harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
4\Lambda^{4}\prod_{a=1}^{n_{c}-2}(x-\phi_{a})^{2}-4\Lambda^{4}\prod_{i=1}^{2n_{c}-4}(x+m_{i})=-4\Lambda^{2}\alpha_{n_{c}-1}^{4}\prod_{a=1}^{n_{c}-3}(x-\alpha_{a})^{2}.
Q\equiv\sum_{i}z_{i}=\sum_{i}\sqrt{z_{i}^{2}}=\sum_{i}\sqrt{1+(z_{i}^{2}-1)}=\sum_{i}\sum_{p=0,l\leqp}^{\infty}\left(\begin{array}{c}{p}\\{l}\\\end{array}\right)c_{p}(-1)^{p-l}z_{i}^{2l},
D=\int_{-Q_{1}}^{Q_{1}}{\rho_{1}(k)dk},\;\;\;D_{m}=\int_{-B_{1}}^{B_{1}}{\sigma_{1}(\Lambda)d\Lambda},\;\;\;E_{1}/N=D^{-1}\int_{-Q_{1}}^{Q_{1}}{k^{2}\rho_{1}(k)dk}.
\left\{p,F\right\}_{D}=-1,\;\;\;\left\{p^{i},n^{j}\right\}_{D}=-\delta_{ij}+n^{i}n^{j},\;\;\;\left\{p^{i},p^{j}\right\}_{D}=\,p^{i}n^{j}-p^{j}n^{i}.
A~=~\mathrm{diag}~\left(~a_{1}~\sigma_{2},~a_{2}~\sigma_{2},\ldots,~a_{\frac{N-1}{2}}~\sigma_{2},~0~\right)~,~\mathrm{with}~|v_{i}|^{2}~=~2~|a_{i}|^{2}~+~c~.
i\dot{a}_{m}(t)=-\sum_{n\neqm}a_{n}(t)e^{i[\gamma_{n}(t)-\gamma_{m}(t)]}e^{-ig\int_{t_{0}}^{t}dt^{\prime}\epsilon_{n}(t^{\prime})-\epsilon_{m}(t^{\prime})}<m;t|i\frac{\partial}{\partialt}|n;t>
\Gamma_{5}\Gamma\Theta\frac{1}{-\Theta^{2}+M^{2}}=\left(\begin{array}{cc}{\mathbb{D}(\mathbb{D}^{\dagger}\mathbb{D}+M^{2})^{-1}}&{0}\\{0}&{\mathbb{D}^{\dagger}(\mathbb{D}\mathbb{D}^{\dagger}+M^{2})^{-1}}\\\end{array}\right),
{\calL}_{class}(\phi_{1},\phi_{2})=-\phi_{1}\Box\phi_{2}+m^{2}\phi_{1}\phi_{2}+{\frac{\eta}{3!}}\phi_{1}^{3}+{\frac{\lambda}{4}}\phi_{1}^{2}\phi_{2}^{2}+{\frac{\alpha}{4!}}\phi_{2}\phi_{1}^{4}+{\frac{\beta}{3!}}\phi_{1}^{2}\Box\phi_{1}+{\frac{\gamma}{6!}}\phi_{1}^{6}+{\frac{\zeta}{3!}}\phi_{2}^{3},
\begin{array}{lll}{X_{33}^{(1)}=\gamma_{\omega_{2}}^{-\frac{1}{n}},}&{X_{33}^{(2)}=\gamma_{\omega_{1}}^{\frac{1}{n}}\gamma_{\omega_{2}}^{\frac{1}{n}},}&{X_{33}^{(3)}=\gamma_{\omega_{1}}^{-\frac{1}{n}},}\\\end{array}
D_{z}=\left(\begin{array}{cc}{(\mathbf{p}_{1})^{\dagger}\mathbf{x}^{1}+(\mathbf{p}_{1}^{\prime})^{\dagger}\mathbf{x}^{2}-\mathbf{a}_{1}}&{0}\\{0}&{(\mathbf{p}_{2})^{\dagger}\mathbf{x}^{1}+(\mathbf{p}_{2}^{\prime})^{\dagger}\mathbf{x}^{2}-\mathbf{a}_{2}}\\{-\lambda_{1}A_{1}}&{-\lambda_{2}A_{2}}\\\end{array}\right)\,,
R_{110}^{0}=\frac{1}{g^{2}}\left(\frac{f^{\prime\prime}}{f}-\frac{f^{\prime}}{f}\frac{g^{\prime}}{g}\right)\quad;\quadR_{220}^{0}=R_{330}^{0}=\frac{1}{g^{2}}\frac{f^{\prime}}{f}\frac{h^{\prime}}{h}
\Psi\left(x\rightarrow-\infty\right)=e^{ipx}\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)+e^{-ipx}\left[B\left(\begin{array}{c}{0}\\{1}\\\end{array}\right)-\frac{im}p\left(\begin{array}{c}{1}\\{0}\\\end{array}\right)\intV\left(x^{\prime}\right){\psi^{+}}^{2}\left(x^{\prime}\right)dx^{\prime}\right].
V(r)\sim-\frac{1}{\pi^{2}M_{5}^{3}\lambdar}\left[\frac{\pi}{2}-\frac{\frac{2R}{\lambda}}{\frac{2R}{\lambda}-\frac{15}{8}}\left\{\frac{\pi}{2}-f\left(\frac{r}{R}\left(\frac{2R}{\lambda}-\frac{15}{8}\right)\right)\right\}\right]
\begin{array}{l}{E_{\mathrm{vib}}=\displaystylehc\omega_{e}(v+{\frac{1}{2}})-hc\omega_{e}x_{e}(v+{\frac{1}{2}})^{2}+hc\omega_{e}y_{e}(v+{\frac{1}{2}})^{3}+\cdots~,}\\{E_{\mathrm{rot}}=BhcJ(J+1)-DhcJ^{2}(J+1)^{2}+HhcJ^{3}(J+1)^{3}+\cdots~.}\\\end{array}
\gamma^{\mu}{\stackrel{\leftrightarrow}{\nabla}}_{\mu}=e_{m}^{\mu}[\gamma^{m}(\stackrel{\leftrightarrow}{\partial}_{\mu}-igA_{\mu}^{a}\tau_{a})\,+{\frac{i}{2}}\Gamma_{n\mup}\{\gamma^{m};\Sigma^{np}\}],
\begin{array}{lcr}{(|X_{0}|^{2}-|X_{1}|^{2})-(|Y_{0}|^{2}-|Y_{1}|^{2})=R_{1}}\\{X_{0}\overline{{Y}}_{0}-X_{1}\overline{{Y}}_{1}=\overline{{X}}_{0}Y_{0}-\overline{{X}}_{1}Y_{1}=0,}\\\end{array}
\widehat{\Omega}=\left(\begin{matrix}{1}&{0}&{0}\\{0}&{-i\gamma^{\lambda}\nabla_{\lambda}\hat{1}}&{0}\\{0}&{0}&{\hat{1}}\\\end{matrix}\right)\,\qquad\widehat{K}=\left(\begin{matrix}{Z}&{q^{\prime}\overline{{\psi}}}&{0}\\{0}&{q\hat{1}}&{0}\\{0}&{0}&{-f\hat{1}}\\\end{matrix}\right),
\left|\frac{\partial(\,a_{1}^{1},a_{1}^{2},a_{1}^{3},a_{2}^{1},a_{2}^{2},a_{2}^{3}\:)}{\partial(x_{1},x_{2},x_{3},\varphi^{1},\varphi^{2},\varphi^{3})}\right|=\frac{1}{8}\:x_{2}\:r^{2}\;,
i\hbar{\frac{\partial\Phi}{\partialt}}(1,\cdots,N)={\frac{1}{2m}}\sum_{p=1}^{N}\{i\hbar\partial_{i}^{(p)}-{\frac{e}{c}}\sum_{r=1(\nep)}^{N}{\calA}_{i}(p,r)\}^{2}\Phi(1,\cdots,N)\,.
\frac{\lambda}{4\pi}\intd^{2}z\left\{\sum_{\alpha_{i}>0}\left[E^{\alpha_{i}}(z)\overline{{E}}^{-\alpha_{i}}(\bar{z})+E^{-\alpha_{i}}(z)\overline{{E}}^{\alpha_{i}}(\bar{z})\right]+\sum_{i=1}^{2}H^{i}(z)\overline{{H}}^{i}(\bar{z})\right\}\.
\begin{array}{rcl}{g}&{=}&{1+\sigma^{I}T_{I}\,,}\\{}&{}&{}\\{h}&{=}&{1-\sigma^{I}W_{I}{}^{i}M_{i}\,,}\\{}&{}&{}\\{x^{\mu\,\prime}}&{=}&{x^{\mu}+\sigma^{I}k_{(I)}{}^{\mu}\,.}\\\end{array}
G_{(1)}^{\mu,ab}=\alphag^{2}N\delta^{ab}{\frac{k^{\mu}}{k\cdotn}}(iT)\sum_{l=-\infty}^{+\infty}\int{\frac{d^{3-\epsilon}q}{(2\pi)^{3-\epsilon}}}{\frac{1}{(q\cdotn)}}
{\calL}=\frac{|\partial_{t}W|^{2}-2|\partial_{z}W|^{2}}{(1+|W|^{2})^{2}}+8\theta_{1}\frac{|\partial_{z}W|^{2}}{(1+|W|^{2})^{4}}(|\partial_{t}W|^{2}-|\partial_{z}W|^{2}),
L=i\left(w_{\alpha}D_{-}\varphi^{\alpha}-\tilde{w}_{\overline{{\alpha}}}D_{-}\tilde{\varphi}^{\overline{{\alpha}}}\right)(\varphi,\tilde{\varphi})-qA_{-}^{a}X_{a}^{(w)}(\varphi,\tilde{\varphi}).
{\frac{\kappa}{4\pi}}\{\xi_{\alphaR}(x),\xi_{\betaL}(z)\}\;=\;\Bigl(R^{T\;-1}(x)F(z)L^{-1}(z)\Bigr)_{\alpha\beta}\;+\;\Bigl(\tilde{f}(x)L^{-1}(z)\Bigr)_{\alpha\beta}\quad,
\Gamma^{\mu}=\left(\begin{array}{cc}{0}&{\sigma_{+}^{\mu}}\\{\sigma_{-}^{\mu}}&{0}\\\end{array}\right)\equiv\gamma^{\mu}\;,\quad\Gamma^{5}=-i\left(\begin{array}{cc}{-1}&{0}\\{0}&{1}\\\end{array}\right)\equiv-i\gamma^{5}\;,\quad\sigma_{\pm}^{\mu}=(1,\pm\sigma^{i})\;,
\left({\frac{J}{M^{2}}}\right)^{2}\leq\frac{1}{4}\left[2-10\left({\frac{Q}{M}}\right)^{2}-\left({\frac{Q}{M}}\right)^{4}+2\left(1+2\left({\frac{Q}{M}}\right)^{2}\right)^{3/2}\right]\;.
L=\bar{\psi}i\gamma^{\mu}\partial_{\mu}\psi+{\bar{\psi}}\gamma^{\mu}(B_{\mu}+A_{\mu}+\epsilon_{\mu\nu}\partial^{\nu}\theta)\psi+\partial^{\mu}\etaB_{\mu}+\frac{1}{2\pi}\partial^{\mu}\theta\partial_{\mu}\theta+\frac{1}{\pi}\epsilon^{\mu\nu}A_{\mu}\partial_{\nu}\theta\quad.
\phi^{(n,n_{1})}=\sqrt{Z}\frac{(-g)^{n}(-ig)^{n_{1}}}{\sqrt{n!n_{1}!}}\left(\frac{p^{+}}{P^{+}}\right)^{\gamma}\prod_{i}\frac{y_{i}}{\sqrt{16\pi^{3}q_{i}^{+}}(\mu^{2}+q_{\perpi}^{2})}\prod_{j}\frac{z_{j}}{\sqrt{16\pi^{3}r_{j}^{+}}(\mu_{1}^{2}+r_{\perpj}^{2})}\,.
a_{i}^{I}({\bfr},t)=\epsilon^{ij}\nabla_{j}\frac{1}{\kappa_{I}}\sum_{p}q_{p}^{I}\int\!d^{2}\!{\bfr}^{\prime}\,G({\bfr}-{\bfr}^{\prime})\rho_{p}({\bfr}^{\prime},t),
\delta\Gamma={\frac{1}{16(4\pi)^{5}4!}}\int_{M^{10}}d^{10}x\epsilon^{A_{1}A_{2}\cdotsA_{10}}\mathrm{tr}\epsilonF_{A_{1}A_{2}}\mathrm{tr}F_{A_{3}A_{4}}F_{A_{5}A_{6}}\mathrm{tr}F_{A_{7}A_{8}}F_{A_{9}A_{10}}.
\begin{array}{rcl}{\Phi(x,y)}&{=}&{\phi(x)+yy\phi_{yy}(x)+yxy\phi_{yxy}(x)}\\{}&{}&{\qquad+xyy\partial_{x}\phi_{yy}(x)+yxxy\phi_{yxxy}(x)+yyyy\phi_{yyyy}(x)+\cdots}\\\end{array}
\langle\phi(x(\tau_{a}))\phi(x(\tau_{b}))\rangle=\int_{0}^{\infty}d\bar{T}e^{-m^{2}\bar{T}}{(4\pi\bar{T})}^{-{\frac{D}{2}}}\mathrm{exp}\Biggl[-{\frac{{(x(\tau_{a})-x(\tau_{b}))}^{2}}{4\bar{T}}}\Biggr]\,
\Phi^{\alpha_{0}}\rightarrowA_{\mu}^{a},\;\epsilon^{\alpha_{1}}\rightarrow\left(\begin{array}{l}{\epsilon^{b}}\\{\epsilon_{\nu}^{b}}\\\end{array}\right),\;Z_{\;\;\alpha_{1}}^{\alpha_{0}}\rightarrow\left(\delta_{\;\;b}^{a}\partial_{\mu},\;\delta_{\;\;b}^{a}\delta_{\;\;\mu}^{\nu}\right),
ds^{2}={\frac{1}{\sqrt{H_{p+r}H_{p+s}}}}ds_{p+1}^{2}-\sqrt{{\frac{H_{p+r}}{H_{p+s}}}}ds_{s}^{2}-\sqrt{{\frac{H_{p+s}}{H_{p+r}}}}ds_{r}^{2}-\sqrt{H_{p+r}H_{p+s}}ds_{9-p-r-s}^{2}\.
\beta=\frac{16\sigma^{2}\eta^{2}}{\alpha}\int_{0}^{\infty}\mathrm{d}\xi\;\xi^{2}\phi_{\sigma}^{2}\left(\xi\right)\,,\quad\phi_{\sigma}\left(\xi\right)=\mathrm{e}^{-2\sigma\xi}\int_{0}^{\infty}\mathrm{d}s\;\mathrm{e}^{-2\xis}\,{\calV}_{\sigma}\left(\eta,\zeta,s\right)
T_{z}^{\prime}C_{\alpha}\otimesT_{z}^{\prime}C_{\beta}\mapstoe^{(\frac{\lambda_{i}-\lambda_{j}}{d_{\alpha}}+\frac{\lambda_{i}-\lambda_{l}}{d_{\beta}})t}T_{z}^{\prime}C_{\alpha}\otimesT_{z}^{\prime}C_{\beta}
S_{bh}~=~lnN^{{\calP}_{0}}~=~{\frac{A_{H}}{4l_{p}^{2}}}~-~{\frac{3}{2}}~ln\left({\frac{A_{H}}{4l_{p}^{2}}}\right)~-~{\frac{1}{2}}~ln\left({\frac{\pi}{8(ln2)^{3}}}\right)~-~O(A_{H}^{-1}).
\mathrm{Im}\frac{d}{ds}\:\zeta_{H}(-(2m+1);ix)=\pi\:\mathrm{Re}\:\frac{{B_{2m+2}(ix)}}{4(m+1)}+\frac{(-1)^{m+1}x^{2m+1}}{2}\mathrm{ln}\:x+\frac{(-1)^{m+1}(2m+1)!}{2(2\pi)^{2m+1}}\sum_{n=1}^{+\infty}\frac{e^{-2n\pi{x}}}{n^{2(m+1)}}
\tilde{A}_{f}^{\dagger}(\vec{p})=({\frac{1}{2(2\pi)^{D}f(\vec{p})}})^{1/2}\int_{x}e^{-i\vec{p}\cdot\vec{x}}[f(\vec{p})(\tilde{\phi}_{x}-\varphi_{x})+i\tilde{\Pi}_{x}]\;.
S_{\mathrm{bulk}}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-G}\left[{\calR}_{G}-{\frac{4}{D-2}}(\partial\phi)^{2}+e^{-2a\phi}\Lambda-{\frac{1}{2\cdot(p+3)!}}e^{2a_{p+1}\phi}F_{p+3}^{2}\right],
Q(s)=\sum_{n}\frac{1}{2\Gamma(s)}\sqrt{\frac{b}{a}}\int_{0}^{\infty}dt\;e^{-t\left(c-\frac{b^{2}}{8a}\right)}\;t^{s-1}\;K_{\frac{1}{4}}\left(\frac{b^{2}t}{8a}\right)
h(\tau,\sigma^{1},\sigma^{2},\xi)=1-\xiK_{a}^{a}+\frac{1}{2}\xi^{2}(K_{a}^{a}K_{b}^{b}-K_{a}^{b}K_{b}^{a})-\frac{1}{3!}\xi^{3}\epsilon_{abc}\epsilon^{def}K_{d}^{a}K_{e}^{b}K_{f}^{c}.\nonumber
F_{1}(T)=\xi^{2}\left[-{\frac{3}{32\,a\,\pi}}+{\frac{5\,{T}}{16}}+{\frac{a\,\pi\,{T^{2}}}{{{\left({e^{8\,a\,\pi\,T}}-1\right)}^{2}}}}+{\frac{\left(5\,T\,+8\,a\,\pi\,{T^{2}}\,\right)}{8\,\left({e^{8\,a\,\pi\,T}}-1\right)}}\right].
0=\frac{\partial{\calL}}{\partialW_{\mu}^{I}}=2\partial_{\nu}\frac{\partial{\calL}}{\partial{\calF}_{\mu\nu}^{I}}=2\partial_{\nu}\left(\frac{\partial{\calL}}{\partial{\calF}_{\mu\nu}^{+I}}+\frac{\partial{\calL}}{\partial{\calF}_{\mu\nu}^{-I}}\right)\,
S_{F}=\intdx^{0}\left(i\overline{{\pi}}_{a}D_{0}\pi_{a}+|l_{a}|^{2}-|E_{a}(\phi)|^{2}-\overline{{\pi}}_{a}{\frac{\partialE_{a}}{\partial\phi_{i}}}\tau_{i}-{\frac{\partial\overline{{E}}_{a}}{\partial\overline{{\phi}}_{i}}}\overline{{\tau}}_{i}\pi_{a}\right)
_{\left(\Gamma^{M}\right)_{ab}\left(\tilde{\Gamma}^{N}\right)^{bc}+\left(\Gamma^{N}\right)_{ab}\left(\tilde{\Gamma}^{M}\right)^{bc}=\left(1|\Gamma^{M}\right)_{ab}\left(1|\tilde{\Gamma}^{N}\right)^{bc}+\left(1|\Gamma^{N}\right)_{ab}\left(1|\tilde{\Gamma}^{M}\right)^{bc}=2\eta^{MN}\delta_{a}^{c}.}
S_{\mathrm{goldst}}\=\{\frac{1}{4}}\intd^{4}xd^{2}\thetaW^{2}\+\{\frac{1}{4}}\intd^{4}xd^{2}\bar{\theta}\bar{W}^{2}\+\{\frac{1}{8}}\intd^{4}xd^{4}\thetaW^{2}\bar{W}^{2}\+\{\calO}(W^{6}).
\intde^{\prime}dn^{\prime}\frac{i}{k}\mathrm{~exp}-\frac{2i\pi}{k}[e^{\prime}(\tilde{n}-1/2)+e(\widetilde{n^{\prime}}-1/2)]\\tau^{-2e_{0}(\widetilde{n_{0}}-1/2)/k}\zeta_{e^{\prime}n^{\prime}}(z_{1},z_{2},w;\tau)
S\to{\frac{aS+b}{cS+d}},\\\\\\left(\begin{matrix}{{\calF}_{\mu\nu}^{i}}\\{\star{\calF}_{\mu\nu}^{i}}\\\end{matrix}\right)\to\omega_{S}^{-1}\left(\begin{matrix}{{\calF}_{\mu\nu}^{i}}\\{\star{\calF}_{\mu\nu}^{i}}\\\end{matrix}\right);\\\\omega=\left(\begin{matrix}{a}&{b}\\{c}&{d}\\\end{matrix}\right),
{\calE}\propto4\frac{v^{2}}{r^{2}}+\frac{2}{3}\left(\frac{w^{\prime}}{r}-\frac{w}{r^{2}}\right)^{2}+4\frac{v^{2}w^{2}}{r^{4}}+\frac{2}{r^{4}}\left(v^{2}-1\right)^{2}\approx\frac{2}{3}\frac{\alpha^{2}(\alpha-1)(3\alpha-1)}{x^{4-2\alpha}}
{\calL}_{eff}^{(0)}(g_{\mu\nu})={\frac{1}{8\pi}}{\frac{\Lambda_{ind}}{G_{ind}}}\,\,\,\,\,{\calL}_{eff}^{(2)}(\partial_{\lambda}\partial_{\tau}g_{\mu\nu},(\partial_{\lambda}g_{\mu\nu})^{2})={\frac{1}{16\piG_{ind}}}R\,.
|\Phi\rangle=a^{\dagger}({\bfk_{1}})a^{\dagger}({\bfk_{2}})...a^{\dagger}({\bfk_{n}})c^{\dagger}({\bfq_{1}})c^{\dagger}({\bfq_{2}})...c^{\dagger}({\bfq_{m}})|0\rangle
\ddot{x}^{\sigma}=\frac{e}{mc}\tilde{g}^{\sigma\rho}F_{\rho\gamma}\dot{x}^{\gamma}\approx\frac{e}{mc}\left(\eta^{\sigma\rho}+\frac{e^{2}}{m^{2}A^{2}}F_{\lambda}^{\sigma}F^{\lambda\rho}\right)F_{\rho\gamma}\dot{x}^{\gamma}\,{.}
\zeta_{0}^{(p)}(\nu)={\frac{2}{(4\pi)^{D/2}\Gamma(\nu)}}\sum_{n=0}^{\infty}{\frac{1}{2\nu+2n-D}}\sum_{m=n}^{2n}{\frac{\Gamma(m-n+1/2)}{\sqrt{\pi}}}\breve{a}_{2(m-n),m}~~~.
\frac{d_{+}}{d_{-}}=\frac{\Gamma(-\frac{2j+1}{k+2})}{\Gamma(\frac{2j+1}{k+2})}\;\sqrt{\frac{\Gamma(\frac{2j}{k+2})\;\Gamma(\frac{2j+2}{k+2})}{\Gamma(-\frac{2j}{k+2})\;\Gamma(-\frac{2j+2}{k+2})}}\;=\;\frac{\gamma_{-}}{\gamma_{+}},
T_{\alpha\beta}^{g_{1}g_{2}}\:=\:\left(T_{\alpha\beta}^{g_{2}}\right)\,\left(g_{2}^{*}T_{\alpha\beta}^{g_{1}}\right)\,\left(\omega_{\alpha}^{g_{1},g_{2}}\right)\,\left(\omega_{\beta}^{g_{1},g_{2}}\right)^{-1}
\langle0|T_{i}^{k}|0\rangle=\delta_{i}^{k}\frac{\pi^{(d-1)/2}}{\Gamma\left(\frac{d-1}{2}\right)}\int_{0}^{\infty}dk\,k^{d}\sum_{\sigma=0,1}\beta_{\sigma}\sum_{n=1}^{\infty}C_{Z}^{2}f_{{\mathrm{em}}}^{(i)}[Z_{i\omega_{Zn}}(k\xi,k\xi_{2})],\quad\beta_{0}=d-2,\quad\beta_{1}=1,
\begin{array}{ccc}{\rho_{1}>\rho_{2}>\rho_{3}>\rho_{4}\ge0}&{\mathrm{Area}_{H}>0}&{\frac{1}{8}\,\mathrm{BPS}}\\{\rho_{1}=\rho_{2}>\rho_{3}=\rho_{4}\ge0}&{\mathrm{Area}_{H}=0}&{\frac{1}{4}\,\mathrm{BPS}}\\{\rho_{1}=\rho_{2}=\rho_{3}=\rho_{4}\ge0}&{\mathrm{Area}_{H}=0}&{\frac{1}{2}\,\mathrm{BPS}}\\\end{array}
\Theta_{\{\Lambda\}}(v)=\sum_{n_{i}\in{\bfZ}}exp\bigl(3\pii(n_{i}+\frac{\lambda_{i}}{3})\tau_{ij}(n_{j}+\frac{\lambda_{j}}{3})+2\pii\sqrt{3}(n_{i}+\frac{\lambda_{i}}{3})v_{i}\bigr)\,
F^{\prime}(z)=N\phi^{\prime}(z,\frac{1}{2}\gamma)+N\phi^{\prime}(z,\gamma)+\sum_{h=1}^{N_{h}}\bar{\phi}^{\prime}(z-\lambda_{h},\gamma)+\sum_{l=1}^{N_{c}}\phi^{\prime}(z-z_{l},\gamma),
S_{1}\,[\psi\,,\,{\overline{\psi}}\,,\,A_{\mu}\,,\,g\,]\,=\,\intd^{2}x\Big(-{\frac{1}{4}}Tr(F^{\mu\nu}\,F_{\mu\nu})+i\overline{\psi}\gamma^{\mu}\big(\partial_{\mu}-i{\tilde{A}}_{\mu}\big)\psi\,\Big)
\frac{m_{i}}{m_{j}}=\frac{M_{c}^{i}}{M_{c}^{j}}=\sqrt{\frac{(\vec{\alpha}_{i}\cdot\Lambda_{+})(\vec{\alpha}_{i}\cdot\Lambda_{-})}{(\vec{\alpha}_{j}\cdot\Lambda_{-})(\vec{\alpha}_{j}\cdot\Lambda_{+})}}.
\begin{array}{rclrcl}{{}[Q^{a},U(z,\bar{z})]}&{=}&{0,}&{{}[Q^{a},V^{b}(z,\bar{z})]}&{=}&{f_{c}^{ab}V^{c}(z,\bar{z}),}\\{{}[Q^{a},\tilde{U}(z,\bar{z})]}&{=}&{0,}&{{}[Q^{a},\tilde{V}^{b}(z,\bar{z})]}&{=}&{f_{c}^{ab}\tilde{V}^{c}(z,\bar{z}).}\\\end{array}
\delta\Theta^{\underline{{\mu}}}(\xi)={\frac{1}{2}}(1+\bar{\Gamma})_{~~\underline{{\nu}}}^{\underline{{\mu}}}\kappa^{\underline{{\nu}}}(\xi),\qquad\deltaX^{\underline{{m}}}(\xi)=-i\bar{\Theta}\Gamma^{\underline{{m}}}\delta\Theta,\quad(\bar{\Gamma})^{2}=1
\begin{array}{lllll}{\phi_{1}=p_{y}}&{}&{}&{\psi_{1}=p_{\gamma}}\\{\phi_{2}=x}&{}&{}&{\psi_{2}=z+\alpha}\\{\phi_{3}=p_{z}}&{}&{}&{\psi_{3}=p_{x}+p_{\beta}}\\{\phi_{4}=\gamma}&{}&{}&{\psi_{4}=y}\\\end{array}
{\bfH}_{\mathrm{Thirring}}=\int\frac{dx}{2\pi}\,\Bigl\{-i\psi_{R}^{\dagger}\partial_{x}\psi_{R}^{}+i\psi_{L}^{\dagger}\partial_{x}\psi_{L}^{}+{\ttg}\,\psi_{R}^{\dagger}\psi_{R}^{}\psi_{L}^{\dagger}\psi_{L}^{}\,\Bigr\}\,.
\begin{array}{rl}{T}&{=X-2s,}\\{2S}&{=\displaystyle\sum_{i=1}^{s}(x_{i}-2)(x_{i}-3)+2\sum_{1\lei<j\les}(x_{i}-1)(x_{j}-1),}\\{}&{=X^{2}-(2s+3)X+s(s+5),}\\{P}&{=X-s.}\\\end{array}
{\cal{X}}_{rs}=X^{\mu}(C\Gamma_{\mu}^{(11)})_{rs}+X_{\bf{O}}^{\mu\nu}(C\Gamma_{\mu\nu}^{(11)})_{rs}=X_{\bfO}^{\mu_{1}\ldots\mu_{5}}(C\Gamma_{\mu_{1}\ldots\mu_{5}}^{(11)})_{rs}\,.
\left[\Gamma^{5}\,S_{[1]}^{c}\right]\,\lambda(p^{\mu})\,=\,\pm\,\lambda(p^{\mu})\,\,,\quad\left[\Gamma^{5}\,S_{[1]}^{c}\right]\,\rho(p^{\mu})\,=\,\pm\,\rho(p^{\mu})\quad,
\begin{array}{lll}{\deltax^{a}=-\varrhon^{a}}&{\deltaz=0}&{\qquad\delta\theta^{\alpha}=0}\\{\deltax^{a}=0}&{\deltaz=0}&{\qquad\delta\theta^{\alpha}=-2i\mun^{\alpha}{}_{\beta}\theta^{\beta}}\\{\delta{x^{a}}=-\eta^{\alpha}(\sigma^{a})_{\alpha\beta}n^{\beta}{}_{\gamma}\theta^{\gamma}}&{\deltaz=0}&{\qquad\delta\theta^{\alpha}=-in^{\alpha}{}_{\beta}\eta^{\beta}\;,}\\\end{array}
\int{\calD}V_{h}\prod_{i=3}^{3}{\calD}\phi_{h}^{i}e^{-S}{\calO}_{1}\cdots{\calO}_{n}\left\{\left(M_{h}\frac{d}{dM_{h}}\frac{1}{g_{h}^{2}(M_{h})}\right)S_{0}+\intd^{4}xd^{2}\thetaM_{h}\mathrm{Tr}\phi_{h}^{i}\phi_{h}^{i}\right\}=0,
J_{\mu\nu}[\xi|s]=\tilde{g}\,\epsilon_{\mu\nu\rho\sigma}\dot{\xi}^{\sigma}(s)[\bar{\psi}(\xi(s))\gamma^{\rho}T^{i}\psi(\xi(s))]\Omega_{\xi}^{-1}(s,0)\tau_{i}\Omega_{\xi}(s,0),
{\calS}_{\sigma}=\frac{1}{4\pi\alpha^{\prime}}\intd^{2}\sigma\left[\sqrt{\gamma}g_{ij}(\vec{x})\partial_{\alpha}x^{i}\partial^{\alpha}x^{j}+i\epsilon^{\alpha\beta}B_{ij}(\vec{x})\partial_{\alpha}x^{i}\partial_{\beta}x^{j}+\alpha^{\prime}\sqrt{\gamma}R^{(2)}\Phi(\vec{x})\right]
m^{2}{\equiv}\frac{e^{2}a(a+1-r^{2})}{a-r^{2}}=\frac{{e^{\prime}}^{2}a^{\prime}(a^{\prime}+1-{r^{\prime}}^{2})}{a^{\prime}-{r^{\prime}}^{2}}{\equiv}{m^{\prime}}^{2}.
\begin{array}{c}{\displaystyle{\sum_{\nu=1/2}^{\nu_{k}}\,F(x,\nu)=\int_{1/2}^{\nu_{0}}\,F(x,\nu)\,d\nu+}}\\{\displaystyle{+\frac{1}{2}\left\{F(x,\nu_{0})+F(x,1/2)\right\}+}}\\{\displaystyle{+\left.\frac{1}{12}\,\partial_{\nu}\left[F(x,\nu)\right]\right|_{\nu=1/2}^{\nu=\nu_{0}}+\dots}\,,}\\\end{array}
{S}=-T_{p}\intd^{p+1}\sigma\,\mathrm{Tr}\,e^{-\phi}\sqrt{-det\left(P\left[E_{ab}+E_{ai}(Q^{-1}-\delta)^{ij}E_{jb}\right]+F_{ab}\right)\,det(Q^{i}{}_{j})}+S_{WZ}\,
\Omega_{2}={\frac{B_{1}D_{2}}{C_{1}B_{2}}}+\theta_{2}{\frac{D_{1}}{C_{1}}}\\mathrm{and}\\tilde{\Omega}_{2}={\frac{B_{2}D_{2}}{C_{2}B_{1}}}+\Gamma\theta_{2}{\frac{D_{1}}{C_{2}}}
*(\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge...\wedge\omega_{i_{q}})=\frac{1}{(m-q)!}\left(\begin{array}{cccccc}{1}&{2}&{...}&{...}&{...}&{m}\\{i_{1}}&{...}&{i_{q}}&{j_{1}}&{...}&{j_{m-q}}\\\end{array}\right)\omega_{j_{1}}\wedge\omega_{j_{2}}\wedge...\wedge\omega_{j_{m-q}},
+\frac{\Omega}{4\pi^{2}}\frac{\left(\alpha\phi_{c}\right)^{4-2s}}{\Gamma\left(s\right)}\sum_{N=1}^{\infty}\frac{K_{s-2}\left(2N\alphaa\phi_{c}\right)}{\left(N\alphaa\phi_{c}\right)^{2-s}},
W=\frac{\Lambda_{1d}^{5}\Omega_{(2,\,r-1)}}{\Omega_{(1,\,r-1)}}+\frac{\Lambda_{rd}^{5}\Omega_{(1,\,r-2)}}{\Omega_{(1,\,r-1)}}\pm2\frac{(\Lambda_{1d}^{5}\Lambda_{rd}^{5}\prod_{i=2}^{r-1}\Lambda_{i}^{4})^{1/2}}{\Omega_{(1,\,r-1)}},
S_{l=1}(V^{\prime},U^{\prime})=\intd^{D_{1}}x\left(-\frac{1}{2g_{2}^{2}}\eta^{\mu\nu}\partial_{\mu}\bar{V}_{i}^{\prime}\partial_{\nu}V_{i}^{\prime}+\bar{U}^{\prime}{_i}(\frac{c}{2}\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}-m^{2})U_{i}^{\prime}\right)~.
n\alpha\hbar=\frac{1}{2\pi}\oint_{C_{1}}\frac{pdy}{1-y^{2}}=\frac{1}{2\pi}\left(\oint_{\gamma_{0}}\frac{pdz}{1-z^{2}}-\oint_{\gamma_{1}}\frac{pdy}{1-y^{2}}-\oint_{\gamma_{2}}\frac{pdy}{1-y^{2}}\right).
\left.+\frac{\partial}{\partialx_{\nu}}((x-x^{\prime})_{\rho}\delta_{\mu\lambda}-(x-x^{\prime})_{\lambda}\delta_{\mu\rho})\right]D_{1}\left((x-x^{\prime})^{2}\right)=\oint_{C}^{}dx_{\mu}\oint_{C}^{}dy_{\mu}G\left((x-y)^{2}\right),
\omega=\sum_{i=1}^{n}\frac{du_{i}^{-}}{u_{i}^{-}}\wedged\beta_{i}^{-}-\sum_{i<j}\frac{d\beta_{i}^{-}\wedged\beta_{j}^{-}}{\beta_{i}^{--}\beta_{j}^{-}}+\sum_{i=1}^{n}\frac{du_{i}^{+}}{u_{i}^{+}}\wedged\beta_{i}^{+}-\sum_{i<j}\frac{d\beta_{i}^{+}\wedged\beta_{j}^{+}}{\beta_{i}^{+}-\beta_{j}^{+}}.
{\calL}=-\frac{i}{2}\epsilon_{\mu\nu\lambda}B_{\mu}^{*}(\partial_{\nu}-iga_{\nu})B_{\lambda}+MB_{\mu}^{*}B_{\mu}-\frac{i}{2}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+\frac{1}{2\rho}(\partial_{\mu}a_{\mu})^{2}+(\partial_{\mu}\bar{c})(\partial_{\mu}c)\;.
\frac{a}{b}=\frac{4k_{\pm}^{2}}{\pim^{2}}\times\frac{1-4\alphak_{\pm}^{2}/M^{2}+2\alpham^{2}/M^{2}}{1-12\alphak_{\pm}^{2}/M^{2}+\alpham^{2}/M^{2}}.
\begin{array}{rcl}{i\Sigma_{ij}^{\mathrm{1PI}}(p)}&{=}&{\displaystylei\lambda^{2}\delta_{ij}\int\frac{d^{4}k_{1}}{(2\pi)^{4}}\int\frac{d^{4}k_{2}}{(2\pi)^{4}}U(k_{i},p),}\\{U(k_{i},p)}&{=}&{\displaystyle\sum_{x=a}^{h}U_{x},}\\\end{array}
\frac{\partial^{2}{\bf\tilde{Z}}(\omega,r)}{\partialr_{*}^{2}}+\left\lbrack\omega^{2}-V(r)\right\rbrack{\bf\tilde{Z}}(\omega,r)=S_{l}(\omega,r)+\frac{i\omega}{(2\pi)^{1/2}}{Z_{0}}_{l}(r)\,,
t_{0}=\gammaK^{-1},\;\;t_{1}=K,\;\;e_{1}=J_{0}^{+},\;\;f_{1}=J_{0}^{-},\;\;e_{0}t_{1}=J_{1}^{-},\;\;t_{1}^{-1}f_{0}=J_{-1}^{+}.
\begin{array}{c}{-\frac{1}{2}(k-1)k(k+1)<O_{k-2}O_{k_{1}}...O_{k_{p-1}}P>_{g-1}}\\{+4t_{0}(k+1)<O_{k}O_{k_{1}}...O_{k_{p-1}}P>_{g}+4t_{2}(k+2)<O_{k+2}O_{k_{1}}...O_{k_{p-1}}P>_{g}}\\{+2\sum_{i=1}^{p-1}(2k+k_{i}+2)<O_{k+k_{i}}O_{k_{1}}...\hat{O}_{k_{i}}...O_{k_{p-1}}P>_{g}=0,}\\\end{array}
\Theta_{\alpha}^{A}\equiv\epsilon^{ABC}\,\left(\,\frac{3}{16}\,\tau_{\alpha\mu\nu\rho}\,\eta^{B}\,D_{\mu}\,T_{\nu\rho}^{C}+\frac{9}{2}\,\tau_{\mu}\,\eta^{B}\,T_{\alpha\mu}^{C}\right)~,
\left\langle0\left|\psi_{n+1}^{\downarrow}(t_{1})\bar{\psi}_{-n}^{\uparrow}(t_{2})\right|0\right\rangle_{E}=\left[\begin{array}{c}{\theta(t_{1}-t_{2})\theta(n+\frac12)}\\{-\theta(t_{2}-t_{1})\theta(-n-\frac12)}\\\end{array}\right]\frac{t_{2}^{n}}{t_{1}^{n+1}},
\gamma^{-1/2}e^{\phi}\left[\frac{\delta\mathcal{S}}{\delta\gamma_{ij}}\frac{\delta\mathcal{S}}{\delta\gamma_{kl}}\gamma_{ik}\gamma_{jl}+\frac{1}{2}(\frac{\delta\mathcal{S}}{\delta\phi})^{2}+\gamma_{ij}\frac{\delta\mathcal{S}}{\delta\gamma_{ij}}\frac{\delta\mathcal{S}}{\delta\phi}\right]=0~~~.
\int_{1}\delta_{11}-\int_{12}S_{12}^{-1}\frac{\deltaW}{\deltaS_{12}^{-1}}=e^{2}\int_{1\cdots6}V_{123}V_{456}D_{36}\left\{\frac{\delta^{2}W}{\deltaS_{12}^{-1}\deltaS_{45}^{-1}}+\frac{\deltaW}{\deltaS_{12}^{-1}}\frac{\deltaW}{\deltaS_{45}^{-1}}\right\},
\Biggl[-\frac{\hbar^{2}}{3\pim_{P}^{2}a}\frac{\partial^{2}}{\partiala^{2}}-\frac{3\pim_{P}^{2}}{4}V_{g}(a)+\frac{\hbar^{2}}{4\pi^{2}a^{3}}\frac{\partial^{2}}{\partial{\phi}^{2}}+2\pi^{2}a^{3}V(\phi)\Biggr]\Psi_{E}(a,\phi)=0.
\left(\mu^{2}\frac{\partial}{\partial\mu^{2}}+\beta_{\rho}\frac{\partial}{\partial\rho}+\beta_{\xi}\frac{\partial}{\partial\xi}-\gamma_{\phi}\phi\frac{\delta}{\delta\phi}+\gamma_{g}g^{\ast\mu\nu}\frac{\delta}{\deltag^{\ast\mu\nu}}\right)V=0
C_{Q}=\frac{r_{+}\Omega_{\tilde{d}+1}[(\tilde{d}+1)r_{+}^{\tilde{d}}+r_{-}^{\tilde{d}}][1-(r_{-}/r_{+})^{\tilde{d}}]^{(2+\tilde{d})/2\tilde{d}}}{4[(\tilde{d}-1)r_{-}^{\tilde{d}}/r_{+}^{\tilde{d}}-1]}.
R^{(d)}=R^{(d_{l})}+R^{(d_{t})}-\frac{2}{\sqrt{-g^{(d_{l})}}}h^{(d_{t})ij}D_{i}D_{j}\sqrt{-g^{(d_{l})}}-\frac{2}{\sqrt{h^{(d_{t})}}}g^{(d_{l})\alpha\beta}D_{\alpha}D_{\beta}\sqrt{h^{(d_{t})}}