harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S(e_{i})=\xi_{i}^{j}e_{j}\,,\\\overline{{S}}(e_{i})=\sigma_{i}^{j}e_{j}\,,\\S(e^{j})=\tilde{\sigma}_{i}^{j}e^{i}\,,\\\overline{{S}}(e^{j})=\tilde{\xi}_{i}^{j}e^{i}\,,
H^{n}({\calH})^{L_{0}}=\left(\bigoplus_{n=N^{c}}H^{0}\left({\calF}(\alpha_{-m}^{0},b_{-m};k^{0})\right)\otimesH^{N^{c}}\left({\calF}(c_{-m}^{})\right)\otimes{\calH}_{K}\right)^{L_{0}},
\left(\begin{matrix}{V^{0}}\\{V^{1}}\\{V^{2}}\\{V^{3}}\\\end{matrix}\right)\,=\,\left(\begin{matrix}{{\itdt}\,{e^{a}}}\\{{\itdw}\,{e^{a}}}\\{-{\itda}}\\{{\itdx}_{\perp}\,{e^{a}}}\\\end{matrix}\right)
\begin{array}{c}{B_{i_{1}\ldotsi_{N_{f}-N_{c}}}M_{j}^{i_{1}}=0,\;\;\bar{B}^{i_{1}\ldotsi_{N_{f}-N_{c}}}M_{i_{1}}^{j}=0,\;}\\{M_{j_{1}}^{i_{1}}\ldotsM_{j_{N_{c}}}^{i_{N_{c}}}\epsilon_{i_{1}\ldotsi_{N_{f}}}\epsilon^{j_{1}\ldotsj_{N_{f}}}=B_{i_{N_{c}+1}\ldotsi_{N_{f}}}\bar{B}^{i_{N_{c}+1}\ldotsi_{N_{f}}}.}\\\end{array}
\begin{array}{ll}{S_{lp}=}&{\int_{0}^{\beta}d\tau\{\bar{\xi}\frac{\partial\xi}{\partial\tau}-H^{(l)}(\bar{\xi},\xi)+\mu\bar{\xi}\xi\}+\int_{0}^{\beta}d\tau\frac{2\piipr}{n\beta}\bar{\xi}\xi}\\{}&{=S_{l}+\deltaS_{p}}\\\end{array}
Q^{(1)}=\sum_{\sigma}\int\frac{d^{3}p}{(2\pi)^{3}}\,\left[a_{\sigma}^{\dagger}(\vec{p})a_{\sigma}(\vec{p})-b_{\sigma}(\vec{p})b_{\sigma}^{\dagger}(\vec{p})+c_{\sigma}^{\dagger}(\vec{p})c_{\sigma}(\vec{p})-d_{\sigma}(\vec{p})d_{\sigma}^{\dagger}(\vec{p})\right]\quad,
H_{nmy}\equiv\langle1|\langle2|\langle3|H\rangle=v(\mu,y)\frac{1}{4}\frac{\alpha}{2}\left(\frac{\omega_{1m}}{\mu\alpha_{1}}+\frac{\omega_{3n}}{\mu\alpha_{3}}\right)\left[2\left(\overline{{N}}_{mn}^{13}\right)^{2}-2\left(\overline{{N}}_{-m,-n}^{13})\right)^{2}\right].
S_{V}(\beta)\equiv\sigma_{T}(\beta)+\sigma_{R}(\beta)=\frac{\Gamma\left(\frac{1}{2}+\frac{\beta}{2i\pi}\right)\Gamma\left(-\frac{\beta}{2i\pi}\right)}{\Gamma\left(\frac{1}{2}-\frac{\beta}{2i\pi}\right)\Gamma\left(\frac{\beta}{2i\pi}\right)},
\frac{\partial^{2}F}{(\partialx^{1})^{2}}=\phi_{11}^{3}+\sum_{j=2}^{3}[2\phi_{1j}^{3}f_{1}^{j}+\phi_{j}^{3}f_{11}^{j}+\sum_{k=2}^{3}(\phi_{kj}^{3}f_{1}^{k})f_{1}^{j}]
\zeta_{\hat{H}}(s)=(\lambda_{0}\kappa^{2})^{-\frac{s}{2}}\frac{2\pi}{\Gamma(\frac{s}{2})}\sum_{n=1}^{\infty}(2n)^{\frac{1-s}{2}}\int_{0}^{\infty}\frac{t^{\frac{s}{2}-1}}{e^{t}-1}J_{\frac{s-1}{2}}(nt)dt+(\lambda_{0}\kappa^{2})^{-\frac{s}{2}}\zeta(s),
\overline{{s}}=\left(\frac12+\frac12\sqrt{1+4\left(\frac{4\overline{{q}}^{2}}{3}\right)^{3}}\right)^{\frac13}+\left(\frac12-\frac12\sqrt{1+4\left(\frac{4\overline{{q}}^{2}}{3}\right)^{3}}\right)^{\frac13},
+\frac{1}{2}\{(\partial_{\mu}\varphi_{1})^{2}+(\partial_{\mu}\varphi_{2})^{2}\}+\lambda\{\varphi_{c}^{4}+6\varphi_{c}^{2}(\varphi_{1}^{2}+\frac{2}{3}\varphi_{c}\varphi_{1})+2\varphi_{c}^{2}\varphi_{2}^{2}+\cdots\},
\psi(\alpha,\beta,\beta^{\prime},s,s^{\prime})=\frac{\varphi_{(\alpha+\beta+\beta^{\prime},s+s^{\prime})}\varphi_{(\alpha+\beta-\beta^{\prime},s-s^{\prime})}}{\varphi_{(-\alpha+\beta+\beta^{\prime},-s+s^{\prime})}\varphi_{(\alpha-\beta+\beta^{\prime},-s-s^{\prime})}}.
n\:C\;\alpha_{1}^{1-p-\gamma}\:\beta_{1}^{1-q}\;\alpha_{i_{0}}^{\gamma}\left(\prod_{j=2}^{p}\alpha_{1}^{a_{j}-\alpha_{j}}\:\alpha_{j}^{-a_{j}}\right)\left(\prod_{k=2}^{q}\beta_{1}^{b_{k}-\beta_{k}}\:\beta_{k}^{-b_{k}}\right)\;,
g_{\mu\nu}=\left(\begin{array}{cccc}{-N^{2}}&{}&{0}&{}\\{}&{}&{}&{}\\{0}&{}&{\frac{1}{8}e^{2\alpha}\chi_{iAB}\chi_{j}^{AB}}&{}\\{}&{}&{}&{}\\\end{array}\right)=\left(\begin{array}{cccc}{-N^{2}}&{}&{0}&{}\\{}&{}&{}&{}\\{0}&{}&{e^{2\alpha}\chi_{i}^{I}\chi_{Ij}}&{}\\{}&{}&{}&{}\\\end{array}\right),
D_{\mu}\deltaq^{i}\equiv{\calD}_{\mu}\deltaq^{i}-{(j_{u})_{\mu}}^{\nu}{\calD}_{\nu}\deltaq^{j}{(J_{u})_{j}}^{i}-{(j_{u})_{\mu}}^{\nu}\partial_{\nu}q^{j}{\calD}_{k}{(J_{u})_{j}}^{i}\deltaq^{k}=0.
\renewcommand{\arraystretch}{1.2}\begin{array}{rclrcl}{c_{[ij}^{m}c_{k]m}^{l}}&{=}&{0,}&{\frac{1}{2}f_{m}^{[ij}f_{l}^{k]m}+c_{lm}^{[i}\varphi^{jk]m}}&{=}&{0,}\\{c_{m[i}^{[k}f_{j]}^{l]m}}&{=}&{0,}&{f_{m}^{[ij}\varphi^{kl]m}}&{=}&{0.}\\\end{array}
{\calE}_{i}=-2\alpha_{2}\left(\widetilde{\nabla}_{a}\;\widetilde{\nabla}_{b}K^{ab}{}_{i}+K^{ac}{}_{j}K_{cb}^{j}K_{a}{}^{b}{}_{i}-{\frac{1}{2}}K_{ab}{}^{j}K^{ab}{}_{j}K_{i}\right)\,.
f_{I_{2}(m)}^{N}(q,\mu)=\sum_{j=1}^{m}g_{|j|}\,g_{|j+N|}\,x_{|j+N|}(\alpha_{j+N}\cdotq,\alpha_{j+N}^{\vee}\!\!\cdot\mu)\,x_{|j|}(\alpha_{j}\cdotq,-\alpha_{j+2N}^{\vee}\!\!\cdot\mu).
e^{iA}=\frac{\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}+\frac{1}{2}]\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}+\frac{1}{2}]}{\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}+1]\Gamma[\frac{ib^{2}}{4v\pi\alpha^{\prime}}]}.
\langleU|S_{\mu}^{\\nu}(\phi)|U\rangle\sim\frac{(E^{2}-4d^{2}M^{2})}{32M^{4}f^{2}}\left(\begin{matrix}{-1}&{0}&{0}&{0}\\{0}&{1/3}&{0}&{0}\\{0}&{0}&{1/3}&{0}\\{0}&{0}&{0}&{1/3}\\\end{matrix}\right)\,
X(z,\bar{z})=\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{k}\left(\frac{\alpha_{k-\vartheta}}{k-\vartheta}z^{-k+\vartheta}+\frac{\tilde{\alpha}_{k+\vartheta}}{k+\vartheta}\bar{z}^{-k-\vartheta}\right),
\begin{array}{lll}{\mathrm{Matrix}}&{}&{\mathrm{QFT}}\\{x_{ij}}&{\Longrightarrow}&{q\quad(\mathrm{external~momentum})}\\{x_{jk}}&{\Longrightarrow}&{p\quad(\mathrm{loop~momentum})}\\{x_{ik}}&{\Longrightarrow}&{q+p}\\{\displaystyle\sum_{k}\cdots}&{\Longrightarrow}&{\rho\displaystyle\intd^{d}p\cdots}\\\end{array}
\mathfrak{s}_{W}^{*}=\mathfrak{s}_{W^{^{\prime}}}\quad\longleftrightarrow\,\,\,\,\,\,\,j_{W}=j_{W^{^{\prime}\,}}\,,\,\,\,\,\delta_{W}^{-1}=\delta_{W^{^{\prime}}}
\hat{T}^{\mu\nu}\equiv{\frac{2}{\sqrt{-\hat{g}}}}{\frac{\delta(\sqrt{-\hat{g}}{\calL}_{\mathrm{mat}})}{\delta\hat{g}_{\mu\nu}}}=\left(\hat{\rho}c_{\mathrm{ph}}^{2}+\hat{p}\right)\hat{U}^{\mu}\hat{U}^{\nu}+\hat{p}\hat{g}^{\mu\nu},
e_{0}\q^{\mu}\Gamma_{\mu}^{(3)}(p,P;q,Q)-\sigmaQ\2M\Gamma^{(4)}(p,P;q,Q,k,k^{\prime})=e_{0}^{2}\left[\frac{1}{G^{(2)}(p^{\prime},P^{\prime})}-\frac{1}{G^{(2)}(p,P)}\right]\.
G_{a}^{(n)}=\sum_{m=0}^{n}\bigl\{T_{a}^{(n-m)},\,A^{(m)}\bigr\}_{(q,p)}+\sum_{m=0}^{n-2}\bigl\{T_{a}^{(n-m)},\,A^{(m+2)}\bigr\}_{(\eta)}+\bigl\{T_{a}^{(n+1)},\,A^{(1)}\bigr\}_{(\eta)}
\Gamma_{\mu_{1}\cdots\mu_{n}}(k_{1},\cdots,k_{n},u)=-\frac{1}{\left(2\pi\right)^{3}}\int\frac{\mathrm{d}^{3}Q}{2|{\bfQ}|}\left\{P_{\alpha\beta}(Q)N\left(Q_{0}\right)\left[A_{\mu_{1}\cdots\mu_{n}}^{\alpha\beta}\left(Q,k_{1},\cdots,k_{n}\right)+(Q\rightarrow-Q)\right]\right\}_{Q_{0}=|{\bfQ}|},
\psi(\rho)=N^{-1/2}(n)(\beta/2\pi)^{-n/4}(2\pi\hbar)^{-n}(1-\beta\rho^{2}/2)^{-1/2-\sqrt{1/4+n/8}}\intd^{n}x~e^{-ix\cdot\rho/\hbar}\psi(x)
{\calM}^{a\,\mu\nu}=\frac{1}{2}\left[{\calP}^{a\mu}X^{\nu}+2\beta\sqrt{-\gamma}\left(K^{ab\,i}-\gamma^{ab}K^{i}\right)n^{\mu}{}_{i}e^{\nu}{}_{b}-(\mu\leftrightarrow\nu)\right]\,.
{\mathcalP}_{n}\,{\mathcalQ}_{(0)}^{0}\,{^{[0]}W}=-\left[\left(\partial_{0}{\mathcalP}_{n}\right)+\sum_{k=0}^{n}{\mathcalQ}_{(k)}^{n}{\mathcalP}_{k}\right]\,{^{[0]}W}\,,\quad,n\geq1.
\varphi_{n_{1}0}=\frac{1}{\pi\sqrt{\hbar}}\frac{1}{\sqrt{n_{1}!}}\Bigl(\frac{2m\omega}{\hbar}\Bigr)^{n_{1}/2}\Bigl(q-i\frac{p}{m\omega}\Bigr)^{n_{1}}exp\Bigl(-\frac{2H(p,q)}{\hbar\omega}\Bigr)
\mathrm{e}^{\phi_{1}}={\frac{\partial\bar{\omega}\bar{\partial}\omega}{(1+\omega\bar{\omega})^{2}}},\quad\mathrm{and}\quad\mathrm{e}^{\phi_{2}}={\frac{\partial\omega\bar{\partial}\bar{\omega}}{(1+\omega\bar{\omega})^{2}}}.
g=\left(\begin{array}{cc}{e^{i\phi/2}}&{0}\\{0}&{e^{-i\phi/2}}\\\end{array}\right)\,\frac{1}{\sqrt{1+|\tau|^{2}}}\,\left(\begin{array}{rr}{1}&{-\bar{\tau}}\\{\tau}&{1}\\\end{array}\right)\,.
\beta_{ln}=\left(\frac{\omega_{ln}^{\mathrm{out}}}{\omega_{ln}^{\mathrm{in}}}\right)^{1/2}\frac{\Gamma(1-\imath\omega_{ln}^{\mathrm{in}}t_{0})\Gamma(\imath\omega_{ln}^{\mathrm{out}}t_{0})}{\Gamma(\imath\omega_{ln}^{-}t_{0})\Gamma(1+\imath\omega_{ln}^{-}t_{0})}.
M=a\left\{1-\frac{a}{L}+\frac{1}{3}\left(\frac{a}{L}\right)^{2}-\left(1-\frac{a}{L}\right)^{2}\frac{\int_{0}^{a}|u_{0}(r)|^{2}dr}{a|u_{0}(a)|^{2}}\right\}.
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\cdot2!}}e^{2a\phi}F_{2}^{2}\right],
S_{1}=\intd\tau\left(\begin{array}{c}{p_{1}\cdot\dot{x}_{1}-p_{1}\cdot\lambda_{2}\,A_{1}}\\{+\bar{\theta}_{1}\gamma^{\mu\nu}\dot{\theta}_{1}\,p_{1\mu}\lambda_{2\nu}-\frac12e_{1}p_{1}^{2}}\\\end{array}\right).
a\left(\eta\right)=\sqrt{\frac{\omegaM}{2}}\,\Big[c_{2}~\mathrm{cosh}\left(2\,\eta\right)-1\Big]^{\frac{1}{2}},~~~~\mathrm{with}~~c_{2}=\sqrt{1+{\frac{3Q^{2}}{4M^{2}}}}\,.
R_{\phantom{uv}{st}}^{ij}{\calU}_{i}^{AI}{\calU}_{j}^{BJ}=\Omega_{st}^{u}{\frac{i}{2}}(\varepsilon^{-1}\sigma_{u})^{AB}{\calC}^{IJ}+{{\calR}^{IJ}}_{st}\varepsilon^{AB},
L^{ab}=\frac12\Omega^{ab}-\deltaL^{ab}\qquadwith\qquad\;\deltaL^{ab}=\frac1{2(k+1)}\left(\begin{array}{cccc}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{-1}&{0}\\{0}&{0}&{0}&{0}\\\end{array}\right)
H^{2}=g{\frac{2\sum_{i,j}V_{i,j}+\rho_{f}-p_{f}}{1-g\sum_{i}(\psi_{i}^{\prime})^{2}-g[n_{f}^{2}(\theta_{f}^{\prime})^{2}/\rho_{f}+p_{f}]}},\\{\frac{\ddot{a}}{a}}=-2H^{2}+3g(2\sum_{i,j}V_{i,j}+\rho_{f}-p_{f}),
{\calL}_{\mathrm{eff}}=\frac{1}{8\pi}\,\mathrm{Im}\!\int\!\!\mathrm{d}^{2}\theta\,W_{a}W_{D}^{a}+\frac{1}{4\pi}\,\mathrm{Im}\!\int\!\!\mathrm{d}^{2}\theta\mathrm{d}^{2}\bar{\theta}\,\bar{A}_{a}A_{D}^{a}
\frac{\delta\Gamma}{\delta<\zeta>}<\zeta_{,\theta}>+\frac{\delta\Gamma}{\delta\Phi_{i}}\Phi_{i,\theta}=Z_{\theta}^{-1}\left[\frac{\delta\Gamma^{R}}{\delta\Phi_{i}^{R}}(\Phi_{i,\theta})^{R}+\frac{\delta\Gamma^{R}}{\delta<\zeta^{R}>}<\zeta_{,\theta}>^{R}\right]
(u_{\sigma}\,p_{*}u_{0}-u_{0}\,p_{*}u_{b})\bigg|_{-\infty}^{\infty}=(\sigma_{0}^{2}-\sigma_{b}^{2})\int_{-\infty}^{\infty}\frac{{\calE}}{{\calA}}\,u_{b}\,u_{0}\,dr^{*}
S_{1}=\intdx\left(C^{\alphaa}R_{\alpha}^{i}\varphi_{ia}^{*}+\varepsilon^{ab}C_{\alphab|a}^{*}B^{\alpha}+B_{\alpha}^{i}\bar{\varphi_{i}}(-1)^{\varepsilon_{i}+\varepsilon_{\alpha}}\right)
\frac{B_{s,N}(\kappa)}{A_{s}(\kappa)}=\left[\frac{2}{N}\right]^{2s-1}\,\frac{\Gamma\Bigl(s+\frac{iN\kappa}{4}\Bigr)\Gamma\Bigl(s-\frac{iN\kappa}{4}\Bigr)}{\Gamma\Bigl(s+\frac{i\kappa}{2}\Bigr)\Gamma\Bigl(s-\frac{i\kappa}{2}\Bigr)}.
E_{BPS}^{SYM}=\frac{\sqrt{\mathrm{det}G}\pi^{2}}{g_{YM}^{2}(1+\mathrm{Pf}\theta)}|\mathrm{tr}(\tilde{\theta}G^{-1}\tilde{\theta}G^{-1})|+\frac{(2\pi)^{2}}{g_{YM}^{2}(1+\mathrm{Pf}\theta)}
(\dot{T}^{M})^{\dot{A}\dot{B}}(\Gamma^{A})^{\dot{B}\ddot{A}}-(\Gamma^{A})^{\dot{A}\ddot{B}}(\ddot{T}^{M})^{\ddot{B}\ddot{A}}=(T^{M})^{AB}(\Gamma^{B})^{\dot{A}\ddot{A}}.
r^{\prime}=r\\,\\p_{r}^{\prime}=p_{r}\\;\\\theta^{\prime}=\theta+\alpha\\,\\p_{\theta}^{\prime}=p_{\theta}\\;\\z^{\prime}=z+\frac{1}{g}\alpha\\,\\p_{z}^{\prime}=p_{z}\\;\\\xi^{\prime}=\xi+\frac{1}{g}\dot{\alpha}\\,
\mathfrak{D}\epsilon\,=\,\nabla\epsilon\,+\,\textstyle{\frac{1}{4}}v_{i}{}^{m}\mathcal{D}v_{mj}\Gamma^{ij}\epsilon\;=\;\nabla\epsilon\,-\,\textstyle{\frac{i}{2}}\mathcal{Q}_{(1)}\sigma^{2}\epsilon\;,
\langle\overline{{n_{m},\ldots,n_{1}}}|=\,(-1)^{\sum_{a}\bar{n}_{a}(a-1)}\langle\bar{n}_{m},\ldots,\bar{n}_{1}|=\langle\bar{0}|(\hat{\eta}^{m})^{n_{m}}\cdots(\hat{\eta}^{1})^{n_{1}}
w^{\prime}\left(1-\frac{4\beta\Phi^{\prime}(1-3w)e^{-2\Phi}}{r}\right)+\frac{{F}}{r}e^{-2\Phi}+rw\Phi^{2}=\frac{(1-w)}{r}\left(1-4\betaw(e^{-2\Phi})^{\prime\prime}\right)\;,
R_{uu}=\frac{3}{4}\frac{S\partial_{a}H_{1}\partial_{a}H_{1}}{H_{5}H_{1}^{3}}-\frac{3}{4}\frac{S\partial_{a}\partial_{a}H_{1}}{H_{5}H_{1}^{2}}+\frac{1}{4}\frac{S\partial_{a}H_{5}\partial_{a}H_{5}}{H_{1}H_{5}^{3}}-\frac{1}{4}\frac{S\partial_{a}\partial_{a}H_{5}}{H_{5}^{2}H_{1}}+\frac{\partial_{a}\partial_{a}S}{H_{1}H_{5}}
f\cdotg\equiv\int_{-i\infty}^{i\infty}\frac{dy}{2\pii}f_{X}(y)g_{X}(y)+\sum_{n=1}^{\infty}\int_{-i\infty}^{i\infty}\prod_{i=1}^{n}\frac{dy_{i}}{2\pii}f_{n}(y_{1},\cdots,y_{n})g_{n}(y_{1},\cdots,y_{n}).
Z_{a_{1}}^{a}=\left(\begin{matrix}{\psi}&{\partial\psi}&{0}&{0}&{0}\\{\nabla^{-3/2}}&{-1/2\partial^{2}h}&{\psi}&{0}&{\partial\psi}\\{0}&{\nabla^{(-5/2)}}&{0}&{\partial\psi}&{\psi}\\\end{matrix}\right)\.
{\calW}_{4}={\frac{1}{8N}}\oint\oint\oint\oint\mathrm{Tr}[{\calP}(T_{x}^{a}T_{y}^{a}T_{z}^{b}T_{w}^{b})]D_{\mu\nu}(x-y)D_{\rho\sigma}(z-w)\;dx^{\mu}dy^{\nu}dz^{\rho}dw^{\sigma}\;\;,
{\calH}=T_{0}^{0}=\frac{1}{2}\dot{\phi_{1}}^{2}+\frac{1}{2}\dot{\phi_{2}}^{2}+\frac{1}{2}{{\phi^{\prime}}_{1}}^{2}+\frac{1}{2}{{\phi^{\prime}}_{2}}^{2}+V(\phi_{1},\phi_{2}),
H^{(2)}\,=\,\frac{1}{2}\,\int\,d^{3}x\left[\sum_{i=1}^{3}\pi_{i}^{2}+\sum_{cyclic}\xi_{i}^{2}\frac{\phi_{j}^{2}+\phi_{k}^{2}}{(\phi_{j}^{2}-\phi_{k}^{2})^{2}}+V(\phi,\chi)\right]\,.
H=-\frac{M^{2}}{\piT_{II}}+\frac{1}{2}N\sum_{n=1}^{\infty}\omega_{n}\left(a_{n}^{\dagger}\cdota_{n}+\tilde{a}_{n}^{\dagger}\,cdot\tilde{a}_{n}+2\,c_{n}^{\dagger}\cdotc_{n}\right).
I_{s}^{(r)}(-u_{r,2s}\Lambda^{2})=\left(\frac{\Lambda_{\epsilon}^{2}}{\nu_{0}^{2}}\right)^{-\epsilon}\left\{a_{s}(u,v,w,x,y,z)\cdot\frac{1}{\epsilon}+[\mathrm{{terms\,\,finite\,\,as\,\,}}\epsilon\rightarrow0]+0(\epsilon)\right\}\Lambda^{2}\quad(s=1\ldotsr)\,,
(E+m)\psi(x)=\sigma\left(\begin{array}{c}{(E+m)\,cos\,k(x-\xi)}\\{k\,sin\,k(x-\xi)}\\\end{array}\right)+\tau\left(\begin{array}{c}{(E+m)\,sin\,k(x-\xi)}\\{-k\,cos\,k(x-\xi)}\\\end{array}\right)
+\frac{1}{2!}\int_{0}^{1}d\tau_{1}\int_{0}^{1}d\tau_{2}\left.\frac{\delta^{2}W[X]}{\deltaX^{a}(\tau_{1})\deltaX^{b}(\tau_{2})}\right|_{X=\overline{{X}}}\xi^{a}(\tau_{1})\xi^{b}(\tau_{2})+{\calO}(\xi^{3})
G(\tau,{\bfx};\tau^{\prime},{\bfx}^{\prime})=\int\frac{d\omega}{2\pi}\,e^{-i\omega(\tau-\tau^{\prime})}\sum_{n}\frac{\varphi_{n}({\bfx})\,\varphi_{n}^{*}({\bfx}^{\prime})}{\omega^{2}+\omega_{n}^{2}},
S=\intd^{4}x~\left[\frac{1}{2m}\epsilon_{ijk}\dot{A}^{i}B^{jk}-\frac{1}{2m}\epsilon_{ijk}\partial^{i}B^{jk}\dot{\theta}+\frac{1}{m}\epsilon_{ijk}\dot{Q}^{i}\partial^{j}A^{k}-\widetilde{H}_{c}^{\prime}\right],
\vec{\xi}_{x}=\frac{1}{\kappa\sqrt{c}}\frac{\partial}{\partialx}\left(\begin{array}{c}{f(r)}\\{\theta}\\\end{array}\right),\:\:\vec{\xi}_{y}=\frac{1}{\kappa\sqrt{c}}\frac{\partial}{\partialy}\left(\begin{array}{c}{f(r)}\\{\theta}\\\end{array}\right),\:\:\vec{\xi}_{\theta}=\sqrt{\frac{\beta}{4\pi}}\frac{\partial}{\partial\theta}\left(\begin{array}{c}{f(r)}\\{\theta}\\\end{array}\right)
2\mathrm{Im}\;S_{\mathrm{ABCD}}^{(IA)}=\frac{16\pi^{2}}{g^{2}}+2\pi^{2}\rho^{2}v^{2}-\frac{96\pi^{2}\rho^{4}}{l^{4}}+O(\rho^{6}/l^{6})\;.
\sum_{B_{1}<B_{2}<\cdots<B_{i}}\Lambda_{B_{1},\cdots,B_{i}}^{[i]}\flat_{B_{i}}^{+}\cdots\flat_{B_{1}}^{+}|0>=\sum_{A_{1},\,\cdots\,A_{i}}X^{[i]i,\,A_{i}}\cdotsX^{[i]1,A_{1}}\flat_{A_{i}}^{+}\cdots\flat_{A_{1}}^{+}|0>,
{\calT}(y)=\left(\begin{array}{cccc}{\mathrm{e}^{i\varphi_{1}}}&{}&{}&{}\\{}&{\ddots}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{\mathrm{e}^{i\varphi_{N}}}\\\end{array}\right),\quad\mathrm{with}\quad\varphi_{i}\equiv\sum_{b=1}^{n}{\frac{h_{b,i}}{L^{b}}}y^{b},\quad\sum_{i=1}^{N}h_{b,i}=0.
I_{n}(\alpha_{1},\alpha_{2},\alpha_{3})=\left(\frac{-\pi\mu}{\gamma(-b^{2})}\right)^{n}\frac{\prod_{j=1}^{n}\gamma(-jb^{2})}{\prod_{k=0}^{n-1}\left[\gamma(2\alpha_{1}b+kb^{2})\gamma(2\alpha_{2}b+kb^{2})\gamma(2\alpha_{3}b+kb^{2})\right]}.
(3a)=-2e^{4}i\int{\calD}q\frac{1}{(p+q)^{2}}\int{\calD}k\;\varepsilon_{\mu\nu\rho}\frac{k^{\rho}}{k^{2}}\;\varepsilon^{\nu\mu\gamma}\frac{(k+q)_{\gamma}}{(k+q)^{2}}
\frac{1}{4g^{2}}\,\int\!\mathrm{d}^{2}\theta\,\mathrm{Tr}\,W^{2}=-\frac{1}{8g^{2}}\left(G_{\mu\nu}^{a}G^{a\mu\nu}-iG_{\mu\nu}^{a}\tilde{G}^{a\mu\nu}\right)+\frac{1}{4g^{2}}D^{a}D^{a}+\frac{i}{2g^{2}}\lambda^{a}\sigma^{\mu}{\calD}_{\mu}\bar{\lambda}^{a}\,.
H=2\pi\sum_{n>0}^{\infty}n\left\{-{\frac{1}{2}}(\psi_{n}^{A\dagger}\psi_{n}^{B}+\psi_{n}^{B\dagger}\psi_{n}^{A}+\tilde{\psi}_{n}^{A\dagger}\tilde{\psi}_{n}^{B}+\tilde{\psi}_{n}^{B\dagger}\tilde{\psi}_{n}^{A})+\psi_{n}^{T\dagger}\psi_{n}^{T}+\tilde{\psi}_{n}^{\daggerT}\tilde{\psi}_{n}^{T})\right\}
\underline{{A}}^{\mu}=\frac1\ell\underline{{\Theta}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu\rho}\underline{{\Omega}}^{\nu\rho}\qquad,\qquad\underline{{\tilde{A}}}^{\mu}=-\frac1\ell\underline{{\Theta}}^{\mu}+{\frac{1}{2}}\epsilon^{\mu}\_{\nu\rho}\underline{{\Omega}}^{\nu\rho}\qquad(\epsilon_{012}=1).
\Gamma_{\delta}(x)\equiv\sum_{w\in\mathrm{Z},n\in\mathrm{Z}}q^{{\frac{1}{2}}({\frac{n}{x}}+{\frac{wx}{2}})^{2}}{\bar{q}}^{{\frac{1}{2}}({\frac{n}{x}}-{\frac{wx}{2}})^{2}}\quad,
\langle0|\mathrm{Tr}\,\lambda^{2}|0\rangle^{2}=\frac{2^{10}\pi^{4}}{5}\,e^{-8\pi^{2}/g^{2}}\,\frac{M_{\mathrm{PV}}^{6}}{g^{4}}=\frac{144}{5}\,\Lambda_{G}^{6}\,,
P^{-}=-g^{2}\int_{-L}^{L}dx^{-}Tr\left(J^{+}{\frac{1}{D_{-}^{2}}}J^{+}\right)+L(\partial_{+}v)^{2}++{\frac{i}{\sqrt{2}}}\int_{-L}^{L}dx^{+}Tr\left(\Psi_{L}\partial_{+}\Psi_{L}+\partial_{+}\Psi_{L}\Psi_{L}\right)\;.
\frac{27i}{20}\int\!\!d^{4}x\,\,\,\eta_{{\mathbfm}{\mathbfi}}[D^{\mathbfm}{}_{{\mathbfj}}{\bar{D}}_{{\mathbfk}{\mathbfl}}+{\bar{D}}^{\mathbfm}{}_{{\mathbfj}}D_{{\mathbfk}{\mathbfl}}]T^{{\mathbfi}{\mathbfj}{\mathbfk}{\mathbfl}}|.
{\calI}_{l}=\intds^{1}ds^{2}\;C^{r}\left(\begin{array}{c}{D_{r}^{(0)}D_{l}^{(0)}\phi^{(0)}}\\{(D_{r}^{(0)}D_{l}^{(0)}\phi^{(0)})^{\ast}}\\{\frac{\partial}{\partials^{r}}F_{lk}^{(0)}}\\\end{array}\right)^{\dagger}\hat{L}(\Psi^{(0)})\Psi^{(2)}.
\delta\eta_{\alpha_{2k+1}}^{*}=Z_{\;\;\alpha_{2k+1}}^{\alpha_{2k}}\Phi_{\alpha_{2k}}^{*}+A_{\alpha_{2k+1}}^{\;\;\alpha_{2k+2}}\Phi_{\alpha_{2k+2}}^{*},\;k=0,\cdots,b,
|a_{n}^{[n\alpha]}(\vec{q},\vec{q})|\sim\frac{(n-1)![2n(1-\alpha)]!}{2^{n-[n\alpha]}(2n-1)!}\left(C_{[2(1-\alpha)/\alpha],0,0}(\vec{q})\right)^{[n\alpha]}\sim\Gamma(n(1-2\alpha)).
\gamma_{13}^{\Psi}\gamma_{24}^{\Psi}(x_{12}^{2})^{-\delta_{1}}(x_{34}^{2})^{-\delta_{3}}\,\sum_{M\in\{l,l-2,l-4,...0\}}2^{-l}\frac{G_{M}^{(l)}}{G_{l}^{(l)}}\,u^{\frac{1}{2}(\delta-\delta_{1}-\delta_{3}-M)}\sum_{n,m=0}^{\infty}\,\frac{u^{n}(1-v)^{m}}{n!m!}\,A_{nm}^{(M)}(\delta_{1},\delta_{3},\lambda,l)
d\pi_{\dot{q}}^{2+}=\frac12\Omega^{(0)}\pi_{\dot{q}}^{2+}-\frac14\Omega_{\dot{q}\dot{p}}\pi_{\dot{p}}^{2+}+\frac12\tilde{\gamma}_{\dot{q}p}^{i}\Omega^{+2i}\pi_{p}^{2-},
\begin{array}{}{t]{rcl}H_{JS\eta}}&{=}&{{\bfb}^{\dagger}{\bfb}+1+4\alpha(\phi_{1}-\phi_{2}\Gamma_{1})+\eta\alphaI_{3}\Gamma_{2}+\alpha^{2}I_{4}}\\{}&{=}&{(N_{-}^{^{\prime}}-16\alpha^{2}+\alpha^{2}I_{4}){\bf1}+\eta\alphaI_{3}\Gamma_{2}}\\\end{array}
R_{\lambda\mu\nu}^{\lambda}(x)=\partial_{\mu}\Gamma_{\lambda\nu}^{\lambda}-\partial_{\nu}\Gamma_{\lambda\mu}^{\lambda}+\Gamma_{\tau\mu}^{\sigma}\Gamma_{\lambda\nu}^{\tau}-\Gamma_{\tau\nu}^{\sigma}\Gamma_{\lambda\mu}^{\tau}.
\sigma_{1,\mathrm{\tiny~field~th}}^{T}=\frac{\kappa^{3}N^{3}\omega^{7}}{6144\pi^{7/2}}\left[1+8\left(\frac{\piT}{\omega}\right)^{2}+48\left(\frac{\piT}{\omega}\right)^{4}+32\left(\frac{\piT}{\omega}\right)^{6}+O\left(e^{-\omega/T}\right)\right]\,.
\begin{array}{rcl}{}&{}&{b^{i}(z)c_{j}(w)=\displaystyle\frac{\delta_{j}^{i}}{z-w}+\cdots~,}\\{}&{}&{c_{i}(z)b^{j}(w)=\displaystyle\frac{\delta_{i}^{j}}{z-w}+\cdots~,}\\\end{array}
\begin{array}{lcl}{{\mathcalL}_{3}^{(0,3)}}&{=}&{p^{3}+u_{2}p+(u_{3}-\thetau_{2}^{\prime})}\\{{\tilde{\mathcalL}}_{-3}^{(-3,-1)}}&{=}&{{v_{0}\starp}^{-3}+v_{-1}\starp^{-2}+v_{-2}\starp^{-1}}\\\end{array}
\langlea,\,{\bfx}|{\calI}|b,\,{\bfy}\rangle\approx\,\frac{\delta<{\bar{\calG}}^{a}({\bfx})>}{\delta\omega^{b}({\bfy})}\vert_{\omega=0}\,=\,\langlea,\,{\bfx}|\Pi^{2}|b,\,{\bfy}\rangle\,
W_{\nu}^{d+1}(Z,Z^{\prime})=w_{\nu}(\zeta)=\frac{e^{-i\pi\frac{d-1}2}}{(2\pi)^{\frac{d+1}2}}(\zeta^{2}-1)^{-\frac{d-1}4}Q_{\nu-\frac12}^{\frac{d-1}2}(\zeta).
ds^{2}=\frac{d\xi^{0}d\xi^{0}}{1+\xi^{0}\xi^{0}}-(1+\xi^{0}\xi^{0})\frac{d{\bfx}(I+{\bfx}^{\prime}{\bfx})^{-1}d{\bfx}^{\prime}}{1+{\bfx}{\bfx}^{\prime}}~,
\begin{array}{l}{\sigma_{per}^{+}=\sigma^{+}-m\sqrt{2},\qquad\sigma_{per}^{-}=\sigma^{-}-n\sqrt{2}}\\{-\frac1{\sqrt{2}}<\sigma_{per}^{\pm}<\frac1{\sqrt{2}},\qquad\mathrm{{periodic.}}}\\\end{array}
{\calL}_{1}\equiv{\calL}_{2}\left(\varphi,\partial_{\mu}\varphi,\partial_{\mu\nu}\varphi;B_{\rho},F^{(1)},F^{(2)},C^{(1)},C^{(2)};B_{\rho\sigma},D^{(1)},D^{(2)},G^{(1)},G^{(2)},G^{(3)}\right).
\begin{array}{lcr}{(|X_{0}|^{2}+|X_{2}|^{2}+2|Y_{1}|^{2})-(|Y_{0}|^{2}+|Y_{2}|^{2}+2|X_{1}|^{2})=R_{1}}\\{X_{0}\overline{{Y}}_{0}+X_{2}\overline{{Y}}_{2}-2X_{1}\overline{{Y}}_{1}=0}\\{\overline{{X}}_{0}{Y}_{0}+\overline{{X}}_{2}{Y}_{2}-2\overline{{X}}_{1}{Y}_{1}=0,}\\\end{array}
d\tilde{A}_{(4)}=-\frac{4}{3L}\sum_{i}X_{i}^{-1}\epsilon_{(1,4)}-\frac{L}{6}\sum_{i}X_{i}^{-2}F_{(2)}^{i}\wedge*_{(1,4)}F_{(2)}^{i}-\frac{L}{2}[Ld\phi_{1}+A^{1}]\wedgeF_{(2)}^{2}\wedgeF_{(2)}^{3}
E_{\mathrm{YM}}=\frac{g_{\mathrm{YM}}^{2}}{NV_{s}}\left[\left(\tilde{m}^{1}\right)^{2}+\left(\frac{V_{s}}{g_{\mathrm{YM}}^{2}}\right)^{2}\left(\tilde{m}_{2}\right)^{2}+\left(\frac{V_{s}}{g_{\mathrm{YM}}^{2}}\right)^{4}\left(\tilde{m}_{5}\right)^{2}\right]
\Pi_{N}^{\mu}=-\frac{i}{2}\left[(\delta^{\mu\nu}-\frac{a^{\mu}a^{\nu}}{a^{\sigma}a^{\sigma}})\partial_{\nu}+\partial_{\nu}(\delta^{\mu\nu}-\frac{a^{\mu}a^{\nu}}{a^{\sigma}a^{\sigma}})+\frac{2ca^{\mu}}{a^{\sigma}a^{\sigma}}\right].
\int_{0}^{2\pi}\mathrm{d}f_{1}\ldots\int_{0}^{2\pi}\mathrm{d}f_{n-1}\theta(f_{2}-f_{1})\ldots\theta(f_{n-1}-f_{n-2})e^{-\sigmaA_{D}\left(x_{1},\ldots,x_{n},f_{1},\ldots,f_{n-1}\right)}
S_{B,\phi,A,\lambda}=\frac{1}{2}\intd^{2}x\left\{m^{2}(B_{\mu}B^{\mu}+\phi^{2})+2m\epsilon^{\mu\nu}(B_{\mu}\partial_{\nu}\lambda+\phi\partial_{\mu}A_{\nu}+2\lambda\partial_{\mu}A_{\nu})\right\}
\int\prod_{i=1}^{s_{+}-1}d^{2}w_{i}|w_{i}|^{-2+8j_{1}/\alpha_{+}^{2}}|1-w_{i}|^{-4(2j_{2}+k)/\alpha_{+}^{2}}\prod_{i<j}^{s_{+}-1}|w_{i}-w_{j}|^{-8/\alpha_{+}^{2}}
\intd^{(p+1)}x\,d^{(d-p-1)}x\frac{\mathrm{e}^{ik_{\bot}\cdotx_{\bot}}}{(d-p-3)\,r^{d-p-3}\,\Omega_{d-p-2}}=\frac{V_{p+1}}{k_{\perp}^{2}}~,
\Pi_{ij}^{\Sigma}={\frac{4\alpha}{\beta}}\int_{0}^{1}dx\int{\frac{d^{2}l}{(2\pi)^{2}}}[2x(1-x)(p_{b}^{2}\delta_{ij}-p_{i}p_{j})S_{2}^{\Sigma}+(1-2x)p_{0b}S_{2}^{*\Sigma}-\delta_{ij}(1-2x)p_{0b}S_{2}^{*\Sigma}]
\left\{\frac{d^{2}}{dr^{2}}+\left[\frac{2M}{\hbar^{2}}\,E+\lambda\,\frac{\theta(a-r)}{a^{2}}\right]-\frac{l^{2}-\lambda\,\theta(r-a)-1/4}{r^{2}}\right\}u_{l}(r)=0\;,
S~=~\intd^{d+1}x\sqrt{g}e^{-\Phi}\big(R+\partial_{\mu}\Phi\partial^{\mu}\Phi-{\frac{1}{12}}H_{\mu\nu\lambda}H^{\mu\nu\lambda}-{\frac{\alpha^{\prime}}{4}}F^{N}{}_{\mu\nu}F^{N}{}^{\mu\nu}+2\Lambda\big)
\tilde{J}_{\alpha\dot{\alpha}}^{\alpha}=-3i\,\frac{T_{G}}{4\pi^{2}}\,\mathrm{Tr}\left[\bar{G}_{\dot{\alpha}\dot{\beta}}\bar{\lambda}^{\dot{\beta}}\right]\,,\qquad\tilde{\vartheta}_{\mu}^{\mu}=-3\,\frac{T_{G}}{16\pi^{2}}\,\mathrm{Tr}\,\left[G_{\rho\sigma}G^{\rho\sigma}\right]\,.
{\calM}_{\mathrm{c}}(\theta)=\frac{ig^{4}}{2m\Theta^{2}}[(T^{a}T^{b}+T^{b}T^{a})\otimesT_{b}T_{a}]\int\frac{d^{2}k}{(2\pi)^{2}}\frac{{\bfk}\cdot({\bfk}+{\bfq})}{{\bfk}^{2}({\bfk}+{\bfq})^{2}}.
\hat{\alpha}_{i}\rightarrow\hat{\alpha}_{i}+\frac{2\pi\mathrm{i}k}{N},\;\;\;\hat{\alpha}_{i}^{*}\rightarrow\hat{\alpha}_{i}^{*}+\frac{2\pi\mathrm{i}l}{N},\;\;\;\;\;I_{n}\rightarrowe^{-2\pi\mathrm{i}k(N-n)/N}I_{n},\;\;\;I_{n}^{*}\rightarrowe^{-2\pi\mathrm{i}l(N-n)/N}I_{n}^{*},
\mathcal{L}_{I}^{lin}=-\frac{i}{4}\left(h^{a\mu}\bar{\psi}\gamma_{a}\frac{1\pm\gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial_{\mu}^{\psi}}\psi+h_{\\mu}^{\mu}\bar{\psi}\gamma^{a}\frac{1\pm\gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial_{a}^{\psi}}\psi\right)
\Omega^{(1)}=\frac{d\zeta}{3\eta},\;\;\Omega^{(2)}=\frac{d\zeta}{3\eta^{2}},\;\;\Omega^{(3)}=\frac{\zetad\zeta}{3\eta^{2}},\;\;\Omega^{(4)}=\frac{\zeta^{2}d\zeta}{3\eta^{2}}.
h_{\mu\nu}^{\mathrm{s.p.}}[y(\xi)]=\frac{im^{3}}{8\pig_{m}^{2}}\intd\sigma_{\mu\nu}(y(\xi^{\prime}))\frac{K_{1}\left(m\left|y(\xi)-y(\xi^{\prime})\right|\right)}{\left|y(\xi)-y(\xi^{\prime})\right|},
{\calH}=\frac{1}{4\pi^{2}g}\bigg\{\frac{R^{2}+B^{2}+R^{2}{R^{\prime}}^{2}}{\sqrt{(1-{\dot{R}}^{2})(R^{2}+B^{2})+R^{2}({R^{\prime}}^{2}-E_{z}^{2})}}-\frac{h}{2}R^{2}\bigg\}+\frac{1}{g}D_{E}A_{0}^{\prime}
{\calM}=\int\frac{d^{D}\!q}{(q-p)^{2}(q\!\cdot\!n)}\longrightarrow^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!ML}\int\frac{d^{D}\!q\,(q\!\cdot\!n^{*})}{(q-p)^{2}\left[(q\!\cdot\!n)(q\!\cdot\!n^{*})+i\epsilon\right]}.
\delta^{ab}(C_{1}C_{3}+{\frac{1}{2}}C_{1}C_{2}){\frac{g^{4}}{8p^{2}}}\Lambda^{3-\omega}\int{\frac{d^{\omega}k}{(2\pi)^{\omega}}}{\frac{k\cdotp+2p^{2}}{|k|(k+p)^{2}}},
e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}\delta\lambda.\frac{\partial}{\partial\lambda}\beta_{\mu\nu}^{b}={\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\left(\delta\lambda.\frac{\partial}{\partial\lambda}\beta_{ij}^{b^{\star}}+\beta_{j}^{w}\deltav_{i}-\beta_{i}^{w}\deltav_{j}\right)
\begin{aligned}{\Xi^{\prime}(u,\lambda)\Xi^{\prime}(u,-\lambda)^{T}}&{{}=\sum_{i=1}^{n}h_{i}(u)^{2}E_{ii},}\\{\lambda\Xi(u,\lambda)\Xi^{\prime}(u,-\lambda)^{T}}&{{}=(h_{1}^{2}(u)\h_{2}^{2}(u)\\ldots\h_{n}^{2}(u)).}\\\end{aligned}