harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_{S}{\cal{F}}_{\chi=2}(g)=\frac{1}{N-2}W_{\mathrm{pert}}(g)=\frac{1}{2N}W_{\mathrm{pert}}^{SU(N)}(f=2g)=\frac{1}{2}\partial_{S}{\cal{F}}_{\chi=2}^{SU(N)}(f=2g)
\frac{l}{2}G_{\\nu}^{\mu}+\chi_{\\nu}^{\mu}+\frac{l^{3}}{8}{\calS}_{\\nu}^{\mu}-\frac{l^{2}}{4}\Box\chi_{\\nu}^{\mu}+C_{\\nu}^{\mu}=\frac{\kappa^{2}}{2}\stackrel{A}{T_{\\nu}^{\mu}}\,
\sum_{i=1}^{M}{\dot{C}}_{a}^{i}(t)\theta_{a}\delta^{3}({\vec{\theta}}-{\vec{\theta}}^{i})\prod_{j\neqi}\theta_{a_{j}}^{j}\theta_{b_{j}}^{j}\theta_{c_{j}}^{j}=\sum_{i=1}^{M}C_{a}^{i}(t){\dot{\theta}}_{a}\delta^{3}({\vec{\theta}}-{\vec{\theta}}^{i})\prod_{j\neqi}\theta_{a_{j}}^{j}\theta_{b_{j}}^{j}\theta_{c_{j}}^{j}\,
\sigma_{{\mathrm{abs}}}~=~{\frac{4\pi}{\omega^{3}}}{\frac{\Phi_{{\mathrm{abs}}}}{\Phi_{{\mathrm{in}}}}}~=~{\frac{2\pi^{2}r_{0}^{2}}{\omega}}(p+q){\frac{1}{|E|^{2}}}~.
S_{\mathrm{eff}}=-\frac{1}{2\kappa^{2}}\intd^{10}X\sqrt{-G}e^{-2\phi}\left\{R^{(10)}+\frac{1}{6}F_{\mu\nu\rho}F^{\mu\nu\rho}-4D_{\mu}\phiD^{\mu}\phi\right\}.
H=\frac{J}{2}\sum_{i=1}^{N}\left[2t^{-1}\sigma_{i}^{+}\sigma_{i+1}^{-}+2t\sigma_{i}^{-}\sigma_{i+1}^{+}+\frac{q+q^{-1}}{2}\sigma_{i}^{z}\sigma_{i+1}^{z}\right]\;,
\,h_{\vec{p}}(t)=\frac1{2\omega_{0}\left(\vec{p}\right)}\left(\omega^{2}\left(\vec{p},t\right)+\omega_{0}^{2}\left(\vec{p}\right)\right)\,\,\,;\,\,\,g_{\vec{p}}(t)\,=\frac1{2\omega_{0}\left(\vec{p}\right)}\left(\omega^{2}\left(\vec{p},t\right)-\omega_{0}^{2}\left(\vec{p}\right)\right)
\begin{array}{llcll}{}&{V_{3}}&{\stackrel{C_{3}^{12}}{\longrightarrow}}&{V_{1}\otimesV_{2}}&{}\\{\Pi_{3}}&{\downarrow}&{}&{\downarrow}&{\Pi_{1}\otimes\Pi_{2}}\\{}&{V_{3}}&{\stackrel{C_{3}^{12}}{\longrightarrow}}&{V_{1}\otimesV_{2}}&{}\\\end{array}
{\calL}=\frac{1}{2\pi\alpha}\epsilon^{\mu\nu\alpha}\,\partial_{\mu}A_{\nu}\,A_{\alpha}+\bar{\psi}(i\not\!\partial-m)\psi+\bar{\psi}\gamma^{\mu}\psiA_{\mu}-G(\bar{\psi}\psi)(\bar{\psi}\psi)+\frac{1}{2\lambda}(\partial_{\mu}A^{\mu})^{2},
\begin{array}{l}{LG_{2X}=X_{R}^{-1/3}X_{L}^{1/3}LG_{2}X_{L}^{-1/3}X_{R}^{1/3}}\\{G_{2X}=X_{R}^{-1/3}X_{L}^{1/3}G_{2}X_{L}^{-1/3}X_{R}^{1/3}.}\\\end{array}
L_{_{\mathrm{GF+FP}}}^{2}=\mathrm{Tr}\left([\delta_{1}+\delta_{2}](-\frac{1}{2}\overline{{\Phi}}[X_{\mu},\psi^{\mu}]+\frac{1}{2}s~e\overline{{\Phi}}[\Phi,\eta]+\frac{i}{4}\chi^{\mu\nu}B_{\mu\nu})\right).
A_{\pm}^{\mu\nuab}=\mathrm{g}^{2}\langlek\mid(\alpha_{1}^{\mu}J_{R1}^{a}\pm\tilde{\alpha}_{1}^{\mu}J_{L1}^{a})\int_{0}^{2\pi}d\sigma\sum_{n,m}(\alpha_{Rm}^{b}\pm\tilde{\alpha}_{-n}^{\nu}J_{L-m}^{a})e^{-ikX(\sigma)}e^{i(n-m)\sigma}\int{\frac{dq}{q^{3-N}}}\midB\rangle,
\mathcal{L}_{m}(g)=\frac{(-1)^{m}}{2^{m}}\delta_{\mu_{1}\nu_{1}\ldots\mu_{m}\nu_{m}}^{\rho_{1}\sigma_{1}\ldots\rho_{m}\sigma_{m}}{R^{\mu_{1}\nu_{1}}}_{\rho_{1}\sigma_{1}}\cdots{R^{\mu_{m}\nu_{m}}}_{\rho_{m}\sigma_{m}}\;,
F(i,j,k,l)={\frac{1}{256}}e^{E_{k}^{(4)}-p_{j,\mu}(\theta_{jk}\sigma^{\mu}\bar{\theta}_{jk})}(\theta_{lk}\theta_{jk})\bar{\theta}_{lk}^{2}\theta_{ik}^{2}\left(-{\frac{1}{2}}\bar{\theta}_{ij}^{2}+{\frac{2}{3}}\bar{\theta}_{ik}^{2}\right).
n_{\mathrm{s}}(k^{2})\;\equiv\;{\frac{2}{\pi}}\int_{0}^{k}{\frac{\sqrt{k^{2}-\zeta^{2}}}{\sqrt{1-\zeta^{2}}}}d\zeta\;=\;{\frac{2}{\pi}}\left[E(k)-(1-k^{2})K(k)\right]\;,
{\calR}_{0}=m\sqrt{u^{2}}+eu^{\mu}A_{\mu}-\frac{u^{\mu}}{2}{\omega}_{\mu}^{ab}S_{ab}+\frac{eg}{4m}F^{ab}S_{ab}\sqrt{u^{2}}-\frac{\kappa}{8m}R^{abcd}S_{ab}S_{cd}\sqrt{u^{2}}\,.
A_{\gamma\gamma}(s,t,u)=\int\prod_{i=1}^{4}dx_{i}\mu(x)\frac{\xi_{1}\cdot\xi_{2}}{(x_{1}-x_{2})^{2}}\frac{\Phi_{3}^{a}\Phi_{4}^{a}}{(x_{3}-x_{4})^{2}}\prod_{i<j}(x_{i}-x_{j})^{k_{i}k_{j}}+\cdots.
H_{A}\rightarrow{\frac{\left[g_{s}N_{A}(\ell_{s}U)^{A-3}\right]}{(V_{b}/\ell_{s}^{b})(\ell_{s}U)^{4-b}}}={\frac{N_{A}}{(V_{b}/\ell_{s}^{b})N_{B}}}{\frac{\left[g_{s}N_{B}(\ell_{s}U)^{B-3}\right]}{(\ell_{s}U)^{4-a}}}.
\begin{array}{ccc}{A\alpha_{0}=\alpha_{1},}&{A\alpha_{n+1}=\alpha_{n+2},}&{A\alpha_{j}=\alpha_{j},\quadj=2,\ldots,n,}\\{A\alpha_{1}=\alpha_{0},}&{A\alpha_{n+2}=\alpha_{n+1},}&{}\\\end{array}
\begin{array}{l}{{\frac{1}{\sqrt{\dot{x}_{1}^{2}+\dot{x}_{2}^{2}}}}(\dot{x}_{1}\gamma_{1}+\dot{x}_{2}\gamma_{2}+\dot{x}_{+}\gamma_{-})(\psi_{0}+(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{-}\gamma_{+})\psi_{1})=}\\{=i\Gamma_{1}(\psi_{0}+(x_{1}\gamma_{1}+x_{2}\gamma_{2}+x_{-}\gamma_{+})\psi_{1})}\\\end{array}
B=\frac{1}{2}(1-\zeta^{2})\partial_{\zeta}^{2}+(\zeta\partial_{\zeta})^{2}+\frac{c^{\prime}+d-5}{2}\zeta\partial_{\zeta}-m^{ij}\zeta^{i}\partial_{\zeta^{j}}-\lambda(\lambda+\frac{c^{\prime}+d-3}{2})\,.
G_{\perp\perp}=\frac{1}{2}\nu(\nu-1)\!\left(\frac{{a^{\prime}}^{2}}{a^{2}}-\frac{\dot{a}^{2}}{n^{2}a^{2}}-\frac{k}{a^{2}}\right)+\nu\!\left(\frac{n^{\prime}a^{\prime}}{na}+\frac{\dot{n}\dot{a}}{n^{3}a}-\frac{\ddot{a}}{n^{2}a}\right).
(\sqrt{\nabla_{\mu}l\nabla^{\mu}l-i\epsilon\nabla^{2}l-\epsilon^{2}\nabla^{2}-2i\epsilon\nabla_{\mu}l\nabla^{\mu}+\epsilon^{2}V(x)}-\lambda)\sum_{m=0}^{\infty}\epsilon^{m}\sigma_{m}=\mathrm{I}
E_{2}=\frac{1}{24\pi}\int_{a}^{b}d\lambda\frac{d^{2}V}{d\lambda^{2}}\frac{1}{\sqrt{2(E_{F}^{(0)}-V)}}-\frac{1}{24}\left[\frac{1}{\pi}\int_{a}^{b}d\lambda\frac{1}{\sqrt{2(E_{F}^{(0)}-V)}}\right]^{-1}
H_{\lambda=1}^{(orb)}=\frac{1}{sin\gamma}\sum_{j=1}^{L}\left(\tau_{j}^{+}\tau_{j+1}^{-}+\tau_{j}^{-}\tau_{j+1}^{+}+\rho_{j}^{+}\rho_{j+1}^{-}+\rho_{j}^{-}\rho_{j+1}^{+}-\sigma_{j}^{0}-\sigma_{j+1}^{0}\right)
\left(^{U[\vec{\varphi}]}a_{1}\right)^{a}\conga_{1}^{a}(x)-\epsilon^{abc}\,\varphi^{b}(x)\,a_{1}^{c}(x)+\partial_{1}\varphi^{a}(x)=a_{1}^{a}(x)-D_{1}^{ab}[a_{1}]\varphi^{b}(x)\;,
4.1I_{11}^{B}=\!\int\!d^{11}x\left[-\frac{\sqrt{g}}{4\kappa^{2}}R(g)-\frac{\sqrt{g}}{48}F^{2}+\frac{2\kappa}{144^{2}}\epsilon^{1\cdots11}F_{1\cdots}F_{5\cdots}A_{..11}\right],
S=-i\left[\intd^{2}\sigmad\theta_{0}^{+}d\theta_{0}^{-}g_{ij}D_{0+}\varphi^{i}D_{0-}\varphi^{j}+\intd^{2}\sigmadtd\theta_{0}^{+}d\theta_{0}^{-}H_{ijk}\partial_{t}D_{0+}\varphi^{j}D_{0-}\varphi^{k}\right],
c_{r,s}^{(l)}=q^{\frac{-\hat{c}}{24}}\sum_{m=0}^{N/2}C_{2m}^{l-1}\left\{\sum_{j\in{\bfZ},m\equivm_{r,s}(j)}q^{\alpha_{r,s}(j)}-\sum_{j\in{\bfZ},m\equivm_{r,-s}(j)}q^{\alpha_{r,-s}(j)}\right\}
b(t)=ln\left[_0F_{1}\left(\frac{q_{0}}{f_{1}},\frac{c}{4f_{1}^{2}}e^{2f_{1}t}\right)+c_{3}{e^{2(f_{1}-q_{0})t}}{_0F_{1}}\left(2-\frac{q_{0}}{f_{1}},\frac{c}{4f_{1}^{2}}e^{2f_{1}t}\right)\right]
\{\{...\{QA_{0}(x)\xi_{0},\underbrace{S\},S\},...\},S\}}_{n}=\left\{\begin{array}{lr}{Q(2i)^{n}\Phi_{n}(p,x,\vec{\xi})\xi_{5}\xi_{0}}&{n-odd}\\{}&{}\\{Q(2i)^{n}\Phi_{n}(p,x,\vec{\xi})\xi_{0}}&{n-even}\\\end{array}\right.+{\calO}(Q^{2}),
H=N\left[-\frac{1}{24}\epsilon_{ijk}(\hat{\theta}^{i}-\frac{1}{2}\theta^{i})(\hat{\theta}^{j}-\frac{1}{2}\theta^{j})(\hat{\theta}^{k}-\frac{1}{2}\theta^{k})-\frac{1}{32}\epsilon_{ijk}\theta^{i}\theta^{j}(\hat{\theta}^{k}-\frac{1}{2}\theta^{k})\right]\,.
\left[L,\left(L^{2}\right)_{\geq1}\right]=\left(D\left((D^{4}\Phi_{0})-D(D\Phi_{0})^{2}-2(D^{2}\Phi_{1})\right)\right)+D^{-1}\left((D^{4}\Phi_{1})+2D^{2}(\Phi_{1}(D\Phi_{0}))\right)
<\phi_{\mathcal{B}}{}_{\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}}^{N_{1}N_{2}N_{3}N_{4}}\ldots\phi_{\mathcal{B}}{}_{\alpha_{10}^{\prime}\alpha_{8}^{\prime}\alpha_{5}^{\prime}\alpha_{1}^{\prime}}^{N_{10}N_{8}N_{5}N_{1}}>\\=\\\lambda\\\delta^{\alpha_{1}\alpha_{1}^{\prime}}\ldots\delta^{\alpha_{10}\alpha_{10}^{\prime}}\\{\mathcal{B}}_{N_{1}\ldotsN_{10}}~~,
M_{q}=m_{q}\left\{1+\frac{4}{3}\frac{\alpha_{s}(M_{q})}{\pi}+K_{q}\left(\frac{\alpha_{s}(M_{q})}{\pi}\right)^{2}+O\left(\left(\frac{\alpha_{s}(M_{q})}{\pi}\right)^{3}\right)\right\},
X_{i}={\frac{1}{2}}\left\{NS_{i}\left({\frac{\theta_{3}}{\eta}}\right)^{m}-\widetilde{NS}_{i}\left({\frac{\theta_{4}}{\eta}}\right)^{m}-R_{i}\left({\frac{\theta_{2}}{\eta}}\right)^{m}+\widetilde{R}_{i}\left({\frac{\theta_{1}}{\eta}}\right)^{m}\right\}\quad,
F_{8}^{{\mathcal{O}}_{0,2}^{0,1}|++------}=\frac{-\langle{\mathcal{O}}_{0,2}^{0,1}\rangle(\sigma_{5}^{-}+\sigma_{1}^{-}(\sigma_{2}^{+})^{2}+\sigma_{3}^{-}\sigma_{2}^{+})/(\sigma_{2}^{+})^{5/2}}{2^{6}\,\mathrm{cosh}(\theta_{12}/2)\,\prod_{3\leqi<j\leq8}\mathrm{cosh}(\theta_{ij}/2)}\prod_{i<j}F_{\mathrm{min}}^{\mu_{i}\mu_{j}}(\theta_{ij}),
HT=\pi+2\bar{k}K(\bar{k})\sqrt{\frac{1-2\bar{k}^{2}}{1-\bar{k}^{2}}}-2K(\bar{k})E(\bar{\phi},\bar{k^{\prime}})+2[K(\bar{k})-E(\bar{k})]F(\bar{\phi},\bar{k^{\prime}}),
\left(\begin{array}{c}{\ddot{\Sigma}_{n}^{t}}\\{\ddot{\Sigma}_{n}^{*}}\\\end{array}\right)+{\calV}\left(\begin{array}{c}{\Sigma_{n}^{t}}\\{\Sigma_{n}^{*}}\\\end{array}\right)={\calG}^{-1}\left(\begin{array}{c}{U_{n}^{t}}\\{U_{n}^{*}}\\\end{array}\right)\equiv\left(\begin{array}{c}{\tilde{U}_{n}^{t}}\\{\tilde{U}_{n}^{*}}\\\end{array}\right),
\int_{0}^{\infty}dw^{\prime\prime}\alpha_{jn,w^{\prime\prime}}\alpha_{j^{\prime}n^{\prime},w^{\prime\prime}}^{*}-\beta_{jn,w^{\prime\prime}}\beta_{j^{\prime}n^{\prime},w^{\prime\prime}}^{*}=\delta_{jj^{\prime}}\delta_{nn^{\prime}},
\frac{\delta\Gamma}{{\delta}A_{\mu}}=\langle\frac{{\delta}S}{\delta\hat{A}_{\mu}}\rangle,\qquad\frac{\delta\Gamma}{\delta{\bar{\psi}}^{\alpha}}=\langle\frac{{\delta}S}{\delta\hat{\bar{\psi}}^{\alpha}}\rangle,\qquad\frac{\delta\Gamma}{\delta{\psi}^{\alpha}}=\langle\frac{{\delta}S}{\delta\hat{\psi}^{\alpha}}\rangle.
S_{eff}={\frac{1}{2}}\,\intd^{2}\sigma\,\left[g^{\alpha\beta}\,\bar{G}_{ab}\,\partial_{\alpha}\bar{x}^{a}\,\partial_{\beta}\bar{x}^{b}+\epsilon^{\alpha\beta}\,\bar{B}_{ab}\,\partial_{\alpha}\bar{x}^{a}\,\partial_{\beta}\bar{x}^{b}\right]\,
0=\frac{\delta\Psi_{k}({\bfA}^{U_{h}})}{\deltah^{a}({\bfx})}|_{h=0}=\intd{\bfy}\frac{\delta\Psi_{k}({\bfA})}{\deltaA_{i}^{b}({\bfy})}\frac{\delta(A_{i}^{b}({\bfy})^{U_{h}})}{\deltah^{a}({\bfx})}|_{h=0}=D_{i}^{ab}({\bfA})\frac{\delta\Psi_{k}({\bfA})}{\deltaA_{i}^{b}({\bfx})}
W_{k\neq0}^{\beta}=\mathrm{}-\frac12\frac{\partial}{\partial\epsilon}\left[\frac{\mu^{2\epsilon}}{\Gamma(\epsilon)}\int_{0}^{\infty}\!\frac{{\mathrm{d}}s}{s^{1-\epsilon}}\,{\mathrmTr}{K}^{\beta}|_{k\neq0}(s)\right]_{\epsilon=0},
\langle0^{(a)}|\varphi(x)\varphi(x^{\prime})|0^{(a)}\rangle=\langle0_{{\mathrm{m}}}|\varphi(x)\varphi(x^{\prime})|0_{{\mathrm{m}}}\rangle+\langle\varphi(x)\varphi(x^{\prime})\rangle_{b}^{(a)},
P=\left(\begin{matrix}{1}&{}&{}&{}&{}\\{}&{q}&{}&{}&{}\\{}&{}&{q^{2}}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{q^{n-1}}\\\end{matrix}\right),Q=\left(\begin{matrix}{0}&{1}&{0}&{\dots}&{0}\\{0}&{0}&{1}&{\dots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\ddots}&{0}\\{0}&{0}&{0}&{\dots}&{1}\\{1}&{0}&{0}&{\dots}&{0}\\\end{matrix}\right)
\left[A_{k}^{\mu}\left(\theta\right)+S^{\mu}\,\!\!_{\nu}B_{k}^{\nu\dagger}\left(\theta\right)\right]\left|B_{X}\left(\theta\right)\right\rangle=0,\qquad\left[A_{k}^{\mu\dagger}\left(\theta\right)+S^{\mu}\,\!\!_{\nu}B_{k}^{\nu}\left(\theta\right)\right]\left|B_{X}\left(\theta\right)\right\rangle=0.
\begin{array}{ccc}{e^{a}}&{\rightarrow}&{\frac{1}{\alpha}e^{a}}\\{\omega^{ab}}&{\rightarrow}&{\omega^{ab}}\\{b}&{\rightarrow}&{\frac{1}{3\alpha}b}\\{\psi_{i}}&{\rightarrow}&{\frac{1}{\sqrt{\alpha}}\psi_{i}}\\{\overline{{\psi}}^{i}}&{\rightarrow}&{\frac{1}{\sqrt{\alpha}}\overline{{\psi}}^{i}}\\{a_{j}^{i}}&{\rightarrow}&{a_{j}^{i}.}\\\end{array}
\eta_{ij}(t)=-m^{-6}\left(1+\frac{1}{6}m^{4}t^{2}\right)\left(e^{-m_{i}^{2}t}-e^{-m_{j}^{2}t}\right)-\frac{1}{2}m^{-4}t\left(e^{-m_{i}^{2}t}+e^{-m_{j}^{2}t}\right)
S_{\mathrm{soft}\,(N)}=\frac{(-ie)^{N}}{8\pi^{2}}\!\int_{0}^{\beta}\!dx_{4}\int\!\frac{d^{3}p_{1}}{(2\pi)^{3}}\!\cdots\!\frac{d^{3}p_{N}}{(2\pi)^{3}}\delta^{3}(p_{1}\!+\!\ldots\!+\!p_{N})\int_{0}^{1}\!dx_{1}\!\cdots\!\int_{0}^{x_{N-1}}\!dx_{N}\!\sum_{\stackrel{{\scriptstylek=0}}{k\,\mathrm{even}}}^{N}\!I_{N}^{(k)}\,
\mathrm{reg}\,\,p=(\mathrm{reg}\,\,p)_{\mathrm{Neumann}}\left[1+4a\left(\frac{1}{\beta_{1}}+\frac{1}{\beta_{2}}\right)\frac{\zeta_{R}(D-1)}{D\zeta_{R}(D+1)}+\cdots\right],
\widehat\Psi^{\mu}=\Psi^{\mu}-\frac{1}{4}\theta^{\nu\rho}\bigg(\left\{\omega_{\nu},(D_{\rho}+\partial_{\rho})\Psi^{\mu}\right\}-2\left\{R_{\nu\rho},\Psi^{\mu}\right\}+2\delta_{\nu}^{\mu}\left\{R_{\tau\rho},\Psi^{\tau}\right\}\bigg)+{\calO}\left(\theta^{2}\right).
{\calL}_{TM}={\calL}_{D}+\chi\,\frac{m}{2}\,\epsilon^{\mu\nu\lambda}\,A_{\mu}\,\partial_{\nu}A_{\lambda}-\beta^{\frac{s-r}{s}}\,\left(\frac{s}{2r-s}\right)\,\left[\frac{m^{2}}{8}\,F_{\mu\nu}\,F^{\mu\nu}+\frac{e^{2}}{4}\,J_{\mu}\,J^{\mu}+\frac{\chi\,e\,m}{2}\,\epsilon^{\mu\nu\lambda}\,J_{\mu}\,\partial_{\nu}A_{\lambda}\right]^{\fracr{s}}\,.\nonumber
J_{\mathrm{CS}}^{\mu}=-{\frac{\deltaS_{\mathrm{CS}}}{\deltaA_{\mu}}}=-\sigma_{xy}f_{L}\epsilon^{\mu\nu\rho}\partial_{\nu}A_{\rho}+{\frac{\sigma_{xy}}{2}}\partial_{2}f_{L}\epsilon^{2\mu\nu}A_{\nu}
L_{~\nu}^{\mu}={\itk}\epsilon^{\mu\alpha\beta\lambda}\left(\bar{Q}_{\alpha\beta\nu}\nabla_{\lambda}+D_{\alpha\nu\beta}\nabla_{\lambda}-g_{\alpha\nu}D_{~\beta\lambda}^{\sigma}\nabla_{\sigma}+\nabla_{\nu}Q_{\alpha\beta\lambda}\right)+O({\itk}^{2})
{\calI}_{L,M,0;L^{\prime},M^{\prime},0}=-C^{2}(k+2)^{2}\sum_{l^{\prime},m^{\prime}}^{R}N_{LL^{\prime}}^{l^{\prime}}\delta_{M-M^{\prime}+m^{\prime}+1}^{(2k+4)}I_{l^{\prime}}^{m^{\prime}}(q)\quad,
Q=\left(\begin{array}{ccc}{q_{1}}&{}&{}\\{}&{\ddots}&{}\\{}&{}&{q_{N}}\\\end{array}\right),\\\\\widetildeQ=\left(\begin{array}{ccc}{\sqrt{q_{1}^{2}+c}}&{}&{}\\{}&{\ddots}&{}\\{}&{}&{\sqrt{q_{N}^{2}+c}}\\\end{array}\right)\,C\,e^{i\beta},
V_{nm}^{<}(B)=\frac{-1}{2\pi^{2}l^{3}}\int_{0}^{1}d\eta\int_{0}^{1}du\,\frac{u^{2}\left[f(u;r)\right]^{n-2m}\left[g(u;r)\right]^{m}}{\left[(ru+\frac{\pi}{2})^{2}+\bar{c}^{2}\right]^{n}}.
\left(1-\frac{wN}{6}e^{2\phi}\right)\left(\delta^{2}\phi-2\delta\phi\,\delta\rho\right)-(\delta\phi)^{2}=\frac{N}{12}e^{2\phi}\left[(\deltaZ)^{2}+2\delta^{2}Z-4\deltaZ\delta\rho\right];
\begin{array}{rcl}{}&{}&{tt^{-1}=1=t^{-1}t~,}\\{}&{}&{tet^{-1}=q^{2}e~,~~~~tft^{-1}=q^{-2}f~,}\\{}&{}&{[e,f]=\displaystyle\frac{t-t^{-1}}{q-q^{-1}}~.}\\\end{array}
\Phi=\left(\begin{array}{ccc}{\Phi_{1}}&{0}&{0}\\{0}&{\Phi_{2}}&{0}\\{0}&{0}&{\Phi_{3}}\\\end{array}\right),\,\delta_{1}=\left(\begin{array}{ccc}{0}&{1}&{0}\\{0}&{0}&{1}\\{1}&{0}&{0}\\\end{array}\right),\,\delta_{-1}=\left(\begin{array}{ccc}{0}&{0}&{1}\\{1}&{0}&{0}\\{0}&{1}&{0}\\\end{array}\right)
\tilde{x}^{0,1}=\frac{1}{\sqrt{\alpha_{\mathrm{eff}}^{\prime}}}x^{0,1},\qquad\tilde{x}^{2,\ldots,5}=\frac{1}{\epsilon\sqrt{\alpha_{\mathrm{eff}}^{\prime}}}x^{2,\ldots,5}
\int_{o}^{\infty}\frac{kdk}{q^{2}+k^{2}}J_{\nu}(kr_{x})J_{\nu}(kr_{y})=\frac{\pii}{2}J_{\nu}(iqr_{x})H_{\nu}^{1}(iqr_{y})=I_{\nu}(qr_{x})K_{\nu}(qr_{y}),
\varphi_{\alpha_{1},\alpha_{2},\alpha_{3}}(z_{1},z_{2},z_{3})\equivz_{1}^{\alpha_{1}-h}z_{2}^{\alpha_{2}-h}z_{3}^{\alpha_{3}-h}z_{12}^{\alpha_{3}}z_{23}^{\alpha_{1}}z_{31}^{\alpha_{2}}\,,
\int[d\delta{\hat{g}}]e^{-|\delta{\hat{g}}|^{2}/2}\equiv\prod_{\sigma}\int[d\delta{\hat{g}}]_{\sigma}e^{-|\delta{\hat{g}}|_{\sigma}^{2}/2}=1=J_{M}({\cal\phi},{\hat{g}})\int[d\delta{\cal\phi}]_{e^{\phi}{\hat{g}}}\int[d\delta\eta]_{\hat{g}}^{\prime}\int_{0}^{\infty}dle^{-|\delta{\hat{g}}|^{2}/2}\quad,
\int_{0}^{\infty}d\nu\\nu\int_{m}^{\infty}dk\[k^{2}-m]^{1/2-s}\frac{\partial}{\partialk}\frac{t^{j}}{\nu^{n}}=-m^{1-2s}\frac{\Gamma(\frac32-s)\Gamma(1+\frac{j-n}2)\Gamma(s+\frac{n-3}2)}{2(mR)^{n-2}\Gamma(\fracj2)},
\sigma(\mathrm{odd})={\frac{4}{k}}{\frac{N_{0}^{2}(a)}{J_{0}^{2}(a)+N_{0}^{2}(a)}}+{\frac{8}{k}}\sum_{n=1}^{\infty}{\frac{N_{n}^{2}(a)}{J_{n}^{2}(a)+N_{n}^{2}(a)}}
{\calP}(M_{3})\Psi^{(0)}=\sum_{j_{1}+\cdots+j_{N}=0}^{\ell}\left(j_{1}+2j_{2}+\cdots+Nj_{N}-\frac{N\ell}{2}\right)\alpha_{j_{1}\ldotsj_{N}}\prod_{k=1}^{N}s_{k}^{[N]}(z_{1},\ldots,z_{N})^{j_{k}}.
{\calL}=\frac{1}{2}(\partial\chi)^{2}+m\psi_{-}^{T}\gamma_{0}e^{\frac{4\pii\chi}{g^{2}}}\psi_{-}+m\psi_{+}\gamma_{0}e^{-\frac{4\pii\chi}{g^{2}}}\psi_{+}+\cdots,
\begin{array}{ccl}{c_{2}^{\prime}\;P_{-}}&{=}&{\phi(c_{2})\;P_{-}=(-q)P_{-}M_{1}K_{1}\tilde{M}_{1}^{-1}\check{R}M_{1}K_{1}\tilde{M}_{1}^{-1}P_{-}}\\{}&{}&{}\\{}&{=}&{(-q)P_{-}M_{1}M_{2}K_{1}\check{R}K_{1}\tilde{M}_{2}^{-1}\tilde{M}_{1}^{-1}P_{-}=det_{q}Mdet_{q}\tilde{M}^{-1}c_{2}\;P_{-}\quad,}\\\end{array}
S=\int_{0}^{1}\left[-\frac{1}{2e}\left(\dot{x}^{\mu}-i\psi^{\mu}\chi\right)^{2}-\frac{e}{2}m^{2}-im\psi^{5}\chi-i\left(\psi_{\mu}\dot{\psi}^{\mu}-\psi_{5}\dot{\psi}_{5}\right)\right]d\tau\;,
F_{(f)1}^{[1,0]}\left(p_{1}^{2},p_{2}^{2},p_{3}^{2}\right)=\frac{1}{\Lambda^{2}}\left[x_{1,6}\left(\Pi_{1}\Pi_{2}+\Pi_{2}\Pi_{3}+\Pi_{3}\Pi_{1}\right)+x_{1,7}\left(\Pi_{1}\Pi_{2}\Pi_{3}\right)\right]\,,
F_{\alpha\beta\gamma}^{\;\;\;\;(\sigma)}=\partial_{\alpha}A_{\beta\gamma}^{\;\;\;(\sigma)}+\partial_{\beta}A_{\gamma\alpha}^{\;\;\;(\sigma)}+\partial_{\gamma}A_{\alpha\beta}^{\;\;\;(\sigma)}\equiv\partial_{\left[\alpha\right.}A_{\left.\beta\gamma\right]}^{\;\;\;\;(\sigma)},
T_{ii}^{(1)}={\frac{N^{2}}{2}}\int{{\frac{d^{3}\vec{p}}{\omega_{\vec{p}}}}(p_{i}^{2}-{\frac{1}{3}}{\omega_{\vec{p}}}^{2})F(\vec{p})e^{-2i\omega_{\vec{p}}t}}+cc,
S[\Phi]={\frac{1}{2}}\int\biggl((\mathrm{e}^{-\Phi}Q_{B}\mathrm{e}^{\Phi})(\mathrm{e}^{-\Phi}\eta_{0}\mathrm{e}^{\Phi})-\int_{0}^{1}dt(\mathrm{e}^{-t\Phi}\partial_{t}\mathrm{e}^{t\Phi})\{(\mathrm{e}^{-t\Phi}Q_{B}\mathrm{e}^{t\Phi}),(\mathrm{e}^{-t\Phi}\eta_{0}\mathrm{e}^{t\Phi})\}\biggr),
m_{\mathrm{b}}=2\sqrt{v}\,\Bigl(1-{\frac{1}{32}}\,\bigl(Ng^{2}\bigr)^{2}+{\frac{1}{32\pi}}\,\bigl(Ng^{2}\bigr)^{3}+{\calO}\bigl(\bigl(Ng^{2}\bigr)^{4},1/N\bigr)\Bigr),
d~=~\left(\begin{matrix}{0}&{1}&{-1}&{0}&{0}&{1}&{-1}&{0}\\{1}&{-1}&{0}&{0}&{1}&{-1}&{0}&{0}\\{-1}&{0}&{0}&{1}&{-1}&{0}&{0}&{1}\\{0}&{0}&{1}&{-1}&{0}&{0}&{1}&{-1}\\\end{matrix}\right)
\begin{array}{cl}{}&{\mathrm{Res}_{z_{2}=z_{1}\tau^{2}}\cdots\mathrm{Res}_{z_{n}=z_{n-1}\tau^{2}}w_{\alpha_{1}\cdots\alpha_{n}}(z_{1},\cdots,z_{n})}\\{=}&{\delta_{\alpha_{1}n}\delta_{\alpha_{2}n-1}\cdots\delta_{\alpha_{n}1}z_{1}^{n-1}(\tau^{2}-1)^{n-1}\tau^{(n-1)^{2}}\left[n-1\right]!~~v^{(n\cdots21)},}\\\end{array}
D_{i\alpha}D_{j\beta}\tilde{\mathrm{A}}^{\gamma\delta}(\theta)=\textstyle{\frac{1}{12}}(\delta_{\alpha}^{~\gamma}\delta_{\beta}^{~\delta}-\delta_{\alpha}^{~\delta}\delta_{\beta}^{~\gamma})D_{i\epsilon}D_{j\eta}\tilde{\mathrm{A}}^{\epsilon\eta}(\theta)
\Psi=(1-{\frac{\rho}{4}}x^{+}x^{-})^{\frac{1}{2}}\left(\begin{array}{c}{e^{-{\frac{\theta}{2}}}e^{\imathv}p(x^{-})}\\{e^{{\frac{\theta}{2}}}e^{\imathu}f(x^{+})}\\\end{array}\right)
\begin{array}{lcl}{\mathrm{NS5}}&{:}&{012345}\\{\mathrm{NS^{\prime}5}}&{:}&{012389}\\{\mathrm{D6}}&{:}&{0123789}\\{\mathrm{D4}}&{:}&{01236}\\{\mathrm{O4}}&{:}&{01236}\\\end{array}
\begin{array}{rl}{\mathrm{I.}}&{\quadd<\infty\,,\qquadR=\infty\,;}\\{\mathrm{II.}}&{\quadd=\infty\,,\qquadR=\infty\,;}\\{\mathrm{IIIa.}}&{\quadd=\infty\,,\qquad1<R<\infty\,;}\\{\mathrm{IIIb.}}&{\quadd=\infty\,,\qquad0<R\leq1\,.}\\\end{array}
N(E)=\frac{N}{\pi}\int_{a}^{b}d\lambda\sqrt{2(E-V)}-\frac{1}{24\piN}\int_{a}^{b}d\lambda\frac{d^{2}V}{d\lambda^{2}}\frac{1}{\sqrt{2(E-V)}}+O\left(\frac{1}{N^{3}}\right)
(A,B)=\intd^{n}x\left(\frac{\delta^{R}{A}}{\delta\phi^{A}(x)}\frac{\delta^{L}{B}}{\delta\phi_{A}^{*}(x)}-\frac{\delta^{R}{A}}{\delta\phi_{A}^{*}(x)}\frac{\delta^{L}{B}}{\delta\phi^{A}(x)}\right).
N_{1ij}=n_{ij}-n_{ik}\!\!\;\left(\frac{\delta^{2}A(\Phi)}{\delta\Phi_{k}\delta\Phi_{l}}-m_{kl}\right)n_{lj}+n_{ik}\!\!\;\left(\frac{\delta^{2}A(\Phi)}{\delta\Phi_{k}\delta\Phi_{l}}-m_{kl}\right)\!\!\;n_{lp}\!\!\;\left(\frac{\delta^{2}A(\Phi)}{\delta\Phi_{p}\delta\Phi_{q}}-m_{pq}\right)\!\!\;n_{qj}-\ldots
J_{\mathrm{asym}}\=-\intd^{2}x\,\,x^{i}\epsilon_{ij}\,\left\{\dot{f}\partial_{j}f+\bigl[f^{2}(\dot{\theta}+A_{0})-\rho_{e}\bigr](\partial_{j}\theta+A_{j})\right\}-\pi\rho_{e}\sum_{a}(-1)_{a}|\vec{q}_{a}|^{2}.
Z=\sum_{N=1}^{\infty}e^{-\beta\frac{N\hbar}{2R}}\int\left(\prod_{k,\lambda}d\Pi_{\lambda}^{k}\right)\left(\prod_{l,\mu}d\theta_{\mu}^{l}\right)<\Pi_{\lambda}^{k}\theta_{\mu}^{a}|e^{-\betaH\left(\Pi_{\lambda}^{k},Y_{\lambda}^{k},\theta_{\mu}^{a}\right)}|\Pi_{\lambda}^{k}\theta_{\mu}^{a}>
R=\frac{\Gamma(1/2+i(\rho+\frac{(2mV_{0})^{1/2}}{\omega}))\Gamma(1/2+i(\rho-\frac{(2mV_{0})^{1/2}}{\omega}))\Gamma(-i\rho)}{\Gamma(1/2+i\frac{(2mV_{0})^{1/2}}{\omega})\Gamma(1/2-i\frac{(2mV_{0})^{1/2}}{\omega})\Gamma(i\rho)}
\textstyle{\frac{1}{2\piR}}\int\mathrm{d}^{6}x\,\textstyle{\frac{1}{4}}H_{5\mu\nu}H^{5\mu\nu}+\textstyle{\frac{1}{24}}\epsilon^{\mu\nu\lambda\rho\sigma}\partial_{5}B_{\mu\nu}H_{\lambda\rho\sigma}\,,
H=\frac{1}{2}Tr\left\{-\frac{1}{N^{2}}\frac{\partial^{2}}{\partial\Phi^{2}}+\frac{1}{4}\left(\frac{\partialW}{\partial\Phi}\right)^{2}-\frac{1}{2N}\frac{\partial^{2}W}{\partial\Phi^{2}}\right\}
\begin{array}{ccccccccc}{}&{}&{}&{}&{\circ_{0}}&{\left(\frac{1}{l_{p}^{3}}\right)}&{}&{}&{}\\{}&{}&{}&{}&{\mid}&{}&{}&{}&{}\\{\left(\frac{R_{1}}{l_{p}^{3}}\right)\;\;\circ_{1}}&{-}&{\circ_{2}}&{-}&{\circ_{3}}&{-}&{\circ_{4}}&{-\dots-}&{\circ_{d-1}\;\;\left(\frac{1}{R_{d}}\right)}\\\end{array}
{\frac{M}{M_{Pl}}}={\frac{(P_{k}^{S})^{1/4}}{2\sqrt{L}u^{1/2}[(1+u^{2})(1+3u^{2})]^{1/8}}}\approx{\frac{2.0\times10^{-4}\left(10^{10}P_{k}^{S}\right)^{1/4}\left({60/L}\right)^{1/2}}{u^{1/2}[(1+u^{2})(1+3u^{2})]^{1/8}}};
G_{\,+}^{\ellk}(q^{0})=-\left(\Phi_{p},\intd\vec{x}j^{\ell}(\vec{x},0){\frac{1}{q^{0}+i\epsilon-(H-E)}}j^{k}(\vec{y},0)-j^{k}(\vec{y},0){\frac{1}{q^{0}-i\epsilon+(H-E_{0})}}\intd\vec{x}j^{\ell}(\vec{x},0)\Phi_{p}\right)
\sum_{i}(\gamma_{A_{i}})_{\alpha\beta}(\gamma^{c}\gamma^{\overline{{A}}_{i}})_{\gamma\delta}=\mathrm{Trace(I)}\;(\gamma^{c})_{\alpha\delta}\delta_{\beta\gamma},\;\;\;\;\sum_{i}(\gamma_{A_{i}})_{\alpha\beta}(\gamma^{\overline{{A}}_{i}}(-1)^{i}\gamma^{c})_{\gamma\delta}=\mathrm{Trace(I)}\;(\gamma^{c})_{\gamma\beta}\delta_{\delta\alpha}.
{\calC}_{0}=i\,\intd{\bf{r}}\,{\textstyle{\calX}^{\alpha}}({\bfr})\,j_{0}^{\alpha}({\bfr})\;\;\;\;\;\mathrm{and}\;\;\;\;\;\;\;{\bar{\calC}}=i\,\intd{\bf{r}}\,\overline{{{\calY}^{\alpha}}}({\bfr})\,j_{0}^{\alpha}({\bfr})
G_{\\nu}^{\mu}=\frac{\kappa^{2}}{l}\frac{1}{1-\Omega^{2}}\left(\stackrel{A}{T_{\\nu}^{\mu}}+\Omega^{2}\stackrel{B}{T_{\\nu}^{\mu}}\right)+\frac{2\Omega^{2}}{1-\Omega^{2}}\left(\delta\phi_{\|\nu}^{|\mu}-\delta_{\nu}^{\mu}\Box\delta\phi\right)\.
Z^{\prime}=\intd[\psi,\bar{\psi},B_{\mu},\theta]\delta(\partial_{\mu}B^{\mu})exp\i\int[\bar{\psi}(i\partial\!\!\!/\,-\lambdaB\!\!\!\!/\,)\psi-\frac{\theta^{2}}{2}+\epsilon_{\mu\nu}B^{\mu}\partial^{\nu}\theta+\epsilon_{\mu\nu}J^{\mu}\partial^{\nu}\theta]d^{2}x
Y_{i}^{a}(\theta+\frac{i\pi}{k})Y_{i}^{a}(\theta-\frac{i\pi}{k})=\left[1+Y_{i}^{b}(\theta-\sigma_{ba})\right]\prod_{j=1}^{k-1}\left[1+Y_{j}^{a}(\theta)^{-1}\right]^{-I_{ij}},\quada\neqb\,.
\begin{array}{rclcl}{\mathcal{H}_{\mathrm{Aut}}}&{=}&{\mathrm{SU}\left(\mathcal{N}\right)\times\mathrm{U}\left(1\right)}&{;}&{\mathcal{N}=1,2,3,4,5,6}\\{\mathcal{H}_{\mathrm{Aut}}}&{=}&{\mathrm{SU}\left(\mathrm{8}\right)}&{;}&{\mathcal{N}=7,8\}\\\end{array}
{\bfG}(x+b)=\frac{b^{1/2-bx}}{\sqrt{2\pi}}\Gamma(bx){\bfG}(x);\quad\quad{\bfG}(x+1/b)=\frac{b^{x/b-1/2}}{\sqrt{2\pi}}\Gamma(x/b){\bfG}(x).
\Theta=\frac{1}{\sqrt{2}}\left(\begin{matrix}{\theta_{1}+i\theta_{2}}\\{\theta_{1}-i\theta_{2}}\\\end{matrix}\right)\,,\quad\bar{\Theta}=\frac{1}{\sqrt{2}}\left(\begin{matrix}{\theta_{1}-i\theta_{2}}\\{\theta_{1}+i\theta_{2}}\\\end{matrix}\right)^{T}{\calC}\,,
Q=\int_{x^{0}=t}\!\!d^{3}x\,\Big(H^{\alpha\beta}(x)_{,\beta}+\frac{1}{4}\Phi(x)^{,\alpha}\Big){\stackrel{\longleftrightarrow}{\partial_{0}^{x}}}u^{\gamma}(x)\eta_{\alpha\gamma}\,.
\gamma_{,\rho}=\frac{\rho}{(2\mathrm{Im}E)^{2}}\left(E_{,\rho}{\bar{E}}_{,\rho}-E_{,\zeta}{\bar{E}}_{,\zeta}\right),\quad\gamma_{,\zeta}=\frac{\rho}{2(\mathrm{Im}E)^{2}}E_{,(\rho}{\bar{E}}_{,\zeta)},
\left(\begin{matrix}{\widetilde{\calF}^{+}}\\{\widetildeG_{+}}\\\end{matrix}\right)={\calS}\left(\begin{matrix}{{\calF}^{+}}\\{G_{+}}\\\end{matrix}\right)=\left(\begin{matrix}{A}&{B}\\{C}&{D}\\\end{matrix}\right)\left(\begin{matrix}{{\calF}^{+}}\\{G_{+}}\\\end{matrix}\right)\.
T_{\nu_{l}}^{ab}(\eta,z)=z\left\{\frac{\bar{J}_{\nu_{l}}^{2}(z)}{\bar{J}_{\nu_{l}}^{2}(\etaz)}\left[A^{2}+B^{2}(\eta^{2}z^{2}-\nu_{l}^{2})\right]-A^{2}-B^{2}(z^{2}-\nu_{l}^{2})\right\}^{-1},\quad\eta=\frac{b}{a},
\tilde{\Psi}_{\sigma}^{(n)}(w|\zeta)=\tilde{\vartheta}_{\sigma}^{(n)}(w|\zeta)\prod_{a^{\prime\prime}\inA}\prod_{j=1}^{2n}x^{-\frac{(v_{a}-u_{j})^{2}}{2r}}\psi\Bigl(\frac{w_{a}}{z_{j}}\Bigr)\prod_{1\leqslantj<k\leqslant2n}x^{-\frac{(u_{j}-u_{k})^{2}}{4r}}.
\partial^{\nu}\Lambda_{\nu\mu}=\varepsilon_{\mu\nu}n^{\nu}\bar{c}^{a}\,\frac{\deltaS}{\delta\phi^{a}}-A_{\mu}^{a}\,\frac{\deltaS}{\deltac^{a}}+\partial_{\mu}\bar{c}^{a}\,\frac{\deltaS}{\deltab^{a}}+\mathrm{total~deriv.}\quad.