harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{S_{eff}}^{\prime}=\int{d^{4}x[-\frac{1}{4}{F^{\prime}}^{2}-\frac{1}{2}{F^{\prime}}^{{\mu}{\nu}}S_{{\mu}{\nu}}^{\lambda}A_{\lambda}-2S_{\lambda}^{{\mu}{\nu}}A^{\lambda}S_{{\mu}{\nu}}^{\beta}A_{\beta}+\frac{1}{2}{m^{2}}_{\gamma}A^{2}]}
V(\boldsymbol{\varphi})=-e^{2}\left[2+4f(|\boldsymbol{\varphi}|)-\frac{1}{2}f^{2}(|\boldsymbol{\varphi}|)\left(|\boldsymbol{\varphi}|^{4}-|\varphi_{1}^{2}+\varphi_{2}^{2}|^{2}\right)\right]
\hat{\Phi}_{d}=\left(\begin{array}{ccc}{\Phi^{\prime}}&{0}\\{0}&{\Phi^{\prime}}\\\end{array}\right)\,;\,\,\,\,\,\,\hat{\Phi}_{nd}=\left(\begin{array}{ccc}{0}&{\Phi^{\prime\prime}}\\{-\Phi^{\prime\prime}}&{0}\\\end{array}\right)\,;
\begin{array}{ccl}{\left(\begin{array}{c}{1}\\{r}\\{\tilde{r}}\\{\tilde{r}}\\\end{array}\right)}&{=}&{\mathcal{S}\,[a,b;E;V,|W|]\,\left(\begin{array}{c}{t}\\{t}\\{\tilde{t}}\\{\tilde{t}}\\\end{array}\right)~,}\\\end{array}
Z(R(S),\langle\cdot\,,\cdot\rangle)=\pi^{\zeta/2}\,det(\widetilde{T})^{-1/2}\,\prod_{k=1}^{N}det(\widetilde{T}_{k}^{*}\widetilde{T}_{k})^{\frac{1}{2}(-1)^{k-1}}\,.
L=\frac{-T_{r}}{4g^{2}}\left[4(\partial_{+}A_{a}\partial_{-}A_{a})-([A_{a},A_{b}])^{2}+\sqrt{2}i\overline{{\psi_{-}}}\left[\left\{\gamma_{-}[D_{0},\psi_{-}]+\gamma_{-}\alpha^{a}[A_{a},\psi_{-}]\right\}\right]\right].
g_{\mathrm{total}}=\frac{4\pi^{2}R}{e^{2}}\,\left(M_{ij}d\vec{x}_{i}\cdotd\vec{x}_{j}+(M^{-1})_{ij}(d\xi_{i}+\vec{v}_{ik}\cdotd\vec{x}_{k})(d\xi_{j}+\vec{v}_{jm}\cdotd\vec{x}_{m})\right).
\dot{P}_{\alpha}(\theta)=\sum_{l=0}^{N}\left(\dot{p}_{\alpha}(l+1)-\dot{p}_{\alpha}(l)\right)\Theta\left(\theta-\theta_{l}\right)-\sum_{l=0}^{N}\left(p_{\alpha}(l+1)-p_{\alpha}(l)\right)\delta(\theta-\theta_{l})\dot{\theta_{l}}
\Sigma_{\alpha}{}^{i}:=\delta_{\alpha}^{i}{\mathcalL}-\frac{\partial{\mathcalL}}{\partialD_{i}\Psi}\,D_{\alpha}\Psi-D_{\alpha}\overline{{\Psi}}\,\frac{\partial{\mathcalL}}{\partialD_{i}\overline{{\Psi}}}
\bar{H}_{\mathrm{D}}(q^{r},p^{r})=\pi_{a}^{s_{a}}\dot{\mu}_{a}^{s_{a}}+p_{a}^{r_{a}}\dot{q}_{a}^{r_{a}}-L_{\mathrm{D}}+\bar{\lambda}_{a}^{(1)s_{a}}\pi_{a}^{s_{a}}+\bar{\lambda}_{a}^{(2)s_{a}}\psi_{a}^{s_{a}}.
\vec{S}\to\frac{\vec{S}+\lambda\SigmaF^{-2(d+2)/(d-2)}(\vec{H}\vec{S})\partial^{2}\vec{S}}{\vert\vec{S}+\lambda\SigmaF^{-2(d+2)/(d-2)}(\vec{H}\vec{S})\partial^{2}\vec{S}\vert}
R_{\beta\\gamma\tau}^{\\alpha}=\delta_{\tau}\Gamma_{\\beta\gamma}^{\alpha}-\delta_{\gamma}\Gamma_{\\beta\tau}^{\alpha}+\Gamma_{\,beta\gamma}^{\varphi}\Gamma_{\\varphi\tau}^{\alpha}-\Gamma_{\,beta\tau}^{\varphi}\Gamma_{\\varphi\gamma}^{\alpha}+\Gamma_{\,beta\varphi}^{\alpha}W_{\\gamma\tau}^{\varphi}.
\tilde{\alpha}=\alpha\beta^{-m}=\left(\begin{array}{ccc}{\omega_{k}^{-2y}\omega_{2d}^{2m}}&{0}&{0}\\{0}&{\omega_{k}^{y}\omega_{2d}^{-m}}&{0}\\{0}&{0}&{\omega_{k}^{y}\omega_{2d}^{-m}}\\\end{array}\right)
f=-\sqrt{2\pi}\left[f_{0}(r)\delta(\omega)\tilde{w_{1}}(k)+\frac{1}{2}f_{+}(r)\delta(\omega-2\omega_{0})\tilde{w_{2}}(k)+\frac{1}{2}f_{-}(r)\delta(\omega+2\omega_{0})\tilde{w_{2}}(k)\right].
e^{-S_{b}^{free}(m;\Lambda,\lambda)}=\int{\calD}A{\calD}\psi{\calD}\overline{\psi}\quadJ(A)\quade^{\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi+m\overline{\psi}e^{-2i\varphi\gamma_{5}}\psi+(A,d\lambda+i\astd\Lambda)}.
\int_{f_{1}}^{f_{2}}f^{3}[4H^{2}+2{\dot{H}}]={\frac{1}{3}}\lambda({\frac{\rho_{2(nr)}}{f_{2}}}+{\frac{\rho_{1(nr)}}{f_{1}}});\,C_{3}={\frac{1}{3}}\lambda{\frac{\rho_{i(nr)}}{f_{i}}}
\mathcal{G}_{\mu\nu}^{\mathrm{MCS}}=\frac1{4\pi}\varepsilon_{\mu\nu\rho}\frac{(x-y)^{\rho}}{\left|x-y\right|^{3}}+\fracm{4\pi}\frac{e^{-m\left|x-y\right|}}{\left|x-y\right|}\left(g_{\mu\nu}-\frac1m\varepsilon_{\mu\nu\rho}\frac{(x-y)^{\rho}}{\left|x-y\right|^{2}}(1+m\left|x-y\right|\right)\;.\;
\left[\left(\begin{array}{c}{\varphi}\\{\Phi}\\\end{array}\right),\left(\begin{array}{c}{\psi}\\{\Psi}\\\end{array}\right)\right]_{{\calK}_{i}}=\langle\varphi\mid\psi\rangle+\langle\Phi\mid\mathrm{sgn}\,M_{i}\Psi\rangle,
{\frac{1}{\Gamma(s)}}\int_{0}^{\infty}t^{s-1}{\frac{e^{-{\frac{a}{t}}}}{t}}e^{-\mut}dt={\frac{1}{(s-1)}}\mu^{1-s}+{\frac{\Gamma(1-s)}{\Gamma(s)}}({a})^{s-1}+{\calO}(\mu^{2-s},\mu)
S_{E}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-g^{E}}\left[{\calR}_{g^{E}}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(p+2)!}}e^{2a_{p}\phi}F_{p+2}^{2}\right]
\nu^{2}{\frac{\theta_{ij}}{(z_{ij})^{2}}}\xi_{i}\epsilon_{j}K_{ij}^{\mathrm{tot}},\quad\nu^{2}{\frac{\theta_{ij}}{(z_{ij})^{2}}}\xi_{i}\epsilon_{j}K_{ij}^{\mathrm{tot}}K_{jk}^{\mathrm{tot}}\,.
=-4i\left({\epsilon}_{1}\cdotp_{1}{\epsilon}_{2}\cdotp_{2}\frac{1}{p^{2}+m^{2}}+{\epsilon}_{2}\cdotp_{1}{\epsilon}_{1}\cdotp_{2}\frac{1}{q^{2}+m^{2}}\right)-2i{\epsilon}_{1}\cdot{\epsilon}_{2}
\frac{\partial}{\partial\theta}ImI^{\prime}=-\pi\int_{0}^{1}dx\delta(B)\frac{\partialB}{\partial\theta}=\frac{-2\pi\thetap_{0}^{2}}{2as^{\prime}}\intdy(\frac{1}{4}-y^{2})\{\delta(y-a)+\delta(y+a)\}
v_{(2)}{}^{(1)}\otimesv_{(1)}(f^{(2)^{\prime}})\,v_{(2)}{}^{(2)^{\prime}}=v^{(1)}{}_{(2)}\otimesv^{(2)^{\prime}}\,\widehat{v^{(1)}{}_{(1)}}(f^{(2)^{\prime}}).
Q_{BRST}=\ointdz\left\{c(-{\frac{1}{2}}\partialX\partialX+{\frac{1}{2}}\psi\partial\psi-{\frac{1}{2}}\partial\beta\gamma-{\frac{3}{2}}\beta\partial\gamma)+c\partialcb+{\frac{1}{2}}\gamma\psi\partialX-{\frac{1}{4}}\gamma^{2}b\right\},
\mathrm{Im}\,\Pi^{\mathrm{(non\mathrm{-}planar)}}=\frac{\theta(p^{2})\pi^{\frac{n}{2}+1}}{(2\pi)^{n}}\int_{0}^{1}dx\,\frac{1}{|M|^{4-n}}\left(\frac{|\bar{\theta}||M|}{2}\right)^{2-\frac{n}{2}}J_{\frac{n}{2}-2}(|\bar{\theta}||M|)
\mathcal{A}(p^{I})\simG_{s}^{n-2}\sum_{\substack{\sigma\mathrm{~cyclically}\,\mathrm{inequivalent}}}\mathcal{F}_{G}(p^{\sigma_{1}},\cdots,p^{\sigma_{n}})e^{-\frac{i}{2}\theta^{ab}\sum_{I>J}p_{a}^{\sigma_{I}}p_{b}^{\sigma_{J}}}
\int\mathrm{d}^{3}y~\nu_{\alpha,y}^{(1)T}(2,x)\frac{\delta}{\delta\xi^{(1)\alpha}(y)}\int\mathrm{d}^{3}z~{\calH}^{(1)}(z)=(d_{1}-1)\partial_{i}\partial^{i}B+d_{4}\frac{\partialg}{\partial\theta},
\hat{H}=W^{-\frac{1}{4}}(x)\hat{\tilde{H}}W^{\frac{1}{4}}(x),\;\phi^{A}(x)=W^{-\frac{1}{4}}(x)\tilde{\phi}^{A}(x),\;\chi^{a}(x)=W^{-\frac{1}{4}}(x)\tilde{\chi}^{a}(x).
S=\intd^{5}x\sqrt{|g|}\,k\,[R-{\textstyle\frac{1}{4}}k^{2}F^{2}(A)-{\textstyle\frac{1}{4}}k^{-2}F^{2}(B)+{\textstyle\frac{\epsilon}{8\sqrt{|g|}}}k^{-1}F(A)F(B)B]\,.
{\calS}\,={\frac{1}{2}}\,\int\,\left\{\left[\,\gamma_{\Lambda\Sigma}(\phi)\,F^{\Lambda}\,\wedge\,\starF^{\Sigma}\,+\,\theta_{\Lambda\Sigma}(\phi)\,F^{\Lambda}\,\wedge\,F^{\Sigma}\,\right]\,+\,g_{IJ}(\phi)\,\partial_{\mu}\phi^{I}\,\partial^{\mu}\phi^{J}\,\mathrm{d}^{4}x\,\right\}\,
F(k)\;=\;e_{N}^{2}\int_{0}^{1}d\alpha\,\int_{0}^{\infty}\frac{dp^{2}}{\pi}\,\frac{1}{\beta}\,\sum_{n=-\infty}^{+\infty}\frac{p^{2}-\alpha(1-\alpha)k^{2}-{\tilde{\omega}}_{n}^{2}}{[p^{2}+\alpha(1-\alpha)k^{2}+{\tilde{\omega}}_{n}^{2}]^{2}}\;,
{\frac{\partialf_{a}^{\prime}}{\partial\phi^{A}}}\delta_{a}\phi^{A}\,+\,{\frac{1}{2}}g_{3}^{\prime}\delta_{2}\,\delta_{1}\phi^{A}\,-\,{\frac{1}{2}}\epsilon^{ab}{\frac{\partialf_{a}^{\prime}}{\partial\phi^{A}}}{\frac{\partialf_{b}^{\prime}}{\partial{\bar{\phi}}^{A}}}\,\,=\,\delta_{2}\delta_{1}B[\phi^{A}]\,\,,
\phi(g_{1},\ldots,g_{4})=\sum_{N_{1}\ldotsN_{4}}\sum_{\Lambda}\left(\phi_{\alpha_{1}\ldots\alpha_{4}}^{N_{1}\ldotsN_{4}}\D_{\alpha_{1}\beta_{1}}^{(N_{1})}(g_{1})\ldotsD_{\alpha_{4}\beta_{4}}^{(N_{4})}(g_{4})\C_{\beta_{1}\ldots\beta_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}\right)\\left({C}_{\gamma_{1}\ldots\gamma_{4}}^{N_{1}\ldotsN_{4}\,\Lambda}w_{\gamma_{1}}\ldotsw_{\gamma_{4}}\right).
{\renewcommand{\arraystretch}{1.4}\begin{array}{ccc}{\partial^{1\dot{1}}f^{\dot{1}\dot{1}}+\partial^{2\dot{1}}f^{\dot{1}\dot{2}}}&{=}&{0,\nonumber}\\{\partial^{1\dot{2}}f^{\dot{1}\dot{1}}+\partial^{2\dot{2}}f^{\dot{1}\dot{2}}}&{=}&{0,\nonumber}\\{\partial_{1\dot{1}}f_{11}+\partial_{1\dot{2}}f_{12}}&{=}&{0,\nonumber}\\{\partial_{2\dot{1}}f_{11}+\partial_{2\dot{2}}f_{12}}&{=}&{0.}\\\end{array}}
{\langleJ(x_{1})J(x_{2})\rangle_{s(\psi)}}=K_{1}\,Y_{1}[s(\psi)]+K_{2}\,{Y}_{2}[s(\psi)]-\sum_{I=1}^{6}Y_{1}[s(\chi^{I})]\,Y_{1}[s(y^{I})]\,Y_{1}[s(\omega^{I})],
L^{-}=\left(\begin{array}{ccccc}{l_{11}^{-}}&{}&{}&{}&{}\\{l_{21}^{-}}&{l_{22}^{-}}&{}&{}&{}\\{l_{31}^{-}}&{l_{32}^{-}}&{l_{33}^{-}}&{}&{}\\{l_{41}^{-}}&{l_{42}^{-}}&{l_{43}^{-}}&{l_{44}^{-}}&{}\\{l_{51}^{-}}&{l_{52}^{-}}&{l_{53}^{-}}&{l_{54}^{-}}&{l_{55}^{-}}\\\end{array}\right)~,
I[l,x,\bar{c}]=I[l,x]+\intd^{2}\xi[\langlel^{-1}\partiall,T^{a}\rangleE_{ab}^{0}+\partialx^{\alpha}F_{\alphab}^{L}-\langlel^{-1}\partiall,T_{b}\rangle]\bar{c}^{b}
\frac{a(\mu_{j})}{d(\mu_{j})}\prod_{k=1\atop{k\nej}}^{N}\frac{f(\mu_{j},\mu_{k})}{f(\mu_{k},\mu_{j})}=1,\qquad\mathrm{or}\qquad\frac{a(\mu_{j})}{d(\mu_{j})}\prod_{k=1}^{N}\frac{h(\mu_{j},\mu_{k})}{h(\mu_{k},\mu_{j})}=(-1)^{N-1}
\left(\begin{array}{ccccc}{\alpha_{1}}&{\beta}&{0}&{0}&{\gamma}\\{-{\bar{\beta}}}&{\alpha_{2}}&{0}&{0}&{\delta}\\{0}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}\\{-{\bar{\gamma}}}&{-{\bar{\delta}}}&{0}&{0}&{-\alpha_{1}-\alpha_{2}}\\\end{array}\right),
e^{-S_{\mathrm{dual}}(A)\;}=\intD\varphiD\varphi^{\dagger}Db_{\mu}Db_{\mu}^{\dagger}D\eta_{\mu}e^{-\left(S(\varphi,(\partial+i\eta)\varphi,b)+\intd^{3}xA_{\mu}\epsilon^{\mu\nu\rho}\partial_{\nu}\eta_{\rho}\right)\;}\;.
\lambda_{1}\,=\mathrm{const.}\times\beta^{0}{,}~~~|\mathrm{const.}|\,>\,2\,{;}\quad{}~~~~~\lambda_{0}\,=\,\frac{(d-2)^{2}\pi^{2}}{36M^{4}\beta^{4}}\,\frac{(\lambda_{1}\,-\,2)^{2}}{\lambda_{1}(\lambda_{1}\,-\,1)^{2}}\,{.}
S_{1}^{AC}=\frac{\pi^{D}(p^{2})^{D-4}\Gamma^{2}(D-3)\Gamma^{2}(\frac{1}{2}D-1)\Gamma(2-\frac{1}{2}D)\Gamma(4-D)}{\Gamma(D-2)\Gamma(\frac{3}{2}D-4)},
\begin{array}{llll}{\phi(x)}&{\rightarrow}&{\phi^{\prime}(x)=\phi(x)e^{i\Lambda(x)},}&{}\\{A_{\mu}(x)}&{\rightarrow}&{A_{\mu}^{\prime}(x)=A_{\mu}(x)-\partial_{\mu}\Lambda(x),}&{}\\{B_{\mu\nu}(x)}&{\rightarrow}&{B_{\mu\nu}^{\prime}(x)=B_{\mu\nu}(x)+\partial_{\mu}\xi_{\nu}(x)-\partial_{\nu}\xi_{\mu}(x),}&{}\\\end{array}
{\frac{c_{jk}}{c_{jm}c_{mk}}}={\frac{\sigma(\nu_{jk})}{\sigma(\mu_{jk})}}{\frac{\sigma(\mu_{jm})}{\sigma(\nu_{jm})}}{\frac{\sigma(\mu_{mk})}{\sigma(\nu_{mk})}}{\frac{\sigma(\nu_{jm}+\nu_{mk}-\nu_{jk})}{\sigma(\mu_{jm}+\mu_{mk}-\mu_{jk})}}.
\int{\mathrm{d}^{2\mu}x}\frac{(x,u)^{n}(z-x,u)^{m}}{x^{2\alpha}(x-z)^{2\beta}}=\pi^{mu}\frac{H_{n}(\alpha)H_{m}(\beta)}{H_{n+m}(\alpha+\beta-\mu)}\frac{(u,z)^{m+n}}{z^{2(\alpha+\beta-\mu)}},
S=\intdt\,dx{\calL}=\intdt\,dx\left[\frac{i\hbar}{2}\sum_{i=1}^{N}(\phi_{i}^{*}\partial_{t}{\phi_{i}}-{\phi_{i}}\partial_{t}{\phi_{i}}^{*})-\frac{\hbar^{2}}{2m}\sum_{i=1}^{N}\left|\left(\partial_{x}+i{\frac{\lambda}{2}}\rho^{2}\right)\phi_{i}\right|^{2}-V\right],
\left[\omega_{WP}^{(n)}\right]^{n-4}\cap\left[\overline{V}^{(k+2)}\times\overline{V}^{(n-k)}\right]=\left[\omega_{WP}^{(k+2)}+\omega_{WP}^{(n-k)}\right]^{n-4}\cap\left[\overline{V}^{(k+2)}\times\overline{V}^{(n-k)}\right],
\Theta_{b}(G)=\left\langle\widehat{\varphi}(x,0)\right\rangle_{B}-\left\langle0|\widehat{\varphi}|0\right\rangle=\frac{b}{(1+b^{2})}\partial_{a}(2\mathcal{G}(a)\mathcal{-}\mathbf{G(}a\mathbf{))|}_{a=0}
\Gamma(\Omega,\Omega^{\prime})=\delta(\Omega,\Omega^{\prime})+\frac{\nu}{4\pi}\int_{0}^{1}d\rho\left(\begin{array}{l}{\rho^{-\nu}(1-\rho\sigma.n\\sigma.n^{\prime})-\rho^{\nu}(\sigma.n\\sigma.n^{\prime}-\rho)}\\\end{array}\right).
\left.\left(-\psi_{+\mu}\delta\psi_{+}^{\mu}+\psi_{-\mu}\delta\psi_{-}^{\mu}\right)\right|_{\sigma=0}=\left.\left(-\psi_{+\mu}\delta\psi_{+}^{\mu}+\psi_{-\mu}\delta\psi_{-}^{\mu}\right)\right|_{\sigma=\pi}=0.
\mathcal{L}=-\frac12\left[\overline{{\Psi}}(x)\left(\Gamma_{\mu}\overrightarrow{D_{\mu}^{(-)}}+m\right)\Psi(x)-\overline{{\Psi}}(x)\left(\Gamma_{\mu}\overleftarrow{D_{\mu}^{(+)}}-m\right)\Psi(x)\right]-\frac14\mathcal{F}_{\mu\nu}
\begin{array}{rcl}{}&{=}&{-(1-q_{+}^{-1})\xi(\infty)J_{+}(\infty)X_{+}^{+}(X_{+}^{-})^{n}(X_{-}^{-})^{m}V_{\alpha}(z)}\\{}&{}&{-(1-q_{-}^{-1})\xi(\infty)J_{-}(\infty)X_{-}^{+}(X_{+}^{-})^{n}(X_{-}^{-})^{m}V_{\alpha}(z)~,}\\\end{array}
{{\bar{R}}_{\mu\nu\lambda}}^{\;\;\;\;\;\;\kappa}=\partial_{\mu}{\bar{\Gamma}}_{\nu\lambda}^{\;\;\;\;\kappa}-\partial_{\nu}{\bar{\Gamma}}_{\mu\lambda}^{\;\;\;\;\kappa}-[\bar{\Gamma}_{\mu},\bar{\Gamma}_{\nu}{]_{\lambda}}^{\kappa}.
M_{ij}=\left(\begin{array}{cc}{{\frac{\partial\beta_{(2i)}^{\lambda}}{\partial\lambda_{(2j)}}}}&{{\frac{\partial\beta_{(2i)}^{\lambda}}{\partial\xi_{(2j)}}}}\\{{\frac{\partial\beta_{(2i)}^{\xi}}{\partial\lambda_{(2j)}}}}&{{\frac{\partial\beta_{(2i)}^{\xi}}{\partial\xi_{(2j)}}}}\\\end{array}\right).
{V_{g}^{(r)}=\zeta_{IJ}^{(r)}(\dot{X}^{I}\dot{X}^{J}-2\dot{X}^{I}S\gamma^{JL}Sk^{L}+2S\gamma^{IL}Sk^{L}S\gamma^{JM}Sk^{M})e^{ik\cdotX},}
S=\int\,d^{4}x\,\left[-\,\frac{1}{4}\,F_{\mu\nu}F^{\mu\nu}+\frac{1}{12}\,H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{1}{2}\,m\,\epsilon_{\mu\nu\rho\lambda}\,A^{\mu}\partial^{\nu}B^{\rho\lambda}\right]
{\calD}\left[\begin{array}{c}{\sum\xi_{n}^{j+}D_{n,-1/2}^{(j)}}\\{\sum\xi_{n}^{j-}D_{n,+1/2}^{(j)}}\\\end{array}\right]=\!\!-\!\!\left[\begin{array}{cc}{0}&{(J_{-}^{(j)})_{-1/2,+1/2}}\\{(J_{+}^{(j)})_{+1/2,-1/2}}&{0}\\\end{array}\right]\left[\begin{array}{c}{\sum\xi_{n}^{j+}D_{n,+1/2}^{(j)}}\\{\sum\xi_{n}^{j-}D_{n,-1/2}^{(j)}}\\\end{array}\right]
H_{c}^{r}\,=\,\int\left[(\pi_{a}\pi_{a})(\phi^{b}\phi^{b})\,-\,(\phi^{a}\pi_{a})(\phi^{b}\pi_{b})\,-\,\phi^{a}\partial_{i}\partial_{i}\phi^{a}{\frac{1}{\phi^{b}\phi^{b}}}\right]dx\,.
(m_{1}m_{2}|m_{3}|m_{4})=\int{\frac{d^{d}k}{(2\pi)^{d}}}\int{\frac{d^{d}q}{(2\pi)^{d}}}{\frac{1}{k^{2}+m_{1}^{2}}}{\frac{1}{k^{2}+m_{2}^{2}}}{\frac{1}{q^{2}+m_{3}^{2}}}{\frac{1}{(q-k)^{2}+m_{4}^{2}}}
\hat{x}^{\mu}=\frac{\ell}{(\alpha^{\prime})^{1/2}}x^{\mu}\,\quad\hat{r}=\frac{\ell^{3}}{(\alpha^{\prime})^{3/2}}r^{\prime}\,\quad\hat{g}=g^{\prime}\frac{\ell^{2}}{\alpha^{\prime}},\quad\ell^{2}=\theta\alpha^{\prime}\,\quad\alpha^{\prime}\rightarrow0\.
K(T)=\textrm{Tr}\;e^{-\frac{i}{\hbar}{\cal{H}}}=\int{\cal{D}}\phi_{a}(x)K(\phi_{a},T|_{a},0)=\int{\cal{D}}\phi_{a}(x)\int_{\phi_{a},0}^{\phi_{a},T}{\cal{D}}\phi(x,t)e^{\frac{i}{\hbar}S[\phi,T]}.
Q_{v}\equivd+i_{v}=\intdt\Bigl(c^{i}\frac{\partial}{\partiala^{i}}+q_{i}\frac{\partial}{\partial\bar{c}_{i}}+(\partial_{t}a^{i}-h^{i})\tau_{i}+(\partial_{t}\bar{c}_{i}+\bar{c}_{j}\partial_{k}h^{j})\pi^{i}\Bigr).
\delta\mathcal{E}_{k}=-\frac{\lambda}{2}\left(e^{i\delta}\mathcal{G}_{\alpha}^{(k)}(\beta)+e^{-i\delta}\mathcal{G}_{-\alpha}^{(k)}(\beta)\right)\quad,\quad\mathcal{G}_{\alpha}^{(k)}(\beta)=\Big\langlek\Big|e^{i\alpha\Phi(0)}\Big|k\Big\rangle\:.
\left(\begin{array}{c}{y_{1}}\\{y_{2}}\\\end{array}\right)\to\left(\begin{array}{rc}{d}&{c}\\{-b}&{a}\\\end{array}\right)\left(\begin{array}{c}{y_{1}}\\{y_{2}}\\\end{array}\right),\quad\left(\begin{array}{c}{\bar{y}_{1}}\\{\bar{y}_{2}}\\\end{array}\right)\to\left(\begin{array}{rc}{a}&{b}\\{-c}&{d}\\\end{array}\right)\left(\begin{array}{c}{\bar{y}_{1}}\\{\bar{y}_{2}}\\\end{array}\right).
F_{\mathrm{D3}}=-N^{2}V_{3}T^{4}\left[\frac{\pi^{2}}{6}+\frac{1}{4}\sum_{i=1}^{3}\omega_{i}^{2}+\frac{1}{32\pi^{2}}\Big(\sum_{i=1}^{3}\omega_{i}^{2}\Big)^{2}-\frac{1}{16\pi^{2}}\sum_{i=1}^{3}\omega_{i}^{4}\right]
h_{12}(x)=(g^{\dagger})^{-1}(\eta_{1}^{\dagger}M\eta_{2}-\eta_{2}^{\dagger}M\eta_{1})g^{-1}+g(\eta_{1}M^{-1}\eta_{2}^{\dagger}-\eta_{2}M^{-1}\eta_{1}^{\dagger})g^{\dagger}.
L_{1}\,=\,-\frac{a}{4}\left(F_{\mu\nu}\right)^{2}+\,b\,\partial_{\mu}F^{\mu\lambda}\,\partial^{\nu}F_{\nu\lambda}\,+\,\theta\,\epsilon^{\mu\nu\rho}\,\partial_{\sigma}\,A_{\mu}\,\partial_{\nu}\partial^{\sigma}\,A_{\rho},
S[\varphi]=\frac{1}{2}\,\delta\,\epsilon^{d}\sum_{\alpha\inZ_{M}}\,\sum_{{\bfj}\in(Z_{N})^{d}}\Biggl\{\left[\frac{\varphi^{{\bfj}}(t_{\alpha+1})-\varphi^{{\bfj}}(t_{\alpha})}{\delta}\right]^{2}-\sum_{\mu=1}^{d}\left[\frac{\varphi^{{\bfj}+{\bfe}_{\mu}}(t_{\alpha})-\varphi^{{\bfj}}(t_{\alpha})}{\epsilon}\right]^{2}-m^{2}\left[\varphi^{{\bfj}}(t_{\alpha})\right]^{2}\Biggr\}\;,
\{\gamma^{\mu},\gamma^{\nu}\}\;=\;2\,g^{\mu\nu}\;\;,\;\;\{\gamma^{\mu},\gamma_{5}\}\;=\;0\;,\;\mathrm{tr}(\gamma^{\mu}\gamma^{\nu}\gamma_{5})\,=\,\frac{2i}{\tau_{2}}\,\epsilon^{\mu\nu}\;,
\frac{\partial\Gamma_{\Lambda}}{\partial\Lambda}=-\frac{1}{2}tr\left[\frac{\partialC^{-1}}{\partial\Lambda}\;.\left(C^{-1}+\gamma\right)^{-2}.\hat{\Gamma}.\left(1+\{C^{-1}+\gamma\}.\hat{\Gamma}\right)^{-1}\right]
\mathrm{P}{\int}_{0}^{\infty}dk_{+}\left(\frac{n_{+}}{(k_{+}-a)^{2}}+\frac{n_{+}}{(k_{+}+a)^{2}}\right)=\mathrm{P}{\int}_{-\infty}^{\infty}dk_{+}\frac{n_{+}}{(k_{+}-a)^{2}}=-\int_{-\infty}^{\infty}dq_{-}\frac{n_{-}}{q_{-}^{\;\;2}},
U=\frac{3\langleN\rangle}{2\beta}\left[1-\frac{\langleN\rangle}{V}\lambda_{T}^{3}\delta(q)+\frac{\langleN\rangle^{2}}{V^{2}}\lambda_{T}^{6}\left(4\delta^{2}(q)-\frac{\Gamma(q)}{12}\right)+...\right],
M_{P}^{2}\sim\frac{M_{6}^{4}}{m_{H}^{2}}\int_{0}^{\infty}dxM^{2}(x){\calL}(x)\equiv\frac{M_{6}^{4}}{2}\int_{-\infty}^{+\infty}M^{2}(w)L^{2}(w)dw
a_{j}^{om}({\mathsfs})=\frac{-\sqrt{\mathsfs}\sqrt{\Gamma_{e}({\mathsfs})\Gamma_{f}({\mathsfs})}}{{\mathsfs}-M_{Z}^{2}+i\sqrt{\mathsfs}\Gamma_{Z}(\mathsfs)}\approx\frac{-M_{Z}B_{ef}\Gamma_{Z}}{{\mathsfs}-M_{Z}^{2}+i\frac{\mathsfs}{M_{Z}}\Gamma_{Z}}=\frac{R_{Z}}{{\mathsfs}-M_{Z}^{2}+i\frac{\mathsfs}{M_{Z}}\Gamma_{Z}}\,,\quadm_{0}^{2}\leq{\mathsfs}<\infty\,.
\begin{array}{l}{0=k_{1}+3k_{3}}\\{{}}\\{0=k_{2}+k_{4}+k_{5}+4k_{6}+4k_{7}}\\{{}}\\{0=k_{2}+k_{3}+3k_{4}+k_{5}+4k_{6}+12k_{7}}\\{{}}\\{0=4c_{2}-6c_{1}-k_{2}+k_{3}+k_{4}-k_{5}-4k_{6}+4k_{7}}\\\end{array}
\langle\,J^{5}(x,y)\,\rangle=-\frac{g_{5}^{2}}{32\pi^{2}}\,F_{\mu\nu}\widetilde{F}^{\mu\nu}\,\sum_{n}\frac{1}{M_{n}}\phi_{n}^{{}_{L}}(y)\phi_{n}^{{}_{R}}(y)\quad.
\nabla_{i}V_{j}={\frac{1}{2}}P_{AB,i}P_{CD,j}P_{,k}^{ABCD}g^{k\bar{l}}\bar{V}_{\bar{l}}={\frac{1}{4}}\epsilon_{ABCDEF}P_{AB,i}P_{CD,j}P_{EF,k}g^{k\bar{l}}\bar{V}_{\bar{l}}.
W[{\psi},A]=\int_{-\infty}^{\infty}dt\int_{0}^{\mathrm{L}}dx\sum_{k=1}^{2}\{[\overline{{\psi}}_{k}{\gamma}^{\mu}(i{\partial}_{\mu}-e_{k}A_{\mu}){\psi}_{k}-m_{k}\overline{{\psi}}_{k}{\psi}_{k}]-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\},
{\calH}=\Pi^{i}\dot{A}_{i}-{\calL}={\frac{m^{2}}{2}}\Pi_{i}\Pi_{i}+({\frac{1}{2}}+{\frac{\theta^{2}}{8m^{2}}})A_{i}A_{i}-{\frac{\theta}{2}}\epsilon_{ij}\Pi_{i}A_{j}+{\frac{1}{4m^{2}}}A_{ij}A_{ij}.
s=-\sum_{i=1}^{N}\frac{\alpha_{i}}{\alpha}-\frac{Q}{\alpha}~,\quad~\alpha_{i}=-\frac{Q}{2}+\frac{1}{2}\sqrt{Q^{2}+4(2-h_{i})},\quad~\alpha=-\frac{Q}{2}+\frac{1}{2}\sqrt{Q^{2}+8}
\left.\frac{d^{2}[TF_{1}(T)]}{dT^{2}}\right|_{T=T_{c}^{+}}=\frac{1}{2\pii}\oint_{C}V(z)\,\frac{d}{dT}\left.\left(\frac{1}{\sqrt{(z-a)(z-b)}}\right)\right|_{T=T_{c}^{+}}\,dz\,,
C_{T}\frac{\Gamma(c)\Gamma(a-b)}{\Gamma(c-b)\Gamma(a)}(\alpha-1)^{b-a}2^{a-b}=\frac{\Gamma(c)\Gamma(b-a)}{\Gamma(c-a)\Gamma(b)}\left(C_{R}+\frac{\Gamma(2-c)\Gamma(c-a)}{\Gamma(c)\Gamma(2-c-a)}\right).
\begin{array}{c}{p_{(1)}\cdotp_{(2)}=\tilde{p}_{(1)}\cdot\tilde{p}_{(2)}=2t/\alpha^{\prime},\qquadp_{(1)}\cdotD\cdot\tilde{p}_{(2)}=\tilde{p}_{(1)}\cdotD\cdotp_{(2)}}\\{\mathrm{and}\qquadp_{(1)}\cdotD\cdot\tilde{p}_{(1)}=p_{(2)}\cdotD\cdot\tilde{p}_{(2)}=2s/\alpha^{\prime},}\\\end{array}
\begin{array}{rl}{3H^{2}\=}&{{\displaystyle{\frac{2\pi\lambda\phi^{4}}{M_{\mathrm{p}}^{2}-8\pi\xi\phi^{2}}}\,}}\\{(1-6\xi)\3H\dot{\phi}\phi\=}&{{\displaystyle-\{\frac{\lambda\phi^{4}M_{\mathrm{p}}^{2}}{M_{\mathrm{p}}^{2}-8\pi\xi\phi^{2}}}\.}}\\\end{array}
{\mathcal{L}_{\lambda,\kappa}}=\frac{m^{2}A^{2}}{2\lambda}-\frac{m^{2}B^{2}}{4\beta\kappa}+\beta^{2}f(\lambda)-\frac{\beta^{2}}{2}l(\kappa)-\frac{\chi\theta}{2}\varepsilon^{\mu\nu\rho\lambda}\partial_{\mu}B_{\nu\rho}A_{\lambda}\;.
\Psi_{1}=b_{\alpha}^{\dot{\alpha}}\left(\bar{D}_{\dot{\alpha}}\sigma^{\alpha}+k\beta_{\dot{\alpha}}^{\alpha}\right)+\left(\bar{\sigma}^{\dot{\alpha}}D_{\alpha}+\bar{k}\bar{\beta}_{\alpha}^{\dot{\alpha}}\right)\bar{b}_{\dot{\alpha}}^{\alpha}\,.
\frac{\hat{\omega}}{i_{v}\cdot\hat{\omega}}\left(\frac{d\hat{\omega}}{i_{v}\cdot\hat{\omega}}\right)^{k}=\frac{\hat{\omega}}{i_{v}\cdot\hat{\omega}}\left(d\,\frac{\hat{\omega}}{i_{v}\cdot\hat{\omega}}\right)^{k}\equiv\rho\left(d\rho\right)^{k}
{\calB}_{\Gamma}^{(\varphi,t)}=\frac{\ell}{8\piG}\int_{\calV}[\partial_{\varphi}(e^{-\Phi}\partial_{[t}Y\partial_{r]}X)+\partial_{t}(e^{-\Phi}\partial_{[r}Y\partial_{\varphi]}X)]dr\,d\varphi\,dt\qquad.
\left(\begin{array}{cccc|cc}{\times}&{\times}&{}&{}&{}&{}\\{\times}&{\times}&{}&{}&{}&{}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\{\times}&{\times}&{\times}&{\times}&{\times}&{\times}\\\hline{\times}&{\times}&{}&{}&{\times}&{}\\{\times}&{\times}&{}&{}&{\times}&{\times}\\\end{array}\right)
\sqrt{-g}\;(R_{\mu\nu\lambda\delta}R^{\mu\nu\lambda\delta}-4R_{\mu\nu}R^{\mu\nu}+R^{2})={\frac{\partial}{\partialr}}\left[4(e^{2U})^{\prime}\left(1-(e^{U}R^{\prime})^{2}\right)\right]\.
\begin{array}{ccc}{R_{AB}}&{=}&{\tilde{R}_{AB}-a^{i}D_{B}\partial_{A}a_{i},}\\{}&{}&{}\\{R_{ij}}&{=}&{-(a_{k}D^{A}\partial_{A}a_{l}+a^{m}a_{k}\partial_{A}a_{m}\partial^{A}a_{l}-\partial_{A}a_{k}\partial^{A}a_{l})\eta_{ij}^{kl}.}\\\end{array}
L_{0}(\xi,\dot{\xi})=\frac{1}{2}\sum_{i=1}^{M}\dot{\xi_{i}}^{2}(t)-\sum_{i=1}^{M}\sum_{k=1,k>i}^{M}\frac{\hbar^{2}}{(\xi_{i}(t)-\xi_{i}(t))^{2}}-\frac{b^{2}}{2}\sum_{i=1}^{M}\xi_{i}^{2}(t).
ds^{2}~=~\alpha^{\prime}~\left(\frac{u^{2}}{R^{2}}[h(u)dx_{0}^{2}+dx_{i}^{2}]~+~\frac{R^{2}}{u^{2}h(u)}du^{2}~+~R^{2}~d\Omega_{5}^{2}\right)~,
\begin{array}{rl}{R}&{\equiv\mathrm{Cof}\,P+p^{0}Q\,,}\\{M}&{\equiv2\mathrm{det}\,P+(p^{0}q_{0}+\mathrm{tr}(PQ))p^{0}\,,}\\{D}&{\equiv2[(\mathrm{tr}PQ)^{2}-\mathrm{tr}(PQ)^{2}]-(p^{0}q_{0}+\mathrm{tr}PQ)^{2}+4(p^{0}\,\mathrm{det}Q-q_{0}\,\mathrm{det}P)\,,}\\\end{array}
\Upsilon_{\alpha\beta\gamma}^{g_{1}g_{2}}\:=\:\left(\Upsilon_{\alpha\beta\gamma}^{g_{2}}\right)\,\left(g_{2}^{*}\Upsilon_{\alpha\beta\gamma}^{g_{1}}\right)\,\left(\Omega_{\alpha\beta}^{g_{1},g_{2}}\right)\,\left(\Omega_{\beta\gamma}^{g_{1},g_{2}}\right)\,\left(\Omega_{\gamma\alpha}^{g_{1},g_{2}}\right)
{\calL}_{eff}=\intd^{4}\thetaZ(\frac{\mu}{\Lambda_{UV}\Phi},\frac{\mu}{\Lambda_{UV}\Phi^{\dagger}})Q^{\dagger}e^{-V}Q+\intd^{2}\thetay_{0}Q^{3}+\tau(\frac{\mu}{\Lambda_{UV}\Phi}){\calW}_{\alpha}^{2})+\mathrm{h.c.}.
({\frac{\delta\tilde{\Gamma}}{\deltaA}},{\frac{\delta\tilde{\Gamma}}{\deltaK}})+({\frac{\delta\tilde{\Gamma}}{\deltac}},{\frac{\delta\tilde{\Gamma}}{\deltaL}})+{\frac{\delta\tilde{\Gamma}}{\deltaV}}\cdot{\frac{\delta\tilde{\Gamma}}{\deltaM}}=0.
N^{\prime}\psi^{\prime}=\frac{2\gamma}{x^{2}}(e^{2\gamma\psi}(N(K^{\prime})^{2}+\frac{(K^{2}-1)^{2}}{2x^{2}})-e^{-2\gamma\psi}\frac{\beta^{2}x^{2}}{4}(H^{2}-1)^{2})\,
\biggl[\biggl(t^{\prime}{\frac{d}{dt^{\prime}}}\biggr)^{24}+\sum_{i=1}^{23}{\bfB}_{i}(t^{\prime})\biggl(t^{\prime}{\frac{d}{dt^{\prime}}}\biggr)^{i}\biggr]\tilde{\Pi}(t^{\prime})=0,
h_{12}(x)={\frac{t_{1}t_{2}}{1-t_{1}t_{2}}}\Big[(g^{\dagger})^{-1}\eta_{1}^{\dagger}M\eta_{2}g^{-1}+g\eta_{1}M^{-1}\eta_{2}^{\dagger}g^{\dagger}\Big]-\mathrm{h.c.},
2\theta^{\tau_{1}\theta_{1}}\theta^{\tau_{2}\theta_{2}}\theta^{\tau_{3}\theta_{3}}\mathrm{Tr}\int_{X}\varepsilon^{\mu\nu\rho\sigma}\left({\calR}_{\mu\nu}{\calR}_{\rho\sigma\tau_{1}\theta_{1}\tau_{2}\theta_{2}\tau_{3}\theta_{3}}^{(3)}+{\calR}_{\mu\nu\tau_{1}\theta_{1}}^{(1)}{\calR}_{\rho\sigma\tau_{2}\theta_{2}\tau_{3}\theta_{3}}^{(2)}\right),
\Gamma_{\mathrm{kl}}^{\mathrm{j}}={\frac{1}{2}}\,g^{jm}\biggl({\frac{\partialg_{\mathrm{ml}}}{\partialx^{\mathrm{k}}}}+{\frac{\partialg_{\mathrm{mk}}}{\partialx^{\mathrm{l}}}}-{\frac{\partialg_{\mathrm{kl}}}{\partialx^{\mathrm{m}}}}\biggr).
\int\Phi_{1}*\ldots*\Phi_{n}=\langle\Phi_{1}|\Phi_{2}*\ldots*\Phi_{n}\rangle=\langlef_{1}^{(n)}\circ\Phi_{1}(0)\ldotsf_{n}^{(n)}\circ\Phi_{n}(0)\rangle_{\mathrm{UHP}},
\left\{\gamma^{5}\frac{1}{1}{i}\frac{\partial}{\partialx}-g^{2}\left(\langle\bar{q}(x)q(x)\rangle\gamma^{0}+\langle\bar{q}(x)\mathrm{i}\gamma^{5}q(x)\rangle\mathrm{i}\gamma^{1}\right)\right\}\varphi_{n}(x)=\omega_{n}\varphi_{n}(x)\.
{\calH}^{(ij)}f_{i_{1}...i...j...i_{r}}=\frac{g^{2}}{8\pi^{2}}\left((T_{i}\otimesT_{j})_{\calH}\\,hat{\calH}_{ij}f_{i_{1}...i...j...i_{r}}+(T_{i}\otimesT_{j})_{\calG}\\\hat{\calG}_{ij}f_{i_{1}...j...i...i_{r}}\right)
\delta\left(\frac{1}{2\kappa_{5}^{2}}\intd^{4}xdz\sqrt{g}\frac{1}{2}{\calK}_{\rho\rho}(\partial\rho)^{2}\right)=\frac{1}{2\kappa_{5}^{2}}\intd^{4}x\\delta\rho(\epsilon)\left[e^{-3y}{\calK}_{\rho\rho}\partial_{z}\rho\right]_{z=\epsilon}\quad.
\frac{\overline{{w}}_{a+\bar{\epsilon}_{i}b+\bar{\epsilon}_{k}}(z,z^{\prime})}{\overline{{w}}_{ab}(z,z^{\prime})}=\frac{\prod_{m=0}^{i-1}(h(u+(\bar{a}_{m}-\bar{b}_{k})w)\prod_{m=i+1}^{n-1}(h(-u-(\bar{a}_{m}-\bar{b}_{k})w)}{\prod_{m=0}^{k-1}(h(-u-(\bar{a}_{i}-\bar{b}_{m}+1)w)\prod_{m=k+1}^{n-1}(h(u+(\bar{a}_{i}-\bar{b}_{m}+1)w)}