+}(w)=-4p_{0}I(\hat{\rho}_{+}^{(0)}(w)+\hat{\rho}_{+}^{(1)}(w)+\dots)+\frac{\tilde{z}h}{i(w-i\epsilon)}\frac{\hat{J}_{-}(0)}{\hat{J}_{+}(w)}+\sum_{j}\frac{\tilde{b}_{j}\tilde{z}H\hat{J}_{-}(0)\hat{J}_{-}(iw_{j})}{iw_{j}\hat{J}_{+}(w)i(w-iw_{j})}e^{-2Bw_{j}}+\dots
b_{i}({\bf{q}})=\sum_{{\bf{k}}}[b_{i}({\bf{q}}),a_{{\bf{k}}}^{\dagger}({\bf{q}})]a_{{\bf{k}}}({\bf{q}})-\sum_{{\bf{k}}}[b_{i}({\bf{q}}),a_{{\bf{k}}}(-{\bf{q}})]a_{{\bf{k}}}^{\dagger}(-{\bf{q}})
K_{F}(x,x^{\prime})=\sum_{\vec{p},s}\frac{m}{EV}\{u^{(s)}(\vec{p}){\bar{u}}^{(s)}(\vec{p})e^{ip\cdot(x-x^{\prime})}\theta(t-t^{\prime})-v^{(s)}(\vec{p}){\bar{v}}^{(s)}(\vec{p})e^{-ip\cdot(x-x^{\prime})}\theta(t^{\prime}-t)\}
{\frac{1}{4\pi}}\mathrm{Im}\left\{{\frac{\partial{\calF}(A)}{\partialA}}\bar{A}\Bigg|_{D}\+\{\frac{1}{2}}{\frac{\partial^{2}{\calF}(A)}{\partialA^{2}}}W_{\alpha}W^{\alpha}\Bigg|_{F}\right\}\,
\frac{d}{dt}\left(\frac{8\pi^{2}}{g_{h}^{2}}\right)=\beta\left(\frac{8\pi^{2}}{g_{h}^{2}}\right);\qquad\beta\left(\frac{8\pi^{2}}{g_{h}^{2}}+2\pii\right)=\beta\left(\frac{8\pi^{2}}{g_{h}^{2}}\right)
S_{eff}=\int\!d^{4}x~[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}(A_{\mu}+\partial_{\mu}\rho)^{2}+A^{\mu}\partial_{\mu}B-\frac{1}{2}\alpha(B)^{2}-\partial_{\mu}{\overline{{\calC}}}\partial^{\mu}{\calC}~],
L=\frac{1}{2e}\left(\dot{x}_{\mu}-\frac{i}{2}v\epsilon_{\mu\nu\lambda}\xi_{a}^{\nu}\xi_{a}^{\lambda}\right)^{2}-\frac{e}{2}m^{2}-i\numv\xi_{1}^{\mu}\xi_{2\mu}-\frac{i}{2}{\xi}_{a\mu}\dot{\xi}{}_{a}^{\mu},
\frac{\delta}{\deltas_{\mu\nu}(t)}exp\left[-{\frac{1}{4}}\int_{0}^{1}d\tau\frac{\dot{x}^{2}(\tau)}{e(\tau)}\right]={\frac{1}{2}}\omega_{\mu\nu}(x(t))\left[-{\frac{1}{4}}\int_{0}^{1}d\tau\frac{\dot{x}^{2}(\tau)}{e(\tau)}\right]
\Gamma_{2}^{(1)}(\theta=0)=-\frac{i}{4\pi}\intd^{3}x\varepsilon^{\mu\nu\lambda}[(\kappa+N\mathrm{sgn}(\kappa))\mathrm{Tr}(A_{\mu}\partial_{\nu}A_{\lambda})-\mathrm{sgn}(\kappa)(\mathrm{Tr}A_{\mu})\partial_{\nu}(\mathrm{Tr}A_{\lambda})].
a\equiv\left(\begin{matrix}{w_{\dot{\alpha}}^{~ui}}\\{{a^{\prime}}_{\alpha\dot{\beta}~li}}\\\end{matrix}\right)~~~,~~~{\calM}^{A}\equiv\left(\begin{matrix}{\mu^{Aui}}\\{{M^{\prime}}_{~~li}^{\betaA}}\\\end{matrix}\right)~~,
\begin{array}{rcl}{\tilde{v}_{1}}&{=}&{{\frac{V_{11}}{B_{1}}}{\frac{cos^{4}\alpha}{cos2\alpha}}-{\frac{V_{22}}{B_{2}}}{\frac{sin^{4}\alpha}{cos2\alpha}}}\\{\tilde{v}_{2}}&{=}&{-{\frac{V_{11}}{B_{1}}}{\frac{sin^{4}\alpha}{cos2\alpha}}+{\frac{V_{22}}{B_{2}}}{\frac{cos^{4}\alpha}{cos2\alpha}}}\\\end{array}
L\left(m^{2}(\beta),\beta\right)=-\frac{3}{2}\frac{1}{(2\pi)^{D/2}}\sum_{n=1}^{\infty}\biggl(\frac{m(\beta)}{\betan}\biggr)^{\frac{D}{2}-2}K_{\frac{D}{2}-2}\left(m(\beta)n\beta\right),
S_{\mathrm{\smallbrane}}^{\mathrm{{\smallqu}}}=-\int{\frac{d^{4}p}{(2\pi)^{4}}}\left({\widetildea}(p^{2})~\mathrm{Tr}\left[{\widehatF}^{\mu\nu}(p)~{\widehatF}_{\mu\nu}(-p)\right]+a~\mathrm{Tr}\left[f^{\mu\nu}(p)~f_{\mu\nu}(-p)\right]\right)~,
r\,=\,\frac{e^{-2ikL}}{\Omega}\frac{(m+E+k)B(2\nu,-\mu-\nu-\lambda)}{(m+E-k)B(-2\nu,1-\mu+\nu+\lambda)}\times\left[1-e^{4ipL}\frac{B(2\mu,-\mu-\nu+\lambda)B(2\mu,-\mu+\nu+\lambda)}{B(-2\mu,\mu+\nu-\lambda)B(-2\mu,\mu-\nu-\lambda)}\right]
{\vec{x}}^{\prime}=\vec{x}+\frac{\vec{V}}{c}\left[\frac{\vec{V}\vec{x}}{c\left(\gamma+\sqrt{\gamma^{2}+\left(\displaystyle\frac{\vec{V}}{c}\right)^{2}}\right)}-\frac{\vec{\sigma}\vec{x}}{c\gamma_{0}}-x^{0}\right]\gamma^{-1},
S=-\frac{1}{4}\,\intd^{4}x\,\left[(1-\frac{1}{2}\,\theta^{\alpha\beta}F_{\alpha\beta})\,F_{\mu\nu}F^{\mu\nu}+2\mathrm{Tr}\,(\thetaF^{3})-\partial_{\beta}\left(\theta^{\alpha\beta}A_{\alpha}F_{\mu\nu}F^{\mu\nu}\right)\right]
L_{n,a}(x)=\left(\begin{array}{cc}{xq^{S_{n}^{3}}-x^{-1}q^{-S_{n}^{3}}}&{(q-q^{-1})S_{n}^{-}}\\{(q-q^{-1})S_{n}^{+}}&{xq^{-S_{n}^{3}}-x^{-1}q^{S_{n}^{3}}}\\\end{array}\right)\quad,
T_{\alpha\beta}=-\left(\nabla_{\alpha}\phi\nabla_{\beta}\phi-\frac12((\nabla\phi)^{2}+m^{2}+\xi\tilde{R}~\phi^{2})\gamma_{\alpha\beta}+\xi(\gamma_{\alpha\beta}\nabla^{2}-\nabla_{\alpha}\nabla_{\beta})\phi^{2}\right)~,
I[X({\xi})]={\frac{1}{2}}\int_{X({\xi})}dx^{\mu}\wedgedx^{\nu}\Pi_{\mu\nu}(x)-\int_{D}d^{2}{\xi}\,N({\xi})\left[{\frac{1}{2}}\Pi_{\mu\nu}\Pi^{\mu\nu}+m^{4}\right]\,
S={\frac{1}{2\pi}}\intd^{2}x\,\sqrt{-g}\left\{e^{-2\Phi}\left(R+4{\left(\nabla\Phi\right)}^{2}+4\lambda^{2}\right)-{\frac{1}{2}}\sum_{i=1}^{N}{\left(\nablaf_{i}\right)}^{2}\right\}\;,
\Sigma_{A}(\bar{p}_{\parallel})~\simeq~e^{2}(4|eH|)\int\frac{d^{4}\hat{q}}{(2\pi)^{4}}\frac{\mathrm{e}^{-\hat{q}_{\perp}^{2}}}{\hat{q}^{2}}\frac{\Sigma_{A}(\bar{p}_{\parallel}-q_{\parallel})}{(\bar{p}_{\parallel}-q_{\parallel})^{2}+\Sigma_{A}^{2}(\bar{p}_{\parallel}-q_{\parallel})},
\int\left[D\hat{h}\,D\hat{\varphi}\,D\hat{A}\,D\hat{\eta}\right]\hat{\Psi}(x)\,\mathrm{e}^{\mathrm{i}S_{0}}\,\mathrm{e}^{\mathrm{i}S_{I}}\,\mathrm{e}^{\mathrm{i}S_{\mathrm{boundary}}}\equiv\left<\hat{\Psi}(x)\,\mathrm{e}^{\mathrm{i}S_{I}}\,\mathrm{e}^{\mathrm{i}S_{\mathrm{boundary}}}\right>~~.
\omega^{\mu,\nu}=\frac{1}{4\pi}\epsilon^{\mu\nu\lambda\rho}(\partial_{\lambda}\partial_{\rho}-\partial_{\rho}\partial_{\lambda})\chi(x)=n\intd\taud\sigma\frac{\partial(y^{\mu},y^{\nu})}{\partial(\tau,\sigma)}\delta^{(4)}(x-y(\tau,\sigma)).
e^{-S_{\mathrm{eff}}}=e^{-S_{0}}\beta^{\mathrm{N}}\int\frac{da_{0}^{\alpha}}{2\pi}\prod_{\alpha<\beta}\prod_{n=-\infty}^{\infty}\left(\frac{\left(\frac{2\pin}{\beta}+\frac{\pi}{\beta}+a_{0}^{\alpha}-a_{0}^{\beta}\right)^{2}+\vert\vec{a}_{\alpha}-\vec{a}_{\beta}\vert^{2}}{\left(\frac{2\pin}{\beta}+a_{0}^{\alpha}-a_{0}^{\beta}\right)^{2}+\vert\vec{a}_{\alpha}-\vec{a}_{\beta}\vert^{2}}\right)^{8}
S=\intd^{2}x\sqrt{-{{h}}}e^{-{\gamma}}\left[{R}_{2}+\frac{1}{2}\left({\nabla}{\gamma}\right)^{2}+\frac{1}{4}\mathrm{Tr}\left({\nabla}L{\nabla}L^{-1}\right)+\frac{1}{4}\mathrm{Tr}\left({\nabla}N{\nabla}N^{-1}\right)\right]
\lbrackM_{(m)}^{\alpha,\beta},M_{(n)}^{\alpha^{\prime},\beta^{\prime}}]\equivM_{(m)}^{\alpha,\beta}M_{(n)}^{\alpha^{\prime},\beta^{\prime}}-(-)^{\alpha\beta^{\prime}+\beta\alpha^{\prime}}e^{2i(\beta^{\prime}m-\betan)x}M_{(n)}^{\alpha^{\prime},\beta^{\prime}}M_{(m)}^{\alpha,\beta}
\left(\begin{matrix}{X^{\Lambda}}\\{-{\frac{i}{2}}F_{\Lambda}}\\\end{matrix}\right)\Longrightarrow\left(\begin{matrix}{X^{0}}\\{-iX^{1}}\\\end{matrix}\right),~~\left(\begin{matrix}{X^{1}}\\{-iX^{0}}\\\end{matrix}\right).
S^{(2)}\sim3\intd^{26}xTA_{\mu}A^{\mu}+2\intd^{26}x\left[\partial_{\mu}\partial_{\nu}TA^{\mu}A^{\nu}-\partial_{\mu}T\partial_{\nu}A^{\mu}A^{\nu}+T\partial_{\nu}A^{\mu}\partial_{\mu}A^{\nu}\right]
<\ldots>_{2}={\frac{1}{2!}}{\frac{1}{2^{2}}}F_{xxyy}+{\frac{1}{2}}F_{xxy}F_{y}+{\frac{1}{4}}F_{xy}F_{xy}+{\frac{1}{2}}F_{x}F_{xy}F_{y}-{\frac{1}{2}}{\frac{\partialF}{\partialt_{0}}}.
\begin{array}{ll}{Y_{1}=X_{2}+\sqrt{-1}\X_{1},\quad}&{Y_{2}=-X_{2}+\sqrt{-1}\X_{1},}\\{Y_{3}=X_{4}+\sqrt{-1}\X_{3},\quad}&{Y_{4}=-X_{4}+\sqrt{-1}\X_{3}.}\\\end{array}
\left((D^{2})^{ab}+g\epsilon^{adc}A^{d}D^{cb}\right)f_{b}=j_{\mathrm{tot},0}^{b};~~~~j_{\mathrm{tot},0}^{b}=g\epsilon^{abc}A_{i}^{a}E_{i}^{c}+j_{0}^{b}~,
Q=a_{J^{\prime}}^{I^{\prime}}(z^{\prime\prime},z^{\prime})z^{J^{\prime}}\frac{\partial}{\partialz^{I^{\prime}}}+b_{J^{\prime}}^{I^{\prime\prime}}(z^{\prime\prime},z^{\prime})z^{J^{\prime}}\frac{\partial}{\partialz^{I^{\prime\prime}}}\nonumber
k_{\Lambda}^{u}=i{\nu}_{e\Lambda}^{(1)}\left(\frac{\partial}{\partialC_{0}}-\frac{\partial}{\partial\bar{C}_{0}}\right)^{u}+i\nu_{e\Lambda}^{(2)}\left(\frac{\partial}{\partialS}-\frac{\partial}{\partial\bar{S}}\right)^{u}.
\psi(x,t)\,=\,\left(\begin{array}{c}{\psi_{1}(x,t)}\\{\psi_{2}(x,t)}\\{\psi_{3}(x,t)}\\\end{array}\right)\,=\,\left(\begin{array}{c}{f_{1}(t)\phi_{1}(x)}\\{f_{2}(t)\phi_{2}(x)}\\{f_{3}(t)\phi_{3}(x)}\\\end{array}\right)
\mathcal{S}=\int{d^{4}x\sqrt{-g}\left(\left[\left\{1+\frac{1}{2}\phi\right\}g^{\mu\nu}-\phi^{\mu\nu}\right]\mathcal{R}_{\mu\nu}+\frac{m^{2}}{4}\left[\phi_{\mu\nu}\phi^{\mu\nu}-\phi^{2}\right]\right)}.
h_{n,m}^{3n,2m}-\frac{c({3n,2m})}{24}+h_{n,m}^{4n,5m}-\frac{c({4n,5m})}{24}=h_{2n,2m}^{6n,5m}-\frac{c({6n,5m})}{24}+h_{n,m}^{3n,4m}-\frac{c({3n,4m})}{24}\,.
\begin{array}{rcl}{{\Psi}}&{=}&{\mathrm{\bf{1}}\psi+\mathrm{\bf{i}}\hat{\psi}\quad\in\Lambda_{-},}\\{\overline{{\Psi}}}&{=}&{\mathrm{\bf{j}}\hat{\overline{{\psi}}}+\mathrm{\bf{k}}\overline{{\psi}}\quad\in\Lambda_{+},}\\\end{array}
{\tau}_{2}^{e}({\pi}(f_{0}),{\pi}(f_{1}),{\pi}(f_{2})){\equiv}Tr{\sigma}_{i}{\sigma}_{j}{\sigma}_{k}.{\tau}_{2}({\pi}(f_{0}^{i}),{\pi}(f_{1}^{j}),{\pi}(f_{2}^{k})),
\begin{array}{cccc}{E_{\pm1}=\gamma_{8127}P_{\pm}\,,~}&{E_{\pm2}=\gamma_{8163}P_{\pm}\,,~}&{E_{\pm3}=\gamma_{8246}P_{\pm}\,,~}&{E_{\pm4}=\gamma_{8347}P_{\pm}\,,}\\{{}}&{{}}&{{}}&{{}}\\{E_{\pm5}=\gamma_{8567}P_{\pm}\,,~}&{E_{\pm6}=\gamma_{8253}P_{\pm}\,,~}&{E_{\pm7}=\gamma_{8154}P_{\pm}\,.}&{{}}\\\end{array}
{\mathrm{Ind}}D=\frac{1}{8\pi^{2}}\int[R^{ab}\mathrm{\tiny~\wedge~}R_{ab}+2(T^{a}\mathrm{\tiny~\wedge~}T_{a}-R_{ab}\mathrm{\tiny~\wedge~}e^{a}\mathrm{\tiny~\wedge~}e^{b})],
\{f,g\}=i{\chi}(x){\theta}^{{\mu}{\nu}}\frac{\partial{f}}{\partial{x^{\mu}}}\frac{\partial{g}}{\partial{x^{\nu}}}-\frac{ia}{2}{\chi}^{2}(x){\theta}^{{\alpha}{\beta}}{\eta}^{{\mu}{\nu}}\frac{{\partial}^{2}{f}}{\partial{x^{\mu}}\partial{x^{\alpha}}}\frac{{\partial}^{2}{g}}{\partial{x^{\nu}}\partial{x^{\beta}}}.
{\bfh_{\varphi}}=\left(\begin{array}{cc}{e^{\varphi/4}}&{{\frac{1}{2}}\,\varphi_{w}e^{-\varphi/4}}\\{0}&{e^{-\varphi/4}}\\\end{array}\right),\qquad{\bf\bar{h}_{\varphi}}=\left(\begin{array}{cc}{e^{\varphi/4}}&{0}\\{-{\frac{1}{2}}\,\varphi_{\bar{w}}e^{-\varphi/4}}&{e^{-\varphi/4}}\\\end{array}\right).
e_{i}={\bf1}\otimes\cdots\otimes\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{q^{-1}}&{-1}&{0}\\{0}&{-1}&{q}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)\otimes\cdots\otimes{\bf1}\;\;\;\in{\bfC}^{\otimes2l}\;,
(e^{-2\phi})_{\mathrm{fix}}={\frac{|q_{0}q_{1}+p^{0}p^{1}|}{(q_{1})^{2}+(p^{0})^{2}}}\qquad(a)_{\mathrm{fix}}={\frac{p^{1}q_{1}-q_{0}p^{0}}{(q_{1})^{2}+(p^{0})^{2}}}\.
b_{\sigma_{0}}=\left({\frac{d\rho_{0}}{d\sigma_{0}}}\right)b_{\rho_{0}}+\left({\frac{d\rho_{0}^{*}}{d\sigma_{0}}}\right)b_{\rho_{0}^{*}}\,,\qquadb_{\rho_{0}}=\oint_{C_{0}}{\frac{d\rho}{2\pii}}b(\rho),\quadb_{\rho_{0}^{*}}=\oint_{C_{0}^{*}}{\frac{d\rho}{2\pii}}b(\rho)
S=\frac{1}{2\pi}\int_{\calM}d^{2}x\sqrt{-g}\left[e^{-2\phi}(R+4(\nabla\phi)^{2}+4\lambda^{2})-\frac12\sum_{i=1}^{N}(\nablaf_{i})^{2}-\frac{\kappa}{4}(R\frac{1}{\nabla^{2}}R+2\phiR)\right]
\Gamma_{\,j}\;=\;\left(\begin{array}{ccc}{0}&{0}&{1}\\{1}&{0}&{0}\\{0}&{1}&{0}\\\end{array}\right)_{\,j},\qquad\sigma_{\,j}\;=\;\left(\begin{array}{ccc}{1}&{0}&{0}\\{0}&{\omega}&{0}\\{0}&{0}&{\omega^{2}}\\\end{array}\right)_{\,j}
\ell=\intdt\Bigl(\partial_{t}a^{i}\frac{\partial}{\partiala^{i}}+\partial_{t}\bar{c}_{i}\frac{\partial}{\partial\bar{c}_{i}}+\partial_{t}c^{i}\tau_{i}+\partial_{t}q_{i}\pi^{i}+\ell_{h}\Bigr),
\Psi_{n,m}(x,y)=x^{p_{1}}y^{p_{2}}L_{n}^{(-1/2+p_{1})}(\omegax^{2})L_{m}^{(-1/2+p_{2})}(\omegay^{2})e^{-\frac{\omegax^{2}}{2}-\frac{\omegay^{2}}{2}}\,
A^{i}(\vec{x},t)=\frac{1}{2\pi\kappa}\intd^{2}y\,\epsilon^{ij}{\frac{(x^{j}-y^{j})}{|\vec{x}-\vec{y}|^{2}}}\,\rho(\vec{y},t)=\frac{e}{2\pi\kappa}\sum_{a=1}^{N}\epsilon^{ij}{\frac{(x^{j}-x_{a}^{j}(t))}{|\vec{x}-\vec{x}_{a}(t)|^{2}}}
\epsilon_{k}(\eta)\sim\frac{k^{2}r^{2}}{1+r^{2}}{\calG}(\eta)\int^{\eta}{\calG}(\tau)\overline{{C}}(\tau)\biggl(\frac{b^{2}-a^{2}}{b^{2}}\biggr)\int^{\tau}\frac{d\eta^{\prime\prime}}{{\calG}(\eta^{\prime\prime})^{2}}
W_{2}(k_{1})=\frac{1}{2\piR_{c}}\left[\int_{0}^{\piR_{c}}dx\,<\frac{1}{N}\mathrm{Tr}\,\left[\,\Omega^{(k_{1})}(x)\,\right]\frac{1}{N}\mathrm{Tr}\,\left[\,\Omega^{(-k_{1})}(0)\,\right]>+\,k_{1}\to-k_{1}\right].
\begin{aligned}{H_{t}^{t}}&{{}=-\partial_{r}\sqrt{f(r)}~,}&{H_{\beta}^{\alpha}}&{{}=-\frac1r\sqrt{f(r)}\delta_{\beta}^{\alpha}~,}\\{R_{tt}}&{{}=R_{t\alpha}=H_{t\alpha}=0~,}&{R_{\beta}^{\alpha}}&{{}=\frac{n-1}r\delta_{\beta}^{\alpha}~,}\\\end{aligned}
\begin{array}{rclcrcl}{\psi_{-}}&{\rightarrow}&{e^{-i\theta}g^{-1}\psi_{-}}&{\quad\quad}&{\psi_{+}}&{\rightarrow}&{\psi_{+}\nonumber}\\{\psi_{-}^{\dagger}}&{\rightarrow}&{\psi_{-}^{\dagger}ge^{i\theta}}&{\quad\quad}&{\psi_{+}^{\dagger}}&{\rightarrow}&{\psi_{+}^{\dagger}}\\\end{array}\quad,
\hat{H}_{\mathrm{eff}}=\frac{1}{2}\hat{p}_{r}^{2}-\frac{\hbar^{2}}{8\hat{r}^{2}}+\frac{1}{2\hat{r}^{2}}\hat{p}_{\theta}^{2}+\frac{1}{2}\hat{p}_{z}^{2}+\frac{1}{2}\omega^{2}\hat{r}^{2}+\hat{\xi}\Big[\hat{p}_{z}+g\hat{p}_{\theta}\Big]+F(\hat{\xi})\hat{p}_{\xi}-i\hbarF^{\prime}(\hat{\xi})c^{1}b_{1}-i\hbarc^{1}b_{2}\\.
\hat{H}_{0}=\hbarg\omega\left[{\alpha_{i}^{a}}^{\dagger}\alpha_{i}^{a}+\frac{1}{2}2d\right]=\hbarg\omega\left[\hat{N}+d\right]\\,\\\hat{\phi}=i\hbar\epsilon^{ab}{\alpha_{i}^{a}}^{\dagger}\alpha_{i}^{b}\\.
{\calL}={\calL}[\varphi_{\mathrm{cl}},\alpha_{\mathrm{cl}}]-\frac{r}{4}\left(\vec{u}\cdot{\calM}_{\mathrm{B}}\cdot\vec{u}+\vec{{\rho}}_{\alpha}^{\,T}\cdot{\calM}_{\mathrm{F}}^{\alpha\beta}\cdot\vec{\rho}_{\beta}\right)
{\calH}={\frac{1}{2}}\vec{\pi}^{2}-\vec{\pi}.\nabla\phi-{\frac{1}{2}}c^{2}(\nabla\times\vec{u})^{2}+\nu\vec{\pi}.\nabla^{2}\vec{u}-\vec{u}.\vec{J}+\phin+\nu\vec{u}.\nablan,
R_{ij}\!\!-\!\frac{1}{2}Rg_{ij}\!=\!8\piGT_{ij}^{m}\!+\!({\calR}_{\mu\nu}\!-\!\frac{1}{2}{\calR}{\calG}_{\mu\nu})Z_{,i}^{\mu}Z_{,j}^{\nu}\!+\!Q_{ij}+S_{ij}
\frac{1}{\phi_{0}m^{2}}\frac{\partial^{2}\phi}{\partialt^{2}}-\frac{1}{\phi_{0}m^{2}}\frac{\partial^{2}\phi}{\partialx^{2}}+\frac{\eta_{a}}{\phi_{0}m^{2}}\frac{\partial\phi}{\partialt}=-\frac{\phi}{\phi_{0}}+\frac{\alpha\phi_{o}}{m^{2}}\frac{\phi^{2}}{\phi_{o}^{2}}-\frac{\lambda\phi_{0}^{2}}{m^{2}}\frac{\phi^{3}}{\phi_{0}^{3}}+\frac{\xi_{a}}{\phi_{0}m^{2}}.
S_{B}=\int_{r\to\infty}d^{4}xr^{5}\left[A_{>}\Phi^{0}\frac{\partial}{\partialr}\Phi^{0}+\sum_{I,J=1}^{\infty}A_{>}^{IJ}\Phi^{I}\frac{\partial}{\partialr}\Phi^{J}+\sum_{>}^{I}\frac{\partial}{\partialr}\left(\Phi^{0}\Phi^{I}\right)\right]
\tilde{S}=-\frac{1}{2}\int\!d\xi^{+}d\xi^{-}[{(E_{0}+\tilde{\Pi})^{-1}}^{\:\:ij}(\partial_{+}\tilde{g}\tilde{g}^{-1})_{i}(\partial_{-}\tilde{g}\tilde{g}^{-1})_{j}+\partial_{+}t\partial_{-}t].
\displaystyle\frac{\vartheta_{\varepsilon}^{(nl)}(x_{1},\cdots,x_{m-1},z_{N-1}\tau|\zeta_{1},\cdots,\zeta_{N-2},\zeta_{N-1},-\sigma\zeta_{N-1}\tau^{-1})}{\vartheta_{\sigma\varepsilon}^{(n-1,l-1)}(x_{1},\cdots,x_{m-1}|\zeta_{1},\cdots,\zeta_{N-2})}=\displaystylec^{(nl)}z_{N-1}^{N-m-1}\prod_{\mu=1}^{m-1}\theta(qx_{\mu}/z_{N-1}|q^{2}).
Q_{\mathrm{naive}}\equiv2L\stackrel{o}{\jmath}^{+}=\sum_{n\geN}a_{n+{\frac{1}{2}}}^{\dagger}a_{n+{\frac{1}{2}}}-\sum_{n<N}a_{n+{\frac{1}{2}}}a_{n+{\frac{1}{2}}}^{\dagger}+N-\hat{q}
\mathrm{E}_{1}\left(\eta_{1},\cdots,\eta_{p}\right)=\eta_{1}^{a_{1}}\cdots\eta_{p}^{a_{p}}\,\,h^{a}(\phi)\partial_{a}\,\,\partial_{a_{1}}^{(1)}\cdots\partial_{a_{p}}^{(p)}F_{p}(\phi,\cdots,\phi)
{\calL}_{GF}^{(2)}~=~\frac{\kappa}{2\pi}\epsilon^{mn}A_{n}{\dot{A}}_{m}+\pi_{a}{\dot{z}}_{a}+\pi_{a}^{*}\dot{z^{*}}_{a}+\Omega_{1}^{(0)}\dot{\alpha}+\Omega_{2}^{(1)}\dot{\beta}+\Omega_{3}^{(1)}\dot{\gamma}+\Omega_{4}^{(2)}\dot{\sigma}-{\calH}^{(2)}
\begin{array}{rl}{x^{i}(\tau,\sigma)=\sqrt{2\alpha^{\prime}}\left[\right.}&{q^{i}+\left(\alpha_{0}^{i}+\bar{\alpha}_{0}^{i}\right)\tau}\\{}&{}\\{}&{\left.+\frac{1}{2}i{\sum_{n}}^{\prime}\frac{1}{n}\left(\alpha_{n}^{i}e^{-2in(\tau-\sigma)}+\bar{\alpha}_{n}^{i}e^{-2in(\tau+\sigma)}\right)\right]\,}\\\end{array}
L_{g.f.}^{+1A}=e_{m}^{+1A}(\phi,\eta,\theta=0)dx^{m}+\theta^{+}e_{+}^{+}(\phi,\eta)f_{+{\frac{1}{2}}\;+{\frac{1}{2}}}^{\;\;+1A}e_{+}^{+}(\phi,\eta)d\theta^{+}\.
D_{\mu\nu}^{(\mathrm{static})}={\frac{1}{(\kappa+\Pi_{\mathrm{odd}})^{2}\,\vec{p}^{2}+\Pi_{1}\Pi_{2}}}\left[P_{\mu\nu}\,\Pi_{2}+Q_{\mu\nu}\,\Pi_{1}-(\kappa+\Pi_{\mathrm{odd}})\,\epsilon_{\mu\nu\lambda}p_{\lambda}\right].
\xi_{l}:={\frac{b_{l}}{a_{l}}}\sim\alpha{\frac{\nu-|l|+\beta-2\beta\,(A_{1}/B_{1})\;(\Phi(A_{1}+1,B_{1}+1;\beta)/\Phi(A_{1},B_{1};\beta))}{{\nu+|l|-\beta+2\beta\,(A_{1}/B_{1})\;(\Phi(A_{1}+1,B_{1}+1;\beta)/\Phi(A_{1},B_{1};\beta)})}}.
\frac{d}{dt}\biggl|_{t=0}Z(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle^{A(t)})=\biggl(\sum_{k=0}^{N}\frac{1}{2}(-1)^{k}Tr(B_{k})\biggr)Z(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)
\frac{1}{8\pi}\int_{r}{[K]}=-\frac{1}{\kappa}\Biggl(\frac{A(r)}{4}\Biggr)\Biggl[\frac{1}{2}\frac{\partial_{r}g_{tt}}{\sqrt{-g_{rr}g_{tt}}}+\frac{2}{r}\sqrt{\frac{g_{tt}}{-g_{rr}}}-\frac{2}{r}\sqrt{g_{tt}}\Biggr]
I({\calM})={\int}_{\calM}{\varepsilon}^{{\mu}_{1}{\mu}_{2}{\mu}_{3}}{\varepsilon}^{{\nu}_{1}{\nu}_{2}{\nu}_{3}}{\omega}_{{\mu}_{1}}^{(1)}(x){\omega}_{{\nu}_{1}}^{(1)}(y){\omega}_{{\mu}_{2}}^{(2)}(x){\omega}_{{\nu}_{2}}^{(2)}(y)G_{{\mu}_{3}{\nu}_{3}}^{(\chi)}(x,y)d^{3}xd^{3}y,
\psi_{\pm}(x,y)=\left(\phi_{1}(x,y),C_{\pm}\phi_{2}(x,y)\right),{C_{\pm}=-\frac{2\pi}{\beta\mu}\left(\alpha_{0}+n+\frac{1}{2}\right)\pm\sqrt{\frac{4\pi^{2}}{\beta^{2}\mu^{2}}\left(a+n+\frac{1}{2}\right)^{2}+1}}
dl^{2}=\frac{1}{2f^{2}}\left|dE+\frac{2\mathrm{Im}\Phi}{\mathrm{Im}z}d\Phi-\left(\frac{\mathrm{Im}\Phi}{\mathrm{Im}z}\right)^{2}dz\right|^{2}-\frac{1}{f\mathrm{Im}z}\left|d\Phi-\frac{\mathrm{Im}\Phi}{\mathrm{Im}z}dz\right|^{2}+\frac{|dz|^{2}}{2(\mathrm{Im}z)^{2}}.
{J}_{(\mu_{1}\dots\mu_{n};\mu_{n+1})}^{(n)}+\frac{\partial{J}_{(\mu_{1}\dots\mu_{n}}^{(n)}}{\partial\psi^{\sigma}}\Gamma_{\mu_{n+1})\lambda}^{\sigma}\psi^{\lambda}=\frac{i}{2}\psi^{\rho}\psi^{\sigma}R_{\rho\sigma\nu(\mu_{n+1}}{{J}^{(n+1)\nu}}_{\mu_{1}\dots\mu_{n})}
\psi^{0}(z)=\sum_{m\in{\bfZ}}d_{m}^{0}z^{-m-\frac{1}{2}}~,\quad\widetilde{\psi}^{0}(\overline{{z}})=\sum_{m\in{\bfZ}}d_{m}^{0}\overline{{z}}^{-m-\frac{1}{2}}~,