harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
eT_{\mu\nu}=e\Theta_{\mu\nu}+\frac{1}{2}g_{\mu\nu}\left({\tilde{\overline{\Psi}}}_{L}\frac{\delta{S}^{-}}{\delta{\tilde{\overline{\Psi}}}_{L}}+{\tilde{\Psi}}_{L}\frac{\delta{S}^{-}}{\delta{\tilde{\Psi}}_{L_{0}}}\right).
H_{P}=\int(\frac{8\pi^{2}}{g^{2}}(\pi_{i}^{a}+\frac{1}{8\pi}\epsilon_{ij}A^{j,a})^{2}+\frac{g^{2}}{64\pi^{2}}G_{ij}^{a}G_{ij}^{a})d^{2}x
W^{ac}\sum_{n=0}^{\infty}(-1)^{n}\,\left(\Lambda^{n}\right)^{cb}=\sum_{p=1}^{\infty}a_{p}\,\left[(g\bar{\theta})^{p}\right]^{ac}\sum_{n=0}^{\infty}c_{n}\,\left[(g\bar{\theta})^{2n}\right]^{cb}\,,
S_{L}[{\hat{\Phi}}]\=\{\frac{4\pi}{L+1}}\,\mathrm{Tr}[{\frac{1}{2}}({\hat{\calL}}_{i}{\hat{\Phi}})^{2}+{\frac{r}{2}}{\hat{\Phi}}^{2}+{\frac{\lambda}{4!}}{\hat{\Phi}}^{4}]\.
\displaystyle{F_{\mu\nu}^{+I}\longrightarrow\tilde{F}_{\mu\nu}^{+I}=U_{J}^{I}\,F_{\mu\nu}^{+J}+Z^{IJ}\,G_{\mu\nuJ}^{+}\;\;G_{\mu\nuI}^{+}\longrightarrow\tilde{G}_{\mu\nuI}^{+}=V_{I}^{J}\;G_{\mu\nuJ}^{+}+W_{IJ}\,F_{\mu\nu}^{+J}},
\sigma_{0}^{2}=\left(\frac{R}{\xi_{1}h^{2}}-\frac{1}{\lambda_{1}h^{2}}+\frac{2R}{3(4\pi)^{2}}\right)\left(\frac{1}{6\lambda_{2}h^{4}}-\frac{11}{12(4\pi)^{2}}\right)^{-1}.
{\calL}_{aux}=f_{vis}(\tilde{q}^{\dagger},\tilde{q})|F_{\Phi}|^{2}+\frac{\partial^{2}f_{vis}}{\partial\tilde{q}\partial\tilde{q}^{\dagger}}|F_{Q}|^{2}+\frac{\partialf_{vis}}{\partial\tilde{q}}F_{Q}F_{\Phi}^{\dagger}+3W_{vis}(\tilde{q})F_{\Phi}+{\frac{\partialW_{vis}}{\partial\tilde{q}}F_{Q}+\mathrm{h.c.}}+...
|B(y,{\bfs})\rangle^{\hat{\mu}}=|B_{X}(y)\rangle\otimes|B_{\mathrm{gh}}\rangle\otimes\psi_{-1/2}^{\hat{\mu}}|0\rangle_{\mathrm{NS}}\otimes{\widetilde{|{\psi_{\bfs}}\rangle}}_{\mathrm{R}},
{\cal{G}}_{\mathrm{ir}}^{\Lambda,\Lambda_{0}}\{\varphi\}=(\varphi,J)-\frac{1}{2}(\varphi,(G_{0}^{\Lambda,\Lambda_{0}})^{-1}\varphi)-{\cal{G}}_{\mathrm{c}}^{\Lambda,\Lambda_{0}}\{J\{\varphi\}\}\,,
\tilde{R}_{11}^{\prime}=\left(\frac{\tilde{R}}{R}\right)^{1/2}\frac{l_{p}^{3}}{R_{8}R_{9}}\\,\\\tilde{l}_{p}=\left(\frac{\tilde{R}}{R}\right)^{1/3}\frac{l_{p}^{2}}{(RR_{8}R_{9})^{1/3}}\\,\\\tilde{R}_{i}^{\prime}=\left(\frac{\tilde{R}}{R}\right)^{1/2}R_{i}.
\begin{array}{rl}{A\circ_{X}B}&{=(AX)(X^{\dagger}B)}\\{}&{=(AX^{2/3}X^{1/3})(X^{-1/3}X^{-2/3}B)}\\{}&{=X^{1/3}[(X^{-1/3}AX^{2/3})(X^{-2/3}B)]}\\{}&{=X^{1/3}[(X^{-1/3}AX^{1/3}X^{1/3})(X^{-1/3}X^{-1/3}B)]}\\{}&{=X^{1/3}\{[(X^{-1/3}AX^{1/3})(X^{-1/3}BX^{1/3})]X^{-1/3}\}}\\{}&{=X^{1/3}[(X^{-1/3}AX^{1/3})(X^{-1/3}BX^{1/3})]X^{-1/3}.}\\\end{array}
\bar{j}_{{}_{\mathrm{REG}}}^{\varphi}(r)=-\frac{sr}{4\pi}\int_{0}^{\infty}dk\,\frac{k^{3}}{\sqrt{k^{2}+m^{2}}}[J_{1-F}^{2}(kr)-J_{2-F}(kr)J_{-F}(kr)+J_{1+F}(kr)J_{-1+F}(kr)-J_{F}^{2}(kr)].
u\in{\mathrm{USp}}(4):\qquadu_{i}^{I}u_{J}^{i}=\delta_{J}^{I}\;,\\u_{i}^{I}\Omega^{ij}u_{j}^{J}=\Omega^{IJ}\;,\\u_{i}^{I}=(u_{I}^{i})^{*}\;.
S_{\mathrm{phys}}(t_{f},t_{i})={\calE}\,S(t_{f},t_{i})\,{\calE}=e^{-\frac{i}{\hbar}(t_{f}-t_{i})\hat{H}}\,{\calE}={\calE}\,e^{-\frac{i}{\hbar}{\calE}\hat{H}{\calE}}\,{\calE}\\\,
E[\sigma]=\sum_{\ell}N_{\ell}\int_{m}^{\infty}\frac{dt}{2\pi}\frac{t}{\sqrt{t^{2}-m^{2}}}\left[\beta_{\ell}(t,0)\right]_{N}+\sum_{i=1}^{N}E_{\mathrm{FD}}^{(i)}+E_{\mathrm{CT}}\,.
Det(1-Q(\nu;x))=1+\sum_{n=1}^{\infty}\frac{1}{(n!)}\int_{1}^{\infty}d\alpha_{1}...\int_{1}^{\infty}d\alpha_{n}(\prod_{j=1}^{n}\frac{D(\nu,\alpha_{j})e^{-2\alpha_{j}x}}{2\pi\alpha_{j}})\prod_{i<j}^{n}\frac{(\alpha_{i}-\alpha_{j})^{2}}{(\alpha_{i}+\alpha_{j})^{2}}
im\sum_{n=3}^{N}b_{n}\varepsilon^{\mu\nu\sigma}h_{\mu|}^{n}{}^{\alpha(n)}e_{\nu|\alpha(2)}h_{\sigma|\alpha(n-2)}^{n-2}+im\sum_{n=1}^{N-2}\lambda_{\alpha(n+2)}^{n+2}e^{\mu|\alpha(2)}h_{\mu|}^{n}{}^{\alpha(n)},
ds_{n+3+d}^{2}=2dx^{+}dx^{-}+\frac{2n+4}{(n+4)^{2}}\frac{1}{{x^{-}}^{2}}\big(n(y^{0})^{2}-\sum_{i=1}^{n}{y^{i}}^{2}\big)(dx^{-})^{2}+\sum_{i=0}^{n}{dy^{i}}^{2}+ds_{\mathbf{R}^{d}}^{2}
p=p_{l}(\epsilon)=\frac{2}{\epsilon}\,\left(\lambda_{l}^{(\ast)}\right)^{1/2}-1=\frac{2}{\epsilon}\,\left(\lambda_{l}^{(\ast)}\right)^{1/2}\,\left[1-\frac{\epsilon}{2}\left(\lambda_{l}^{(\ast)}\right)^{-1/2}\right]\;,
Y_{i_{1}j_{1},\ldots,i_{p}j_{p}}^{k_{1}l_{1},\ldots,k_{q}l_{q}}=\bar{a}_{i_{1}\ldotsi_{p}}\bar{a}_{j_{1}\ldotsj_{p}}b^{k_{1}\ldotsk_{q}}b^{l_{1}\ldotsl_{q}}\mathrm{~-~traces~}
V\Big[\begin{array}{c}{\alpha}\\{\beta}\\\end{array}\Big]=\sum_{n_{i},m_{i}=0}^{1}C\Big[\begin{array}{c}{\alpha+n_{1}b_{1}+m_{1}b_{2}}\\{\beta+n_{2}b_{1}+m_{2}b_{2}}\\\end{array}\Big]\prod_{f\inF^{\prime}}\Theta^{1/2}\Big[\begin{array}{c}{(\alpha+n_{1}b_{1}+m_{1}b_{2})(f)}\\{(\beta+n_{2}b_{1}+m_{2}b_{2})(f)}\\\end{array}\Big].
K=\sum_{anc}(b_{1}^{+})^{n_{1}}\cdots(b_{r}^{+})^{n_{r}}\;C_{n_{1},\cdots,n_{r}}^{m_{1},\cdots,m_{r}}(b_{1}^{+}b_{1},b_{2}^{+}b_{2},\cdots,b_{r}^{+}b_{r})\;(b_{1})^{m_{1}}\cdots(b_{r})^{m_{r}},
T_{+p\pm2}^{-q}=-\frac12\gamma_{q\dot{q}}^{i}\Omega_{+p}^{-2i}\psi_{\pm2\dot{q}}^{2+},\quadT_{-p\pm2}^{-q}=\frac12\gamma_{q\dot{q}}^{i}\Omega_{-p}^{-2i}\psi_{\pm2\dot{q}}^{2+}.
{\frac{\partial_{\mu}\tilde{T}\partial^{\mu}\tilde{T}}{(\tilde{T}^{\prime})^{2}}}=\sum_{i,j}{\frac{\partial_{\mu}a_{i}\partial^{\mu}a_{j}}{(x-a_{i})(x-a_{j})}}/\left(\sum_{k}{\frac{1}{(x-a_{k})}}\right)^{2}\.
\frac{M_{F1\otimesF1\otimesF1}^{2}}{K}=\frac{M_{F1}^{2}(K_{1})}{K_{1}}+\frac{M_{F1}^{2}(K_{2})}{K_{2}}+\frac{M_{F1}^{2}(K_{3})}{K_{3}},
\left(\begin{matrix}{\tilde{a}_{1}^{1}}&{\;\tilde{a}_{1}^{2}}&{\;\tilde{a}_{1}^{3}}\\{\tilde{a}_{2}^{1}}&{\;\tilde{a}_{2}^{2}}&{\;\tilde{a}_{2}^{3}}\\\end{matrix}\right)=\left(\begin{matrix}{a_{1}^{1}}&{\;a_{1}^{2}}&{\;a_{1}^{3}}\\{a_{2}^{1}}&{\;a_{2}^{2}}&{\;a_{2}^{3}}\\\end{matrix}\right)\Big(U^{ab}(\vec{\varphi})\Big)
eT_{\mu\nu}=e\Theta_{\mu\nu}+\frac{1}{2}g_{\mu\nu}\left({\tilde{\overline{\Psi}}}_{L_{0}}\frac{\delta{\tilde{S}}_{F}}{\delta{\tilde{\overline{\Psi}}}_{L_{0}}}+{\tilde{\Psi}}_{L_{0}}\frac{\delta{\tilde{S}}_{F}}{\delta{\tilde{\Psi}}_{L_{0}}}\right).
\begin{array}{c}{A^{\left(*\right)}(p)=\sum_{m}v(p,m)a^{*}(p,m)}\\{B^{\left(*\right)}(p,)=\sum_{m}v(p,m)b^{*}(p,m)}\\{A(p)=\sum_{m}u(p,m)a(p,m)}\\{B(p)=\sum_{m}u(p,m)b(p,m)}\\\end{array}
ds^{2}~=~\alpha^{\prime}~R^{2}~\left(\frac{1-\pi^{4}T^{4}v^{4}}{v^{2}}dx_{0}^{2}~+~\frac{1}{v^{2}}dx_{i}^{2}~+~\frac{1}{v^{2}(1-\pi^{4}T^{4}v^{4})}dv^{2}~+~d\Omega_{5}^{2}\right)~.
P_{\bar{M}_{1}}^{\left[pq\right]_{+}}V_{m}^{\bar{M}_{1}}=Y_{\bar{M}_{1}\bar{M}_{2}\bar{M}_{3}}V_{m}^{\bar{M}_{3}}=Z_{\bar{M}_{1}\bar{M}_{2}\cdots\bar{M}_{5}}^{\left[pq\right]_{+}}V_{m}^{\bar{M}_{5}}=0.
S_{WZ}^{NF1}=\mu_{1}\int_{R^{1+1}}\mathrm{Tr}\left(B^{(2)}DXDX+i(2\pi\alpha^{\prime})i_{\Phi}i_{\Phi}i_{k}C^{(5)}-\frac12(2\pi\alpha^{\prime})^{2}(i_{\Phi}i_{\Phi})^{2}i_{k}N^{(7)}+\dots\right)\,,
\Phi_{1}^{G}=x_{0}-\kappa\tau,\quad\,\Phi_{2}^{G}=\chi,\quad\,\Phi_{3}^{G}=e-\frac{\kappa}{p_{0}},\quad\,\Phi_{4\nu}^{G}=b_{\nu},\quad\,\Phi_{5}^{G}=\xi_{0},
(L_{4})_{ikms}^{~jlnp}=\delta_{i}^{~l}\delta_{s}^{~j}\delta_{k}^{~n}\delta_{m}^{~p}+\delta_{i}^{~n}\delta_{k}^{~j}\delta_{s}^{~l}\delta_{m}^{~p}-\delta_{i}^{~l}\delta_{m}^{~j}\delta_{k}^{~p}\delta_{s}^{~n}-\delta_{i}^{~p}\delta_{k}^{~j}\delta_{m}^{~l}\delta_{s}^{~n}.
{\vertO_{k}\rangle}_{\nu\bar{\nu}}=\epsilon^{l_{1}..l_{k-2}}\,\epsilon^{t_{1}\cdotst_{k-2}}\,(\Gamma_{A_{1}B_{1}}^{z}\,\nu_{l_{1}}^{A_{1}}\,\nu_{t_{1}}^{B_{1}})\cdots(\Gamma_{A_{k-2}B_{k-2}}^{z}\,\nu_{l_{k-2}}^{A_{k-2}}\,\nu_{t_{k-2}}^{B_{k-2}})\,\vert\Phi\rangle
TrK_{B_{3}^{\alpha}}={\frac{1}{(4\pis)^{3/2}}}\left(V_{B_{3}^{\alpha}}+(-{\frac{1}{l^{2}}}V_{B_{3}^{\alpha}}+{\frac{\xi}{l^{2}}}V_{B_{3}^{\alpha}}+A_{+}~(2\pi\alpha)c_{2}(\alpha))~s~+O(s^{2})~\right)
e^{-S_{\mathrm{dual}}(A)\;}=\intDB_{\mu}DB_{\mu}^{\dagger}D\eta_{\mu}e^{-\left(S(B,(\partial+i\eta)B)+\intd^{3}xA_{\mu}\epsilon^{\mu\nu\rho}\partial_{\nu}\eta_{\rho}\right)\;}\;,
\begin{array}{l}{x^{i}x^{j}=qx^{j}x^{i}~,~~~~~\mathrm{for}~i<j~~\mathrm{and}~~i\not=-j~,}\\{qx^{+2}x^{-2}-q^{-1}x^{-2}x^{+2}=\frac{q^{1/2}-q^{-1/2}}{q-1+q^{-1}}\frac{1}{a^{2}}~,}\\{qx^{+1}x^{-1}-q^{-1}x^{-1}x^{+1}=(1-q^{2})x^{+2}x^{-2}+q\frac{q^{1/2}-q^{-1/2}}{q-1+q^{-1}}\frac{1}{a^{2}}~.}\\\end{array}
=-n(U_{0}^{n}|_{i_{n}\leftrightarrowi_{2k}^{\prime}}+U_{0}^{n}|_{i_{n}\leftrightarrowi_{2k-1}^{\prime}})Str\left(\ldots\underbrace{\phi^{i_{2k}^{\prime}}\phi^{i_{2k-1}^{\prime}}}\ldots\right)
=\sum_{b^{\prime\prime}}w(a_{i}^{\prime\prime},a_{i},b^{\prime\prime},a_{i}^{\prime}|v)w(a_{i},a_{i+1},a_{i+1}^{\prime\prime},b^{\prime\prime}|u+v)w(b^{\prime\prime},a_{i+1}^{\prime\prime},a_{i+1}^{\prime},a_{i}^{\prime}|u)~.
\begin{array}{lcr}{S_{PB}}&{=}&{{\frac{1}{2\pi\alpha^{\prime}}}\int((D_{\mu}X)^{2}+\theta^{T}\not{D}\theta+{g_{s}}^{2}Tr{F_{\mu\nu}}^{2})d\alphad\betad\sigmad\tau}\\{}&{}&{{}-\int(({\frac{1}{{g_{s}}^{2}}}\{X^{i},X^{j}\})^{2}-{\frac{1}{g_{s}}}\theta^{T}\gamma_{i}\{X^{i},\theta\})d\alphad\betad\sigmad\tau.}\\\end{array}
(-\frac{d^{2}}{dt^{2}}-q^{2})g_{ab}\bar{\phi}_{b}({\bfq},t)+\intdt^{\prime}\frac{\partial^{2}S_{\mathrm{int}}}{\partial\varphi_{a}(-{\bfq},t)\partial\varphi_{b}({\bfq},t^{\prime})}\bar{\phi}_{b}({\bfq},t^{\prime})=0,
B^{T}{\frac{C^{3}}{C_{3}^{2}}}B=-{\frac{2}{\pi\mu\alpha_{3}}}\left({\frac{\alpha_{3}}{\alpha_{1}\alpha_{2}}}\right)^{2}\left(\mathrm{coth}\pi\mu\alpha_{1}+\mathrm{coth}\pi\mu\alpha_{2}\right)^{-1}.
S=\frac{1}{16\pi}\intd^{4}x\sqrt{-g}\biggl\{-R+\frac{1}{2}\frac{\partial_{\mu}\lambda\partial^{\mu}\overline{{\lambda}}}{({\mathrm{Im}}\;\lambda)^{2}}-\sum_{n=1}^{2}F_{\mu\nu}^{(n)}{}^{\star}\tilde{F}^{(n)\mu\nu}\biggr\}\;.
T_{\mu\nu}=g_{\mu\nu}L_{M}-2\frac{\partialL_{M}}{\partialg^{\mu\nu}}=2\mathrm{Tr}(F_{\mu\alpha}F_{\nu\beta}g^{\alpha\beta}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta})\,
\begin{array}{lll}{SO(3,2)\oplusSO(N)}&{\mathrm{con}\N=1,2,...}&{\qquadSO(4,1)\oplusU(1)}\\{SO(4,2)\oplusU(N)}&{\mathrm{con}\N\neq4}&{\qquadSO(4,2)\oplusSU(4)}\\{SO(5,2)\oplusSU(2)}&{}&{\qquadSO(6,1)\oplusSU(2)}\\{SO(6,2)\oplusSU(N,H)}&{\mathrm{con}\N=1,2,...}&{}\\\end{array}
S_{\alpha\beta}=\left(\begin{array}{ccccc}{-\frac{1+f^{2}}{1-f^{2}}}&{\frac{2f}{1-f^{2}}}&{}&{}&{}\\{\frac{2f}{1-f^{2}}}&{-\frac{1+f^{2}}{1-f^{2}}}&{}&{}&{}\\{}&{}&{1}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{1}\\\end{array}\right)~~.
H=\sum_{i=1}^{N-1}\sigma_{i}^{x}\sigma_{i+1}^{x}+\sigma_{i}^{y}\sigma_{i+1}^{y}+\frac{q+q^{-1}}{2}\sigma_{i}^{z}\sigma_{i+1}^{z}+\frac{q-q^{-1}}{2}(\sigma_{1}^{z}-\sigma_{N}^{z})
q^{(j+n)(j-m)}\left[\begin{array}{l}{j+m}\\{j-n}\\\end{array}\right]_{q^{2}}^{1/2}\left[\begin{array}{l}{j-n}\\{m-n}\\\end{array}\right]_{q^{2}}^{1/2}a^{-m-n}c^{m-n}P_{j+n}^{m-n,-m-n}(c^{*}c;q^{2}),
V_{T}(|\lambda\rangle_{+})_{0}c_{-{\frac{1}{2}}}\hat{\rho}=2^{-\lambda^{2}}\left((1-2{\lambda^{1}}^{2})c_{-{\frac{1}{2}}}-2\lambda^{1}\lambda^{2}d_{-{\frac{1}{2}}}\right)\gamma_{\lambda}\hat{\rho}\,,
\textrm{tr}\left[IV^{d}\right]=\textrm{tr}\left[m\,\sigma^{3}\otimes\left(\hat{n}(\vec{x})\cdot\vec{\tau}\right)\left(m^{2}\,\partial_{i}\hat{n}^{a}\,\partial_{j}\hat{n}^{b}\,\sigma^{i}\,\sigma^{j}\otimes\tau^{a}\,\tau^{b}\right)^{\frac{d}{2}}\right]
na_{n}(q^{\prime},q)+(q^{\prime}-q)\frac{\partiala_{n}(q^{\prime},q)}{\partialq^{\prime}}=\frac{1}{2}\frac{\partial^{2}a_{n-1}(q^{\prime},q)}{\partialq^{2}}-V(q^{\prime})a_{n-1}(q^{\prime},q)
\langlem\rangle=-{\frac{q_{c}^{\prime}(h)}{q_{c}(h)}}={\frac{\left(e^{2h}-1\right)\left(3+2e^{2h}+\sqrt{7}e^{2h}\right)}{1+(-1+{\sqrt{7}})e^{2h}+(2+{\sqrt{7}})e^{4h}}}\,
\overline{{w}}_{1}\Gamma^{(1)}S_{F}^{(1)}\Gamma^{(2)}\ldots\Gamma^{(n)}w_{2}=(-1)\cdot\overline{{w_{2}^{c}}}\,\Gamma^{\prime\,(n)}\ldotsS_{F}^{\prime\,(1)}\Gamma^{\prime\,(1)}w_{1}^{c},
\delta\psi_{\mua}\;=\;D_{\mu}\,\varepsilon_{a}\,+\,i\,\gamma_{\mu}\,T\left(\Gamma_{45}\right)_{ab}\varepsilon^{b}\;+\,i\;\frac{1}{6}\;\sqrt{\,\frac{1}{2}}\,\left(\,\gamma_{\mu}^{\;\nu\rho}\;-\;4\,\delta_{\mu}^{\nu}\,\gamma^{\rho}\,\right)\left(\,H_{\,\nu\rhoab}\;+\;\sqrt{\,\frac{1}{2}}\,h_{\,\nu\rhoab}\,\right)\varepsilon^{b},
\int_{r_{0}}^{r}f(r^{\prime})dr^{\prime}=\int_{t_{0}}^{t}{\frac{\sqrt{n^{2}(z)-b^{2}(z)\dot{y}^{2}-d^{2}(z)\dot{z}^{2}}}{a(z)}}dt\equiv\int_{t_{0}}^{t}{\calL}\left[y(t),\dot{y}(t),z(t),\dot{z}(t);t\right]dt
\lbrack\mathbf{M_{\varphi_{j}}},p_{k}]=i\epsilon_{jkl}p_{l},\qquad\lbrack\mathbf{M_{\psi_{j}}},p_{k}]=i\delta_{jk}p_{0},\qquad\lbrack\mathbf{M_{\psi_{j}}},p_{0}]=ip_{j}.
{\calL}=\intd^{4}\theta\:\overline{\Phi}e^{-2V}\Phie^{2V}~+~\frac{1}{4}\intd^{2}\theta\:W^{\alpha}W_{\alpha}~+~\frac{1}{4}\intd^{2}\overline{\theta}\:\overline{W}_{\alpha}\overline{W}^{\alpha}~.
\begin{array}{l}{e^{+2}\nabla\psi_{+2\dot{q}}^{1-}+e^{-2}\nabla\psi_{-2\dot{q}}^{1-}+e^{+}\nablah_{+\dot{q}}^{1-}+e^{-}\nabla\chi_{\dot{q}}^{1}+T^{+2}\psi_{+2\dot{q}}^{1-}+T^{-2}\psi_{-2\dot{q}}^{1-}+T^{+}h_{+\dot{q}}^{1-}+T^{-}\chi_{\dot{q}}^{1}}\\{=-\frac12\tilde{\gamma}_{\dot{q}p}^{i}\Omega^{-2i}\pi_{p}^{1+},}\\\end{array}
{L}_{{BI}}^{{SUSY}}=\frac{\beta^{2}}{4e^{2}}\left[\frac{1}{\beta^{2}}\intd^{2}\thetaW^{2}+\frac{1}{\beta^{2}}\intd^{2}\bar{\theta}\bar{W}^{2}\right]+\sum_{r,s,t=0}^{\infty}a_{rst}\intd^{4}\theta\left(W^{2}\bar{W}^{2}\right)^{r}X^{s}Y^{t}
\begin{aligned}{[m]ds^{2}}&{{}=2dx^{+}dx^{-}+K(dx^{-})^{2}+\sum_{i}dx^{i}dx^{i}}\\{F}&{{}=\mu\,dx^{-}\wedgedx^{1}\wedgedx^{2}\wedgedx^{3}~,}\\\end{aligned}
\frac{ig^{2}}{2k_{1}\cdotk_{2}}\left[\varepsilon(k_{1})\cdot\varepsilon(k_{2})(k_{1}-k_{2})\cdotp_{1}-2p_{1}\cdot\varepsilon(k_{1})k_{1}\cdot\varepsilon(k_{2})+2p_{1}\cdot\varepsilon(k_{2})k_{2}\cdot\varepsilon(k_{1})\right]C_{a_{1}a_{2}f}C_{b_{1}b_{2}f}\,
\mathrm{Tr}^{(q)}(\cdots)\equiv\frac{1}{2}\mathrm{tr}^{(LR)}[\frac{1}{3}\mathrm{tr}^{(3)}\{\frac{1}{2}\mathrm{tr}^{(2)}(\cdots)\}],\quad\mathrm{Tr}^{(l)}(\cdots)\equiv\frac{1}{2}\mathrm{tr}^{(LR)}\{\frac{1}{2}\mathrm{tr}^{(2)}(\cdots)\},
\left[\hat{x}^{\alpha},\hat{p}_{\beta}\right]_{-}=i\left\{x^{\alpha},p_{\beta}\right\}_{D(\Phi_{3n}^{(1)})}=\delta_{\beta}^{\alpha},\,\,\,\,\,\left[\hat{\psi}^{m},\hat{\psi}^{n}\right]_{+}=i\left\{\psi^{m},\psi^{n}\right\}_{D(\Phi_{3n}^{(1)})}=-\frac{1}{2}\eta^{mn}.
:{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}:~=~{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}{}~~\mathrm{if}~n>0~~~~~~~~:{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}:~=~-\hat{\psi}_{n}{\hat{\psi}_{m}}^{\dagger}~~\mathrm{if}~n\le0
\begin{array}{ccccc}{}&{}&{h^{0,0}}&{}&{}\\{}&{h^{1,0}\!}&{}&{\!h^{0,1}\!}&{}\\{h^{2,0}\!}&{\!}&{h^{1,1}}&{\!}&{\!h^{0,2}}\\{}&{h^{2,1}}&{}&{h^{1,2}}&{}\\{}&{}&{h^{2,2}}&{}&{}\\\end{array}=\begin{array}{ccccc}{}&{}&{1}&{}&{}\\{}&{0\!}&{}&{\!0}&{}\\{1\!}&{\!}&{20}&{\!}&{\!1}\\{}&{0\!}&{}&{\!0}&{}\\{}&{}&{1}&{}&{}\\\end{array}\.
R(s)=2\sum_{n}\int_{-\infty+i\epsilon}^{\infty+i\epsilon}\left(e^{-2\piiz}-1\right)^{-1}\left[\left(\left(\frac{2\piz}{\beta}\right)^{2}-\mu^{2}+E_{n}^{2}\right)^{2}+4\mu^{2}\left(\frac{2\piz}{\beta}\right)^{2}\right]^{-s}dz
i\frac{\partial}{\partialt}\langle\vec{q}\,^{\prime},t\mid\vec{q},0\rangle=-\frac{1}{2}\sum_{\nu=1}^{3}\frac{\partial^{2}}{\partialq_{\nu}^{2}}\langle\vec{q}\,^{\prime},t\mid\vec{q},0\rangle+V(\vec{q}\,^{\prime})\langle\vec{q}\,^{\prime},t\mid\vec{q},0\rangle
f_{a}^{(i)}(z)=\zeta^{\frac{n-1}{n}\frac{r-1}{r}-\frac{2\bar{a}_{i}}{r}}\frac{g(\zeta)p_{a}^{(i)}(\zeta)p_{a}^{(i)}(x^{2}\zeta^{-1})}{g(\zeta^{-1})p_{a}^{(i)}(\zeta^{-1})p_{a}^{(i)}(x^{2}\zeta)},
{\frac{\lambda_{1}}{\lambda_{2}}}={\frac{X_{1}X_{1}^{\prime}}{X_{2}X_{2}^{\prime}}}\qquad{\frac{\lambda_{2}}{\lambda_{3}}}={\frac{X_{2}X_{2}^{\prime}}{X_{3}X_{3}^{\prime}}}\qquad{\frac{\lambda_{3}}{\lambda_{1}}}={\frac{X_{3}X_{3}^{\prime}}{X_{1}X_{1}^{\prime}}}
B^{\mu\nu}(s;q)\simeq\frac{1}{32\pi^{2}}(g^{\mu\nu}-\frac{q^{\mu}q^{\nu}}{q^{2}})(\frac{\Lambda}{q^{2}})^{s/4\pi}\frac{\Gamma(s/4\pi)\Gamma(-s/4\pi)\Gamma(1-s/4\pi)^{2}}{\Gamma(2-s/2\pi)}
\gamma=\frac{1}{2}c_{jk}^{i}\psi_{i}\chi^{j}\chi^{k}+\frac{1}{2}f_{k}^{ij}\psi_{i}\psi_{j}\chi^{k}+\frac{1}{3!}\varphi^{ijk}\psi_{i}\psi_{j}\psi_{k}.
E^{(2)}{}_{4}=3{\frac{L_{*}{}^{\prime}}{\kappa_{1}}}\kappa_{3}\kappa_{4}+2L_{*}\kappa_{4}({\frac{\kappa_{3}}{\kappa_{1}}})^{\prime}+{\frac{L_{*}}{\kappa_{1}}}\kappa_{3}\kappa_{4}{}^{\prime}=0\,.
=m\biggl[\varphi_{m}^{*}(x)\partial_{0}E_{m}(x)-E_{m}^{*}(x)\partial_{0}\varphi_{m}(x)+\varphi_{0}^{*}(x)\partial_{0}\varphi(x)-\varphi^{*}(x)\partial_{0}\varphi_{0}(x)
A_{k}\,=\,\left(\begin{array}{ccc}{-6\,i\,\gamma_{k}-\frac{4}{3}\,a_{5}\,\gamma_{ijl}\,\epsilon^{ijlm}\,h_{mk}}&{,}&{(\mu_{3}\,\mathcal{F}^{lm}\,+\,\mu_{4}\,{\tilde{F}}^{lm})\,h_{mk}\,\gamma_{l}}\\{(\mu_{1}\,\mathcal{F}^{lm}\,+\,\mu_{2}\,{\tilde{F}}^{lm})\,h_{mk}\,\gamma_{l}}&{,}&{-6\,i\,\gamma_{k}+\frac{4}{3}\,a_{5}\,\gamma_{ijl}\,\epsilon^{ijlm}\,h_{mk}\}\\\end{array}\right)
y=C_{+}e^{-\gamma_{+}(t-\tau)}+C_{-}e^{-\gamma_{-}(t-\tau)}\,,\\gamma_{\pm}=\frac{1\pmK}{2}\,,\K=\sqrt{1-4\varepsilon}\,,\C_{\pm}=\mp\frac{\tilde{w}+\gamma_{\mp}\tilde{u}}{K}\,.
\!\!S\!=\!-\!\!\displaystyle\int\!d\zeta_{A}^{(-4)}du\left({\widetilde{q_{1}^{+}}}D^{++}q_{1}^{+}+{\widetilde{q_{2}^{+}}}D^{++}q_{2}^{+}+V^{++}({\widetilde{q_{1}^{+}}}q_{2}^{+}-{\widetilde{q_{2}^{+}}}q_{1}^{+}+\xi^{++})\right)
\langle\varphi^{2}(x)\rangle^{(ab)}\approx-\frac{(b-a)^{1-D}}{(4\pi)^{D/2}\Gamma(D/2)}\int_{m_{0}}^{\infty}dy\,\frac{(y^{2}-m_{0}^{2})^{D/2-1}}{c_{b}(y)e^{2y}/c_{a}(y)-1}F(y,r),\quada,b\to\infty,b-a={\mathrm{const}}.
{\calF}(z)=\left(\begin{array}{cc}{\bar{\epsilon}^{-1}}&{0}\\{0}&{-\zeta}\\\end{array}\right)\tilde{{\calF}}(-\tilde{z})^{t}\left(\begin{array}{cc}{\epsilon^{-1}}&{0}\\{0}&{-\bar{\zeta}}\\\end{array}\right)
J_{\mu}^{(j,0)}=i\sqrt{j(j+1)}\frac{(\bar{u}\partial_{\mu}u-u\partial_{\mu}\bar{u})}{(1-|u|^{2})^{j+1}}\sum_{n=0}^{j-1}\tilde{\gamma}_{n}^{(j,0)}|u|^{2n},
g=\left(\begin{array}{cc}{1}&{}\\{x^{-}}&{1}\\\end{array}\right)\left(\begin{array}{cc}{e^{\rho}}&{}\\{}&{e^{-\rho}}\\\end{array}\right)\left(\begin{array}{cc}{1}&{x^{+}}\\{}&{1}\\\end{array}\right)=\left(\begin{array}{cc}{e^{\rho}}&{e^{\rho}x^{+}}\\{e^{\rho}x^{-}}&{e^{-\rho}+e^{\rho}x^{+}x^{-}}\\\end{array}\right)
\delta\psi_{A\mu}=\mathcal{D}_{\mu}\,\epsilon_{A}-\frac{1}{3}\,\mathcal{T}_{AB}^{\rho\sigma}\left(g_{\mu\rho}\,\gamma_{\sigma}-\frac{1}{8}\,\epsilon_{\mu\rho\sigma\lambda\nu}\,\gamma^{\lambda\nu}\right)\,\epsilon^{B}\,+\,S_{AB}\,\epsilon^{B}
\mathcal{S}_{cs}=-\frack{4\pi}\intd^{3}x\;\varepsilon^{\mu\nu\rho}\left(A_{\mu}^{a}\partial_{\nu}A_{\rho}^{a}+\frac13\;f_{abc}\mathrm{~}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}\right)\;,
+r_{2}\left(-\frac{Mc_{0}}{2\lambdar_{0}}\right)^{l_{1}-\frac{1}{2}}\frac{(l_{1}-\frac{3}{2})!}{u^{[\frac{1}{2}(l_{1}-\frac{1}{2})(l_{1}-\frac{3}{2})]}}\left(1+{\calO}\left(\frac{1}{l_{1}}\right)\right),
S_{[1/2]}^{c}\,=\,e^{i\vartheta_{[1/2]}^{c}}\,\left(\begin{array}{cc}{0}&{i\,\Theta_{[1/2]}}\\{-\,i\,\Theta_{[1/2]}}&{0}\\\end{array}\right)\,{\calK}\,\equiv\,{\calC}_{[1/2]}\,{\calK}\quad,
[j_{+}{}^{n},(j_{+}{}^{m})^{\dagger}]\prod_{p=1}^{k}(j_{+}{}^{i_{p}})^{\dagger}\left|{\calE}_{N},2N\right\rangle=n\delta_{n,m}\prod_{p=1}^{k}(j_{+}{}^{i_{p}})^{\dagger}\left|{\calE}_{N},2N\right\rangle,
\vec{\lambda}_{j}=\sum_{1}^{j}\vec{e}_{k},\quadk\leqn-2,\quad\vec{\lambda}_{n-1}={\frac{1}{2}}\left(\sum_{1}^{n-1}\vec{e}_{k}-\vec{e}_{n}\right),\quad\vec{\lambda}_{n}=\sum_{1}^{n}{\frac{\vec{e}_{k}}{2}}.
{\check{\omega}}^{u}=[{\check{I}}^{(k)\,-1}(q^{(k)},\mu^{(k)})]^{uv}{\check{S}}_{q}^{v}-{\check{A}}_{\mu}^{(k)u}(q^{(k)},\mu^{(k)})\,{\dot{q}}^{(k)\mu}.
I_{m}=\intd^{4}x\sqrt{{g}}[-\frac{1}{2}D_{\mu}\phi(D^{\mu}\phi)^{*}-\frac{1}{2}D_{\mu}\Sigma(D^{\mu}\Sigma)^{*}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{4}H_{\mu\nu}H^{\mu\nu}-V(|\phi|,|\Sigma|,\Lambda)],
R={\frac{1}{12}}\left(\begin{matrix}{s}&{0}\\{0}&{s}\\\end{matrix}\right)+\left(\begin{matrix}{0}&{B}\\{B^{T}}&{0}\\\end{matrix}\right)+\left(\begin{matrix}{W^{+}}&{0}\\{0}&{0}\\\end{matrix}\right)+\left(\begin{matrix}{0}&{0}\\{0}&{W^{-}}\\\end{matrix}\right),
\left\{K(\Lambda),K(\bar{\Lambda})\right\}~=~\sum_{I=1}^{n_{1}}k_{I}^{1}\int_{\partialD_{1}}\Lambda_{I}^{1}d\bar{\Lambda}_{I}^{1}~+\sum_{J=1}^{n_{2}}k_{J}^{2}\int_{\partialD_{2}}\Lambda_{J}^{2}d\bar{\Lambda}_{J}^{2}~,
\[(\ell_{j}^{+}\ell_{j}^{-})^{n},(\ell_{k}^{+}\ell_{k}^{-})^{m}]=-2\omega\tilde{g}\sum_{t=0}^{n-1}\left((\ell_{j}^{+}\ell_{j}^{-})^{t+m}m_{jk}(\ell_{j}^{+}\ell_{j}^{-})^{n-t-1}-(\ell_{j}^{+}\ell_{j}^{-})^{t}m_{jk}(\ell_{j}^{+}\ell_{j}^{-})^{m+n-t-1}\right).
\overline{{{\cal{A}}_{i}^{\gamma}}}({\bf{r}})=\left({\delta}_{i\,{\gamma}}-\frac{r_{i}\,r_{\gamma}}{r^{2}}\right)\left(\frac{\overline{{\cal{N}}}}{gr}+{\varphi}_{A}\right)+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\varphi}_{C}
{\calL}=\sum_{i}\frac{1}{2}\partial_{\mu}\Phi_{i}\partial^{\mu}\Phi_{i}-\sum_{i}\frac{1}{2}m_{i}^{2}\Phi_{i}^{2}+\frac{g}{\sqrt{2}}\left(\Phi_{1}^{2}+\Phi_{2}^{2}\right)\Phi_{3}.
\left(u_{\underline{{m}}}^{~\underline{{a}}}\right)=\left(u_{\underline{{m}}}^{a},u_{\underline{{m}}}^{~i}\right)~~\in~~SO(1,D-1)~~~~\Leftrightarrow~~~~u_{\underline{{m}}}^{~\underline{{a}}}\eta^{\underline{{m}}\underline{{n}}}u_{\underline{{n}}}^{~\underline{{b}}}=\eta^{\underline{{a}}\underline{{b}}}
\left\{\begin{array}{cc}{v_{\alpha,j}}&{\left|\begin{array}{c}{v_{\alpha,j}\equivv_{\alpha}+(2j-l-1)\frac{\eta}{2}+\frac{\nu}{2}\quad\bmod({\bfZ}\oplus\frac{\tau}{2}{\bfZ})}\\{\qquad\mathrm{for}\quadj=1,2,\cdots,l,\quad\mathrm{and}\quadv_{\alpha}:\,\mathrm{pure~imaginary}}\\\end{array}\right.}\\\end{array}\right\}.
\int_{\mathfrak{A}^{b}}\alpha_{a}=\int_{\mathcal{M}}\alpha_{a}\wedge\beta^{b}=\delta_{a}^{b}\qquad\int_{\mathfrak{B}^{a}}\beta^{b}=\int_{\mathcal{M}}\beta^{b}\wedge\alpha_{a}=-\delta_{a}^{b}.
{\calM}_{T}\to\omega_{T}^{T}{\calM}_{T}\omega_{T},\\\{\calM}_{U}\to\omega_{U}^{T}{\calM}_{U}\omega_{U},\\\{\calF}_{\mu\nu}\to(\omega_{T}^{-1}\otimes\omega_{U}^{-1}){\calF}_{\mu\nu},
G({\bfr},{\bfr}^{\prime};-iT)={\frac{1}{2\pi}}\int_{0}^{\infty}k\,dk\,e^{-{\frac{\hbar\alpha^{2}Tk^{2}}{2m}}}\sum_{n}J_{{\frac{|n|}{\alpha}}}(kr)\,J_{{\frac{|n|}{\alpha}}}(kr^{\prime})\,e^{in(\theta^{\prime}-\theta)}.
\mathcal{G}\left({\calP}_{0}^{(1)}\oplus{\calP}_{0}^{(2)},{\calP}_{i}^{(1)}\oplus{\calP}_{i}^{(2)}\right)=\mathcal{G}\left({\calP}_{0}^{(1)},{\calP}_{i}^{(1)}\right)\,\mathcal{G}\left({\calP}_{0}^{(2)},{\calP}_{i}^{(2)}\right).
\dot{Q}_{B}=\frac{q^{2}\;\dot{c}\;(p^{2}-m^{2})}{1+q^{2}}(1-q^{2})\quad\mathrm{and}\quad\dot{Q}_{r}=\frac{d}{d\tau}(\lambda\;e)\;\frac{p^{2}-m^{2}}{1+q^{2}}(1-q^{2}).
\int_{-\infty}^{\infty}dk\,\left(\rho_{\perp}(k;\kappa^{2})-\rho_{0}(k)\right)\Phi(k^{2};\beta_{T},\mu)=\frac{1}{2\pi}\int_{-\infty}^{\infty}dk\,\Phi(k^{2};\beta_{T},\mu)\frac{d}{dk}\delta_{\perp}(k;\kappa^{2})
G\left(x,\bar{x};y,\bar{y}\right)\equiv\left<\mathrm{tr}{\,}\left(\bar{\psi}\left(\bar{x}\right)\Phi\left(\bar{x},x\right)\psi(x)\right)\left(\bar{\psi}(y)\Phi\left(y,\bar{y}\right)\psi\left(\bar{y}\right)\right)\right>,
f_{m}(k)\=\\frac{2(\partial_{r}J_{m}(kr)|_{r=R}H_{m-\beta}^{(1)}(kR)-\partial_{r}H_{m-\beta}^{(1)}(kr)|_{r=R}J_{m}(kR))}{H_{m-\beta}^{(2)}(kR)\partial_{r}H_{m-\beta}^{(1)}(kr)|_{r=R}-H_{m-\beta}^{(1)}(kR)\partial_{r}H_{m-\beta}^{(2)}(kr)|_{r=R}}\,
s_{n}(u)=\frac{(u_{n}e_{\bar{z}},e_{\bar{z}})}{(e_{\bar{z}},e_{\bar{z}})}=\frac{(u_{n}e_{\bar{z}})(z)}{e_{\bar{z}}(z)}=\frac{u_{n}(1+z\bar{z})^{n-1}}{(1+z\bar{z})^{n-1}}.
\Gamma_{A^{\mu}A^{\nu}}(p,-p)\equiv\Gamma_{\mu\nu}(p,-p)=(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}})\Gamma^{T}(p^{2})+\frac{p_{\mu}p_{\nu}}{p^{2}}\Gamma^{L}(p^{2})\;.
{\left(\rho_{ab}^{(2)}\right)}_{ij}=\left(\begin{array}{cccc}{0}&{-\delta_{ij}}&{0}&{-\partial_{i}^{x}}\\{\delta_{ij}}&{0}&{-\partial_{i}^{x}}&{0}\\{0}&{\partial_{j}^{x}}&{0}&{0}\\{\partial_{j}^{x}}&{0}&{0}&{0}\\\end{array}\right)\delta_{ab}\delta(\vec{x}-\vec{y}),
S^{\mathrm{grav.}}=-{\frac{T_{KK11}}{2}}l_{11}\intd^{6}\xi\sqrt{|\gamma|}\\left[\hat{k}^{2/3}e^{-\frac{2}{3}\hat{\phi}}\gamma^{ij}D_{i}\hat{X}^{\hat{\mu}}D_{j}\hat{X}^{\hat{\nu}}\hat{g}_{\hat{\mu}\hat{\nu}}-4\right]\,,\,
G^{ij}={\bfe}^{*i}\cdot{\bfe}^{*j}\mathrm{~and~}G_{ij}={\bfe}_{i}\cdot{\bfe}_{j},\mathrm{~where~}G^{ij}G_{jk}={\bfe}^{*i}\cdot{\bfe}_{k}=\delta_{k}^{i}.
Q=\sum_{\sigma}\int\frac{d^{3}{\bfp}}{(2\pi)^{3}}\,\left[a_{\sigma}^{\dagger}({\bfp})a_{\sigma}({\bfp})-b_{\sigma}({\bfp})b_{\sigma}^{\dagger}({\bfp})+c_{\sigma}^{\dagger}({\bfp})c_{\sigma}({\bfp})-d_{\sigma}({\bfp})d_{\sigma}^{\dagger}({\bfp})\right]\quad,
f^{\lambda}=\left(f_{1}\delta_{\sigma}^{\lambda}g^{\mu\nu}+f_{2}g^{\mu\lambda}\delta_{\sigma}^{\nu}+f_{3}g^{\nu\lambda}\delta_{\sigma}^{\mu}\right)\bar{\Gamma}_{~\mu\nu}^{\sigma}+G_{~~~~\sigma}^{\lambda\rho\lambda~\alpha\beta}\nabla_{\rho}\nabla_{\lambda}\bar{\Gamma}_{~\alpha\beta}^{\sigma}
{\calW}_{a}\star_{N}{\calW}_{b}\,(\psi^{\prime},\bar{\psi}^{\prime},N)=\frac{1}{N}\,{\calW}_{a}\star_{1}{\calW}_{b}\,(\psi^{\prime},\bar{\psi}^{\prime},1)+\frac{N-1}{N}\,{\calW}_{a}\,(\psi^{\prime},\bar{\psi}^{\prime})\,{\calW}_{b}\,(\psi^{\prime},\bar{\psi}^{\prime})\,.