harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{Tr}(\hat{A}_{1}(\xi_{0}^{(1)},\xi)\star\hat{A}_{2}(\xi_{0}^{(2)},\xi)\star\hat{A}_{3}(\xi_{0}^{(3)},\xi))\sim\langle\Psi_{1}|\langle\Psi_{2}|\langle\Psi_{3}|V_{3}\rangle\,.
\rho_{c}(z)\;=\;{\frac{\sqrt{(b_{c}-z)z}}{\pi}}\left[{\frac{1}{4\beta_{c}}}-{\frac{d-2}{2}}\int_{0}^{b_{c}}{\frac{dy}{y+z}}{\frac{\rho_{c}(y)}{\sqrt{(b_{c}+y)y}}}\right]\;.
\bar{f}_{\pm}=-\frac{\kappa_{5}^{2}e^{-2A_{\pm}-\alpha_{\pm}^{(0)}}}{\int_{w_{-}}^{w_{+}}dw^{\prime}e^{-3A(w^{\prime})}}\left[e^{-2A_{+}-\alpha_{+}^{(0)}}\bar{\tau}_{+(LL)}+e^{-2A_{--}\alpha_{-}^{(0)}}\bar{\tau}_{-(LL)}\right].
(\delta_{\epsilon}\psi_{I\mu})^{\prime}=e^{\frac{i}{2}\gamma^{5}Arg(S)}\delta_{\epsilon}\psi_{I\mu}\,\qquad(\delta_{\epsilon}\lambda)_{I}^{\prime}=e^{\frac{3i}{2}\gamma^{5}Arg(S)}\delta_{\epsilon}\lambda_{I}\.
N_{f}\,\mathrm{Tr}\,\sum_{n=2}^{\infty}\frac{1}{(p-m+i\epsilon)^{n}}\,f_{n}(A)=N_{f}\sum_{n=2}^{\infty}\int\frac{dp}{2\pi}\,\frac{1}{(p-m+i\epsilon)^{n}}\intdt\,f_{n}(A)
\phi(p^{2})=-\frac{e^{2}}{4}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{\phi(q^{2})}{q^{2}A^{2}(q^{2})+B^{2}(q^{2})}G(p^{2},q^{2})tr\lbrack\gamma_{\mu}\gamma_{\nu}D_{\mu\nu}(p-q)\rbrack.
K^{*}\left(\left.a-\bar{\varepsilon}_{\mu}\begin{array}{c}{a}\\{a^{\prime}}\\\end{array}\right|z\right)=K_{+}^{(n-1)}\left(\left.a+\displaystyle\sum_{\nu=1\atop\nu\neq\mu}^{n}\bar{\varepsilon}_{\nu}\begin{array}{c}{a}\\{a^{\prime}}\\\end{array}\right|z\right).
(-1)^{F_{L}}\left|B,\pm\right\rangle_{\mathrm{RR}}=\left|B,\mp\right\rangle_{\mathrm{RR}},\qquad(-1)^{F_{R}}\left|B,\pm\right\rangle_{\mathrm{RR}}=\left|B,\mp\right\rangle_{\mathrm{RR}}.
\begin{array}{cccc}{R_{1}:}&{\left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right)}&{\to}&{\left(x^{2,3},x^{4,5},-x^{6,7},-x^{8,9}\right)}\\{R_{2}:}&{\left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right)}&{\to}&{\left(-x^{2,3},-x^{4,5},x^{6,7},x^{8,9}\right)}\\{R_{3}:}&{\left(x^{2,3},x^{4,5},x^{6,7},x^{8,9}\right)}&{\to}&{\left(-x^{2,3},x^{4,5},-x^{6,7},x^{8,9}\right)}\\\end{array}
\left\{\prod_{X_{i}\in({\itG}_{1})_{X_{i^{\prime}}^{\prime}}}\;e^{\eta_{1}^{\prime}\sum_{\stackrel{i_{\alpha_{1}}<j_{\alpha_{1}}<k_{\alpha_{1}}}{\left\{i_{\alpha_{1}},j_{\alpha_{1}},k_{\alpha_{1}}\right\}\inJ_{X_{i}}}}t_{i_{\alpha_{1}}}t_{j_{\alpha_{1}}}t_{k_{\alpha_{1}}}{\bf1}_{(w(t_{i_{\alpha_{1}}})=w(t_{j_{\alpha_{1}}})=w(t_{k_{\alpha_{1}}}))}}\right\}.
\hat{E}^{A}=\hat{E}^{A}{}_{M}dX^{M}=\{e^{0}=dt\,e^{i}=e^{i}{}_{\underline{i}}dx^{\underline{i}}\,\;E^{a}=E^{a}{}_{{\beta}}A^{(1){\beta}}{}_{\underline{i}}dx^{\underline{i}}+E^{a}{}_{\alpha}dx^{\alpha}\}
T_{ab}=\frac{1}{16\pig^{2}G_{10}}\int_{r\to\infty}d\Omega_{8-p}r^{8-p}n^{i}[\eta_{ab}(\partial_{i}h_{\c}^{c}+\partial_{i}h_{\j}^{j}-\partial_{j}h_{\i}^{j})-\partial_{i}h_{ab}],
\frac{1}{2\pii}\oint_{0}\tilde{\xi}^{j}\frac{d\zeta}{\zeta^{m+2j-1}}=\frac{1}{2\pii}\oint_{\sum_{i=1}^{k}0_{i}}\frac{\partial\tilde{\eta}}{\partial\tilde{\alpha}^{j}}\tilde{\chi}\frac{d\zeta}{\zeta^{m+2j-1}}
\mathrm{GL}(\gamma_{1},\gamma_{2})={\textstyle{\frac{1}{4\pi}}}\oint_{\gamma_{1}}ds\oint_{\gamma_{2}}dt\,dot{\gamma_{1}}^{a}(s)\dot{\gamma_{2}}^{b}(t)\epsilon_{abc}{\frac{(\gamma_{1}^{c}(s)-\gamma_{2}^{c}(t))}{|\gamma_{1}(s)-\gamma_{2}(t)|^{3}}}
{\calE_{C}}^{(2)}=-{\frac{\hbarc}{4a^{2}\pi\sqrt{\pi}\sigma_{r}}}G_{44}^{33}\left(0,\frac{1}{2},1,2;\frac{3}{2},\frac{3}{2},\frac{5}{2},\frac{1}{2};\frac{16}{{\sigma_{r}}^{2}}\right).
A_{\epsilon}=l^{2}\int\!\!\!\!\int_{D^{\epsilon}}dxdy{\frac{1}{\epsilon^{2}}}e^{2\phi}+{\frac{l^{2}}{2}}\int\!\!\!\!\int_{D^{\epsilon}}dxdy\left(\left({\frac{\partial\phi}{\partialx}}\right)^{2}+\left({\frac{\partial\phi}{\partialy}}\right)^{2}\right).
[\![A_{l}({\bfx}),\psi({\bfy})]\!]^{\mathrm{D}}=[A_{l}({\bfx}),\psi({\bfy})]=-\frac{ie}{m}\,\epsilon_{ln}\frac{(x-y)_{n}}{2\pi|{\bfx-y}|^{2}}\,\psi({\bfy}),
\sum_{m>0}\int\frac{d^{2}\rho}{\pi}\e^{-|\rho|^{2}}(\rho|e^{\sqrt{\alpha^{\prime}/2}k_{\mu}\alpha_{-n}^{\mu}z^{n}/n}\left[z^{-m-1}\alpha_{m}^{\mu}+z^{m-1}\alpha_{-m}^{\mu}\right]e^{-\sqrt{\alpha^{\prime}/2}k_{\mu}\alpha_{n}^{\mu}z^{-n}/n}|q^{n}\rho)
x_{f}^{i}(\sigma,t)=\sum_{n=1}^{\infty}W(n+\frac{\epsilon}{\tau})\Biggl[\frac{c_{n}^{i}}{\sqrt{n}}\,x_{n}^{+}+\frac{\tilde{c}_{n}^{i}}{\sqrt{n}}\,x_{n}^{-}+h.c.\Biggr],
W_{c}^{\,\alpha,\,\alpha_{0}}(J):=L^{\,\alpha,\,\alpha_{0}}(\Phi)\,\vert_{\,\Phi=\tilde{C}_{\alpha}^{\,\alpha_{0}}\!J}+I^{\,\alpha,\,\alpha_{0}}-\frac{1}{2}<J,\tilde{C}_{\alpha}^{\,\alpha_{0}}J>\quad,
F(s+{\frac{3}{2}},s+1,s+2,{\frac{5}{9}})={\frac{\Gamma(s+2)}{\Gamma(1)\\Gamma(s+1)}}\\int_{0}^{1}t^{s}\\left(1-{\frac{5t}{9}}\right)^{-s-{\frac{3}{2}}}\dt
\lambda_{\alpha(\dot{\beta})}^{\{\gamma\delta\}}=g^{-1}x_{\dot{\beta}}^{\beta}G_{\alpha\beta}^{\{\gamma\delta\}}\,,\qquad\left\langle\lambda_{(\dot{1})}\,\bigg|\,\lambda_{(\dot{2})}\right\rangle=\frac{64\pi^{2}\rho^{2}}{g^{2}}\,.
{\tilde{\hat{\calY}}}_{J}^{R}=\sum_{K^{1},K^{2}}\left\langle\begin{array}{l}{\frac{P}{2}}\\{K^{1}}\\\end{array}\right|{\tilde{\hat{\calY}}}_{J}^{R}\left|\begin{array}{l}{\frac{P}{2}}\\{K^{2}}\\\end{array}\right\rangle\left|\begin{array}{l}{\frac{P}{2}}\\{K^{1}}\\\end{array}\right\rangle\left\langle\begin{array}{l}{\frac{P}{2}}\\{K^{2}}\\\end{array}\right|.
V_{F}(p_{0}^{\prime})\,=-2i\pi\,\sum_{k^{\prime}}\,V_{i^{\prime}j^{\prime}}^{0}\,\delta(p_{k^{\prime}0}^{\prime}-E_{k^{\prime}}^{\prime})\,\beta_{k^{\prime}}\,\big[G_{k^{\prime}}^{0}(p_{k^{\prime}0}^{\prime})\big]^{-1}.
S=\frac{1}{32\piG_{4}}\int\sqrt{-g}\,d^{4}x\Big[R-2g_{A\bar{B}}\partial^{\mu}z^{A}\partial_{\mu}\bar{z}^{B}-\frac{1}{4}F_{\mu\nu}^{I}({^{\star}G_{I}})^{\mu\nu}\Big],
\delta_{\hat{\chi}}\hat{L}_{\hat{\mu}_{1}\ldots\hat{\mu}_{r}}=m\hat{\lambda}_{\hat{\mu}_{1}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\nu}\hat{\mu}_{2}\ldots\hat{\mu}_{r}}+\ldots+m\hat{\lambda}_{\hat{\mu}_{r}}\hat{k}^{\hat{\nu}}\hat{L}_{\hat{\mu}_{1}\ldots\hat{\mu}_{r-1}\hat{\nu}}\,.
{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[\frac{1}{2}\frac{\partial\Gamma^{(2)}}{\partialw^{i}}\right]=-{e}_{\hat{0}}^{\;\;0}{\bar{e}}_{\alpha}^{\;\;i}\left[B_{i}^{w(2)}\right]+{e}_{\alpha}^{\;\;\mu}{e}_{\hat{0}}^{\;\;\nu}K_{\mu\nu}^{b},
f=\sum_{{a_{0}+a_{1}+a_{2}=3\atopb_{0}+b_{1}=2}}A_{a_{0}a_{1}a_{2},b_{0}b_{1}}\,x_{0}^{a_{0}}x_{1}^{a_{1}}x_{2}^{a_{2}}y_{0}^{b_{0}}y_{1}^{b_{1}}.
\deltaS_{2d}=-\int_{\calM}{\frac{d^{2}\zeta}{2\pi\alpha^{\prime}}}\\deltaX^{\mu}\partial_{\alpha}\partial^{\alpha}X_{\mu}+\int_{\partial\calM}{\frac{d\zeta^{\alpha}}{2\pi\alpha^{\prime}}}\\deltaX^{\mu}\varepsilon_{\alpha\beta}\partial^{\beta}X_{\mu}
{{[}{\{}\lambda{\}};{\{}\mu{\}}{]}^{a_{1}a_{2}\ldotsa_{\ell}}}_{b_{1}b_{2}\ldotsb_{\ell}}=\frac{1}{\ell!^{2}}\sum_{\rho,\sigma\inS_{\ell}}\chi^{\lambda}(\rho\sigma^{-1}){(E^{\mu_{1}})^{a_{\sigma1}}}_{b_{\rho1}}{(E^{\mu_{2}})^{a_{\sigma2}}}_{b_{\rho2}}\ldots{(E^{\mu_{\ell}})^{a_{\sigma\ell}}}_{b_{\rho\ell}}.
G(x)=\prod_{i=1}^{n_{f}}(\sqrt{x}-im_{i})=\sqrt{x}\sum_{k=0}^{n_{c}}(-1)^{k}s_{2k}(m)x^{n_{c}-k}-i\sum_{k=0}^{n_{c}}(-1)^{k}s_{2k+1}(m)x^{n_{c}-k}.
e^{2\rho}=u\\frac{e^{-\lambda^{2}\int^{x^{+}}\frac{dx^{+\prime}}{A(x^{+\prime})}}\}{A(x^{+})}\\frac{e^{-\lambda^{2}\int^{x^{-}}\frac{dx^{-\prime}}{B(x^{-\prime})}}}{B(x^{-})}\.
\frac{1}{2}\pi^{mn}\varepsilon_{\hat{\alpha}\hat{\beta}}(\tilde{A}_{\mu\nu})_{m}{}^{\hat{\alpha}}(A_{\mu\nu})_{n}{}^{\hat{\beta}}+\frac{2}{3}M_{\hat{\alpha}}^{mnp}\varepsilon_{\hat{\beta}\hat{\gamma}}(A_{\mu})_{p}{}^{\hat{\alpha}}(A_{\nu})_{m}{}^{\hat{\beta}}(\tilde{A}_{\mu\nu})_{n}{}^{\hat{\gamma}}
\gamma_{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right)\;\;,\;\;\gamma_{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right)\;\;,\;\;\gamma_{5}=i\gamma_{2}\gamma_{1}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)
\big\langle\,T\,{\calO}_{s,n}(x,t)\,{\calO}_{s,n}^{\dagger}(0,0)\,\big\rangle=\Big(\frac{t-x}{t+x}\Big)^{\frac{sn}{2}}\,{\calZ}_{s,n}\big(\sqrt{x^{2}-t^{2}}\,\big)\,.
\begin{array}{lll}{c(T)}&{=}&{{\frac{1}{2}}L\bigl(L-1-d(T)\bigr)+\mathrm{\sf~I}(T,L)}\\{{\bfp}(T)}&{=}&{{\frac{1}{2}}L\bigl(L-1-d(T)\bigr)-\mathrm{\sf~I}(T,L)}\\\end{array}.
{\calL}~=~-\frac{1}{4g_{0}^{2}}G_{\mu\nu}^{a}G_{\mu\nu}^{a}~+~\frac{i}{g_{0}^{2}}\lambda_{\dot{\alpha}}^{\dagger}D^{\dot{\alpha}\beta}\lambda_{\beta}~+~\frac{i\theta}{32\pi^{2}}G_{\mu\nu}^{a}\tilde{G}_{\mu\nu}^{a}.
S\,=\,\intd^{3}x\left[-\frac{1}{4}(\frac{1}{2\rho})F_{\mu\nu}(A)F^{\mu\nu}(A)+2\piA^{\mu}{\tilde{J}}_{\mu}+\frac{1}{4\rho}(\partial\rho)^{2}-V(\rho)\right]
a_{i}^{(\mathrm{eff})}(x)={\frac{\theta_{i}}{L_{i}}}-(Q+1)\tilde{x}_{i}+\epsilon_{ij}\partial_{x^{j}}\int\!d\vec{y}G(\vec{x},\vec{y})b^{(1)}(y)-{\frac{e}{\muc}}\epsilon_{ij}\partial_{x^{j}}\int\!d\vec{y}G_{p}(\vec{x},\vec{y})J_{0}(y)\,.
(\hat{M})_{l^{\prime},m^{\prime};l,m}={\frac{N_{l}}{N_{l^{\prime}}}}(M)_{l^{\prime},m^{\prime};l,m},\quad\mathrm{and}\quad(\hat{M^{\prime}})_{l^{\prime},m^{\prime};l,m}={\frac{N_{l}}{N_{l^{\prime}}}}(M^{\prime})_{l^{\prime},m^{\prime};l,m},
\Gamma_{a,b}(t_{2})=\left[\int_{b_{t_{2}}}^{\infty}\int_{x}^{\infty}-\int_{b_{t_{1}}}^{\infty}\int_{a_{t_{1}}}^{\infty}+\int_{a_{t_{1}}}^{\infty}\int_{a_{t_{1}}}^{\infty}-\int_{a_{t_{2}}}^{\infty}\int_{x}^{\infty}\right]f(x)f(y)dydx.
\dot{x}^{AA^{\prime}}=o^{A}\bar{o}^{A^{\prime}}-\frac{\rho^{\sigma}}{\rho^{\tau}}\acute{x}^{AA^{\prime}},\,\,\,\,\acute{x}^{AA^{\prime}}=o^{A}\bar{\iota}^{A^{\prime}}+\iota^{A}\bar{o}^{A^{\prime}}.
\bar{\Xi}_{\lambda_{2}}^{\lambda_{1}}\otimesf_{\dot{J}}^{\dot{I}}\otimes\Xi_{\lambda_{4}}^{\lambda_{3}}\left(\bar{\phi}^{\rho_{1}}\otimess^{\dot{K}}\otimes\phi^{\rho_{2}}\right)=\delta_{\lambda_{2}}^{\rho_{1}}\delta_{\dot{J}}^{\dot{K}}\delta_{\lambda_{4}}^{\rho_{2}}\bar{\phi}^{\lambda_{1}}\otimess^{\dot{I}}\otimes\phi^{\lambda_{3}}.
I_{11}=\frac{1}{2\kappa_{11}^{2}}\intd^{11}x\sqrt{-G}\left[R_{G}-\frac{1}{2\cdot4!}K_{4}{}^{2}\right]-\frac{1}{12\kappa_{11}^{2}}\intC_{3}\wedgeK_{4}\wedgeK_{4}
\tilde{\Gamma}_{5}(p,p^{\prime})\to\gamma_{5}\left(1-\frac{\alpha}{2\pi}\right)-\gamma_{5}\frac{\alpha}{2\pi}\,P^{2}\int_{4m^{2}}^{\infty}\frac{dM^{2}}{M^{2}}\,\frac{a(M^{2})}{M^{2}+P^{2}},
S={\frac{1}{4\pi}}\int_{0}^{2\pi}d\sigma\intd\tau\left[\sqrt{g}g^{\alpha\beta}G_{ij}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}+\epsilon^{\alpha\beta}B_{ij}\partial_{\alpha}X^{i}\partial_{\beta}X^{j}-{\frac{1}{2}}\sqrt{g}\phi{\calR}^{(2)}\right],
\frac{dE}{d\omega}=\frac{2^{2-D}\pi^{-(D-5)/2}{\calG}\,(D-3)D}{\Gamma[(D-1)/2](D^{2}-1)(D-2)}\,\omega^{D+2}{\biggl[}(D-1)D_{ij}^{*}(\omega)D_{ij}(\omega)-|D_{ii}(\omega)|^{2}{\biggr]}\,,
\left[\nabla^{\mu}\nabla_{\nu}\phi\right]^{(1)}={\frac{1}{a^{2}}}\left[\phi_{\|\nu}^{|\mu}+2{\frac{y}{l}}e^{\phi}\phi^{|\mu}\phi_{|\nu}-{\frac{y}{l}}e^{\phi}\delta_{\nu}^{\mu}\phi^{|\alpha}\phi_{|\alpha}\right].
\sqrt{\frac{\pig_{\Lambda}}{2}}\sum_{l=0}^{\infty}(2l+1)I_{l+1/2}(1/g_{\Lambda})\,P_{l}({\vec{S}}\,{\vec{S}^{\prime}})\equiv\sum_{l=0}^{\infty}(2l+1)F_{l}^{\Lambda}(1/g_{\Lambda})\,P_{l}({\vec{S}}\,{\vec{S}^{\prime}})\\,\,\,\,\
L=t\int_{\Sigma}d^{2}z(g_{IJ}\partial_{z}\phi^{I}\partial_{\bar{z}}\phi^{J}+i\eta^{\bar{i}}(D_{z}\rho_{\bar{z}}^{i}+D_{\bar{z}}\rho_{z}^{i})g_{i{\bar{i}}}+i\theta_{i}(D_{\bar{z}}\rho_{z}^{i}-D_{z}\rho_{\bar{z}}^{i})+R_{i{\bar{i}}j{\bar{j}}}\rho_{z}^{i}\rho_{\bar{z}}^{j}\eta^{\bar{i}}\theta_{k}g^{k{\bar{j}}})
I_{\beta\mu\lambda\rho}(q)=\intd^{d}k\frac{2k_{\beta}k_{\mu}(q_{\nu}p_{\rho}+q_{\nu}q_{\rho})+k_{\beta}(q_{\nu}q_{\rho}p_{\mu}-q_{\nu}q_{\mu}p_{\rho})}{(k+p)^{2}(k-q)^{2}k^{2}}\;.
L=c_{0}s+c\sqrt{{\dot{\bfe}}_{3}^{2}-k_{2}^{2}}+{\bfp}({\dot{\bfx}}-s{\bfe}_{1})+\sum_{i}{\bfp}_{i-1}({\dot{\bfe}}_{i-1}-k_{i-1}{\bfe}_{i}+k_{i-2}{\bfe}_{i-2})-\sum_{i,j}d_{ij}({\bfe}_{i}{\bfe}_{j}-\delta_{ij}),
\phi(x)=\langleIN|\tilde{T}(e^{-i\int^{t}d^{4}x^{\prime}~J(x^{\prime})\Phi(x^{\prime})})\Phi(x)T(e^{i\intd^{4}x^{\prime\prime}~J(x^{\prime\prime})\Phi(x^{\prime\prime})})|IN\rangle|
\begin{array}{c}{<O_{k}O_{l}P>_{c}=(\frac{1}{\rho^{2}}\frac{\partial\rho}{\partialt_{l}}+\frac{1}{\rho^{2}}\frac{\partial\rho}{\partial\mu}\frac{\partial\mu}{\partialt_{l}})<O_{k}P>}\\{-\frac{1}{\rho}(<O_{k}O_{l}P>+\frac{\partial\mu}{\partialt_{l}}<O_{k}PP>).}\\\end{array}
\frac{dI_{\epsilon}}{dR}(R)=-\frac{4\pi}{R_{0}}\left(\frac{R^{2}-R_{0}^{2}}{RR_{0}}\right)\left(\frac{R^{2}+R_{0}^{2}}{R^{2}}\right)\int_{\epsilon}^{\infty}\frac{d\tau_{2}}{\tau_{2}^{1/2}}e^{-\pi\tau_{2}\left(\frac{R^{2}-R_{0}^{2}}{RR_{0}}\right)^{2}}
{\sfH}=\left(\begin{array}{cc}{p^{2}}&{\lambda\bar{\eta}}\\{-i\lambdap\llap/\eta}&{p^{2}}\\\end{array}\right)=\left(\begin{array}{cc}{p_{\mu}}&{0}\\{-\frac{i\lambda}2\gamma_{\mu}\eta}&{p_{\mu}}\\\end{array}\right)^{2}+\left(\begin{array}{cc}{0}&{\lambda\bar{\eta}}\\{-\frac{\lambda}2\gamma_{\nu}\eta_{,\nu}}&{0}\\\end{array}\right),
\tilde{\Gamma}_{t}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})=\int_{0}^{t}d\taug_{t-\tau}^{2}e^{-(D-4)\tau}I_{\tau}^{(4)}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4})\;.
\xi=\rho^{2}\quad\Rightarrow\quadd\xi=2\rhod\rho\quad\Rightarrow\quad\frac{1}{\rho}\frac{d}{d\rho}=2\frac{d}{d\xi}\quad\textrm{y}\quad\rho\frac{d}{d\rho}=2\xi\frac{d}{d\xi},
V_{w}^{(1)}=-\frac{eH}{8\pi^{2}}\sum_{l=1}^{\infty}\int_{0}^{\infty}\frac{ds}{s^{2}}exp(-m^{2}s-l^{2}\beta^{2}/4s)[\frac{3}{sinh(eHs)}+4sinh(eHs)]cosh(\betal\mu).
S={\frac{1}{2\kappa^{2}}}\intd^{10}x\sqrt{-g}\bigg[R+\frac{1}{4}tr\nabla_{\mu}{\calM}\nabla^{\mu}{\calM}^{-1}-\frac{1}{12}{\calH}_{\mu\nu\rho}^{T}{\calM}H^{\mu\nu\rho}\bigg].
S_{2}=-\frac13\intd^{D}x\left(f_{abc}A_{\mu}^{a}A_{\nu}^{b}A_{\rho}^{c}+A_{\mu}^{a}F_{a\nu\rho}+A_{\rho}^{a}F_{a\mu\nu}+A_{\nu}^{a}F_{a\rho\mu}\right)^{2}.
\tilde{\calA}_{m}^{n}=\frac{3}{4}\left(\begin{array}{cccc}{z^{-2}}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{z^{-2}}\\\end{array}\right)+\mathrm{~mean~field~terms}
\begin{array}{l}{<\chi,\chi^{\prime}>=\displaystyle\prod_{r=1}^{N}\delta_{n_{r}n_{r}^{\prime}}\cdot\displaystyle\prod_{r=1}^{M}\delta_{m_{r}m_{r}^{\prime}}\cdot\displaystyle\prod_{r=1}^{Q}\delta_{q_{r}q_{r}^{\prime}}\cdot\displaystyle\prod_{r=1}^{T}\delta_{t_{r}t_{r}^{\prime}}\cdot\displaystyle\prod_{r=1}^{4}\delta_{\nu_{r}\nu_{r}^{\prime}}.}\\\end{array}
(l+1)N\gamma_{c}w^{c}f^{a_{1}...a_{l}}=\sum_{i=1}^{l}w^{a_{i}}\left[(\gamma_{c}w^{c}-N)\gamma_{b}f^{a_{1}...{\nota_{i}}...a_{l}b}\right]+d\left[\left(\frac{l}{l+2}\gamma_{c}w^{c}+\frac{2}{l+2}N\right)\gamma_{b}j^{a_{1}...a_{l}b}\right],
J^{(N)}\left(N;1,\ldots,1\right)=2\;\mathrm{i}^{1-2N}\pi^{N/2}\frac{\Gamma\left({\textstyle{\frac{N}{2}}}\right)}{\prodm_{i}}\int_{0}^{\infty}\ldots\int_{0}^{\infty}\prod\mathrm{d}\alpha_{i}\;\delta\left(\alpha^{T}\|c\|\alpha-1\right).
\left({\frac{\partial\sigma^{\prime+}}{\partial\sigma^{+}}}\right)^{2}T_{++}^{\prime}(\sigma^{\prime+})=T_{++}(\sigma^{+})-{\frac{\mathrm{c}}{12}}\left\{{\frac{2\partial_{\sigma}^{3}\sigma^{\prime}\partial_{\sigma}\sigma^{\prime}-3\partial_{\sigma}^{2}\sigma^{\prime}\partial_{\sigma}^{2}\sigma^{\prime}}{2\partial_{\sigma}\sigma^{\prime}\partial_{\sigma}\sigma^{\prime}}}\right\}\,
\begin{array}{ccc}{(\hat{\alpha}\hat{t}f)(\bar{q})}&{=}&{\alpha[\pi(\bar{q})](\hat{t}f)(\bar{q})}\\{}&{=}&{\alpha[\pi(\bar{q})]f(\bar{q}t)\,}\\{(\hat{t}\hat{\alpha}f)(\bar{q})}&{=}&{(\hat{\alpha}f)(\bar{q}t)\quad\quad}\\{}&{=}&{\alpha[\pi(\bar{q}t)]f(\bar{q}t)}\\{}&{=}&{\alpha[\pi(\bar{q})]f(\bar{q}t)}\\{}&{=}&{(\hat{\alpha}\hat{t}f)(\bar{q}).\quad}\\\end{array}
\begin{array}{lll}{\langleq_{i0}-Q_{i0}\rangle}&{=}&{0\,,}\\{}&{}&{}\\{\langle\dot{q}_{i0}\rangle}&{=}&{0\,,}\\{}&{}&{}\\{m\omega_{i}\omega_{j}\langle(q_{i0}-Q_{i0})(q_{j0}-Q_{j0})\rangle}&{=}&{k_{B}T\delta_{ij}\,,}\\{}&{}&{}\\{m\langle\dot{q}_{i0}\dot{q}_{j0}\rangle}&{=}&{k_{B}T\delta_{ij}\,,}\\{}&{}&{}\\{\langle\dot{q}_{i0}(q_{j0}-Q_{j0})\rangle}&{=}&{0\,.}\\\end{array}
\begin{aligned}{\Theta:\qquad}&{{}(z^{1},z^{2},z^{3})\rightarrow(iz^{1},-iz^{2},z^{3}),}\\{\omega:\qquad}&{{}(z^{1},z^{2},z^{3})\rightarrow(z^{1},-z^{2},-z^{3}),}\\\end{aligned}\nonumber
\begin{array}{lllllr}{R_{12}^{*}x^{+}}&{=}&{x^{+}}&{R_{23}^{*}x^{+}}&{=}&{qx^{+}+y^{+}}\\{R_{12}^{*}y^{+}}&{=}&{-qx^{+}+qy^{+}}&{R_{23}^{*}y^{+}}&{=}&{y^{+}}\\\end{array}
\Gamma(\omega)=\frac{4}{3}\intd^{3}x\frac{\xir}{\sqrt{(\xi^{2}-\Omega_{0}^{2}r^{2})}}\left(\frac{\omega^{2}}{\xi^{2}-\Omega_{0}^{2}r^{2}}-\mu^{2}\right)^{3/2}.
S_{cl}(e_{\mu}^{i},\omega_{\mu})=\int{d^{2}}x\;e\bigg(\frac{1}{16\alpha}R_{\mu\nu}{}^{ij}R^{\mu\nu}{}_{ij}-\frac{1}{8\beta}T_{\mu\nu}{}^{i}T^{\mu\nu}{}_{i}-\gamma\bigg),
\displaystyle\prod_{\mu=1}^{m}\oint_{C^{(N)}}{\frac{dx_{\mu}}{2\pii}}=\displaystyle\prod_{\mu=1}^{m-1}\oint_{C^{\prime}{}^{(N)}}{\frac{dx_{\mu}}{2\pii}}\left(\oint_{C^{\prime}{}^{(N)}}{\frac{dx_{m}}{2\pii}}+m\mathrm{Res}_{x_{m}=z_{N}\tau^{-}1}\right),
Z_{K}(V)=\frac{C^{K}}{K!}\int_{0}^{\bar{V}}\!\!dv\int\prod_{i=1}^{K}d{\bfa}_{i}d\rho_{i}\prod_{j=1}^{K}\rho_{j}^{\alpha}\,Theta(\{{\bfa}_{k}\},\{\rho_{k}\})\,\delta(v-\tau\sum_{l}\rho_{l}^{d})\,,
\left(\begin{array}{c}{A_{k}^{\mu}\left(\theta\right)}\\{\tilde{A}_{k}^{\mu\dagger}\left(\theta\right)}\\\end{array}\right)={\calB}_{k}\left(\begin{array}{c}{A_{k}^{\mu}}\\{\tilde{A}_{k}^{\mu\dagger}}\\\end{array}\right),
\omega_{_\mathrm{phys}}^{2}(k_{_\mathrm{phys}})=k_{_\mathrm{phys}}^{2}e^{-k_{_\mathrm{phys}}^{2}/k_{\mathrm{c}}^{2}}\Longleftrightarrow\omega^{2}(k)=k^{2}e^{-k^{2}/(a^{2}k_{\mathrm{c}}^{2})},
S[q^{n};\lambda]=\intd\tau\,\left[L\left(q^{n},\frac{dq^{n}}{d\tau}\right)+\Lambda\right]=\intdt\,\lambda(t)\,\left[L\left(q^{n},\frac{1}{\lambda(t)}\frac{dq^{n}}{dt}\right)+\Lambda\right]\.
\beta_{ab}=\left(1+\frac{1}{2}F^{r}F^{r}-\beta^{-1}\Pi_{r}\Pi_{r}\right)^{-1}\left(\partial_{a}X^{m}\partial_{b}X_{m}+\beta^{cd}F_{ac}^{r}F_{bd}^{r}-\beta^{-1}\Pi_{ra}\Pi_{rb}\right)
\tau\equivC^{(0)}+ie^{-\phi}=C^{(0)}+{\frac{i}{\kappa}}e^{-\kappa\tilde{\phi}}=\tau_{0}+\tilde{C}^{(0)}-i\tilde{\phi}+{\frac{i}{2}}\kappa\tilde{\phi}^{2}+\cdots.
K_{d+1}(s)\sim{\frac{T}{\sqrt{\pi}i(4\pis)^{(d+1)/2}}}\sum_{n=0}^{\infty}\sum_{m=0}^{[n/2]}(-1)^{m}\Gamma\left(m+\frac12\right)\left(a_{2m,n}s^{n-m}+b_{2m,n}s^{n-m+1/2}\right).
\begin{array}{c}{{\tilde{\calZ}}\equivG(z,\bar{z})+iG_{A\bar{B}}(z,\bar{z})\theta^{A}{\bar{\theta}}^{\bar{B}},}\\{{\calR}_{+}=i\theta^{C}K_{C;A}{\tilde{g}}^{A\bar{B}}G_{\bar{B};D}{\theta}^{D},}\\{{\calR}_{-}=\bar{\calR}_{+}\;\;.}\\\end{array}
E_{\tilde{n}_{s},K}=\frac{g_{YM}^{2}}{2V}(\tilde{n}^{s}+\tilde{\theta}^{sd}K)g_{sr}(\tilde{n}^{r}+\tilde{\theta}^{rd}K)+\frac{V}{4g_{YM}^{2}}\phi_{\mu\nu}\phi^{\mu\nu}+\|K\|
\left\{\begin{array}{c}{\partial_{\mu}(LA_{\nu}^{a})-\partial_{\nu}(LA_{\mu}^{a})+\epsilon^{abc}A_{\mu}^{b}A_{\nu}^{c}L^{2}\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)}\\\end{array}\right\}\left(\begin{array}{c}{e^{a}}\\{f^{a}}\\\end{array}\right)=0,
\hat{x}_{1}=\xi_{1}+i\frac{\theta}{2}\,\left(\partial_{\xi_{2}}-\partial_{\xi_{3}}\right)\quad,\quad\hat{x}_{2}=\xi_{2}+i\frac{\theta}{2}\,\left(\partial_{\xi_{3}}-\partial_{\xi_{1}}\right)\quad,\quad\hat{x}_{3}=\xi_{3}+i\frac{\theta}{2}\,\left(\partial_{\xi_{1}}-\partial_{\xi_{2}}\right)\quad.
={\calE}+\left(\begin{array}{cccc}{((p^{2}-q^{2})a+2pqb)}&{(pa+qb)}&{(pb-qa)}&{((p^{2}-q^{2})a+2pqb)}\\{(pa+qb)}&{0}&{0}&{(pa+qb)}\\{(pb-qa)}&{0}&{0}&{(pb-qa)}\\{((p^{2}-q^{2})a+2pqb)}&{(pa+qb)}&{(pb-qa)}&{((p^{2}-q^{2})a+2pqb)}\\\end{array}\right).
ds_{IIA}^{2}=\rho^{2}e^{-3t/2}\mu^{-1}\sqrt{1-r^{2}}\left[\left(e^{t}+e^{-3t}\frac{r^{2}}{1-r^{2}}\right)dr^{2}+e^{t}r^{2}d\Omega_{5}^{2}\right].
T_{\hat{\mu}\hat{\nu}}=\mathrm{diag}\left(-\frac{n(n+1)}{2l^{2}}+\frac{{\calQ}^{2}}{a^{2n}},\frac{n(n+1)}{2l^{2}}+\frac{{\calQ}^{2}}{a^{2n}},\cdots,\frac{n(n+1)}{2l^{2}}+\frac{{\calQ}^{2}}{a^{2n}},\frac{n(n+1)}{2l^{2}}-\frac{{\calQ}^{2}}{a^{2n}}\right).
H_{\theta}=\Bigl({\frac{\Omega_{\rho}}{\Omega^{2}}}\Bigr){\frac{g_{ab}\pi_{0}^{a}\pi_{b}^{0}}{2}}+{\frac{\pi_{0}^{a}}{\Omega}}\int\rho^{-2}g_{ab}\tilde{\pi}_{0}^{b}+\tilde{H}
{\bfg}_{\mu\nu}=\eta_{\mu\nu}-\kappa\gamma_{\mu\nu}+\kappa^{2}{\gamma_{\mu}}^{\alpha}\diamondsuit{\gamma_{\alpha}}_{\nu}-\kappa^{3}{\gamma_{\mu}}^{\alpha}\diamondsuit\gamma_{\alpha_{\beta}}\diamondsuit{\gamma_{\nu}}^{\beta}+O(\kappa^{4}).
(1-\mid{\tilde{Z_{p}}}\mid^{2})=\frac{(1-\mida_{p}\mid^{2})(1-\midZ_{p}\mid^{2})}{1+\mida_{p}\mid^{2}\midZ_{p}\mid^{2}+2\mida_{p}\mid\midZ_{p}\midcos\chi_{p}}\quad\geq0
{\calS}_{\mathrm{quartic}}=-\,\frac{1}{g_{0}^{2}}\,\mathrm{tr}\Bigg\{\frac{1}{4}\left[a_{\mu},a_{\nu}\right]^{2}\,+\,\frac{1}{2}\left[a_{\mu},\chi_{a}\right]^{2}\,+\,\frac{1}{4}\left[\chi_{a},\chi_{b}\right]^{2}\Bigg\}~~.
T^{\ell}(g)|\ell,m\rangle=N_{m}(\alphaz_{1}+\betaz_{2})^{m}(\gammaz_{1}+\deltaz_{2})^{2\ell-m}=\sum_{n}t_{mn}^{\ell}(g)N_{n}z_{1}^{n}z_{2}^{2\ell-n}\,.
f_{\bfk}({\bfx},t)={\frac{n_{I}^{(m-2)/2}}{\left[2\omega(2\pi)^{m}\right]^{1/2}}}\left\{\Theta(-x)\left[e^{in_{I}k_{1}x_{1}}+R_{rg}e^{-in_{I}k_{1}x_{1}}\right]+\Theta(x)T_{rg}e^{i\betax_{1}}\right\}e^{in_{I}(k_{2}x_{2}+\cdots+k_{m}x_{m})}e^{-i\omegat},
\mathrm{Re}\,\Pi^{\mathrm{(non\mathrm{-}planar)}}=\frac{2\pi^{\frac{n}{2}}}{(2\pi)^{n}}\left(\frac{|\bar{\theta}|}{2}\right)^{2-\frac{n}{2}}\int_{0}^{1}dx\,(x(1-x))^{\frac{n}{4}-1}\int_{0}^{\infty}dz\,\frac{1}{z^{2}-p^{2}}\,z^{\frac{n}{2}-1}J_{\frac{n}{2}-2}(|\bar{\theta}|\sqrt{x(1-x)}z)
\begin{array}{c}{D(\beta,L)\k(t,x;t^{\prime},x^{\prime})=\delta(x-x^{\prime})\delta(t-t^{\prime})}\\{Be^{\mu\betat}k(t,x;t^{\prime},x^{\prime})=0,\mathrm{~for~}x=0,1,}\\{k(1,x;t^{\prime},x^{\prime})+e^{-\mu\beta}k(0,x;t^{\prime},x^{\prime})=0.}\\\end{array}
V(\rho)=\frac{1}{\lambda_{0}}\rho^{2}-\frac{R}{(4\pi)^{2}}f(\xi_{1},\xi_{2})\left[\rho^{2}\mathrm{Ei}\left(-\frac{\rho^{2}}{\Lambda^{2}}\right)\frac{}{}+\Lambda^{2}(e^{-\rho^{2}/\Lambda^{2}}-1)\right],
\begin{array}{ccc}{\phi^{i}=\left(\begin{array}{cc}{0}&{\bar{\phi}{}^{at}}\\{\phi_{a}}&{0}\\\end{array}\right)~~~}&{\Longrightarrow}&{~~~\bar{\phi}_{i}=\left(\begin{array}{cc}{\bar{\phi}^{a}}&{0}\\{0}&{-\phi_{a}^{t}}\\\end{array}\right)}\\\end{array}
{\hat{H}}\psi_{s}=\frac{1}{r^{2}}\frac{d}{dr}r^{2}{\rho}^{\prime}\left(\begin{array}{c}{\begin{array}{cc}{-1-E_{c}^{2}}&{E_{c}^{2}}\\{E_{c}^{2}}&{1-E_{c}^{2}}\\\end{array}}\\\end{array}\right)\left(\begin{array}{c}{1}\\{1}\\\end{array}\right)=\frac{1}{r^{2}}\frac{d}{dr}r^{2}{\rho}^{\prime}\left(\begin{array}{c}{-1}\\{1}\\\end{array}\right)
d^{3}F(\xi^{(1)},\xi^{(2)},\xi^{(3)})=\frac{d\xi^{(1)}d\xi^{(2)}d\xi^{(3)}}{(\xi^{(1)}-\xi^{(2)})(\xi^{(2)}-\xi^{(3)})(\xi^{(3)}-\xi^{(1)})}~.
\Pi_{\mu\nu\rho\sigma}(p)=\frac{1}{2}\kappa^{2}\intd^{4}q{\calU}_{\mu\nu\alpha\beta\gamma\delta}(p,-q,q-p){\bar{D}}^{\mathrm{grav}\,\alpha\beta\kappa\lambda}(q)\times{\bar{D}}^{\mathrm{grav}\,\gamma\delta\tau\xi}(p-q){\calU}_{\kappa\lambda\tau\xi\rho\sigma}(q,p-q,-p)V(q,p-q,p),
u_{t}^{i}-J_{2}^{ik}\delta_{u^{k}}H_{0}(u)={\calO}_{j}^{i}(r^{l})\,\Big[r_{t}^{j}-J_{1}^{jk}\delta_{r^{k}}H_{0}\Big|_{u^{m}=M^{m}(r^{n})}\Big]
\intd^{3}p\sum_{\lambda}\left[U_{\vec{p},\lambda}(t,\vec{x})U_{\vec{p},\lambda}^{+}(t,\vec{x}^{\,\prime})+V_{\vec{p},\lambda}(t,\vec{x})V_{\vec{p},\lambda}^{+}(t,\vec{x}^{\,\prime})\right]=e^{-3\omegat}\delta^{3}(\vec{x}-\vec{x}^{\,\prime})\,.
\alpha_{R}^{1}=-i/2\left(\begin{matrix}{0}&{-1}&{0}&{0}\\{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right),\;\;\alpha_{R}^{2}=-i/2\left(\begin{matrix}{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{-1}\\{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\\end{matrix}\right),\;\;\alpha_{R}^{3}=-i/2\left(\begin{matrix}{0}&{0}&{0}&{-1}\\{0}&{0}&{1}&{0}\\{0}&{-1}&{0}&{0}\\{1}&{0}&{0}&{0}\\\end{matrix}\right).
\begin{array}{ll}{I(g,A)=I(g)+\frac{k}{2\pi}\intd^{2}ztr[{\bf{A}}\overline{{\partial}}gg^{-1}-\overline{{\bf{A}}}g^{-1}\partialg+{\bf{A}}\overline{{\bf{A}}}-{\bf{A}}g\overline{{\bf{A}}}g^{-1}]}&{}\\{+\frac{k}{2\pi}\intd^{2}z[\frac{4c}{k}(A\overline{{\partial}}f+\overline{{A}}\partialf)+\frac{8c^{2}}{k}A\overline{{A}}]}&{}\\\end{array}
\intd\mu(x)\;V_{m\bar{m}}^{j}(x)V_{m^{\prime}\bar{m}^{\prime}}^{j^{\prime}}(x)=\delta_{jj^{\prime}}\delta_{mm^{\prime}}\delta_{\bar{m}\bar{m}^{\prime}}\frac{1}{\mathrm{dim}\;D_{j}},
iD_{\mu\nu}(p)=-i\frac{1}{p^{2}-\mu^{2}}\left(g_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)+\mu\frac{1}{p^{2}-\mu^{2}}\frac{1}{p^{2}}\varepsilon_{\mu\nu\rho}p^{\rho}-i\alpha\frac{p_{\mu}p_{\nu}}{p^{4}}.
{\frac{dM^{(1)}}{dm}}={\frac{d(\deltaM)}{dm}}+{\frac{1}{2}}\sum_{\mathrm{ds}}{\frac{d\omega_{\mathrm{ds}}}{dm}}+{\frac{1}{2m}}\int_{-\infty}^{\infty}{\frac{dk}{2\pi}}\sqrt{k^{2}+m^{2}}\,\delta^{\prime}(k).
g_{\mu\nu}(t,\vec{x})\equiv\left[\begin{matrix}{-(c^{2}-v^{2})}&{\vdots}&{-v\;\hat{r}_{j}}\\{\cdots\cdots\cdots\cdots}&{\cdot}&{\cdots\cdots}\\{-v\;\hat{r}_{i}}&{\vdots}&{\delta_{ij}}\\\end{matrix}\right].
\rho_{+}=\frac\ell2(M+\sqrt{M^{2}-\frac{J^{2}}{\ell^{2}}})\qquad\mathrm{and}\qquad\rho_{-}=\frac\ell2(M-\sqrt{M^{2}-\frac{J^{2}}{\ell^{2}}})\qquad