harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
L_{n}^{(0)}=\frac{1}{2}\sum_{k}\alpha_{k}^{\mu}\alpha_{n-k}^{\nu}\eta_{\mu\nu}{,}\quad\bar{L}{}_{n}^{(0)}=\frac{1}{2}\sum_{k}\bar{\alpha}{}_{k}^{\mu}\bar{\alpha}{}_{n-k}^{\nu}\eta_{\mu\nu}
(D\theta)^{I}=\left(d+{\frac{1}{4}}(\omega^{ab}\gamma_{ab}+\omega^{a^{\prime}b^{\prime}}\gamma_{a^{\prime}b^{\prime}})\right)\theta^{I}-{\frac{e}{2}}i\epsilon^{IJ}(e^{a}\gamma_{a}+ie^{a^{\prime}}\gamma_{a^{\prime}})\theta^{J}
S=\intd^{4}xe^{-\frac{a{\chi}}{4}{\frac{\partial}{{\partial}x^{\mu}}}{\frac{\partial}{{\partial}x_{\mu}}}}\bigg[<x|\frac{1}{2}[D,\hat{\Phi}]_{\times}{\times}[D,\hat{\Phi}]_{\times}-\frac{m^{2}}{2}{\hat{\Phi}}{\times}{\hat{\Phi}}-\frac{g^{4}}{4!}({\hat{\Phi}}{\times}{\hat{\Phi}}){\times}({\hat{\Phi}}{\times}{\hat{\Phi}})|x>\bigg].
\begin{array}{c}{\partial_{t}B_{t}=\partial_{x}B_{x}+\left[A_{x},B_{x}\right],}\\{-\partial_{t}A_{x}+\left[B_{x},B_{t}\right]=0,}\\{\partial_{x}B_{t}-\partial_{t}B_{x}+\left[A_{x},B_{t}\right]=0.}\\\end{array}
\langle\phi_{\ell_{1}}(1)\phi_{\ell_{2}}(2)j^{\mu}(3)\rangle=(x_{12}^{2})^{\frac{D-2-\ell_{1}-\ell_{2}}{2}}(x_{13}^{2})^{\frac{\ell_{2}-\ell_{1}-D+2}{2}}(x_{23}^{2})^{\frac{\ell_{1}-\ell_{2}-D+2}{2}}\left(\frac{x_{13}^{\mu}}{x_{13}^{2}}-\frac{x_{23}^{\mu}}{x_{23}^{2}}\right)\;.
\begin{array}{ll}{h_{3}=h_{4}~~~}&{~~~h_{2}+h_{3}+2h_{8}+2c_{1}=0}\\{{}}&{{}}\\{h_{5}=h_{6}~~~}&{~~~h_{6}+h_{7}+2h_{8}+2c_{1}=0}\\{{}}&{{}}\\{h_{9}=h_{10}~~~}&{~~~h_{9}=h_{8}+2c_{1}}\\\end{array}
S_{\mathrm{topol}}\approx12\pi^{2}\int_{0}^{r_{w}}dr{\frac{1}{g^{2}(r)}}R^{\prime\prime}(r)(1-(R^{2}(r)))\equiv{\frac{12\pi^{2}}{g_{w}^{2}}}\int_{0}^{r_{w}}drR^{\prime\prime}(r)(1-R^{2}(r))\,
{\calL}=R-\frac{1}{2}\partial_{\mu}{\varphi}^{\hat{\imath}}\delta_{\hat{\imath}\hat{\jmath}}\partial^{\mu}{\varphi}^{\hat{\jmath}}-{\frac{1}{2}}\sum_{\vec{\alpha}}e^{\alpha_{\hat{\imath}}\varphi^{\hat{\imath}}}\partial_{\mu}\chi^{\vec{\alpha}}\partial^{\mu}\chi^{\vec{\alpha}}+\dots\,,
G^{AB}=\left(\begin{matrix}{\lambda^{\mu\nu}}&{\lambda^{\mu\sigma}y^{k}A_{\sigma}^{kj}}\\{\lambda^{\nu\sigma}y^{k}A_{\sigma}^{ki}}&{\delta^{ij}+y^{k}y^{l}A_{\sigma}^{ik}A_{\rho}^{jl}\lambda^{\sigma\rho}}\\\end{matrix}\right)
{\calL}^{vec}=g_{A\bar{B}}\partial_{\mu}X^{A}\partial^{\mu}\bar{X}^{B}+g_{A\bar{B}}\bar{\lambda}^{iA}\gamma^{\mu}\partial_{\mu}\lambda_{i}^{\bar{B}}+\mathrm{Im}(F_{AB}{\calF}_{\mu\nu}^{-A}{\calF}^{-B\,\mu\nu})+...,
\left(\begin{array}{ccc}{\overline{{D}}^{2}}&{0}&{0}\\{0}&{\overline{{D}}^{2}}&{0}\\{0}&{0}&{\overline{{D}}^{2}}\\\end{array}\right)\left(\begin{array}{c}{\phi^{3}}\\{W_{0}^{3}}\\{W_{i}^{3}}\\\end{array}\right)=0,
\psi_{{\bfq}}({\bfr})=\phi_{{\bfq}}({\bfr})+\frac{2M}{\hbar^{2}}\intd^{d}{\bfr^{\prime}}\,{\mathcalG}_{d}^{(+)}({\bfr}-{\bfr^{\prime}};q)\,V({\bfr^{\prime}})\,\psi_{{\bfq}}({\bfr^{\prime}})\;
+\lambdae^{3}\int_{\alpha}^{1}dx\int{\frac{d^{2}k_{\perp}}{(2\pi)^{3}}}{\frac{\bar{u}(\bar{p},\sigma)2x(1-x)[(1-x)\not{p}p^{+}-(1-x)m^{2}\gamma^{+}-p^{+}m]u(p,s)}{[k_{\perp}^{2}+m^{2}x^{2}][k_{\perp}^{2}+g_{1}m^{2})]}}\;,
\gamma^{0}=\left(\begin{array}{cc}{\sigma_{3}}&{0}\\{0}&{\sigma_{3}}\\\end{array}\right)\gamma^{1}=\left(\begin{array}{cc}{i\sigma_{2}}&{0}\\{0}&{i\sigma_{2}}\\\end{array}\right)\gamma^{2}=\left(\begin{array}{cc}{-i\sigma_{1}}&{0}\\{0}&{i\sigma_{1}}\\\end{array}\right)\gamma^{3}=\left(\begin{array}{cc}{0}&{i\sigma_{1}}\\{i\sigma_{1}}&{0}\\\end{array}\right)
I_{3}=\frac{1}{q_{0}}\int\frac{dr_{0}d^{D-1}\vec{k}}{(\frac{r_{0}^{2}}{q_{0}^{2}}-\vec{k}^{2}+r_{0}-y)(\frac{r_{0}^{2}}{q_{0}^{2}}-\vec{k}^{2}-r_{0}-y)(\frac{r_{0}^{2}}{q_{0}^{2}}-(\vec{k}-\vec{p})^{2})}.
V_{eff}(\sigma)={\frac{\sigma^{2}}{2\lambda}}-\left[\frac{\Lambda\sigma^{2}}{2\pi^{3/2}}+\frac{\sqrt{2}}{\pi}(eB)^{3/2}\zeta\left(-\frac{1}{2},1+\frac{\sigma^{2}}{2eB}\right)+\frac{1}{2\pi}eB\sigma\right]
{\bfD_{3}}(0,\sigma,0,\nu_{1},\nu_{2},\nu_{3})=\frac{1}{\mathrm{i}\pi^{n/2}\Gamma\left(3-\frac{n}{2}\right)}\int\frac{\mathrm{d}^{n}p}{(p^{2}-m^{2})^{\nu_{3}}}\left.J_{011}(\sigma,\nu_{1},\nu_{2};p^{2},m)\right|_{m=1}\;.
R_{acbd}=-\frac{U_{B}^{2}}{16}\left(g_{ab}^{0}g_{cd}^{0}-g_{ad}^{0}g_{cb}^{0}\right)+\frac{1}{4}\left(\frac{\partialU_{B}}{\partial\phi}\right)^{2}(p_{a}g_{[cd}^{0}p_{b]}-p_{c}g_{a[d}^{0}p_{b]})
\tilde{S}_{d-h-1}=\int{\frac{e^{2}}{4}}d\tilde{\phi}_{d-h}\wedge*d\tilde{\phi}_{d-h}+{\frac{\Lambda^{2}}{4}}\left(\tilde{\phi}_{d-h}-d\tilde{\omega}_{d-h-1}\right)\wedge*\left(\tilde{\phi}_{d-h}-d\tilde{\omega}_{d-h-1}\right)\,
\chi=\left[\begin{matrix}{-{\frac{r+\hat{P}}{r}}}&{0}&{\cdot}&{{\frac{|P|}{r}}}&{0}\\{0}&{-{\frac{r+2\beta-\hat{Q}}{r+2\beta}}}&{\cdot}&{0}&{{\frac{|Q|}{r+2\beta}}}\\{{\cdot}}&{{\cdot}}&{{\bfI}}&{{\cdot}}&{{\cdot}}\\{{\frac{|P|}{r}}}&{0}&{\cdot}&{{\frac{r+2\beta-\hat{P}}{r}}}&{0}\\{0}&{{\frac{|Q|}{r+2\beta}}}&{\cdot}&{0}&{{\frac{r+\hat{Q}}{r+2\beta}}}\\\end{matrix}\right],
\tilde{\psi}_{i}(u_{i})=\sum_{n=0}^{\infty}\frac{\lambda^{n}}{2^{n}n!}\oint_{C_{1}}\frac{dx_{1}}{2\pi}\dots\oint_{C_{n}}\frac{dx_{n}}{2\pi}\psi_{i}(_{i})\psi(x_{n})\dots\psi(x_{1})\.
S_{t}=\int\eta_{A}F^{A}+\intd\tau\left[p_{a}(D_{\tau}q)^{a}-{\frac{1}{2}}N(p^{2}+m^{2})+q_{a}A_{\mu}^{A}\dot{x}^{\mu}-{\frac{1}{2}}q^{a}\epsilon_{ab}q^{b}\right]
G(\vec{\alpha}^{\ast},\vec{\gamma},t)={\frac{\mathrm{exp}\(-|\vec{\alpha}|^{2}/2-|\vec{\gamma}|^{2}/2)}{\sqrt{detM_{1}}}}\mathrm{exp}\biggl(-{\frac{1}{2}}\vec{\alpha}^{\ast}M_{1}^{-1}M_{2}\vec{\alpha}^{\ast}+\vec{\alpha}^{\ast}M_{1}^{-1}\vec{\gamma}+{\frac{1}{2}}\vec{\gamma}M_{3}M_{1}^{-1}\vec{\gamma}\biggr)\\.
\left(\begin{array}{c}{\{M_{Aq}^{f}\}^{Y^{1}q}}\\{\{M_{Aq}^{f}\}^{Y^{2}q}}\\\end{array}\right)=\left(\begin{array}{cc}{0}&{i}\\{-i}&{0}\\\end{array}\right)\left(\begin{array}{c}{\{M_{Aq}^{b}\}^{Y^{1}q}}\\{\{M_{Aq}^{b}\}^{Y^{2}q}}\\\end{array}\right).
\phi({\bfx},x^{-})=\int_{0}^{\infty}{\frac{dP^{+}}{\sqrt{4\piP^{+}}}}(a({\bfx},P^{+})e^{-iP^{+}x^{-}}+a^{\dagger}({\bfx},P^{+})e^{iP^{+}x^{-}})
S=\intd^{2}x\left[\pi^{-}{\dot{A}_{-}}+{\varphi}^{\prime}{\dot{\varphi}}+({\pi^{-}}^{\prime}+2e\varphi^{\prime}+M^{2}A_{-})A_{+}-\frac{1}{2}(\pi^{-}+2e\varphi)^{2}\right],
<.....>=\sum_{n_{1}=0}^{r}\sum_{n_{2}=n_{1}}^{r}\sum_{m_{1}=0}^{s}\sum_{m_{2}=0}^{s}\sum_{m_{3}=0}^{s-m_{1}}c_{m_{1},m_{2},m_{3}}^{n_{1}n_{2}}{\tilde{a}}_{m_{1},m_{2},m_{3}}^{n_{1}n_{2}}(l),
D_{l}A_{j_{1}j_{2}j_{3}j_{4}}=\frac{\partialA_{j_{1}j_{2}j_{3}j_{4}}}{\partialu^{l}}-A_{kj_{2}j_{3}j_{4}}\Gamma_{j_{1}l}^{k}-A_{j_{1}kj_{3}j_{4}}\Gamma_{j_{2}l}^{k}-A_{j_{1}j_{2}kj_{4}}\Gamma_{j_{3}l}^{k}-A_{j_{1}j_{2}j_{3}k}\Gamma_{j_{4}l}^{k}.
S\!=\!\!\int\!\!\!\sqrt{-g}d^{D}\!x\!\left[-\sum_{k=1}^{N}\bar{\psi}_{k}\gamma^{\alpha}\nabla_{\alpha}\psi_{k}+\frac{\lambda_{0}}{2N}\left(\sum_{k=1}^{N}\bar{\psi}_{k}\psi_{k}\right)^{2}\right],
F^{\prime}(0)=-\sum_{n,n_{1}}\frac{Z(n/\mu^{2}+n_{1}/\mu_{1}^{2})}{(2n+2n_{1}+3)!n!n_{1}!}\frac{(g/\mu)^{2n}(g/\mu_{1})^{2n_{1}}}{(16\pi^{2})^{n+n_{1}}}\,.
\partial_{\alpha}\,\Gamma^{\,\alpha,\,\alpha_{0}}(\Phi_{c})=\Bigl[\,\partial_{\alpha}\,\left(\,L^{\,\alpha,\,\alpha_{0}}(\Phi)+I^{\,\alpha,\,\alpha_{0}}\right)\Bigr]_{\,\Phi=\tilde{C}_{\alpha}^{\,\alpha_{0}}\!J(\Phi_{c})}
{\phi^{(+)}}_{j,m}=\left(\begin{matrix}{{({\frac{l+{\frac{1}{2}}+m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m-{\frac{1}{2}}}}}\\{{({\frac{l+{\frac{1}{2}}-m}{2l+1}})^{\frac{1}{2}}}Y_{l,{m+{\frac{1}{2}}}}}\\\end{matrix}\right),
\sum_{a=1}^{J_{1}-1}D_{x_{1}x_{2},y_{1}y_{2}}^{J_{1}J_{2},K_{1}K_{2};a}=D_{x_{1}x_{2},y_{1}y_{2}}^{J_{1}J_{2},K_{1}K_{2};0}\sum_{a=1}^{J_{1}-1}q^{a}=D_{x_{1}x_{2},y_{1}y_{2}}^{J_{1}J_{2},K_{1}K_{2};0}\,\frac{q^{J_{1}}-q}{q-1}
\langle\prod_{i=1}^{N}e^{\alpha_{i}\Phi(Z_{i})}\rangle=\left(\frac{\mu}{2\pi}\right)^{s}\frac{\Gamma(-s)}{b}\langle\left(\int\hat{E}e^{b\Phi^{\prime}}\right)^{s}\prod_{i=1}^{N}e^{\alpha_{i}\Phi^{\prime}(Z_{i})}\rangle_{S_{SL}^{\prime}}
\int_{0}^{\infty}dy^{\prime}\\chi(y,y^{\prime})e^{-ty^{\prime}}=\sum_{k=0}^{\infty}\frac{2^{k}(ty)^{k}}{(2k-1)!!}\binom{-\frac{1}{2}}{k}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}(ty)^{k}=e^{-ty},
{\bf\alpha}^{\mu}=({\bfm}_{T^{\mu}}\,{\bfF}_{\mu})^{-1}d({\bfm}_{T^{\mu}}\,{\bfF}_{\mu}),\qquad{\bf\bar{\alpha}}^{\mu}=({\bf\bar{F}}_{\bar{\mu}}\,{\bf\bar{m}}_{\bar{T}^{\mu}})d({\bf\bar{F}}_{\bar{\mu}}\,{\bf\bar{m}}_{\bar{T}^{\mu}})^{-1},
{\calC}_{k}(\eta)=\frac{1}{\sqrt{k}}\sqrt{\frac{2}{\pi}}e^{-i\frac{k\eta}{\sqrt{1+r^{2}}}+i\frac{\pi}{4}}\biggl[\frac{2ik\eta}{\sqrt{1+r^{2}}}\biggr]^{\frac{3}{2}}U[\frac{3}{2}-\frac{i}{2}r\epsilon,3,2i\frac{k\eta}{\sqrt{r^{2}+1}}].
A_{i}=\left[\begin{array}{cccc}{1}&{0}&{\cdots}&{0}\\{0}&{\omega_{i}}&{\ddots}&{0}\\{\cdots}&{\cdots}&{\ddots}&{0}\\{0}&{0}&{\cdots}&{\omega_{i}^{p_{i}-1}}\\\end{array}\right];\quad\quadB_{i}=\left[\begin{array}{cccc}{0}&{1}&{\cdots}&{0}\\{0}&{0}&{\ddots}&{0}\\{\cdots}&{\cdots}&{\ddots}&{1}\\{1}&{0}&{\cdots}&{0}\\\end{array}\right]
V=\sum_{\alpha,\beta\in\Xi_{0}^{\prime}}\sum_{n,m,\tilde{n},\tilde{m}=0,1}V\Big[\begin{array}{c}{\alpha+nb_{3}+{m}b_{4}}\\{\beta+\tilde{n}b_{3}+\tilde{m}b_{4}}\\\end{array}\Big].
0=\frac{\partial}{\partialx^{i}}\left[\frac{C^{i}(x)}{\chi(x)}\right]=\left[\frac{\partial}{\partialx^{i}}C^{i}(x)\right]/\chi(x)-\frac{\partial}{\partialx^{i}}\chi(x)C^{i}(x)/\chi(x)^{2}.
\begin{array}{c}{L\int_{0}^{1}dx\int_{0}^{1}dt\tr\left\{\gamma^{0}e^{\mu\beta(t-t^{\prime})}\gamma^{1}\left(S_{+}+S_{-}\right)\right\}_{(t^{\prime},x^{\prime})=(t,x)}=}\\{\frac\mu\piL\beta.}\\\end{array}
S_{HG}=\intd^{4}x\sqrt{-g^{L}}M^{3}g_{L}^{\mu\nu}\partial_{5}G_{\mu\nu}\Big|_{y=L}-\intd^{4}x\sqrt{-g^{0}}M^{3}g_{0}^{\mu\nu}\partial_{5}G_{\mu\nu}\Big|_{y=0}\.
\langle0,0\vert\sum_{\alpha}\Bigl(\hat{N}_{\alpha}^{(B)}(t)+\hat{N}_{\alpha}^{(D)}(t)\Bigr)\vert0,0\rangle=\sum_{\alpha}\Bigl(f_{\alpha}^{(B+)*}(t)f_{\alpha}^{(B+)}+g_{\alpha}^{(D+)*}(t)g_{\alpha}^{(D+)}\Bigr).
I(g,{\bfA})=I_{WZNW}+\frac{k}{2\pi}\intd^{2}ztr({\bfA}\overline{{\partial}}gg^{-1}-\overline{{{\bfA}}}g^{-1}\partialg+{\bfA}\overline{{{\bfA}}}-g^{-1}{\bfA}g\overline{{{\bfA}}})
T=\left.\frac{1}{4\pi}\left(e^{-2\lambda}\right)^{\prime}\right|_{r=r_{+}}=\frac{(d-1)r_{+}^{4}+(d-3)kl^{2}r_{+}^{2}+(d-5)\tilde{\alpha}k^{2}l^{2}}{4\pil^{2}r_{+}(r_{+}^{2}+2\tilde{\alpha}k)}.
C_{1}=i\,e\left(\begin{array}{cccc}{0}&{0}&{0}&{0}\\{0}&{0}&{-\,\phi_{1}}&{-\,\phi_{2}}\\{0}&{\phi_{1}}&{0}&{A_{1}}\\{0}&{\phi_{2}}&{-\,A_{1}}&{0}\\\end{array}\right)
\{f,\,g\}\,=\,\sum_{a=1}^{2}\left(\frac{\partialf}{\partialp_{a}^{\mu}}\,\frac{\partialg}{\partialq_{a\mu}}\,-\,\frac{\partialf}{\partialq_{a}^{\mu}}\,\frac{\partialg}{\partialp_{a\mu}}\right)\,{.}
j_{\ell}(kr)={\frac{(kr)^{\ell}}{(2\ell+1)!!}}\;\Big(1+o((kr)^{2})\Big)\;,\quad{\frac{dj_{\ell}(kr)}{dr}}={\frac{\ellk^{\ell}r^{\ell-1}}{(2\ell+1)!!}}\;\Big(1+o((kr)^{2})\Big)
ds^{2}=e^{2\phi_{0}}\left(-(1-\epsilonr_{s}/\tilde{r})dt^{2}+\frac{{(\tilde{r}/r_{s}+1)}^{2}d\tilde{r}^{2}}{(1-\epsilonr_{s}/\tilde{r}){(\tilde{r}/r_{s})}^{2}}+{(\tilde{r}+r_{s})}^{2}d\Omega_{2}^{2}\right)
\intdq^{\prime\prime}e^{iF(q^{\prime},q^{\prime\prime},t)}\widetilde{\rho}(q^{\prime\prime},t)e^{-iG(q^{\prime\prime},q^{\prime\prime\prime},t)}={\frac{\delta(q^{\prime}-q^{\prime\prime\prime})}{\rho(q^{\prime})}}.
\beta(r)_{Nut}^{2}=\frac{\lambda\,c\,\,2^{n}\,r\,(r-L)\,L^{n}}{(n+1)(r+L)^{n}}\,\mathrm{{Appell}}F_{1}[n+1,2,-n,n+2,1-\frac{r}{L},\frac{L-r}{2L}].
\hat{\Psi}(m,z)=N_{m}\sqrt{\frac{g(z)}{k}}\left[\left\{\begin{array}{c}{1}\\{B}\\{e^{2iqz_{l}}}\\{e^{2iqz_{l}}B}\\\end{array}\right\}H_{2}^{+}\left(\frac{m}{k}g(z)\right)+\left\{\begin{array}{c}{A}\\{C}\\{e^{2iqz_{l}}A}\\{e^{2iqz_{l}}C}\\\end{array}\right\}H_{2}^{-}\left(\frac{m}{k}g(z)\right)\right]
D_{\mu\rho\nu\sigma}=\delta_{\mu\rho}^{\mu^{\prime}\rho^{\prime}}\,\delta_{\nu\sigma}^{\nu^{\prime}\sigma^{\prime}}\,\delta_{\rho^{\prime}\sigma^{\prime}}\,\frac{\partial}{\partialx^{\mu^{\prime}}}\frac{\partial}{\partialy^{\nu^{\prime}}}
S={\textstyle\frac{1}{16\piG_{N}^{(11)}}}\,C_{U}{\displaystyle\int}d^{8}x\sqrt{|g|}\,\big[R+{\textstyle\frac{1}{4}}\mathrm{Tr}\left({\calD}{\calM}{\calM}^{-1}\right)^{2}+{\textstyle\frac{1}{2}}(\partial\varphi)^{2}-{\textstyle\frac{1}{4}}e^{-\varphi}F^{m}{\calM}_{mn}F^{n}-{\calV}\big]\,,
H(x,y):=\int{\frac{d^{4}p}{(2\pi)^{4}}}\left({\frac{\chi}{p^{2}+M_{1}^{2}}}-{\frac{\chi}{p^{2}+M_{2}^{2}}}\right){\frac{1}{p^{2}}}{\frac{{\frac{-1}{\kappa}}+{\frac{p^{2}}{M^{2}}}\left(1+{\frac{p^{2}}{\tilde{M}^{2}}}\right)}{1+{\frac{p^{2}}{M^{\prime}{}^{2}}}}}e^{ip(x-y)}.
ds_{r^{2}<0}^{2}=(\frac{k_{3}}{r_{1}^{6}}-1)^{-2/3}dx^{\mu}dx^{\nu}\eta_{\mu\nu}+(\frac{k_{3}}{r_{1}^{6}}-1)^{1/3}(dr_{1}^{2}+r_{1}^{2}d\Omega_{7}^{2})
H(z_{2}^{\prime},z_{3}^{\prime},P_{2}^{\prime},P_{3}^{\prime})=-\int_{z_{0}}^{z_{A}^{\prime}}\frac{\partial\beta_{A}}{\partial\mu}(z_{2}^{\prime},z_{3}^{\prime},z_{A}^{\prime\prime})~dz_{A}^{\prime\prime}+f(z_{2}^{\prime},z_{3}^{\prime})
\omega_{\mathrm{\tiny~bhb}}(\phi_{0}(x_{1})\phi_{0}(x_{2}))\doteq(2\pi)^{-3}\int\!dp\,e^{-i(x_{1}-x_{2})p}\varepsilon(p_{0})\delta(p^{2})\frac{1}{1-e^{-\eta(x_{1}+x_{2})p}},
\begin{array}{cc}{\bar{\varepsilon}_{i}=\left(\begin{array}{cc}{0}&{\varepsilon^{a}}\\{-\bar{\varepsilon}_{a}^{t}}&{0}\\\end{array}\right)~~~~}&{~~~~\bar{\rho}_{i}=\left(\begin{array}{cc}{\bar{\rho}^{a}}&{0}\\{0}&{-\rho_{a}^{t}}\\\end{array}\right)}\\\end{array}
I_{\omega}=(-)^{r_{5}}(\omega^{1/2})^{r_{3}^{~2}+r_{3}r_{4}-r_{1}r_{3}-r_{2}r_{3}-r_{3}r_{5}-r_{4}r_{5}}\Bigg[\frac{\Phi(r_{1})\Phi(r_{2})}{\Phi(r_{3})\Phi(r_{4})}\Bigg]^{1/2}
{\mathcalL}_{G}^{(1)}|=\frac{1}{\gamma^{2}}\,\,\frac{\partial_{z}v_{1}\partial_{\bar{z}}v_{1}+\partial_{z}v_{2}\partial_{\bar{z}}v_{2}}{1+v_{1}^{2}+v_{2}^{2}},~~~~~~~~{\mathcalL}_{G}^{(2)}|=\frac{1}{\gamma^{2}}\,\,\frac{\partial_{z}c\partial_{\bar{z}}c+\partial_{z}v_{0}\partial_{\bar{z}}v_{0}}{c^{2}+v_{0}^{2}-1}.
{\calL}_{1}\rightarrow{\calL}_{2}\,=\,{\calL}_{1}+{\frac{1}{1-\lambda_{++}\lambda_{--}}}\left[\lambda_{--}A_{+}^{2}+\lambda_{++}A_{-}^{2}\,+\,\left(1+\lambda_{++}\lambda_{--}\right)A_{+}A_{-}\right]
{\calF}_{5}={\calF}_{5}^{(3)},\quad{\calF}_{6}={}^{4}{\calF}_{6}^{(0)},\quad{\calF}_{7}={\calF}_{7}^{(1)},\quad{\calF}_{8}={}^{1}{\calF}_{8}^{(0)},\quad{\calF}_{9}={}^{2}{\calF}_{9}^{(0)}.
+{\frac{1}{2}}\Bigl[\eta^{\nu_{1}\mu_{2}}\eta^{\nu_{2}\mu_{3}}\eta^{\nu_{3}\mu_{4}}\eta^{\nu_{4}\mu_{1}}+(1\leftrightarrow2),(2\leftrightarrow3)+\mathrm{anti-}(\mu_{j}\leftrightarrow\nu_{j})\Bigr]\,.
(ds)^{2}=a^{2}(dX^{0})^{2}+\frac{K^{2}(X^{0})}{(1+\frac{1}{4}\epsilonr^{2})^{2}}\left[(dX^{1})^{2}+(dX^{2})^{2}+...+(dX^{D})^{2}\right]\,
M^{2}=p_{\alpha}p_{\beta}g^{\alpha\beta}=\sum_{m\ge0}\frac{1}{4}p_{-}^{2}[\overline{x}_{m}^{i}\mu_{ij}^{2}x_{m}^{j}]+\sum_{m>0}\big(\overline{p}_{m}^{i}\delta_{ij}p_{m}^{j}+m^{2}\overline{x}_{m}^{i}\delta_{ij}x_{m}^{j}\big),
g_{{\bf{k}},{\bf{q}}}({\bf{x}})=-e^{-i\mathrm{~}{\bf{q.x}}}(\frac{1}{2\mathrm{~}N\mathrm{~}\epsilon_{{\bf{q}}}})\Lambda_{{\bf{k}}}(-{\bf{q}})\omega_{{\bf{k}}}({\bf{q}})+i\mathrm{~}U_{{\bf{q}}}({\bf{x}})\Lambda_{{\bf{k}}}(-{\bf{q}})=-f_{{\bf{k}},{\bf{q}}}^{*}({\bf{x}})
\begin{array}{rcl}{S_{\mathrm{twisted}}}&{\sim}&{\intd^{6}y\sqrt{|\mathrm{det}g|}\left(1+({\tilde{\calF}}^{(4)})^{2}+\dots\right)-\intd^{6}y\sqrt{|\mathrm{det}g|}=}\\{}&{}&{}\\{}&{=}&{\intd^{6}y\sqrt{|\mathrm{det}g|}({\tilde{\calF}}^{(4)})^{2}+\dots\,,}\\\end{array}
\ell_{1}^{\mu}(\tau)=\bar{\ell}_{0}^{\mu}+{\frac{1}{2}}\delta^{\mu}+V^{\mu}\tau,\,\ell_{2}^{\mu}(\tau)=\bar{\ell}_{0}^{\mu}-{\frac{1}{2}}\delta^{\mu}+V^{\mu}\tau,
(2\pi\tau_{2})^{-23/2}|\eta(\tau)|^{-46}\cdot\left\{|\eta(\tau)|^{-2}\left[\sum_{n,w}q^{{\frac{1}{2}}{\frac{\alpha^{\prime}p_{L}^{2}}{2}}}{\bar{q}}^{{\frac{1}{2}}{\frac{\alpha^{\prime}p_{R}^{2}}{2}}}+\left(|\Theta_{3}\Theta_{4}|+|\Theta_{2}\Theta_{3}|+|\Theta_{2}\Theta_{4}|\right)\right]\right\}=\mathrm{Z}_{23}\cdot\mathrm{Z}_{\mathrm{orb.}},
\rho_{\pm}=\frac{(1+\alpha^{2})(M\pm\sqrt{M^{2}-\left(1-\alpha^{2}\right)Q^{2}})}{\left(1\pm\alpha^{2}\right)},\;\;\;a=\frac{2(1+\alpha^{2})J}{(1+\alpha^{2})\rho_{+}+(1-\alpha^{2}/3)\rho_{-}}.
{\calL}=\frac{1}{2}(\partial_{\mu}\Phi_{1}^{\prime})^{2}+\frac{1}{2}(\partial_{\mu}\Phi_{2}^{\prime})^{2}+\frac{1}{2}(\partial_{\mu}\Phi_{3}^{\prime})^{2}+\frac{m}{\pi}\mucos(2\sqrt{\pi}a\Phi_{1}^{\prime}),
\Phi_{p}=\prod_{k<l}\frac{(z_{k}-z_{l})^{2p}}{|z_{k}-z_{l}|^{2p}}\Psi_{0}(z_{1},...,z_{N})\equivU_{p}({\bfr}_{1},...,{\bfr}_{N})\Psi_{0}(z_{1},...,z_{N})
T^{+2}=-2i\pi_{q}^{1+}\pi_{q}^{1+}\!-\!2i\pi_{q}^{2+}\pi_{q}^{2+},\quadT^{-2}=-2i\pi_{\dot{q}}^{1-}\pi_{\dot{q}}^{1-}\!-\!2i\pi_{\dot{q}}^{2-}\pi_{\dot{q}}^{2-}
S_{B,\psi}=-\frac{\sigma}{2}\sum_{i=1}^{2}(-1)^{i}\Sigma((-1)^{i}\bar{R}_{i})\bar{\gamma}_{i}^{-2}\int\frac{d\nu}{2\pi}\tilde{\psi}_{b,i}^{*}\left(\frac{\partial\tilde{\psi}_{B}}{\partialr}\right)\Bigg|_{r=(-1)^{i}\bar{R}_{i}}+\intdt\hat{L}_{\psi}\,.
\begin{array}{lcl}{a_{1}}&{=}&{0}\\{a_{2}}&{=}&{\frac{2-n}{2n}u_{2}}\\{a_{3}^{\prime}}&{=}&{\frac{7\theta(n-2)(n-3)}{10n}u_{2}^{\prime\prime}-\frac{n-3}{2n}u_{3}^{\prime}-\frac{(n-2)(n-3)(5n+7)}{20\thetan(n^{3}-n)}u_{2}^{2}-2\thetaa_{2}^{\prime\prime}}\\{}&{}&{\vdots}\\\end{array}
{\tilde{X}}_{{\bf{q}}}=(-\frac{1}{2\mathrm{~}i\mathrm{~}N\mathrm{~}\epsilon_{{\bf{q}}}})\sum_{{\bf{k}}}[\Lambda_{{\bf{k}}}(-{\bf{q}})\omega_{{\bf{k}}}({\bf{q}})a_{{\bf{k}}}({\bf{q}})-\Lambda_{{\bf{k}}}({\bf{q}})\omega_{{\bf{k}}}(-{\bf{q}})a_{{\bf{k}}}^{\dagger}(-{\bf{q}})]
\psi_{-,m}^{(n)}(x)=\frac{1}{{\calN}_{m,-}^{(n)}}{\binom{(d_{+})^{m}\chi_{1}^{(n)}(z,\bar{z})}{-i\sqrt{m}(d_{+})^{m-1}\chi_{1}^{(n)}(z,\bar{z})}}
\left(\begin{array}{c}{\phi_{A}}\\{\varphi_{A}}\\\end{array}\right)=\left(\begin{array}{cc}{\frac{3}{4}}&{-\frac{\sqrt{7}}{4}}\\{-\frac{\sqrt{7}}{4}}&{-\frac{3}{4}}\\\end{array}\right)\left(\begin{array}{c}{\phi_{B}}\\{\varphi_{B}}\\\end{array}\right).
\left\vertT_{i}(\theta,g,A/e)\right\vert^{2}=\frac{\tilde{a}_{0}(\theta,g)+a_{2}(\theta,g)A^{2}+a_{4}(\theta,g)A^{4}}{a_{0}(\theta,g)+a_{2}(\theta,g)A^{2}+a_{4}(\theta,g)A^{4}},
\begin{array}{rcl}{U_{c}/m^{4}}&{=}&{\displaystyle+D_{m}^{2}(k_{1})D(k_{1}+p)[9D_{m}(k_{2})D_{m}(k_{1}+k_{2})}\\{}&{}&{\displaystyle+D(k_{2})D(k_{1}+k_{2})](1+K_{12})}\\{}&{}&{\displaystyle+4D^{2}(k_{1})D_{m}(k_{1}+p)D_{m}(k_{2})D(k_{1}+k_{2})}\\{}&{}&{\displaystyle+4(N-1)D_{m}^{2}(k_{1})D(k_{1}+p)D(k_{2})D(k_{1}+k_{2}),}\\\end{array}
{z}=y_{(1)}={\frac{-i}{\pi_{C^{\prime}}{\bar{\pi}}_{C}t^{C{C^{\prime}}}}}[{\frac{1}{\sqrt{2}}}({\bar{\alpha}}_{A}{\bar{\beta}}^{B}+{\bar{\beta}}_{A}{\bar{\alpha}}^{B}){\bar{\pi}}_{B}\omega^{A}-{\frac{1}{\sqrt{2}}}(\alpha_{A^{\prime}}\beta^{B^{\prime}}+\beta_{A^{\prime}}\alpha^{B^{\prime}})\pi_{B^{\prime}}{\bar{\omega}}^{A^{\prime}}],
{\calL}=-{\frac{1}{12}}[2g^{\mu\nu}h_{ij}{\calD}_{\mu}q^{i}{\calD}_{\nu}q^{j}+F_{\Lambda}^{ab}F_{\Lambda}^{ab}+2g^{2}{\calP}_{\Lambda}^{u}{\calP}_{\Lambda}^{u}]\,\varepsilon_{cdef}e^{c}e^{d}e^{e}e^{f}
k^{2}\varepsilon^{\mu\nu}-k^{\mu}k_{\alpha}\varepsilon^{\alpha\nu}-k^{\nu}k_{\alpha}\varepsilon^{\alpha\mu}+k^{\mu}k^{\nu}\varepsilon+\eta^{\mu\nu}(-k^{2}\varepsilon+k_{\alpha}k_{\beta}\varepsilon^{\alpha\beta})=0.
z_{1}a_{1}^{\dagger}+z_{2}a_{2}^{\dagger}=z_{+}a_{+}^{\dagger}+z_{-}a_{-}^{\dagger}\\\,\\\|z_{1}|^{2}+|z_{2}|^{2}=|z_{+}|^{2}+|z_{-}|^{2}\\\.
\mathcal{L}=\frac{1}{2}\left[J_{x^{\prime}}\,\partial_{x^{\prime}}^{-1}(\phi_{x^{\prime}})_{t^{\prime}}-\left(\frac{\phi_{x^{\prime}}}{J_{x^{\prime}}}+\frac{J_{x^{\prime}}}{\phi_{x^{\prime}}}\right)\right]=\frac{1}{2}\left(J_{x^{\prime}}\,\partial_{x^{\prime}}^{-1}(\phi_{x^{\prime}})_{t^{\prime}}-H\right)
{\tilde{\psi}}_{l}^{j}\left(y_{\alpha}+L_{1}\right)=e^{i2\pi\omegax_{\alpha}/L_{1}}e^{i2\pi\frac{j}{\kappa}-i\frac{\pi}{\kappa}\left(N_{A}-1\right)}{\tilde{\psi}}_{l}^{j+1}\left(y_{\alpha}\right)
q_{b}\approx-\frac{2^{-2\nu_{0}}a^{-D-1}\sigma^{1-D}}{\pi\nu_{0}S_{D}\Gamma^{2}(\nu_{0})}\frac{A+B\nu_{0}}{A-B\nu_{0}}\left(\frac{a}{r}\right)^{2\nu_{0}+D+1}\int_{0}^{\infty}dzz^{2\nu_{0}+2}F_{\nu_{0}}^{(q)}[K_{\nu_{0}}(z)],
M_{NSM}^{2}=g^{2}v^{2}\left(\begin{array}{ccccc}{2}&{-1}&{0}&{\cdots}&{+1}\\{-1}&{2}&{-1}&{\cdots}&{0}\\{0}&{-1}&{2}&{\cdots}&{0}\\{}&{}&{}&{\cdots}&{}\\{+1}&{0}&{\cdots}&{-1}&{2}\\\end{array}\right).
\begin{array}{llll}{\epsilon^{i}=\gamma_{5}\epsilon^{i}}&{Q^{i}=-\gamma_{5}Q^{i}}&{\theta^{i}=\gamma_{5}\theta^{i}}&{}\\{\Psi^{i}=-\gamma_{5}\Psi^{i}}&{\Lambda^{i}=-\gamma_{5}\Lambda^{i}}&{\Omega^{i}=-\gamma_{5}\Omega^{i}}&{\phi^{i}=\gamma_{5}\phi^{i}}\\\end{array}
I_{mn}^{\prime}=\frac{1}{A^{4}}\int\frac{d^{d}k}{{(2\pi)}^{d}}\frac{k^{2}k_{m}k_{n}}{(k^{2}+a_{2})(k^{2}+a_{3})({(k+p)}^{2}+a_{2})({(k+p)}^{2}+a_{3})}
\{\tilde{\Theta}_{\alpha}(q,\phi),\tilde{\Theta}_{\beta}(q,\phi)\}=0~~;~~\tilde{\Theta}_{\alpha}(q,\phi)=\Theta_{\alpha}(q)+\Sigma_{n=1}^{\infty}\tilde{\Theta}_{\alpha}^{(n)}(q,\phi)~~;~~\tilde{\Theta}^{(n)}\approxO(\phi^{n})
{\calP}^{a\mu}=\alpha_{2}\sqrt{-\gamma}\left[\left(K_{i}K^{i}\gamma^{ab}-2K_{i}K^{ab\,i}-2{\calG}^{ab}\right)e^{\mu}{}_{b}+2(\widetilde\nabla^{a}K^{i})n^{\mu}{}_{i}\right]\,.
\begin{array}{l}{\tau_{1}\equiv\pi_{0}+\eta^{1}\approx0}\\{\tau_{2}\equivE^{\prime}+\phi^{\prime}+\pi+A_{1}+\eta^{2}\approx0}\\{\tau_{3}\equivE+\eta^{3}\approx0}\\{\tau_{4}\equiv-\pi-\phi^{\prime}-2A_{1}+A_{0}+\eta^{4}\approx0}\\\end{array}
{e^{-icT_{1}}}f(r)=-{\frac{2}{c}}\int_{0}^{\infty}dxf(x){({\frac{x}{r}})^{\frac{1}{2}}}{J_{2\delta+1}({\frac{4}{c}}(xr)^{\frac{1}{2}}){e^{{\frac{2i}{c}}(x+r)}e^{-i\pi\delta}}}
S(z_{\alpha},z_{\beta})=\left(\frac{d^{3}z_{\alpha}}{dz_{\beta}^{3}}\right)/\left(\frac{dz_{\alpha}}{dz_{\beta}}\right)-\frac{3}{2}\left(\frac{d^{2}z_{\alpha}}{dz_{\beta}^{2}}\right)^{2}/\left(\frac{dz_{\alpha}}{dz_{\beta}}\right)^{2}
\times\left(\frac{m}{2}\right)^{n}\,\left[\,\sum_{a=1}^{3}\left(A^{a}-\frac{1}{m}\frac{\partial}{\partialA^{a}}\right)^{2}\,\right]^{n}\,e^{-\frac{1}{2}m(\sum_{a=1}^{3}A^{a})^{2}}\\\.
T_{1}=-i\left(\begin{array}{ccc}{0}&{0}&{0}\\{0}&{0}&{1}\\{0}&{-1}&{0}\\\end{array}\right)\\,\\T_{2}=-i\left(\begin{array}{ccc}{0}&{0}&{-1}\\{0}&{0}&{0}\\{1}&{0}&{0}\\\end{array}\right)\\,\\T_{3}=-i\left(\begin{array}{ccc}{0}&{1}&{0}\\{-1}&{0}&{0}\\{0}&{0}&{0}\\\end{array}\right)\\,
\Bigl\{A,B\Bigr\}(x)\equiv{\frac{1}{\sqrt{G}}}\left[\left(-{\frac{1}{d-1}}\,G_{\mu\nu}G_{\rho\sigma}+G_{\mu\rho}G_{\nu\sigma}\right){\frac{\deltaA}{\deltaG_{\mu\nu}}}\,{\frac{\deltaB}{\deltaG_{\rho\sigma}}}+{\frac{1}{2}}\,L^{ij}(\phi)\,{\frac{\deltaA}{\delta\phi^{i}}}\,{\frac{\deltaB}{\delta\phi^{j}}}\right].
G_{r}^{(2)}=x^{\frac{r-1}{2}}\cdot\prod_{m=1}^{\nu-r}\left(\frac{U_{m}^{2}}{U_{m}^{2}-1}\right)^{r}\times\prod_{m=\nu-r+1}^{\nu-1}\left(\frac{U_{m}^{2}}{U_{m}^{2}-1}\right)^{\nu-m}.
L_{f}^{m}=q^{1/2}p_{\mu}\dot{x}^{\mu}+\frac{i}{2}(\psi_{\mu}\;\dot{\psi}^{\mu}-\psi_{5}\dot{\psi}_{5})-\;\frac{e}{1+q^{2}}(p^{2}-m^{2})+i\chi(\psi_{\mu}p^{\mu}-\psi_{5}m),
\begin{array}{rl}{ds^{2}=\alpha^{\prime}g_{6}\sqrt{N_{1}N_{5}}[}&{u^{2}(dx_{0}^{2}+dx_{1}^{2})+{\frac{du^{2}}{u^{2}}}+d\Omega_{3}^{2}}\\{\nonumber+}&{\beta(N_{1},N_{5})(dx_{2}^{2}+...+dx_{5}^{2})],}\\\end{array}
\mathrm{Im}\left(E_{\mathrm{NP}}^{(+)}({\calN},N)\right)=\mathrm{Im}\left(E_{\mathrm{NP}}^{(-)}({\calN},N+{\calN})\right)=\mathrm{Im}\left(E_{\varsigma,\mathrm{NP}}({\calN},N)\right).
{\int_{-\infty}^{+\infty}\frac{dp}{1+\betap^{2}}}\psi^{*}(p)i\hbar(1+\betap^{2})\partial_{p}\phi(p)={\int_{-\infty}^{+\infty}\frac{dp}{1+\betap^{2}}}\left(i\hbar(1+\betap^{2})\partial_{p}\psi(p)\right)^{*}\phi(p)
\Upsilon_{I}^{i}{}^{(1)}\;=\;-\oint_{z_{I}}\frac{dz}{\pii}\;\frac{1}{\rho^{\prime}(z)}\left(\frac12\frac{\rho^{\prime\prime}(z)}{\rho^{\prime}(z)}+\frac{1}{z-z_{I}}\right)\partial_{z}\chi^{i}(z)\,
R_{\mu\nu}(W)=\partial_{\mu\nu}^{\beta}-\frac{1}{2}(\partial_{\nu}W_{\mu\beta}^{\beta}+\partial_{\nu\beta}^{\beta})-W_{\alpha\nu}^{\beta}W_{\mu\beta}^{\alpha}+W_{\alpha\beta}^{\beta}W_{\mu\nu}^{\alpha},
\int\frac{dp_{+}dp_{-}}{(2\pi)^{2}}\frac{{(p_{-}^{2}k_{+}A_{+}-p_{+}^{2}k_{-}A_{-})}}{{(2p_{+}p_{-}-m^{2})}^{2}}={(k_{+}A_{+}-k_{-}A_{-})}I,
{\frac{1}{2}}(1-\gamma)^{-3}\left(\sum_{k=0}^{\infty}(-1)^{k}{\frac{(k+1)(k+2)}{2}}\beta^{k}\right)\left(-\sum_{r=0}^{3}\sigma_{3,2r}\right)=0\;\mathrm{from}\;W_{\infty}^{5,D},
\begin{array}{ll}{\lambda_{\theta}=p_{n}\sigma^{n}\bar{\psi},}&{\qquad\lambda_{\bar{\theta}}=\psi\sigma^{n}p_{n},}\\{\lambda_{\rho}=-2p_{n}\sigma^{n}\lambda_{\bar{\theta}}\approx0,}&{\qquad\lambda_{\bar{\rho}}=-2\lambda_{\theta}\sigma^{n}p_{n}\approx0.}\\\end{array}
\langle\psi_{N^{\prime}(\theta)}({\bfx})\rangle=\frac{1}{\sqrt{2\bar{\omega}_{k}(M+\bar{\omega}_{k})}}\left(\begin{array}{c}{e^{i\theta/2}ipJ_{N}(kr)e^{iN(\varphi+\theta)}}\\{e^{-i\theta/2}(M+\bar{\omega}_{k})J_{N+1}(kr)e^{i(N+1)(\varphi+\theta)}}\\\end{array}\right)