harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
W_{0}=\frac{{\partial}_{2}({\partial}_{1}c/{\partial}_{2}c)\cdot{\partial}_{1}c-{\partial}_{1}({\partial}_{1}c/{\partial}_{2}c)\cdot{\partial}_{2}c}{({\partial}_{1}c)^{2}+({\partial}_{2}c)^{2}}\cdot{\partial}_{2}c,
V(m)=\frac{N}{4\pi}\int(1+\frac{g}{4\pi}\mathrm{ln}(\frac{\Lambda^{2}}{m^{2}}+1))(\mathrm{ln}(\frac{\Lambda^{2}}{m^{2}}+1)-\frac{2}{1+\frac{m^{2}}{\Lambda^{2}}})dm^{2}.
\tilde{\Xi}_{(t=0)}^{a}=\tilde{\xi}^{a}=\xi^{\mu}e_{\mu}{}^{a}\,,\qquad\tilde{\Xi}_{(t=0)}^{\alpha}=\tilde{\epsilon}^{\alpha}=\epsilon^{\dot{\alpha}}{\calK}(x)_{\dot{\alpha}}{}^{\alpha}\,,\qquad\tilde{L}_{(t=0)}^{ab}=\ell^{ab}+\xi^{\mu}\omega_{\mu}^{ab}\,,
V_{vis}(\tilde{q})=\frac{|\frac{\partialW_{vis}}{\partial\tilde{q}}+\frac{\partialf_{vis}}{\partial\tilde{q}}\overline{{F_{\Phi}}}|^{2}}{\frac{\partial^{2}f_{vis}}{\partial\tilde{q}\partial\tilde{q}^{\dagger}}}-6\mathrm{Re}~W_{vis}(\tilde{q})F_{\Phi}-f_{vis}|F_{\Phi}|^{2}.
{\calM}(x^{-},y^{-})=\sum_{a=1}^{N}\psi_{a}^{\dagger}(x^{-},x^{+})\psi_{a}(y^{-},x^{+}),\{\calC}(x^{-},y^{-})=\sum_{a=1}^{N}\psi_{a}^{\dagger}(x^{-},x^{+})\chi_{a}(y^{-},x^{+}),
<x|e^{-\beta(\frac{1}{2}K^{ij}v_{i}v_{j}-J^{i}v_{i})}|y>=\sum_{n=0}^{\infty}<x|e^{\varphi^{i}v_{i}}(-\frac{1}{2}\betaK^{ij}v_{i}v_{j})^{n}|y>/n!
R\equiv\frac{\sigma(e^{+}e^{-}\rightarrowq\bar{q})}{\sigma(e^{+}e^{-}\rightarrow\mu^{+}\mu^{-})}=2\tilde{N}_{c}N_{f}\left(\frac{N_{c}}{\tilde{N}_{c}}\right)^{2}=2N_{c}N_{f}\frac{N_{c}}{N_{f}-N_{c}}
(\partial_{m}^{2}-\partial_{t}^{2})w_{2}={\frac{2\pi^{2}\alpha^{\prime}}{r_{0}}}\delta^{3}(Im(ze^{i\omegat}))\delta(Re(ze^{i\omegat})-r_{0})\epsilon_{ijk}z^{i}dz^{j}\wedgedz^{k}e^{i\omegat},
{v_{4}}\int_{0}^{\infty}{\frac{dt}{2t^{2}}}\left[\frac{f_{2}(e^{-\pi/2t})}{f_{1}(e^{-\pi/2t})}\right]^{8}\left[\prod_{i}v_{i}\tilde{M}_{i}+\sum_{j}v_{j}\tilde{M}_{j}\prod_{i\neqj}{\frac{\tilde{W}_{i}}{v_{i}}}\right]
T_{\pm2-}^{+2}=T_{\pm2+}^{+2}=T_{\pm2-}^{-2}=T_{\pm2+}^{-2}=0,T_{\pm\pm2}^{+}=T_{+-2}^{+}=T_{\pm\pm2}^{-}=T_{-+2}^{-}=0,
S_{\alpha\beta}=\left(\begin{array}{ccccc}{-\frac{1+f^{2}}{1-f^{2}}}&{\frac{2f}{1-f^{2}}}&{}&{}&{}\\{-\frac{2f}{1-f^{2}}}&{\frac{1+f^{2}}{1-f^{2}}}&{}&{}&{}\\{}&{}&{1}&{}&{}\\{}&{}&{}&{\ddots}&{}\\{}&{}&{}&{}&{1}\\\end{array}\right)
(\psi_{i}^{1})_{L}=\frac{g}{f}\phi^{+}(x_{5})(\psi_{i}^{+})_{L}(x^{\mu}),(\psi_{i}^{1})_{R}=\frac{g}{f}\phi^{-}(x_{5})(\psi_{i}^{-})_{R}(x^{\mu}),
\{\omega^{i_{0},\alpha_{0}}\,;\,\omega^{i_{1},\alpha_{1}}\}=\{\omega^{i_{0},0},\omega^{i_{0},1},\ldots,\omega^{i_{0},N_{0}};\omega^{i_{1},0},\omega^{i_{1},1},\ldots,\omega^{i_{1},N_{1}}\}
\mathrm{Vol}=\int_{T^{4}}\Omega_{0}\wedge\bar{\Omega}_{0}=\mathrm{det}\,\tau+\mathrm{det}\,\bar{\tau}-\tau_{11}\bar{\tau}_{22}-\tau_{22}\bar{\tau}_{11}+\tau_{12}\bar{\tau}_{21}+\tau_{21}\bar{\tau}_{12}\,.
A=\left(\begin{array}{ccccc}{0}&{A_{12}}&{A_{13}}&{...}&{A_{1K}}\\{A_{12}^{\dagger}}&{0}&{A_{23}}&{...}&{A_{2K}}\\{...}&{...}&{...}&{...}&{...}\\{A_{1K}^{\dagger}}&{A_{2K}^{\dagger}}&{...}&{A_{K,K-1}^{\dagger}}&{0}\\\end{array}\right)
\tilde{\calL}_{ren}=-Z\varphi_{1}\Box\varphi_{2}+Z(m^{2}+\deltam^{2})\varphi_{1}\varphi_{2}+{\frac{\lambdaZ_{\lambda}}{4}}Z^{2}\varphi_{1}^{2}\varphi_{2}^{2}+J_{1}\varphi_{1}+J_{2}\varphi_{2}+Z_{\alpha}Z^{1/2}K\varphi_{1}^{2},
p(r)=-\rho_{o}\frac{\sqrt{1-(\beta+\frac{\gamma}{R^{3}})r^{2}}-\sqrt{1-(\beta+\frac{\gamma}{R^{3}})R^{2}}}{\sqrt{1-(\beta+\frac{\gamma}{R^{3}})r^{2}}-\omega\sqrt{1-(\beta+\frac{\gamma}{R^{3}})R^{2}}}\,,
{\calF}(\Lambda_{24})=\wedge_{i=1}^{3}d\lambda_{i}\frac{a_{12}a_{15}^{2}a_{25}^{2}a_{35}^{2}a_{36}a_{46}}{a_{45}a_{56}}\prod_{k<l}a_{kl}^{-3}\Theta(\Lambda_{24}),
\left[{\frac{1}{2m}}\hat{p}^{2}+U_{1}(x)\right]\varphi_{1}(x)=E\varphi_{1}(x),\quad\left[{\frac{1}{2m}}\hat{p}^{2}+U_{2}(x)\right]\varphi_{2}(x)=E\varphi_{2}(x).
\frac{\partialS_{\mathrm{\scriptsize~red}}}{\partialu_{n}}=\frac{\partial\tilde{S}_{\mathrm{\scriptsize~red}}}{\partial\xi_{0}}\frac{\partialH}{\partialu_{n}}+\frac{\partial\tilde{S}_{\mathrm{\scriptsize~red}}}{\partialu_{n}}
\prod_{n=1}^{\infty}\left(\alpha_{-n}^{+}\right)^{a_{n}}\prod_{m=1}^{\infty}\left(\alpha_{-m}^{-}\right)^{b_{m}}\prod_{k=1}^{\infty}\left(s_{-k}\right)^{c_{k}}\,|p^{+},p^{-},s_{0}>
y^{2}=x^{2\tilde{n}_{c}}\left[\prod_{k=1}^{n_{c}-\tilde{n}_{c}}(x-\Phi_{k})^{2}+4\Lambda^{2n_{c}-n_{f}}x^{n_{c}-\tilde{n}_{c}}\right]=x^{2{\tilde{n}}_{c}}(x^{n_{c}-{\tilde{n}}_{c}}-\Lambda^{2})^{2}.
\langlei(\bar{\psi}_{1}\gamma_{5}\psi_{1}-\bar{\psi}_{2}\gamma_{5}\psi_{2})\rangle_{A}=\int_{-\infty}^{\infty}d\lambda\rho(\lambda)\Bigl[{\frac{i}{\lambda+ih}}-{\frac{i}{\lambda-ih}}\Bigr].
R_{12}\tilde{A}_{1}R_{21}\tilde{A}_{2}-\tilde{A}_{2}R_{12}\tilde{A}_{1}R_{21}=\kappa(P_{12}\tilde{A}_{1}R_{21}-R_{12}\tilde{A}_{1}P_{12})\;,\;\;[H,\;\tilde{A}_{j}^{i}]=0\;,
\begin{array}{l}{s(a,b;c,d)=\overline{{\psi}}_{a}\psi_{b}\overline{{\psi}}_{c}\psi_{d}}\\{v(a,b;c,d)=\overline{{\psi}}_{a}\gamma_{\mu}\psi_{b}\overline{{\psi}}_{c}\gamma^{\mu}\psi_{d}}\\{t(a,b;c,d)=-\frac{1}{2}\overline{{\psi}}_{a}\gamma_{\mu\nu}\psi_{b}\overline{{\psi}}_{c}\gamma^{\mu\nu}\psi_{d}.}\\\end{array}
{\calL}\,=\,\frac{1}{2}g^{2}L_{0}^{a}L_{0}^{a}-\frac{1}{2}g^{2}L_{1}^{a}L_{1}^{a}+g\theta^{a}\partial_{0}L_{1}^{a}-g\theta^{a}\partial_{1}L_{0}^{a}+g^{2}\theta^{a}f^{abc}L_{0}^{b}L_{1}^{c}\;.
S_{BFV}=\intd^{4}x\biggl({\dot{A}}_{\mu}\pi^{\mu}+i\psi^{\dagger}{\dot{\psi}}+{\dot{C}}{\calP}+{\dot{\bar{C}}}{\bar{\calP}}\biggr)-\intdt\biggl(H_{0}+[Q,\Psi]\biggr)\,.
\phi_{(0)}^{\prime\prime}+3\alpha_{(0)}^{\prime}\phi_{(0)}^{\prime}+\beta_{(0)}^{\prime}\phi_{(0)}^{\prime}-\gamma_{(0)}^{\prime}\phi_{(0)}^{\prime}=-\sqrt{2}b\left(b^{2}-\frac{2}{3}\right)\sigma^{2}e^{2(\gamma_{(0)}-\sqrt{2}b\phi_{(0)})}.
T(x)=\left(\begin{array}{ll}{a(x)}&{b(x)}\\{c(x)}&{d(x)}\\\end{array}\right)=\left(\begin{array}{ll}{a+x^{i}\partial_{i}a}&{b+x^{i}\partial_{i}b}\\{c+x^{i}\partial_{i}c}&{d+x^{i}\partial_{i}d}\\\end{array}\right),~i=1,2.
K_{i}^{j}=\left(\begin{array}{cc}{{-{\frac{1}{2}}}(u\wedge\bar{u}-v\wedge\bar{v})}&{-u\wedge\bar{v}}\\{\bar{u}\wedgev}&{{{\frac{1}{2}}}(u\wedge\bar{u}-v\wedge\bar{v})}\\\end{array}\right)
\begin{array}{rl}{\}&{\Phi(E,x)=\}\\{\}&{E^{2}-\left(\omega^{2}+\frac{\lambda^{2}}{4}+\lambdaE\right)\left(2x+2u-1\right)^{2}+\frac{\lambda^{2}}{4}\left(2x+2u-1\right)^{4}.}\\\end{array}
B_{N}(\lambda)+\frac{N}{2}\frac{\mathrm{d}V^{(\alpha)}(\lambda)}{\mathrm{d}\lambda}=\frac{\lambda}{2c_{N}}A_{N}(\lambda)-\frac{\alpha}{2}(B_{\mathrm{sing}}^{(N-1)}-B_{\mathrm{sing}}^{(N)})\.
\begin{array}{rclrclrcl}{s_{12}^{\pm}\cdots_{12}^{\pm}}&{=}&{-2}&{s_{12}^{-}\cdots_{12}^{+}}&{=}&{-2}&{s_{12}^{\pm}\cdott_{12}}&{=}&{0}\\{s_{12}^{\pm}\cdots_{23}^{\pm}}&{=}&{1}&{s_{12}^{-}\cdots_{23}^{+}}&{=}&{2}&{s_{12}^{\pm}\cdott_{23}}&{=}&{1}\\{s_{12}^{-}\cdots_{31}^{+}}&{=}&{0}&{s_{12}^{\pm}\cdott_{31}}&{=}&{-1}&{t_{ij}\cdott_{kl}}&{=}&{0}\\\end{array}
[V_{A}^{B},\bar{\Omega}](-1)^{\varepsilon_{B}}+[\bar{V}_{A}^{B},\Omega](-1)^{\varepsilon_{B}}+i\hbar\delta_{A}^{B}=i\hbarV_{A}^{C}\bar{V}_{C}^{B}+i\hbar\bar{V}_{A}^{C}V_{C}^{B}.
\beta^{\Lambda}\equiva\frac{\partial\phantom{a}}{\partiala}\phi^{\Lambda}=a\frac{\partialx^{5}}{\partiala}\frac{\partial\phi^{\Lambda}}{\partialx^{5}}=\mp3g\frac{a}{a^{\prime}}g^{\Lambda\Sigma}\partial_{\Sigma}W=-3g^{\Lambda\Sigma}\frac{\partial_{\Sigma}W}{W}\,.
\overline{{\Psi}}_{\kappa}(x)=\Psi_{\kappa}(x)-\int_{0}^{\infty}d\mu\frac{<\Psi_{\mu}|2\kappa^{2}\sigma|\Psi_{\kappa}>}{\kappa^{2}-\mu^{2}}\Psi_{\mu}(x)+O(\sigma^{2})
\Omega=q^{-\sum_{i=1}^{r}\,H^{i}H_{i}-dc-cd-2h_{\rho}}+\sum_{t}\,S(b_{t}^{\prime})\;q^{-\sum_{i=1}^{r}H^{i}H_{i}-dc-cd}\cdota_{t}^{\prime}\;q^{-2h_{\rho}}
\frac{1}{2}(S^{+}+S^{-})=r_{1}+Mr_{2},\qquad\frac{1}{2}(S^{+}-S^{-})=t_{1}+Mt_{2},\qquad\mathrm{where}\qquadM={\calN}^{2}\sum_{i,j}^{N}a_{i}a_{j}.
\begin{array}{l}{\Gamma\left(\partial_{\bar{Z}^{1}}\right)\Psi_{++}=\Gamma\left(\partial_{\bar{Z}^{2}}\right)\Psi_{++}=0,\;\;\Gamma\left(\partial_{Z^{3}}\right)\Psi_{++}=0}\\{\Gamma^{\tau}\Gamma^{5}\Psi_{++}=-i\Psi_{++}}\\\end{array}
{\frac{1}{\omega}}\left(-{\frac{9}{20}}-{\frac{z_{0}^{2}}{l^{2}}}\right)<{\frac{1}{32}}\left(-32\omegaz_{0}^{2}+9l^{2}-3\sqrt{-64\omegaz_{0}^{2}l^{2}+9l^{4}-64l^{2}z_{0}^{2}\omega^{2}}\right)/(\omega^{2}l^{2})\,,
{R^{\lambda}}_{\mu\nu\rho}=\partial_{\rho}{\Gamma_{\mu\nu}}^{\lambda}-\partial_{\nu}{\Gamma_{\mu\rho}}^{\lambda}+{\Gamma_{\mu\nu}}^{\alpha}\star{\Gamma_{\rho\alpha}}^{\lambda}-{\Gamma_{\mu\rho}}^{\alpha}\star{\Gamma_{\nu\alpha}}^{\lambda}.
\tilde{w}={\hat{T}_{k}}w{\hat{T}_{k}}^{-1}=-e^{-\lambda_{0}^{(k)}}+2\lambda_{1}^{(k)}(\hat{L}_{0}-\lambda_{-1}^{(k)}w)-\lambda_{1}^{(k)2}e^{\lambda_{0}^{(k)}}(\hat{L}_{1}+2\lambda_{-1}^{(k)}\hat{L}_{0}-{\lambda_{-1}^{(k)2}}w),
A_{\mathrm{sym}}^{0}=(a_{ij}^{\mathrm{sym}})=\left(\begin{array}{cc}{(\alpha,\alpha)}&{(\alpha,\beta)}\\{(\beta,\alpha)}&{(\beta,\beta)}\\\end{array}\right)=\left(\begin{array}{cc}{2}&{-1}\\{-1}&{2}\\\end{array}\right)
\deltaD_{pk}({\bfr}^{\prime\prime},{\bfr}^{\prime},i\omega)=D_{pj}(\varepsilon,{\bfr}^{\prime\prime},{\bfr}_{2},i\omega)\delta\varepsilon(i|\omega|)D_{jk}(\varepsilon,{\bfr}_{2},{\bfr}^{\prime},i\omega).
S_{4}=(2\pi)^{4}\int\left[\prod_{j=1}^{2}\frac{d^{\,4}P_{j}}{(2\pi)^{4}}\frac{d^{\,4}Q_{j}}{(2\pi)^{4}}\right]G_{4}\,H\,\delta^{4}\left(\sum_{l}P_{l}+\sum_{j=1}^{2}(Q_{j}-P_{j})\right).
\mu^{2}(\varphi,\beta)=\int_{-\infty}^{\infty}{\frac{d\alpha}{2\sqrt{\pi}}}e^{-{\frac{\alpha^{2}}{4}}}V^{(2)}({\frac{\alpha}{2}}\sqrt{J_{1}(g)-I_{1}(Q^{2})}+\varphi)
\begin{array}{lll}{A_{0}^{+3}|h,\ell_{3}^{+},\ell_{3}^{-},u\rangle}&{=}&{\ell_{3}^{+}|h,\ell_{3}^{+},\ell_{3}^{-},u\rangle}\\{\}&{\}\\{A_{0}^{-3}|h,\ell_{3}^{+},\ell_{3}^{-},u\rangle}&{=}&{\ell_{3}^{-}|h,\ell_{3}^{+},\ell_{3}^{-},u\rangle}\\\end{array}
=\int\frac{\mathrm{d}^{D-1}q}{\left(2\pi\right)^{D-1}}\frac{N\left(|\vec{q}|\right)}{2|\vec{q}|}\left.\left[\frac{1}{\left(q+p\right)^{2}}+\frac{1}{\left(q-p\right)^{2}}\right]\right|_{q_{0}=|\vec{q}|}
\left[M^{2}-4(\vec{k}\,^{2}+m^{2})\right]{\mit\Phi}_{\sigma_{2}\sigma_{1}}(\vec{k},\vec{n})=\frac{m^{2}}{2\pi^{3}}\int\sum_{\sigma_{1}^{\prime}\sigma_{2}^{\prime}}K_{\sigma_{2}\sigma_{1}}^{\sigma_{2}^{\prime}\sigma_{1}^{\prime}}(\vec{k},\vec{k}\,^{\prime},\hat{n},M^{2}){\mit\Phi}_{\sigma_{2}^{\prime}\sigma_{1}^{\prime}}(\vec{k}\,^{\prime},\hat{n})\frac{d^{3}k^{\prime}}{\varepsilon_{k^{\prime}}}
\eta_{\nu^{\pm},\lambda}(r,\varphi)=N_{\nu^{\pm},\lambda}\left(\begin{matrix}{\zeta^{\pm1}J_{n-1}(qr)e^{i(n-1)\varphi}}\\{-iJ_{n}(qr)e^{in\varphi}}\\{\pm\zeta^{\pm1}J_{n-1}(qr)e^{i(n-1)\varphi}}\\{\pmiJ_{n}(qr)e^{in\varphi}}\\\end{matrix}\right)
\hat{X}_{a}\hat{W}\equiv\frac{1}{\sqrt{\pi}}e_{a}^{\mu}\frac{\partial}{\partialp_{\nu}}(j_{0}(\frac{1}{2}\bigtriangleup)-ij_{1}(\frac{1}{2}\bigtriangleup))R_{\mu\nu}^{bc}\{\sigma_{bc},\hat{W}\}
\begin{array}{rcl}{S}&{=}&{-T_{M_{2}}l\intd^{2}\xi\sqrt{|{\tilde{g}}_{ij}-\tilde{k}^{2}(2\pi\alpha^{\prime})^{2}\tilde{\calG}^{(1)}{}_{i}\tilde{\calG}^{(1)}{}_{j}|}}\\{}&{}&{}\\{}&{}&{-\frac{T_{M_{2}}l}{2}\intd^{2}\xi\epsilon^{ij}\left[\tilde{B}^{(2)}{}_{ij}+2(2\pi\alpha^{\prime})\tilde{\calG}^{(1)}{}_{i}\tilde{B}^{(1)}{}_{j}\right]\,,}\\\end{array}
\langleT_{\mu}^{\mu}\rangle_{\mathrm{QFT}}=\frac{\hbarH^{3}}{4\pic^{2}}\left(\frac{mc^{2}}{\hbarH}\right)^{2}\left\lbrack(1-6\zeta)-\left(\frac{mc^{2}}{\hbarH}\right)^{2}\right\rbrack^{\frac{1}{2}}\cdot\mathrm{ctg}\:\pi\left\lbrack(1-6\zeta)-\left(\frac{mc^{2}}{\hbarH}\right)^{2}\right\rbrack^{\frac{1}{2}}
\langle\sigma_{n_{1}}...\sigma_{n_{s}}\rangle_{g}=\int_{{\calM}_{g,s}}\inte^{-S}\prod_{k}e^{\alpha_{k}\pi(z_{k})}\prod_{a,{\bar{a}}=1}^{3g-3}G(\chi_{a}){\bar{G}}({\bar{\chi}}_{\bar{a}})\prod_{i=1}^{s}\int_{\Sigma}\sigma_{n_{i}}^{(2)}
\theta_{1}(\tau)=\sum_{n=-\infty}^{\infty}q^{(n+1/2)^{2}/2},\quad\theta_{2}(\tau)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n^{2}/2},\quad\theta_{3}(\tau)=\sum_{n=-\infty}^{\infty}q^{n^{2}/2}.
R_{\sftt}=-\frac{d-1}{a}\left(\frac{d^{2}a}{d{\sft}^{2}}\right)\;,~~~R_{ij}=\left[a\left(\frac{d^{2}a}{d{\sft}^{2}}\right)+(d-2)\left(\frac{da}{d\sft}\right)^{2}+(d-2)k\right]\gamma_{ij}\;.
\Pi_{\mu_{1}\mu_{2}}(k,u)=-\frac{1}{\left(2\pi\right)^{3}}\int\frac{\mathrm{d}^{3}Q}{2|{\bfQ}|}\left\{P_{\alpha\beta}(Q)N(Q_{0})\left[A_{\mu_{1}\mu_{2}}^{\alpha\beta}\left(Q,k,-k\right)+(Q\rightarrow-Q)\right]\right\}_{Q_{0}=|{\bfQ}|}.
\left\langle{\calO}(\omega,k^{2}){\calO}(\omega^{\prime},k^{2})\right\rangle={\frac{\delta^{2}}{\delta\hat{\varphi}_{0}(\omega,k^{2})\delta\hat{\varphi}_{0}(\omega^{\prime},k^{2})}}K_{SG}[\varphi_{0}],
\begin{array}{l}{\qquadA_{1}^{\mu}(\tau,\sigma)=Y^{\mu}(\tau)+\sum_{n\ne0}y_{n}^{\mu}(\tau)e^{i2n\sigma},}\\{\qquadp_{1}^{\mu}(\tau,\sigma)=P_{y}^{\mu}(\tau)+\sum_{n\ne0}p_{n}^{\mu}(\tau)e^{i2n\sigma}.}\\\end{array}
L_{11}^{(\pm)}=q^{\pm\frac{1}{2}}q^{\pmJ}=q^{\pm\left(\begin{array}{cc}{1}&{0}\\{0}&{0}\\\end{array}\right)},\;\;L_{22}^{(\pm)}=q^{\pm\frac{1}{2}}q^{\mpJ}=q^{\pm\left(\begin{array}{cc}{0}&{0}\\{0}&{1}\\\end{array}\right)},\;\;
e_{\alpha}^{--\beta}={\frac{4g}{(1-gr^{2})}}\left(X^{--}\{(1-{\frac{gr^{2}}{2}})^{2}-{\frac{1}{2}}-4g^{2}\rho^{2})\}+\sigma_{3}({\frac{1}{2}}\rho^{--}I-2g\rhoX^{--})+2g\rho\rho^{--}I\right)_{\alpha}^{\beta}\,
\bar{\epsilon}j^{\mu}=\,\mathrm{tr}\left[-\sqrt{2}\bar{\epsilon}\PsiD^{\mu}\phi+i{\frac{1}{\sqrt{2}g}}\epsilon^{\nu\lambda}F_{\nu\lambda}\bar{\epsilon}\gamma^{\mu}\Psi+\sqrt{2}\bar{\epsilon}\gamma_{5}\Psi\epsilon^{\mu\nu}D_{\nu}\phi\right].
A_{\mu}(x)=\left(\begin{matrix}{n_{1}{\calA}_{\mu}^{0}(x)}&{0}&{\ldots}&{0}\\{0}&{n_{2}{\calA}_{\mu}^{0}(x)}&{\ldots}&{0}\\{0}&{\ldots}&{\ldots}&{0}\\{\vdots}&{\vdots}&{\ddots}&{\vdots}\\{0}&{0}&{\ldots}&{n_{N}{\calA}_{\mu}^{0}(x)}\\\end{matrix}\right)
\begin{array}{ll}{{\displaystyleR_{\mathrm{cr}}=2e^{2\gamma}m_{0}^{2}}\,}&{;D=2\,,}\\{{\displaystyleR_{\mathrm{cr}}=\frac{96}{\pi^{2}}m_{0}^{2}}\,}&{;D=3\,,}\\{{\displaystyleR_{\mathrm{cr}}=12m_{0}^{2}}\,}&{;D=4\,,}\\\end{array}
\nabla_{j}=\left(\begin{array}{cc}{\tilde{\nabla}_{j}\otimes{\bf1}_{m\timesm}+ic_{j}}&{0}\\{0}&{\partial_{j}\otimes{\bf1}_{n\timesn}}\\\end{array}\right)\,,\quadX^{I}=\left(\begin{array}{cc}{x^{I}}&{0}\\{0}&{0}\\\end{array}\right)\,.
H_{gonihedric}^{3d}=-2k\sum_{\vec{r},\vec{\alpha}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}}+\frac{k}{2}\sum_{\vec{r},\vec{\alpha},\vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}+\vec{\beta}}-\frac{1-k}{2}\sum_{\vec{r},\vec{\alpha},\vec{\beta}}\sigma_{\vec{r}}\sigma_{\vec{r}+\vec{\alpha}}\sigma_{\vec{r}+\vec{\alpha}+\vec{\beta}}\sigma_{\vec{r}+\vec{\beta}},
B=\frac{(n-1)^{2}}{4\pi^{2}a^{4}}\frac{1}{4}\sum_{l=1}^{\infty}\int_{0}^{x_{0}}x^{2}\frac{3\nu^{9}-42x^{2}\nu^{7}+18x^{4}\nu^{5}-2x^{6}\nu^{3}}{(x^{2}+\nu^{2})^{6}},
V_{0}(F,\phi;F_{T},t)=-F\,\mathrm{ln}\left(\frac{\phi^{(N_{c}-N_{f})}\mathrm{det}(t)}{\Lambda^{(3N_{c}-N_{f})}}\right)-\phi\mathrm{Tr}\left[F_{T}t^{-1}\right]+\mathrm{h.c.}\.
+\frac{1}{2e^{2}\mid\varphi\mid^{2}}\{(\partial_{\nu}\Lambda^{\nu\mu})+\frac{e}{2}(\varphi^{\dagger}\partial_{\mu}\varphi-\partial_{\mu}\varphi^{\dagger}\varphi)\}^{2}+\frac{1}{2}\mid\partial_{\mu}\varphi\mid^{2}+\lambda(\mid\varphi\mid^{2})^{2}].
\langleJ|J\rangle_{SU(2)}=(1_{-}2_{-})^{2J}=e^{-iJ{(\phi_{1}+\phi_{2})}}\frac{(\tau_{1}-\tau_{2})^{2J}}{(1+|\tau_{1}|^{2})^{J}(1+|\tau_{2}|^{2})^{J}}\,.
\left.\nabla_{m}G\right|=e_{m}^{\\mu}\nabla_{\mu}\left.G\right|-\frac{1}{2}\psi_{m}^{Aa}\left.\nabla_{Aa}G\right|-\frac{1}{2}\psi_{m}^{\dot{A}a}\left.\nabla_{\dot{A}a}G\right|
\hat{A}=-\sum_{k=1}^{n}{\frac{\varepsilon_{\mu\nu}(x-a_{k})^{\mu}d(x-a_{k})^{\nu}}{\zeta+(x-a_{k})^{2}}},\quad\hat{F}=-\zeta\sum_{k=1}^{n}{\frac{\varepsilon_{\mu\nu}d(x-a_{k})^{\mu}d(x-a_{k})^{\nu}}{(\zeta+(x-a_{k})^{2})^{2}}}.
S_{\mathrm{YM/\Phi\,}}^{(\mathrm{q})}\toS_{\mathrm{BI\,}}^{(\mathrm{q})}=\frac{M^{4}\,\sqrtV}{2\pi}\,\int_{\Sigma}d^{2}\sigma\,\left[\,\sqrt{\left(\,1+\frac{1}{4g^{2}\,M^{4}\,V}\,\mathrm{det}[\,\gamma_{mn}\,]\,\right)}-1\,\right]
\sigma_{\mathrm{abs}}^{(1,0)}=\frac{{\calA}_{H}}{1+e^{2\alpha_{n}}}\frac{e^{\frac{\omega}{T_{H}}}+1}{(e^{\frac{\omega}{2T_{R}}}+1)(e^{\frac{\omega}{2T_{L}}}-1)}\frac{\omega}{4T_{L}}.
S={\widehatM}_{P}^{D-3}\int_{\Sigma}d^{D-1}x\sqrt{-{\widehatG}}\left[{\widehatR}-{\widehat\Lambda}\right]+M_{P}^{D-2}\intd^{D}x\sqrt{-G}\left\{R+\zeta\left[R^{2}-4R_{MN}^{2}+R_{MNST}^{2}\right]\right\}~.
\begin{array}{rcl}{G_{\sigma}^{(n)}(\zeta^{\prime\prime},\zeta_{2n-1},\epsilonx^{-1}\zeta_{2n-1})^{\varepsilon^{\prime\prime}\,ss^{\prime}}}&{=}&{s^{(1-\epsilon)/2}\delta_{s+s^{\prime},0}G_{\epsilon\sigma}^{(n-1)}(\zeta^{\prime\prime})^{\varepsilon^{\prime\prime}}~~~~(s,s^{\prime}=\pm).}\\\end{array}
\frac{1}{2\pi}c^{(b)}:e^{\mpi2\sqrt{\pi}\sqrt{\frac{\pi}{\pi+gN}}U_{1b}\Phi^{(1)}(x)}:_{M^{(1)}}\;\prod_{I=2}^{N}:e^{\mpi2\sqrt{\pi}U_{Ib}\Phi^{(I)}(x)}:_{M^{(I)}}e^{\pmi\frac{\theta}{N}}\;,
D\equiv\mathrm{{det}}\left(\begin{matrix}{-k^{2}}&{\omegak_{x}}&{\omegak_{y}}&{\omegak_{z}}\\{\omegak_{x}}&{k_{y}^{2}+k_{z}^{2}-\omega^{2}}&{-k_{x}k_{y}}&{-k_{x}k_{z}}\\{\omegak_{y}}&{-k_{x}k_{y}}&{k_{x}^{2}+k_{z}^{2}-\omega^{2}}&{-k_{y}k_{z}}\\{\omegak_{z}}&{-k_{x}k_{z}}&{-k_{y}k_{z}}&{k_{x}^{2}+k_{y}^{2}-\omega^{2}+{\frac{1}{\alpha}}}\\\end{matrix}\right)=0.
{\Upsilon}(s)=\sum_{\lambda}\int{\frac{d^{-\epsilon}k_{\omega}}{(2\pi)^{-\epsilon}}}\left({\frac{k_{\omega}^{2}+\lambda}{\mu^{2}}}\right)^{-s}\nonumber\,=(1/4\pi)^{-\epsilon/2}{\frac{\Gamma(s+\epsilon/2)}{\Gamma(s)}}\zeta_{1}(s+\epsilon/2),
{\frac{\sqrt{2}\vartheta_{2}^{2}\widetilde\vartheta_{3}}{\vartheta_{3}}}+\sqrt{\vartheta_{3}^{2}\widetilde\vartheta_{3}^{2}+\vartheta_{2}^{2}\widetilde\vartheta_{4}^{2}-\vartheta_{4}^{2}\widetilde\vartheta_{2}^{2}}=0,
\left[\mu\frac{\partial}{\partial\mu}~+~\sum_{\kappa=0}^{\infty}\tilde{\beta}_{(\kappa)}(g_{\alpha})\frac{\partial}{\partialg_{\kappa}}~+~\tilde{\gamma}_{m}(g_{\alpha})m\frac{\partial}{\partialm}~+~\frac{n}{2}\tilde{\gamma}(g_{\alpha})\right]G^{(n)}(p,m,\mu,g_{\beta})~=~0
{\calL}=2\sqrt{-1}\intd\theta^{+}d\overline{{\theta}}^{+}\Phi^{\overline{{i}}}\partial_{-}\Phi^{i}-\intd\theta^{+}d\overline{{\theta}}^{+}\Lambda^{k}\Lambda^{\overline{{k}}}+\intd\theta^{+}F_{k}\Lambda^{k}+\intd\overline{{\theta}}^{+}F_{\overline{{k}}}\Lambda^{\overline{{k}}},
\Lambda_{ij}(\xi)=4\zeta_{ij}-{\frac{2}{3}}\bar{\theta}\Gamma_{i}\Gamma_{j}\Gamma^{\sigma}\partial_{\sigma}\zeta+{\frac{1}{4}}\bar{\theta}\Gamma_{\nu}\Gamma_{ij}\theta\partial_{\sigma}\partial^{\sigma}\zeta^{\nu}+{\frac{1}{3}}\eta_{ij}\partial_{\mu}\zeta^{\mu}.
\mathrm{det}\left[{{\calY}^{\sigma}}_{\alpha}\right]=\frac{(j^{2})^{2}}{({\tilde{j}}^{2})^{2}}\,\mathrm{det}\left[{{\calY}^{\sigma}}_{\alpha}\right]\,=\frac{({\calJ}^{2})^{2}}{({\tilde{j}}^{2})^{2}}\,\mathrm{det}\left[{{\calY}^{\sigma}}_{\alpha}\right]\,=a\frac{({\calX}^{2})^{4}}{({\tilde{j}}^{2})^{2}}.
\langleN_{\mathrm{bag}}\rangle(\beta,R,\mu;\theta)={\displaystyle-\frac{\partialG}{\partial\mu}(\beta,R,\mu;\theta)}={\displaystyle\frac1\beta}Tr\{-i\left(\gamma^{0}\otimes{\calI}_{I}\right)k(t,x;t^{\prime},x^{\prime})\},
\tilde{\calH}={\calH}_{T}+\frac{1}{2}m^{2}(\partial_{i}\theta)^{2}+\frac{1}{2m^{2}}\pi_{\theta}^{2}+\partial_{i}^{2}\theta\tilde{\Omega}_{1}-\frac{1}{m^{2}}\pi_{\theta}\tilde{\Omega}_{2}.
{\frac{1}{2}}\left\{\left[\begin{array}{c}{Q_{\alpha}}\\{\tilde{Q}_{\alpha}}\\\end{array}\right],\left[Q_{\beta}\\tilde{Q}_{\beta}\right]\right\}=\left[\begin{array}{cc}{\delta_{\alpha\beta}}&{0}\\{0}&{\delta_{\alpha\beta}}\\\end{array}\right]M+\left[\begin{array}{cc}{0}&{(\Gamma^{0}Z)_{\alpha\beta}}\\{-(\Gamma^{0}Z^{\dagger})_{\alpha\beta}}&{0}\\\end{array}\right]\,
\Gamma_{\alpha\beta\gamma}(p_{1},p_{2},p_{3})\equiv\Bigl(\bbox{e}_{\alpha}(p_{1}){\otimes}\bbox{e}_{\beta}(p_{2}){\otimes}\bbox{e}_{\gamma}(p_{3})\Bigr)\bullet\Gamma(p_{1},p_{2},p_{3})\qquad(\alpha,\beta,\gamma=R,A)\,.
\langle\psi({\bfp}_{1})\phi^{+}({\bfp}_{2})\rangle=\overline{{\langle\phi({\bfp}_{2})\psi^{+}({\bfp}_{1})\rangle}}=\frac{1}{z_{1}-z_{2}}\,f_{2}\!\left(|{\bfp}_{1}||{\bfp}_{2}|\frac{(z_{1}-z_{2})(\bar{z}_{1}-\bar{z}_{2})}{(1+z_{1}\bar{z}_{1})(1+z_{2}\bar{z}_{2})}\right)
\frac{\partial^{r-1}}{\partialx_{i_{1}}^{r-1}}\frac{\partial^{r-2}}{\partialx_{i_{2}}^{r-2}}\dots\frac{\partial}{\partialx_{i_{r-1}}}f(x_{1},x_{2},\dots)\biggr|_{x_{i_{1}}=x_{i_{2}}=\dots=x_{i_{r-1}}=x_{i_{r}}=0}=0.
T_{AB}=T_{\underline{{A}}\underline{{B}}}-\frac{1}{2}\varepsilon_{AB}T_{\\C}^{C},\,\,\,T_{\dot{A}\dot{B}}=T_{\underline{{\dot{A}}}\underline{{\dot{B}}}}-\frac{1}{2}\varepsilon_{\dot{A}\dot{B}}T_{\\\dot{C}}^{\dot{C}}
0=Q_{B}(|\mathcal{T}\rangle+|\mathcal{T}_{\Phi}\rangle)=|\mathcal{T}_{\mathcal{B}}\rangle+|\mathcal{T}_{(Q_{B}+\tilde{Q}_{B})\Phi}\rangle=|\mathcal{T}_{\mathcal{B}+(Q_{B}+\tilde{Q}_{B})\Phi}\rangle.
\int_{0}^{1}\!\mathrm{d}t\,{\calL}_{\mathrm{ghost}}=\int_{0}^{1}\!\mathrm{d}t\,\frac{1}{2}g(x(t))_{\mu\nu}\big(a(t)^{\mu}a(t)^{\nu}+2\,b(t)^{\mu}c(t)^{\nu}\big)\,,
e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}\left.\frac{\partial\Gamma_{R}}{\partialg^{\mu\nu}}\right|_{b_{\mu\nu}}={\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\frac{\partial\Gamma_{R}}{\partial{\bar{g}}^{ij}},
I_{E}^{\mathrm{bulk}}=-\frac{1}{16\piG}\int_{\cal{M}}d^{D}x\sqrt{g}\Big(R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2(p+2)!}e^{a\phi}F_{p+2}^{2}\Big)
V_{0}(B)={\frac{1}{2}}\mathrm{Tr}(M_{1}^{2}+M_{2}^{2})+{\frac{ge^{B}}{N}}\mathrm{Tr}M_{1}^{4}+{\frac{ge^{-B}}{N}}\mathrm{Tr}M_{2}^{4}
\frac{1}{2}\delta\Lambda_{I}=-\gamma^{\mu}\epsilon_{I}\partial_{\mu}\phi+\frac{1}{\sqrt2}\sigma^{\mu\nu}\left(e^{-\phi}F_{\mu\nu}\alpha_{IJ}-e^{\phi}\tilde{G}_{\mu\nu}\beta_{IJ}\right)^{-}\epsilon^{J}=0\,
{\hat{F}}[P^{3}]~=~{\alpha}^{-1}~~\sum_{m,\ell,R}{\frac{(-)^{6\epsilon_{\ell}-3\epsilon_{R}}{S_{0\Lambda_{\ell}}~S_{0\Lambda_{R}}~S_{\Lambda_{\ell}\Lambda_{m}}~S_{\Lambda_{\ell}\Lambda_{m}}~~S_{\Lambda_{R}\Lambda_{m}}^{*}~q^{-5C_{\ell}+{\frac{3}{2}}C_{R}}}}{S_{00}~S_{0\Lambda_{m}}}}~.
\partial_{\bar{\imath}}F_{g}~=~\sum_{a,b=1}^{3g-3}\int_{M_{g}}(-)^{(a+b)}\langle\prod_{c\nea,d\neb}(\mu_{c}G^{-})(\bar{\mu}_{d}{\bar{G}}^{-})(\mu_{a}T)(\bar{\mu}_{b}\bar{T})\intd^{2}z\Psi_{\bar{\imath}}(z,\bar{z})\rangle\.
R[f(z),g(z)]=\pii\left[\sum_{k}{\mathrm{Res}}_{z=z_{g,k}}g(z)+\sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_{f,k}>0}f(z)-\sum_{k}{\mathrm{Res}}_{{\mathrm{Im}}z_{f,k}<0}f(z)\right].
\dot{\epsilon}\approx{\frac{|p-7|}{16}}Q_{p}^{\frac{1}{p-7}}\sqrt{E^{2}-1}\epsilon^{\frac{8-p}{7-p}}\\\\\Longrightarrow\\\\\epsilon\approx4^{7-p}Q_{p}(E^{2}-1)^{\frac{p-7}{2}}\eta^{p-7},
=-i\sum_{k=1}^{n}\{f({\bf{x}}_{k})a(\sigma_{k})\nabla_{s}^{k}+\frac{1}{2}[\nabla_{s}^{k}f({\bf{x}}_{k})]a(\sigma_{k})\}\varphi_{n}({\bf{x}}_{1}\sigma_{1},{\bf{x}}_{2}\sigma_{2},...,{\bf{x}}_{n}\sigma_{n})
\phi_{n,m}(z,\bar{z})={\frac{1}{\sqrt2}}\left[{\frac{f_{n,m}(z)}{\overline{{f}}_{n,m}(z)}}+{\frac{\overline{f}_{n,m}(z)}{{f}_{n,m}(z)}}\right],\qquad\qquad\;({n,m})\in{\bfZ}^{2},
\hat{r}_{s}^{\mu\nu}(k)=\frac{i}{2\pi}(k^{\mu}k^{\nu}-k^{2}g^{\mu\nu})\int_{-\infty}^{+\infty}dt\frac{t^{2}B_{1}(t^{2}k^{2})}{(t-i0)^{2}(1-t+i0)},
\begin{array}{rcl}{\delta_{\epsilon}\,q^{n}}&{=}&{\left\{q^{n}\,,\,\phi_{\epsilon}(q,p)\,\right\}\\\,}\\{\delta_{\epsilon}\,p_{n}}&{=}&{\left\{p_{n}\,,\,\phi_{\epsilon}(q,p)\,\right\}\\\,}\\{\delta_{\epsilon}\,\lambda^{\alpha}}&{=}&{\dot{\epsilon}^{\alpha}\,+\,\lambda^{\gamma}\epsilon^{\beta}\,{C_{\beta\gamma}}^{\alpha}\,-\,\epsilon^{\beta}\,{C_{\beta}}^{\alpha}\\\,}\\\end{array}
V=q\left({{\calA}_{0}\left({\bfy}\right)-{\calA}_{0}\left({{\bfy}^{\prime}}\right)}\right)=-\frac{{q^{2}}}{\pi}\frac{1}{{1+e^{2}a\left({1-12\alpha^{2}g^{2}\partial^{2}}\right)}}K_{0}\left({\chi|{\bfy}-{\bfy}^{\prime}|}\right).
\langle{\calO}\rangle_{A}=\langle{\calO}\rangle_{L}+\int_{0}^{1}d\kappa\intD\phi\biggl(\tilde{\delta}_{1}[\phi]+\kappa\tilde{\delta}_{2}[\phi]\biggr)\Theta^{\prime}[\phi]{\frac{\delta{\calO}}{\delta\phi}}e^{iS_{\mathrm{eff}}^{M}}.
V_{ferm.}^{(1)}=\frac{1}{4\pi^{2}}\sum_{l=1}^{\infty}(-1)^{l+1}\int_{0}^{\infty}\frac{ds}{s^{3}}exp(\frac{-\beta^{2}l^{2}}{4s}-m^{2}s)(eHs)coth(eHs)cosh(\betal\mu),
D_{\mu\nu}(x-x^{\prime})=\int\frac{d^{4}q}{(2\pi)^{4}}\frac{\mathrm{e}^{iq\cdot(x-x^{\prime})}}{q^{2}-i\epsilon}\left(g_{\mu\nu}-(1-\xi)\frac{q_{\mu}q_{\nu}}{q^{2}}\right).
\begin{array}{lcl}{T_{\scriptscriptstyle}}&{=}&{\frac{1}{4}\left(T_{1}+T_{10}+T_{19}+T_{28}+T_{37}+T_{46}+T_{55}+T_{64}\right.}\\{}&{}&{\left.+i\left(T_{65}+T_{74}+T_{83}+T_{92}+T_{101}+T_{110}+T_{119}-T_{128}\right)\right).}\\\end{array}
\chi_{1,1}^{NS}(q)=q^{-7/240}(1+q^{\frac{3}{2}}+q^{2}+q^{\frac{5}{2}}+q^{3}+2q^{\frac{7}{2}}+2q^{4}+2q^{\frac{9}{2}}+{\calO}(q^{5}))