harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S=-T_{p}V_{p}\intdt\,e^{-\Phi}[h(r)r^{2}\rho^{2}]^{\frac{p}{2}}\sqrt{g_{tt}-g_{rr}(\partial_{t}r)^{2}-g_{\rho\rho}(\partial_{t}\rho)^{2}-g_{\phi\phi}(\partial_{t}\phi)^{2}}+N\intdt\,\rho^{p+1}\partial_{t}\phi.
\Omega_{m}^{n}\,t_{j}^{m}\,\overline{{t}}{}_{n}^{i}=\Omega_{j}^{i}\,,\\\\\,overline{\Omega}{}_{m}^{n}\,\overline{{t}}{}_{j}^{m}\,t_{n}^{i}=\overline{{\Omega}}{}_{j}^{i}\,.
L_{b}={\frac{1}{g^{2}}}\int_{{\calB}\timesR}dtd^{d}\sigma\tr\left\{-{\frac{1}{4}}F^{2}+{\frac{1}{2}}(D_{\alpha}X^{i})^{2}-{\frac{1}{4}}([X^{i},X^{j}])^{2}\right\}\quad,
\begin{aligned}{}&{{}U_{2}:}&{}&{{}\qquadx_{2}(t)=0,}&{}&{{}\qquady_{2}(t)=t^{d},}&{}&{{}\qquadu_{2}(t)=0,}&{}&{{}\qquadv_{2}(t)={\frac{\mu}{t^{d}}}}\\{}&{{}U_{3}:}&{}&{{}\qquadx_{3}(t^{\prime})=0,}&{}&{{}\qquady_{3}(t^{\prime})=t^{d},}&{}&{{}\qquadu_{3}(t^{\prime})=0,}&{}&{{}\qquadv_{3}(t^{\prime})={\frac{\mu}{t^{d}}}.}\\\end{aligned}
|\alpha\rangle=\sum_{j\in\mathrm{Exp}(G)}\frac{\psi_{\alpha}^{j}}{\sqrt{S_{1j}}}|j\rangle\rangle,\;\;\langle\beta|=\sum_{j\in\mathrm{Exp}(G)}\langle\langlej|\frac{(\psi_{\beta}^{j})^{*}}{\sqrt{S_{1j}}},
R_{\;cm}^{ae}R_{\;df}^{bm}g^{cd}=g^{ab}\delta_{\;f}^{e},\;\;\;\;R_{\;\;\;\;\;\;mc}^{-1\;ea}R_{\;\;\;\;\;\;fd}^{-1\;mb}g^{cd}=g^{ab}\delta_{\;f}^{e}.
\begin{array}{c}{(\nu^{2}-(l+\frac{d-3}{2})^{2})R_{\nu,\nu^{\prime},l}=-(l+1)(l+2)R_{\nu,\nu^{\prime},l+2}}\\{(\nu^{2}-(l+\frac{d-3}{2})^{2})R_{\nu,\nu^{\prime},l}=-(l+1)(l+2)R_{\nu,\nu^{\prime},l+2}.}\\\end{array}
\begin{aligned}{R_{mn}{}^{pq}}&{{}=W_{mn}{}^{pq}-\frac{4}{d-2}\,\delta_{[m}{}^{[p}R_{n]}{}^{q]}+\frac{2}{(d-1)(d-2)}R\,\delta_{m}{}^{[p}\delta_{n}{}^{q]}}\\{}&{{}=W_{mn}{}^{pq}-\frac{4}{d-2}\,\delta_{[m}{}^{[p}S_{n]}{}^{q]}-\frac{2}{d(d-1)}R\,\delta_{m}{}^{[p}\delta_{n}{}^{q]}\,.}\\\end{aligned}
S_{+}=\intd^{4}x\sqrt{-g^{0}}\frac{1}{2}V_{0}^{+}\Phi\Bigr|_{y=0}+\intd^{4}x\sqrt{-g^{\pir_{c}}}\frac{1}{2}V_{\pi}^{+}\Phi\Bigr|_{y=\pir_{c}}\,
\displaystyle{\frac{\phi^{(1)}(a,b+1,c|u,v,w+1/4)}{\phi^{(1)}(a,b-1,c|u,v,w+1/4)}}=\displaystyle{\frac{\phi^{(0)}(a,b+1,c|u-\eta,v-\eta,w+1/4)}{\phi^{(0)}(a,b-1,c|u-\eta,v-\eta,w+1/4)}}.
\begin{array}{c}{\{Q^{a\alpha},\bar{Q}_{b\beta}\}=2\delta^{a}{}_{b}\gamma^{\mu}{}^{\alpha}{}_{\beta}P_{\mu}\,,}\\{{}}\\{{}[M_{\mu\nu},Q^{a}]=i\textstyle{\frac{1}{2}}\gamma_{[\mu}\gamma_{\nu]}Q^{a}\,,}\\{{}}\\{{}[{P}_{\mu},Q^{a\alpha}]=0\,.}\\\end{array}
D_{\mu,x}^{ab}iG^{[s]}(x,x^{\prime})_{\nu\rho}^{bc}={\frac{1}{2}}\int_{0}^{\infty}dT\int_{x(0)=x^{\prime},x(T)=x}Dx(t)\dot{x}_{\mu}(T)exp\left(-{\frac{1}{4}}\int_{0}^{T}dt\,\dot{x}^{2}\right)\Phi_{\nu\rho}^{[1]}[C^{xx^{\prime}}]U[C^{xx^{\prime}}]^{ac}
A_{(1)}=\left(\begin{matrix}{a_{(1)}}&{-b_{(1)}}\\{b_{(1)}}&{a_{(1)}}\\\end{matrix}\right)\;\;,\;\;A_{(2)}=\left(\begin{matrix}{a_{(2)}}&{-b_{(2)}}\\{b_{(2)}}&{a_{(2)}}\\\end{matrix}\right)\;\;,...
\displaystyle{\prod_{n=N}^{\infty}\Bigl(1+\frac{\alpha-\varepsilon}{n}\Bigr)}<\prod_{n=N}^{\infty}\Bigl(1+\frac{q_{n}}{n}\Bigr)<\displaystyle{\prod_{n=N}^{\infty}\Bigl(1+\frac{\alpha+\varepsilon}{n}\Bigr)}\,.
W_{c}^{\,\alpha,\,\alpha_{0}}(J):=i\,L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=i\tilde{C}_{\alpha}^{\,\alpha_{0}}\!J}+i\,I^{\,\alpha,\,\alpha_{0}}-\frac{1}{2}<J,\tilde{C}_{\alpha}^{\,\alpha_{0}}J>\quad.
F(h_{+},h_{-},\frac{d}{2},\frac{1+P}{2})\sim\left({\frac{D^{2}}{4}}\right)^{1-d/2}\frac{\Gamma(\frac{d}{2})\Gamma({\frac{d}{2}}-1)}{\Gamma(h_{+})\Gamma(h_{-})},
(\frac{1}{2}\sigma\cdotF)_{ij}^{2}=\frac{1}{2}\left\{(F_{\mu\nu}^{a}F_{\mu\nu}^{b})+\gamma_{5}(F_{\mu\nu}^{a}{*}F_{\mu\nu}^{b})\right\}\frac{1}{4}(\sigma^{a}\sigma^{b})_{ij}
{\cal{A}}=\sum_{\eta=1}^{\infty}{\textstyle\frac{ig^{\eta}}{\eta!}}{\int}d{\bfr}\;\{\,\psi_{(\eta)i}^{\gamma}({\bf{r}})\,+\,f_{(\eta)}^{\vec{\alpha}\beta\gamma}\,{\int}d{\bfr}\,{\cal{M}}_{(\eta)}^{\vec{\alpha}}({\bf{r}})\,\overline{{{\cal{B}}_{(\eta)i}^{\beta}}}({\bf{r}})\,\}\;\Pi_{i}^{\gamma}({\bf{r}})\;.
\left(\begin{array}{ccc}{a(1-\alphapq/2)}&{b(1-\alphapq/2)}&{\sqrt{\alpha}p}\\{c(1-\alphapq/2)}&{d(1-\alphapq/2)}&{\sqrt{\alpha}q}\\{\sqrt{\alpha}(-aq+cp)}&{\sqrt{\alpha}(dp-bq)}&{(1+\alphapq)}\\\end{array}\right)
\sum_{\mu,\nu=1}^{M}\left(e^{i(\omega_{\mu}+\omega_{\nu})}|z_{\mu+}^{+}z_{\nu+}^{-}|^{2}+e^{i(\omega_{\mu}-\omega_{\nu})}|z_{\mu+}^{+}z_{\nu-}^{-}|^{2}+e^{-i(\omega_{\mu}-\omega_{\nu})}|z_{\mu-}^{+}z_{\nu+}^{-}|^{2}+e^{-i(\omega_{\mu}+\omega_{\nu})}|z_{\mu-}^{+}z_{\nu-}^{-}|^{2}\right)\\\.
{\calS}^{(1)}\sim\frac{V_{26-D}}{(\sqrt{\alpha^{\prime}\pi})^{26-D}}e^{-2a}(1+\frac{D}{2}+2a)\frac{1}{(\sqrt{8\pi})^{26}}\Bigl(\prod_{i=1}^{D}\frac{1}{\sqrt{u_{i}\alpha^{\prime}/2}}\Bigr)\int_{0}^{\infty}\frac{dt}{2\pi}e^{2t}f^{-24}(t)
A=2\int_{\Sigma}d\xi^{1}\wedged\xi^{2}\g^{1/2}\left(1+{\frac{\delta^{2}}{2}}\g^{ab}\\partial_{a}y^{\perp}.\partial_{b}y^{\perp}+\delta^{2}\{\calR}+{\calO}(\delta^{4})\right).
\sigma_{V}(p^{2})=\frac{1}{m^{2}}\frac{1+\frac{\Sigma_{A}}{m}}{\epsilon\left(1+\frac{\Sigma_{A}}{m}\right)^{2}+2\left(1+\frac{\Sigma_{A}+\Sigma_{B}}{2m}\right)\frac{\Sigma_{B}-\Sigma_{A}}{m}},
\Phi_{k}(\tau,\boldsymbol{z}+\boldsymbol{\lambda}\tau+\boldsymbol{\mu})={\bfe}\left[-\left(\frac{\langle\boldsymbol{\lambda},\boldsymbol{\lambda}\rangle}{2}\tau+\langle\boldsymbol{\lambda},\boldsymbol{z}\rangle\right)\right]\Phi_{k}(\tau,\boldsymbol{z})\,.
\begin{array}{ll}{g_{_L}(\widehat{\mathbf{f}}_{\mu},\widehat{\mathbf{f}}_{\nu})=g_{\mu\nu}}\\{g_{_L}(\widehat{\mathbf{f}}_{\mu},\mathbf{e}_{k})=0}\\{g_{_L}(\mathbf{e}_{k},\mathbf{e}_{l})=g_{0\,kl}\,,}\\\end{array}
G_{2m+2}=\frac{\mathrm{sn}^{2m-1}[u,\kappa]\mathrm{cn}[u,\kappa]\mathrm{dn}[u,\kappa]+2m(1+\kappa^{2})G_{2m}+(1-2m)G_{2m-2}}{(2m+1)\kappa^{2}},
\tau_{2}(a^{0},a^{1},a^{2})=2\pii\sum_{n_{0}+n_{1}+n_{2}=0\atopm_{0}+m_{1}+m_{2}=0}(n_{1}m_{2}-n_{2}m_{1})\,a_{n_{0},m_{0}}^{0}a_{n_{1},m_{1}}^{1}a_{n_{2},m_{2}}^{2}\,.
B^{\mu}(p)=\frac{1}{\sqrt2}\left[\left(\begin{array}{c}{0}\\{1}\\{i}\\\end{array}\right)+\frac{p^{x}+ip^{y}}{M(E+M)}\left(\begin{array}{c}{E+M}\\{p^{x}}\\{p^{y}}\\\end{array}\right)\right]\phi(p)\;.
S_{0}^{L}\left[\left(A_{(p_{a})}^{\mu_{1}\ldots\mu_{p_{a}}}\right)\right]=\sum_{a}S_{0_{p_{a}}}^{L}\left[A_{(p_{a})}^{\mu_{1}\ldots\mu_{p_{a}}}\right]+S_{I}^{L}\left[\left(A_{(p_{a})}^{\mu_{1}\ldots\mu_{p_{a}}}\right)\right],
DX_{\mu}(\delta_{\mu\nu}I-i[X_{\mu},X_{\nu}])^{-1}DX_{\nu}=\left(I-\frac{1}{2}[X_{\mu},X_{\mu}]^{2}\right)^{-1}\left[DX_{\mu}DX_{\mu}-\left(\frac{1}{2}\varepsilon_{\mu\nu\rho}X_{\mu}[X_{\nu},X_{\rho}]\right)^{2}\right].
\sum_{(n_{j})_{0}^{k}}\sum_{(X_{i})_{0}^{n}}\prod_{i=0}^{n-1}b(X_{i+1},X_{i})\prod_{j=0}^{k}\prod_{i=m_{j-1}+1}^{m_{j}}\chi(X_{i}\inLX_{j}^{\prime})=
\frac{3}{2}F^{\prime\prime}-\frac{15}{2}\frac{K^{\prime}}{K}F^{\prime}+6\bigg[2\bigg(\frac{K^{\prime}}{K}\bigg)^{2}-\frac{K^{\prime\prime}}{K}\bigg]F+\bigg[\sum_{i}\frac{\Lambda_{i}}{K}\delta(z-z_{i})+(\phi_{0}^{\prime})^{2}\bigg]F+\phi_{0}^{\prime}\varphi^{\prime}=0,
H_{LC}^{(AdS)}=\frac{1}{2p_{-}}\int_{0}^{l}d\sigma\,\left(\pi_{i}\pi_{i}+\frac{1}{(2\pi\alpha^{\prime})^{2}}\frac{R^{4}}{Y^{4}}Z^{i\,\prime}Z^{i\,\prime}\right)\.
W(u_{1},u_{2})=\mathrm{det}\left(\begin{array}{cc}{u_{1}(r)}&{u_{2}(r)}\\{u_{1}^{\prime}(r)}&{u_{2}^{\prime}(r)\}\\\end{array}\right)=\mathrm{const}\,\times\,\frac{1}{p(r)}=\mathrm{const}\,\times\,\frac{1}{r^{3}\,g(r)^{2}}
\hat{h}^{\mathrm{e}}=-g_{\mathrm{eff}}^{2}\left\{\frac{1}{J}\frac{\delta}{\delta\varphi_{1}}J\frac{\delta}{\delta\varphi_{1}}+\frac{1}{J}\frac{\delta}{\delta\varphi_{2}}J\frac{\delta}{\delta\varphi_{2}}\right\},
H_{5}={\frac{P}{|\vec{z}-\vec{z}_{0}|^{3-n}}},\\\\H_{W}=1+{\frac{Q_{W}}{[x^{2}+{\frac{4P}{(n-1)^{2}}}|\vec{z}-\vec{z}_{0}|^{n-1}]^{\frac{n+1}{n-1}}}}.
A_{a}=-i\kappa\\xi\\bar{\psi}^{1}\gamma_{a}\psi^{2}+\kappa\\theta\\bar{\psi}^{1}\gamma_{5}\gamma_{a}\psi^{2}-{\frac{1}{2}}\kappa\\varphi\(\bar{\psi}^{1}\gamma_{5}\gamma_{a}\psi^{1}-\bar{\psi}^{2}\gamma_{5}\gamma_{a}\psi^{2})+...\,\,
N_{i}=a_{i}^{+}a_{i}+\sum_{k=1}^{\infty}\sum_{\pi\inS_{k}}\sum_{j_{1}...j_{k}}d_{\pi(j_{1}...j_{k}),j_{1}...j_{k}}\,[2mm]a_{\pi(j_{k})}^{+}.......a_{\pi(j_{1})}^{+}a_{i}^{+}a_{i}a_{j_{1}}....a_{j_{k}}
{\frac{d^{2}t}{d\tilde{\lambda}^{2}}}=\sqrt{h}\left[{\frac{h^{\prime}}{2}}\left({\frac{dt}{d\tilde{\lambda}}}\right)^{2}-y\sum_{j}\left({\frac{dx^{j}}{d\tilde{\lambda}}}\right)^{2}-{\frac{h^{\prime}}{h}}\right]{\frac{dt}{d\tilde{\lambda}}},
0=-\frac{\partialJ(0,\hat{T}_{c})}{\partial\hat{T}}\hat{T}_{c}\left(1-\frac{\hat{T}}{\hat{T}_{c}}\right)+\frac{\partialJ(0,\hat{T}_{c})}{\partial\hat{m}_{T}^{2}}~\hat{m}_{T}^{2},
\frac{p^{\mu}k^{\nu}+p^{\nu}k^{\mu}}{p\cdotk}-\frac{p^{2}k^{\mu}k^{\nu}}{(p\cdotk)^{2}}=p_{\lambda}\frac{\partial}{\partialk_{\lambda}}\left(\frac{k^{\mu}k^{\nu}}{p\cdotk}\right)
S_{5}^{\mathrm{\tiny{A}}}=\int\!d^{5}x\left[\frac{1}{4g^{2}}\intd^{2}\theta\;W^{\alpha}W_{\alpha}+\mathrm{h.c.}+\intd^{4}\theta\frac{1}{g^{2}}(\partial_{5}V-\frac{1}{\sqrt{2}}(\phi+\bar{\phi}))^{2}\right]
\mathcal{L}_{I}^{lin}=-\frac{i}{4}\left(h^{a\mu}\bar{\psi}\gamma_{a}\frac{1\pm\gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial_{\mu}^{\psi}}\psi+h_{\\mu}^{\mu}\bar{\psi}\gamma^{a}\frac{1\pm\gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial_{a}^{\psi}}\psi\right)=-\frac{1}{2}h_{\mu\nu}T^{\mu\nu}\,.
\begin{array}{rl}{}&{X^{++}X^{++}=X^{+-}X^{++}=X^{++}X^{-+}=0}\\{}&{X^{+-}X^{-+}=X^{-+}X^{+-}=0}\\{}&{X^{++}X^{+-}=-{\frac{r^{2}}{2}}X^{++}=-X^{-+}X^{++}}\\{}&{X^{+-}X^{+-}=-{\frac{r^{2}}{2}}X^{+-}\,\qquadX^{-+}X^{-+}={\frac{r^{2}}{2}}X^{-+}}\\{}&{X^{+-}-X^{-+}=-{\frac{r^{2}}{2}}I.}\\\end{array}
(C_{\mu})_{\rho}^{\sigma}(C_{\nu})_{\lambda}^{\rho}+\Big[(C_{\mu})_{\rho}^{\sigma}(\tilde{C}_{\nu})_{\lambda}^{\rho}-(\tilde{C}_{\nu})_{\rho}^{\sigma}(C_{\mu})_{\lambda}^{\rho}\Big]=C_{\mu\nu}^{\rho}(C_{\rho})_{\lambda}^{\sigma}.
{\textstyle\frac{1}{2}}\delta_{\epsilon}\Psi_{\hat{\imath}I}=\partial_{\hat{\imath}}\epsilon_{I}-{\textstyle\frac{1}{2}}\omega_{\hat{\imath}}^{+ab}\sigma_{ab}\epsilon_{I}-{\textstyle\frac{i}{4}}e^{2\phi}\partial_{\hat{\imath}}a\epsilon_{I}-{\textstyle\frac{1}{2\sqrt{2}}}e^{-\phi}\sigma^{ab}F_{ab}^{+}\gamma_{\hat{\imath}}\alpha_{IJ}\epsilon^{J}=0\.
\delta{\calL}=\mu\left(\psi_{R}^{(i)\dagger}\psi_{R}^{(i)}+\psi_{L}^{(i)\dagger}\psi_{L}^{(i)}\right)+\mu_{5}\left(\psi_{R}^{(i)\dagger}\psi_{R}^{(i)}-\psi_{L}^{(i)\dagger}\psi_{L}^{(i)}\right)
{\calO}_{j_{1}m_{1}}^{\pm}(z){\calO}_{j_{2}m_{2}}^{\pm}(z^{\prime})\sim\frac{1}{z-z^{\prime}}F_{j_{1}m_{1},j_{2}m_{2}}^{j_{3}m_{3}}{\calO}_{j_{3}m_{3}}^{\pm}(z^{\prime})
\[(\ell_{j}^{+}\ell_{j}^{-})^{n},(\ell_{k}^{+}\ell_{k}^{-})^{m}]=+2\omega\tilde{g}\sum_{t=0}^{m-1}\left((\ell_{k}^{+}\ell_{k}^{-})^{t+n}m_{jk}(\ell_{k}^{+}\ell_{k}^{-})^{m-t-1}-(\ell_{k}^{+}\ell_{k}^{-})^{t}m_{jk}(\ell_{k}^{+}\ell_{k}^{-})^{m+n-t-1}\right)
{\calO}_{2}{}^{i}{}_{j}\left[\begin{matrix}{\psi_{n1}^{j}}\\{\bar{\psi}_{n}^{j\dot{2}}}\\\end{matrix}\right]=im_{n}^{(2)}\left[\begin{matrix}{\bar{\psi}_{n}^{i\dot{1}}}\\{\psi_{n2}^{i}}\\\end{matrix}\right],
(-1)^{M+(p-1)h_{\psi}}\sum_{s=0}^{h_{\psi}}(-1)^{s}\binom{h_{\psi}}{s}\sum_{L\geq0}\binom{(p-1)h_{\psi}+M-s+L}{L}V_{(h_{\psi}+h_{\chi}-1+t-L)}(\chi)V_{(L)}(\psi)\,.
\stackrel{(2)}{\delta\Sigma_{\\nu}^{\mu}}=\frac{l^{2}}{4}\left(\frac{y}{a^{4}}+\frac{l}{2a^{2}}\right){\calS}_{\\nu}^{\mu}-\frac{l^{2}}{8}\left(\frac{1}{a^{6}}+\frac{1}{a^{2}}\right)\Box\chi_{\\nu}^{\mu}+\frac{C_{\\nu}^{\mu}}{a^{4}},\quadC_{\\mu}^{\mu}=0\.
{\calL}_{\mathrm{RCS}}=\frac{1}{4}\frac{\dot{\rho}^{2}}{\rho}-\kappaB\dot{\omega}-\frac{\kappa}{2}\epsilon^{ij}A_{i}\dot{A}_{j}-\left[|D_{\pm}\phi|^{2}+\frac{\kappa^{2}}{4\rho}\left(B\mp\frac{2}{\kappa^{2}}\rho(\rho-v^{2})\right)^{2}\right]\pmv^{2}B
\sqrt{-G}W_{\mu\nu}^{5}\Big|_{y=0}=\sqrt{-g^{0}}\frac{1}{2}V_{0}e^{b_{0}\Phi}g_{\mu\nu}^{0}\Big|_{y=0},\quad\sqrt{-G}W_{\mu\nu}^{5}\Big|_{y=L}=-\sqrt{-g^{L}}\frac{1}{2}V_{L}e^{b_{L}\Phi}g_{\mu\nu}^{L}\Big|_{y=L}
\begin{array}{rcl}{{\calS}({\bfx_{1}},x_{1}^{4};\dots;{\bfx_{n}},x_{n}^{4})}&{:=}&{{\calW}(-\mathrm{i}x_{1}^{4},{\bfx_{1}};\dots;-\mathrm{i}x_{n}^{4},{\bfx_{n}})=}\\{}&{=}&{\langle\,0|\phi(0,{\bfx_{1}})\mathrm{e}^{-(x_{1}^{4}-x_{2}^{4})H}\dots\mathrm{e}^{-(x_{n-1}^{4}-x_{n}^{4})H}\phi(0,{\bfx_{n}})|0\,\rangle\,.}\\\end{array}
ds^{2}={\frac{r^{2}}{l^{2}}}\left[\left(1-{\frac{r_{0}^{p+1}}{r^{p+1}}}\right)d\tau^{2}+(dx^{i})^{2}-dt^{2}\right]+\left(1-{\frac{r_{0}^{p+1}}{r^{p+1}}}\right)^{-1}{\frac{l^{2}}{r^{2}}}dr^{2},
\mathbf{B}=\Big[\varepsilon\Phi(x,y,\xi+\varepsilonz)\mathrm{sin}\left(\frac{\pmz}{l(x,y)}+b(x,y)\right);-\varepsilon\Phi(x,y,\xi+\varepsilonz)\mathrm{cos}\left(\frac{\pmz}{l(x,y)}+b(x,y)\right);0\Big],
(\omega^{2\ell-1})^{n_{f}}=(\omega^{2\ell-1})^{2\tilde{n}_{c}+(n_{c}-\tilde{n}_{c})}=(\omega^{2\ell-1})^{2\tilde{n}_{c}}e^{\pii(2\ell-1)}=-(\omega^{2\ell-1})^{2\tilde{n}_{c}}.
\begin{array}{lcl}{f_{i}^{L}}&{=}&{(\nu_{i}^{L},e_{i}^{L},u_{i}^{L},d_{i}^{L})}\\{f_{i}^{R}}&{=}&{(e_{i}^{R},u_{i}^{R},d_{i}^{R})}\\\end{array}\qquad\begin{array}{lcl}{\psi_{f_{i}}^{R}}&{=}&{(\psi_{\nu_{i}}^{R},\psi_{e_{i}}^{R},\psi_{u_{i}}^{R},\psi_{d_{i}}^{R})}\\{\psi_{f_{i}}^{L}}&{=}&{(\psi_{e_{i}}^{L},\psi_{u_{i}}^{L},\psi_{d_{i}}^{L})}\\\end{array}\qquad
K={\frac{(d-2+\sum_{i}{\frac{1}{a_{i}^{2}}})}{(\sum_{i}{\frac{z_{i}^{2}}{a_{i}^{4}}})^{1/2}}}-{\frac{(\sum_{i}{\frac{z_{i}^{2}}{a_{i}^{6}}})}{(\sum_{i}{\frac{z_{i}^{2}}{a_{i}^{4}}})^{3/2}}}~~.
K_{l}(x)=\left(\displaystyle\frac{\pi}{2l}\right)^{\frac{1}{2}}\left(\displaystyle\frac{2l}{ex}\right)^{l}\left[1+\left(\displaystyle\frac{1}{12}-\displaystyle\frac{1}{4}x^{2}\right)\displaystyle\frac{1}{l}+\left(\displaystyle\frac{1}{288}-\displaystyle\frac{13}{48}x^{2}+\displaystyle\frac{1}{32}x^{4}\right)\displaystyle\frac{1}{l^{2}}+0\left(\displaystyle\frac{1}{l^{3}}\right)\right],
\frac{1}{M_{6}^{4}}f_{0}(\rho)=-{\frac{1}{M_{6}^{4}}\Lambda}-\frac{3}{4}\frac{\sigma^{\prime\prime}}{\sigma}-\frac{3}{16}\frac{{\sigma^{\prime}}^{2}}{\sigma^{2}}-\frac{{\left(\sigma^{3/4}\gamma^{1/2}\right)}^{\prime\prime}}{\sigma^{3/4}\gamma^{1/2}}\,
\left[\delta_{1},\delta_{2}\right]V_{+}=\left(\varepsilon_{1}^{\prime\beta}\sigma_{\beta\dot{\alpha}}^{\mu}\bar{\varepsilon}_{2}^{\prime\dot{\alpha}}-\varepsilon_{2}^{\prime\beta}\sigma_{\beta\dot{\alpha}}^{\mu}\bar{\varepsilon}_{1}^{\prime\dot{\alpha}}\right)\left(-\frac1{\left|A\right|^{2}}\tilde{\partial}_{+}\tilde{\partial}_{-}\right)\partial^{\mu}\tilde{\partial}_{+}V_{+}
\begin{array}{lcr}{\sum_{j}q_{a}^{j}(|X_{j}|^{2}-|Y_{j}|^{2})}&{=r_{a}^{2}\qquad}&{(a)}\\{\sum_{j}q_{a}^{j}(X_{j}\overline{{Y}}_{j})}&{=0\qquad}&{(b)}\\{\sum_{j}q_{a}^{j}(\overline{{X}}_{j}{Y_{j}})}&{=0.\qquad}&{(c)}\\\end{array}
{\calL}=\bar{\psi}\left(i\partial\!\!\!\!\!\!\not\;\;-m-g\sqrt{\frac{\pi}{N_{C}}}{\vec{\gamma}}_{\perp}{\vec{A}}_{\perp}\right)\psi-\frac{1}{2}\mathrm{tr}\!\!\left({\vec{A}}_{\perp}\Box{\vec{A}}_{\perp}+\lambda^{2}{\vec{A}}_{\perp}^{2}\right).
\Im\mathrm{m}\;\Pi_{\mathrm{reg}}^{(2)}(k^{2})\rightarrow\Im\mathrm{m}\;\Pi^{(2)}(k^{2})=-\frac{e^{2}}{4\sqrt{k^{2}}}\;\mathrm{sgn}(k_{0})\Theta(k^{2}-4m^{2})\;.
\vert\partial_{p}^{\omega}\,G_{l,s}^{\,\alpha}(\vec{\alpha},\vec{\lambda},\vec{p})\vert\\leq\e^{\,-m^{2}\sum_{j=1}^{s}\alpha_{j}}\,P(\vert\vec{p}\vert)\,Q(\sqrt\alpha_{1},\ldots,\sqrt\alpha_{s})\quad,
\phi_{a}=\left(\begin{array}{c}{2(x_{1}x_{3}+x_{2}x_{4})}\\{2(x_{2}x_{3}-x_{1}x_{4})}\\{x_{1}^{2}+x_{2}^{2}-x_{3}^{2}-x_{4}^{2}}\\\end{array}\right),\quad|\phi|=r^{2},\quadn=\phi/r^{2}\equivn_{\mathrm{H}}.
S=\sum_{i,j,k,l}(\sum_{\alpha}e_{i}^{\alpha}e_{j}^{\alpha}e_{k}^{\alpha}e_{l}^{\alpha})\phi_{i}\phi_{j}\phi_{k}\phi_{l}-\frac{2n}{n-1}{\mu}^{2}(\sum_{i}\phi_{i}^{2})+n{\mu}^{4}.
ds_{D}^{2}=f^{\frac{1}{D-2}}(r)\left[-f^{-1}(r)[1-({\frac{r_{0}}{r}})^{D-3}]dt^{\prime2}+[1-({\frac{r_{0}}{r}})^{D-3}]^{-1}dr^{2}+r^{2}d\Omega_{D-2}\right]
\sum_{l=0}^{N-1}\frac{w(v_{2},a-l)}{w(v_{1},-l)\gamma(b,l)}=\varphi_{1}(v_{1},v_{2})\frac{w(v_{2}^{\prime},-b)w(v_{2}/(\omegav_{1}),a)}{w(v_{1}^{\prime},a-b)},
{\calL}_{\pm}^{MWY}=\sum_{k=0}^{\infty}\left[\mp\dot{\phi}_{k}\phi_{k}^{\prime}-{\calG_{\pm}}\phi_{k}^{2}-2\left(\dot{\phi}_{k}+{\calG_{\pm}}\phi_{k}^{\prime}\right)\sum_{m=k+1}^{\infty}\phi_{m}^{\prime}\right]
\tilde{T}=\left(\begin{matrix}{\bar{z}^{m}}\\{-1}\\\end{matrix}\right)+\left(\begin{matrix}{1}\\{z^{m}}\\\end{matrix}\right)\frac{-2i}{1+\bar{z}^{m}z^{m}}\{m\bar{z}^{m-1}t+h(\bar{z})\},
V_{m}^{\prime}(\phi)=\sum_{n=1}^{\infty}v_{mm_{2}\ldotsm_{n}}^{(n)}\phi_{m_{2}}\ldots\phi_{m_{n}},\;\;\;\;\;\;\tilde{V}_{m}^{\prime}(\tilde{\phi})=\sum_{n=1}^{\infty}v_{mm_{2}\ldotsm_{n}}^{(n)}\tilde{\phi}_{m_{n}}\ldots\tilde{\phi}_{m_{2}}
\intdk\\left[\frac{1}{k}\right]_{\mathrm{C}PV}\\left[\frac{1}{k-p_{1}}\right]_{\mathrm{C}PV}\ldots\left[\frac{1}{k-p_{n}}\right]_{\mathrm{C}PV}=0\\{\mathrm{f}orall}p_{i}\neq0.
Z^{u}=f_{T}\left(\begin{matrix}{1}&{1/3}&{0}\\{1/3}&{1/8}&{(2/3)d^{2}}\\{0}&{(2/3)d^{2}}&{(1/3)d^{2}}\\\end{matrix}\right);\,\,\,Z^{d}=f_{T}\left(\begin{matrix}{1}&{4/5}&{0}\\{4/5}&{9/10}&{0}\\{0}&{0}&{1/4}\\\end{matrix}\right).
j(\tau)={\frac{(\theta_{1}^{8}(\tau)+\theta_{2}^{8}(\tau)+\theta_{3}^{8}(\tau))^{3}}{\eta(\tau)^{24}}}={\frac{4.(24f)^{3}}{27g^{2}+4f^{3}}}={\frac{55296a^{3}}{27+4a^{3}}}\.
\hat{S}^{N=1}={\textstyle\frac{1}{2}}\intd^{10}x\,sqrt{|\hat{g}|}\e^{-2\hat{\phi}}\left[-\hat{R}+4\left(\partial\hat{\phi}\right)^{2}-{\textstyle\frac{3}{4}}\hat{H}^{2}\right]
I_{D}=4\left\lbrace-\frac{e^{\frac{z}{\lambda}}}{2z^{2}}-\frac{e^{\frac{z}{\lambda}}}{2\lambdaz}+\frac{1}{2\lambda^{2}}E_{i}\left(\frac{z}{\lambda}\right)\right\rbrace_{z_{1}}^{z_{2}}
A_{j}=\left(\begin{array}{cc}{-\fraci2a_{j}^{0}}&{\frac{l}{2\tau}a_{j}^{-}}\\{}&{}\\{\frac{\tau}{2l}a_{j}^{+}}&{\fraci2a_{j}^{0}}\\\end{array}\right)\,,\qquad{\bar{A}}_{i}=\left(\begin{array}{cc}{-\fraci2{\tilde{a}}_{j}^{0}}&{\frac{\tau}{2l}{\tilde{a}}_{j}^{-}}\\{}&{}\\{\frac{l}{2\tau}{\tilde{a}}_{j}^{+}}&{\fraci2{\tilde{a}}_{j}^{0}}\\\end{array}\right)\,,
{\bfS}=\intd^{8}zE^{-1}\left[\frac{1}{2}{\bfD}_{\alpha\dot{\alpha}}L{\bfD}^{\alpha\dot{\alpha}}L-\frac{\imath}{12}\left(\nabla_{\alpha}T_{\dot{\alpha}}+\nabla_{\dot{\alpha}}T_{\alpha}\right){\bfD}^{\alpha\dot{\alpha}}L\right]
\mathcal{D}_{\xi}(p_{1},\ldots,p_{N})=\sum_{q\in\mathcal{I}}S_{0q}^{(2-2g-N)\mu_{\xi}}\prod_{i=1}^{N}\prod_{\zeta\in\mathcal{O}_{i}^{\xi}}\Lambda_{q}^{p_{i}(\zeta)}\left(\frac{\mu_{\xi}\kappa_{\zeta}}{|\xi|}\right)
Z_{j}[j]=\int{\calD}A{\calD}\psi{\calD}\bar{\psi}{\frac{\delta\^{*}(F(A))}{Vol({\calG}_{A})}}e^{\intd^{2}x\sqrt{g}\bar{\psi}(-i{\partial\!\!/}+{j\!\!/}+{A\!\!/})\psi},
\begin{array}{rcl}{z_{1}}&{=}&{54(1-2\sqrt{7})}\\{z_{2}}&{=}&{-1458}\\{z_{3}}&{=}&{6(10+7\sqrt{7})}\\{z_{4}}&{=}&{2(262-11\sqrt{7})}\\{z_{5}}&{=}&{18(-16+5\sqrt{7})}\\{z_{6}}&{=}&{2(784-83\sqrt{7})\.}\\\end{array}
{\calS}_{\mathrm{open}}=\frac{1}{4\pi}\intdxdy~\eta^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}~\eta_{MN}+\int_{-\infty}^{+\infty}\frac{dx}{a}\intd^{26}k{\tilde{T}}(k)e^{ik_{M}X^{M}}
\int_{-\infty}^{\infty}dq_{-}\frac{1}{q_{-}^{2}}\mathrm{e}^{iq_{-}x^{-}}=2\int_{0}^{\infty}dq_{-}\frac{\mathrm{cos}q_{-}x^{-}}{q_{-}^{2}}=2\int_{0}^{\infty}dq_{-}\frac{1}{q_{-}^{2}}-{\pi}|x^{-}|
\int_{\left|m_{eff}^{2}\right|^{1/2}\beta}^{\infty}\,dy\,\left(y^{2}+\left|m_{eff}^{2}\right|\beta^{2}\right)^{1/2}e^{-2\pirny\lambda/\beta}=\frac{\left|m_{eff}^{2}\right|^{1/2}\beta^{2}}{4\pirn\lambda}\times\left[H_{1}\left(\eta\lambda\right)-N_{1}\left(\eta\lambda\right)\right],
G_{l}(v)_{I_{1}\ldotsI_{s}}^{\mu_{1}\ldots\mu_{s}}={\frac{1}{(v\cdotl)^{d-3+s}}}\left[\hat{\delta}_{I_{1}}^{\mu_{1}}\cdots\hat{\delta}_{I_{s}}^{\mu_{s}}-\mathrm{traces}_{(I)}\right]
\theta^{\mu\nu}=\left(\begin{array}{cccccc}{0}&{\dots}&{\dots}&{\dots}&{\dots}&{0}\\{0}&{0}&{\theta_{1}}&{0}&{\dots}&{0}\\{0}&{-\theta_{1}}&{0}&{\dots}&{\dots}&{0}\\{\dots}&{\dots}&{\dots}&{\dots}&{\dots}&{\dots}\\{0}&{\dots}&{\dots}&{\dots}&{0}&{\theta_{n}}\\{0}&{\dots}&{\dots}&{0}&{-\theta_{n}}&{0}\\\end{array}\right)\.
\left(\begin{array}{cc}{u_{1}^{+}}&{u_{1}^{-}}\\{u_{2}^{+}}&{u_{2}^{-}}\\\end{array}\right)\inSU(2)\quad\Rightarrow\quadu_{i}^{-}=(u^{+i})^{*}\;,\quadu^{+i}u_{i}^{-}=1
A_{m}^{\alpha}A_{n}^{\beta}-(-1)^{\alpha\cdot\beta}A_{n}^{\beta}A_{m}^{\alpha}=\left\{\begin{array}{ll}{A_{m+n}^{\alpha+\beta}}&{if\\\alpha\cdot\beta=-1}\\{\alpha\cdotH_{m+n}+m\delta_{m+n,0}}&{if\\\alpha\cdot\beta=-2}\\{0}&{if\\\alpha\cdot\beta\geq0}\\\end{array}\right\}
{\frac{1}{8}}{(m_{R}^{(k)})}^{2}={\frac{1}{2}}{\sum_{i=3}^{8}}{(p_{R}^{i})}^{2}{{\delta}_{k,0}}+{\frac{1}{2}}{\sum}_{i=1}^{4}{(p^{t}+ku^{t})}^{2}+N_{R}^{(k)}-\frac{1}{2}+c_{k},\;i=1,\dots,6
\begin{array}{ll}{n\left(\frac{1-k}{1+k}\right)=2G\left(U-T-J^{2}-\nuS^{2}\right)}\\{B^{2}=1-8G\left(U+\frac{(J^{2}+\rhoS^{2})}{2}\right)}\\{m^{2}=4G(J^{2}+\nuS^{2}).}\\\end{array}
ds^{2}=[\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}}]dt^{2}-\frac{dr^{2}}{[\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}}]}-r^{2}[d\phi-\frac{J^{2}}{2r^{2}}dt]^{2}
l^{*}(r)={\frac{r}{d}}\left(1-{\frac{1}{d}}{\frac{a^{2}}{r^{2}}}-{\frac{(d^{2}-2)}{d^{2}(d+2)}}{\frac{a^{4}}{r^{4}}}+O(a^{6}/r^{6})\right)~~.
\frac{Q^{\prime}}{Q}+\frac{a^{\prime}}{a}+\frac{c^{\prime}}{c}-\frac{c^{2}}{2a}-2a=0,\quad-\frac{P^{\prime}}{P}=3\frac{a^{\prime}}{a}-\frac{3c^{2}}{2a}=\frac{a^{\prime}}{a}+2\frac{c^{\prime}}{c}-2a.
R_{L}^{2}\equiv\int_{0}^{P^{+}/T_{0}}{\frac{T_{0}d\sigma}{P^{+}}}\langle0|\big[x^{-}(\sigma)-x^{-}(0)\big]^{2}|0\rangle\sim{\frac{d}{3P^{+2}}}\sum_{n=1}^{\alpha^{\prime}P^{+}/\epsilon}n\sim{\frac{d}{6}}\left({\frac{\alpha^{\prime}}{\epsilon}}\right)^{2}\;,
=\dot{x}_{n}^{(m_{n}-1)}p_{n,m_{n}-1}-L_{0}\left(x_{n},\dot{x}_{n},\cdots,\dot{x}_{n}^{(m_{n}-1)}\right)+\sum_{n}\sum_{i_{n}=1}^{m_{n}-1}x_{n}^{(i_{n})}p_{n,i_{n}-1}\,
\partial_{i}{\frac{\delta\Gamma_{\mathrm{eff}}}{\deltaA_{i}^{a}}}+gf^{abc}A_{0}^{b}{\frac{\delta\Gamma_{\mathrm{eff}}}{\deltaA_{0}^{c}}}+gf^{abc}A_{i}^{b}{\frac{\delta\Gamma_{\mathrm{eff}}}{\deltaA_{i}^{c}}}=0
\begin{aligned}{\big(W^{(8)},W^{(8)},Q\big)}&{{}\rightarrow\beta=\tilde{m}(\tilde{m}-1)\,,}\\{\big(W^{(5)},W^{(5)},Q\big)}&{{}\rightarrow\delta=\tilde{n}(\tilde{n}-1)\,,}\\\end{aligned}
\ddot{h}=\left(\frac{\left(2+r\right)r}{\left(1+r\right)^{2}}\right)^{2}h^{\prime\prime}+\left(\frac{2}{\left(1+r\right)^{3}}+\frac{2+r}{\left(1+r\right)^{2}}\right)h^{\prime}-\kappa^{2}f^{2}\left(1+r^{2}h\right),
K_{j}^{a}\partial_{a}(-\gamma^{cd}\partial_{c}X^{\mu}\partial_{d}X^{\nu}g_{\mu\nu}+\frac{\varepsilon^{a_{1}a_{2}...a_{d+1}}}{\sqrt{-\gamma}}\partial_{[a_{1}}A_{a_{2}...a_{d+1}]})=0
\sigma_{\gamma}(x,x)=\sigma_{\tilde{\gamma}}(x_{0},x_{0})+(\deltat\;V^{\mu})\left.\left[\nabla_{\mu}^{x}\sigma_{\tilde{\gamma}}(x,y)+\nabla_{\mu}^{y}\sigma_{\tilde{\gamma}}(x,y)\right]\right|_{(x_{0},x_{0})}+O(\deltat^{2}).
V(y_{i},\theta)=-V_{p+1}(i)^{p}\int_{0}^{\infty}\frac{dt}{t}(8\pi^{2}\alpha^{\prime}\,t)^{-\frac{p+1}{2}}\sum_{m_{i}}e^{-{\frac{t}{2\pi\alpha^{\prime}}}\sum_{i}(y_{i}+m_{i}L_{i})^{2}}\frac{\theta_{11}^{4}(it/2,it)}{e^{\pit}\eta^{12}(it)}\,.
\sum_{N=1}^{\infty}\left[N^{2}+B^{2}\right]^{-p}=-\frac{1}{2}B^{-2p}+\frac{\pi^{\frac{1}{2}}}{2B^{2p-1}\Gamma\left(p\right)}\left[\Gamma\left(p-\frac{1}{2}\right)+4\sum_{N=1}^{\infty}\frac{K_{p-\frac{1}{2}}\left(2\piNB\right)}{\left(N\piB\right)^{\frac{1}{2}-p}}\right],
\large{\begin{array}{l}{p_{0}=\frac{1}{2}(p_{11}+p_{22})=\frac{1}{2},}\\{p_{x}=\frac{1}{2}(p_{12}+p_{21})=\frac{H_{x}}{2\kappa},}\\{p_{y}=\frac{1}{2}i(p_{12}-p_{21})=\frac{H_{y}}{2\kappa},}\\{p_{z}=\frac{1}{2}(p_{11}-p_{22})=\frac{H_{z}}{2\kappa}.}\\\end{array}}
{}_{2}F_{1}\left(\frac{1+n}{2},1+\frac{n}{2},1+n,\frac{1}{(1-2x)^{2}}\right)=\left(1-\frac{1}{2x}\right)^{1+n}{}_{2}F_{1}\left(1+n,\frac{1}{2}+n,1+2n,\frac{1}{x}\right).
{\calA}_{5}^{\mathrm{odd/odd}}\Big|_{\mathrm{left/right},\Gamma^{[7]}}{}=C\,t_{8}^{s_{1}\cdotss_{8}}(\bar{\psi}_{m}^{(1)}\Gamma^{mr_{1}\cdotsr_{6}}\psi_{s_{7}s_{8}}^{(2)})\,W_{r_{1}r_{2}s_{1}s_{2}}^{(3)}W_{r_{3}r_{4}s_{3}s_{4}}^{(4)}W_{r_{5}r_{6}s_{5}s_{6}}^{(5)}\,,
T_{\nu\rho}=\frac{1}{8\pi}\left(F_{\nu\rho}^{2}-\frac14g_{\nu\rho}F^{2}+G_{IJ}\partial_{\nu}X^{I}\partial_{\rho}X^{J}-\frac12g_{\nu\rho}G_{IJ}\partial_{\lambda}X^{I}\partial^{\lambda}X^{J}\right)
{\frac{1}{g_{1}^{2}}}+{\frac{1}{g_{2}^{2}}}={\frac{1}{4\pig_{s}}};\qquad{\frac{1}{g_{1}^{2}}}-{\frac{1}{g_{2}^{2}}}={\frac{1}{4\pi^{2}g_{s}}}\left(\frac{1}{2\pi\alpha^{\prime}}\int_{S^{2}}B_{(2)}-\pi\right).
r\widetilde{F}\left(\beta\right)\approx-\frac{\pi^{4}}{45}\frac{r^{4}}{\beta^{4}}-\frac{1}{240}-\frac{\pi^{2}\left|\mu_{eff}^{2}\right|}{12}\frac{r^{2}}{\beta^{2}}-\frac{3\left|\mu_{eff}^{2}\right|}{16},
S^{(0)}=\sum_{C}\sum_{x(\inC)}\bigg|\left(\frac{\delta}{\delta\sigma^{t\mu}(x)}-igA_{t\mu}(x)\right)\Phi[C]\bigg|^{2}+\sum_{x}-\frac{\varepsilon}{2\cdot3!}F_{\mu\nu\lambda}F^{\mu\nu\lambda}-V(|\Phi[C]|^{2}),