harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\epsilon_{D}(a,b)=-\frac{1}{2}\frac{1}{(4\pi)^{\frac{D-1}{2}}}\left(\frac{32V_{0}}{b^{2}}\right)^{\frac{D-1}{4}}\left[\int_{1}^{\infty}dt\,t^{-\frac{(1+D)}{2}}\,\frac{e^{t(1-q)}}{e^{t}-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}B_{n}(q)}{n!}\frac{1}{n-\frac{D+1}{2}}\right].
{\vec{v}}_{E}^{\prime}=\frac{\gamma_{E}\vec{v}_{E}+\vec{V}_{E}\left[\displaystyle\frac{\vec{V}_{E}\vec{v}_{E}}{c^{2}}\left(\gamma_{E}+1\right)^{-1}-1\right]}{\displaystyle1-\frac{\vec{V}_{E}\vec{v}_{E}}{c^{2}}},
{\frac{L^{2}}{H_{2}H_{6}}}\left[\partial_{r}^{2}\Psi_{3}+\left({\frac{L^{\prime}}{L}}+{\frac{2}{r}}\right)\partial_{r}\Psi_{3}\right]+{\frac{L(r_{\mathrm{e}})}{K(r_{\mathrm{e}})}}\omega^{2}\Psi_{3}=0
\delta\Gamma=2\pi\int_{\Sigma_{2}^{1}}(\chi_{2}^{1}(N_{1})+\epsilon_{1}\chi_{2}^{1}(N^{\prime}))+2\pi\int_{\Sigma_{2}^{2}}(\chi_{2}^{1}(N_{2})+\epsilon_{2}\chi_{2}^{1}(N^{\prime})).
\int_{-1}^{1}P_{r}^{{\bfV}_{+}}(x)P_{s}^{{\bfV}_{+}}(x)\,(1-x)^{-1}(1+x)^{-1}\,dx\,\propto\int_{-1}^{1}P_{r-2}^{(1,1)}(x)P_{s-2}^{(1,1)}(x)(1-x)(1+x)\,dx\,\propto\delta_{r\,s}.
S_{\scriptscriptstyle{l,L}}=\delta_{\scriptscriptstyle{l,L}}+\sqrt2\pi\,i^{-l-1}\sum_{m}\int_{0}^{\infty}dr\,r\,J_{\scriptscriptstyle{l}}(kr)\,V_{\scriptscriptstyle{l-m}}(r)\,\psi_{\scriptscriptstyle{m,L}}(k,r)\,,
e^{i\alpha\frac{e}{2\pi}\phi_{\hat{n}}(x)}V_{\eta}^{1/2}(y,\hat{n}^{\prime})V_{\eta}^{*1/2}(y,-\hat{n}^{\prime})=V_{\eta}^{1/2}(y,\hat{n}^{\prime})V_{\eta}^{*1/2}(y,-\hat{n}^{\prime})e^{i\alpha\frac{e}{2\pi}\phi_{\hat{n}}(x)}e^{i\frac{\alpha}{2}l(C_{1},C_{2})}
\Phi=\left(\begin{matrix}{0}&{h}&{0}&{\ldots}&{0}\\{{\frac{a_{1}}{h}}}&{\ddots}&{\ddots}&{\ddots}&{\vdots}\\{{\frac{a_{2}}{h^{2}}}}&{\ddots}&{\ddots}&{h}&{0}\\{\vdots}&{\ddots}&{{\frac{a_{1}}{h}}}&{0}&{h}\\{{\frac{a_{n-1}}{h^{(n-1)}}}}&{\ldots}&{{\frac{a_{2}}{h^{2}}}}&{{\frac{a_{1}}{h}}}&{0}\\\end{matrix}\right)~dz\quad,
\left.\begin{array}{ll}{\textrm{D3}_{(0,0,0)}:}&{U(1)_{0,(0,0,0)}}\\{\widetilde{\textrm{D7}}_{3,(0)}:}&{U(1)_{1,(0)}}\\{\textrm{D7}_{4,(0)}:}&{U(2)=U(1)_{2,(0)}\timesSU(2)}\\{\textrm{D7}_{5,(0)}:}&{U(3)=U(1)_{3}\timesSU(3)}\\\end{array}\right.
{\calL}=\frac{2}{3}\left(\bfH^{\mathrm{2}}-\bfD^{\mathrm{2}}\right)+2\;\partial_{\mu}\vec{B}\cdot\partial^{\mu}\vec{B}+2\:g^{2}C_{\mu}C^{\mu}\left(B_{1}^{2}+B_{2}^{2}\right)+W(\vec{B}).
\left(\begin{array}{c}{\tilde{\calS}_{D}}\\{\tilde{\calS}}\\\end{array}\right)=\left(\begin{array}{c}{A}\\{C}\\\end{array}\begin{array}{cc}{B}\\{D}\\\end{array}\right)\left(\begin{array}{c}{{\calS}_{D}}\\{{\calS}}\\\end{array}\right).
A={\frac{1}{\sqrt2}}\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{0}&{-1}&{-1}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{1}&{-1}\\\end{matrix}\right),\qquadB={\frac{1}{\sqrt2}}\left(\begin{matrix}{0}&{-1}&{0}&{0}\\{0}&{0}&{0}&{0}\\{0}&{-1}&{0}&{0}\\{0}&{0}&{0}&{0}\\\end{matrix}\right).
\varepsilon^{ijk}{^{\pm}}A_{i}^{AB}\partial_{j}{^{\pm}}A_{kAB}=\varepsilon^{ijk}\big({\frac{1}{2}}A_{i}^{AB}\partial_{j}A_{kAB}\mp{\frac{i}{2}}\varepsilon^{ABCD}A_{iAB}\partial_{j}A_{kCD}\big),
S={\frac{kA_{H}}{4\ell_{P}^{2}}}+{\frac{1}{T_{H}}}\int_{\Sigma}\;e^{_\phi}\;\left(\varrho_{L}-{\calL}_{E}\right)d^{3}r+\int_{\Sigma}{\frac{s}{\sqrt{1-(b/r)}}}d^{3}r.
\tilde{\phi}\equiv\frac{g}{\Lambda}\phi,\quad\tilde{W}\equiv{\frac{g^{2}}{\Lambda^{3}}}W,\quad\tilde{V}\equiv{\frac{g^{2}}{\Lambda^{4}}}V,\quad\tilde{y}\equiv\Lambday,\quad\alpha\equiv{\frac{\Lambda}{gM_{\mathrm{P}}}},
\frac{i}{4}\left[(\gamma_{d})_{\alpha\beta}(\gamma^{ed})_{\gamma\delta}+(\gamma^{ed})_{\alpha\beta}(\gamma_{d})_{\gamma\delta}\right]\theta_{\betaA}(\theta_{B}\gamma^{a})_{\gamma}\theta_{\deltaC}X_{eD}{d^{BC}}_{E}f^{DAE}.
S=\int\mathrm{d}^{4}x\sqrt{-G}\mathrm{e}^{-\Phi}[{\calR}_{G}+\partial_{\mu}\Phi\partial^{\mu}\Phi-{\frac{1}{12}}H_{\mu\nu\rho}H^{\mu\nu\rho}-F_{\mu\nu}^{a}(LML)_{ab}(F^{b})^{\mu\nu}+{\frac{1}{8}}\mathrm{Tr}(\partial_{\mu}ML\partial^{\mu}ML)]\,
\tilde{\psi}({\bfr})=V_{\cal{C}}^{-1}({\bf{r}})\,\psi({\bfr})\;\;\;\;\;\;\;\;\;\;\;\;\mathrm{and}\;\;\;\;\;\;\;\;\;\;\;\;\tilde{\psi}^{\dagger}({\bf{r}})=\psi^{\dagger}({\bf{r}})\,V_{\cal{C}}({\bf{r}})\,,
\left(\begin{array}{c}{a_{D}(u)}\\{a(u)}\\\end{array}\right)\rightarrowM_{i}^{Higgs}\left(\begin{array}{c}{a_{D}(u)}\\{a(u)}\\\end{array}\right)=\left(\begin{array}{c}{a_{D}(ue^{2\pii})}\\{a(ue^{2\pii})=-a}\\\end{array}\right),
f_{2}^{*}(L,m,d,z)|_{d=3}=\frac1{4L}h_{2}(3)\left(\int_{mL}^{2\pi}du\frac{e^{\frac{z}{L}u}}{e^{u}-1}+\int_{mL}^{2\pi}du\frac{e^{-\frac{z}{L}u}}{e^{u}-1}\right).
{\calL}~=~-{\scriptstyle\frac{1}{2}}\,\partial_{\mu}\,\varphi\,\partial^{\mu}\,\varphi-{\scriptstyle\frac{1}{2}}m^{2}\varphi^{2}-{\scriptstyle\frac{1}{4}}\,\lambda\,\left(\partial_{\mu}\,\varphi\,\partial^{\mu}\,\varphi\right)^{2}
\left<T^{\mu}{}_{\mu}(x)\right>={\frac{a^{-N}}{2^{N-1}\pi^{N/2}\Gamma\left(\fracN2\right)}}\sum_{n=0}^{\fracN2-1}c_{2n+1}^{N}\left[\frac{(-1)^{n+1}}{2(n+1)}4^{-n-1}-2H_{n}(0)\right].
\frac12{\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}\frac{\partial\Gamma^{(2)}}{\partial\bar{g}^{ij}}={\bar{e}}_{\alpha}^{\;\;i}{\bar{e}}_{\beta}^{\;\;j}B_{ij}^{\bar{g}(2)}+e_{\alpha}^{\;\;\mu}e_{\beta}^{\;\;\nu}K_{\mu\nu}^{g}.
g^{2}R_{2}^{(\pm)}(\nu;0)=g^{2}\int_{1}^{\infty}d\alpha(\sum_{\ell=1}^{7}\frac{\Phi_{\ell}^{(3)}(\alpha)}{\alpha^{\ell}})[\alpha^{\frac{\nu}{2}}+\alpha^{\frac{-\nu}{2}}]=2\xi(\frac{1}{2}+\nu).
(D\Theta)^{I}=\left(d+{\frac{1}{4}}(\omega^{ab}\gamma_{ab}+\omega^{a^{\prime}b^{\prime}}\gamma_{a^{\prime}b^{\prime}})\right)\Theta^{I}-{\frac{1}{2}}i\epsilon^{IJ}(e^{a}\gamma_{a}+ie^{a^{\prime}}\gamma_{a^{\prime}})\Theta^{J}
\varrho=\varrho_{m}(1+\zetae^{-2\psi})+\varrho_{r}e^{-2\psi}+\varrho_{\psi},\\\\\\\\\\\\\wp=\textstyle{\frac{1}{3}}\varrho_{r}e^{-2\psi}c^{2}+\varrho_{\psi}c^{2}.
\begin{array}{ll}{\partialX_{cl}(z)=a\omega(z),}&{\partial\bar{X}_{cl}(z)=\bar{a}\omega^{\prime}(z),}\\{\bar{\partial}X_{cl}(\bar{z})=b\bar{\omega}^{\prime}(\bar{z}),}&{\bar{\partial}\bar{X}_{cl}(\bar{z})=\bar{b}\bar{\omega}(\bar{z}),}\\\end{array}
(\frac{\lambda}{N})^{n}T^{C}M^{3C-2L}V_{n}\sim(\frac{\lambda}{N})^{n}(1/\sqrt{\lambda})^{C}(\frac{\lambda}{N^{2/3}})^{3C-2L}V_{n}=\lambda^{1}(\frac{1}{N})^{\frac{1}{3}n+2}V_{n}.
\sigma_{c}^{\mathrm{np}}=\frac{N_{f}^{2}}{4}\Lambda_{\mathrm{np}}^{2}\frac{d\Sigma}{dp}\bigg|_{p=\Lambda_{\mathrm{np}}}\simeq\frac{N_{f}^{2}}{4}\Lambda_{\mathrm{np}}^{2}\Sigma_{0}\left(\frac{d}{dp}F({\frac{1}{4}}-{\frac{i\nu}{2}},{\frac{1}{4}}+{\frac{i\nu}{2}},{\frac{3}{2}};-\frac{p^{2}}{\Sigma_{0}^{2}})\right)\bigg|_{p=\Lambda_{\mathrm{np}}},
\tilde{A}_{i}({\bfk})=\Gamma(\nu)\left({\frac{|{\bfk}|}{2}}\right)^{-\nu}\left({\frac{1}{2}}a_{i}({\bfk})-i{\frac{k_{i}}{|{\bfk}|^{2}}}ca_{0}({\bfk})\right).
\phi_{\alpha_{1}\alpha_{2}}\phi_{\alpha_{2}\alpha_{3}}\phi_{\alpha_{3}\alpha_{1}}=f^{(\alpha_{1})}f^{(\alpha_{2})}{}^{-1}f^{(\alpha_{2})}f^{(\alpha_{3})}{}^{-1}f^{(\alpha_{3})}f^{(\alpha_{1})}{}^{-1}=1
{\calP}_{i_{1}i_{2}i_{3}}^{*}\equiv{\frac{1}{2}}\varepsilon_{i_{1}i_{2}i_{3}i_{4}i_{5}}{\frac{\deltaL}{\delta\partial_{0}B^{i_{4}i_{5}}}}={\frac{1}{4}}\partial_{[i_{1}}B_{i_{2}i_{3}]}.
{\calN}^{(n)}(\{x\})=\left(\frac{1}{2\pi}\right)^{-Nn}\!\left[\frac{e^{2}N}{4\pi}e^{2\gamma}\right]^{-\frac{Nn^{2}}{2}}\prod_{a=1}^{N}\prod_{1\leqi<j\leqn}\!\left(x_{i}^{(a)}\!-\!x_{j}^{(a)}\right)^{-2}e^{\frac{1}{2}V\left(x_{i}^{(a)}-x_{j}^{(b)}\right)}\;.
j_{\mathrm{DS}}:=\left(\begin{array}{cccccc}{v_{1}}&{}&{}&{}&{}&{}\\{v_{2}}&{}&{}&{}&{}&{}\\{\vdots}&{}&{}&{}&{}&{}\\{v_{p-1}}&{}&{}&{}&{}&{}\\{v_{p}}&{-\tilde{v}_{p-1}}&{\cdots}&{-\tilde{v}_{2}}&{-\tilde{v}_{1}}&{z_{+}}\\{z_{-}^{t}}&{{}}&{{}}&{{}}&{{}}&{{}}\\\end{array}\right),
\int_{0}^{\mu}d\mu^{\prime}{\calJ}_{0}(m,\mu^{\prime};B)=\frac{N}{2\pi}\,|eB|^{3/2}\left[l(|\mu|-m)\theta(|\mu|-m)+2\sum_{n=1}^{\infty}l(|\mu|-{\calE}_{n})\theta(|\mu|-{\calE}_{n})\right].
S=\frac{1}{2}\intdt\,\left[{\calG}_{\alpha\beta}\left({\dot{z}}^{\alpha}{\dot{z}}^{\beta}+i{\overline{\psi}}^{\alpha}\gamma^{0}D_{t}\psi^{\beta}\right)+\frac{1}{6}R_{\alpha\beta\gamma\delta}{\overline{\psi}}^{\alpha}\psi^{\gamma}{\overline{\psi}}^{\beta}\psi^{\delta}\right]-\frac{4\piv}{e^{2}}k\,,
E_{F}^{K}=-\frac{\hbarm}{2}\underset{1}{\overset{M_{+}}{\sum}}\sqrt{(k_{n}^{+})^{2}+1}-\frac{\hbarm}{2}\underset{0}{\overset{M_{-}}\sum}\sqrt{(k_{n}^{-})^{2}+1}+\deltaM_{F}
\alpha=\left(-\frac{g_{t\phi}}{g_{\phi\phi}}-\Omega_{0}\right)\left(\frac{g_{\phi\phi}^{2}}{-{\calD}}\right)^{1/2},~~~~~\bar{\mu}^{2}=\left(\frac{-{\calD}}{g_{\phi\phi}}\right)\mu^{2},~~~~~\Gamma=\Phi_{0}+A_{t}+\Omega_{0}A_{\phi}.
nY_{n}(q^{\prime},q)+(q^{\prime}-q)\frac{\partialY_{n}(q^{\prime},q)}{\partialq^{\prime}}=\frac{1}{2}\frac{\partial^{2}Y_{n-1}(q^{\prime},q)}{\partialq^{2}}-V(q^{\prime})Y_{n-1}(q^{\prime},q),
U^{(m)}(y)=\mathrm{exp}{\frac{2\piiy}{L}}\left(\begin{array}{cccc}{{\frac{m}{N}}}&{}&{}&{}\\{}&{{\frac{m}{N}}}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{-{\frac{(N-1)m}{N}}}\\\end{array}\right).
ds^{2}={\frac{r^{2}}{L^{2}}}\left[-\left(1-{\frac{r_{0}^{4}}{r^{4}}}\right)dt^{2}+d\vec{x}^{2}\right]+{\frac{L^{2}}{r^{2}}}\left(1-{\frac{r_{0}^{4}}{r^{4}}}\right)^{-1}dr^{2}+L^{2}d\Omega_{5}^{2}\,
A=\frac{\Gamma(\frac{1}{2}-\frac{i\omega}{2\piT_{H}})\Gamma(l+2)\Gamma(l+2)}{\Gamma(\frac{l+2}{2}-\frac{i\omega}{4\piT_{L}})\Gamma(\frac{l+3}{2}-\frac{i\omega}{4\piT_{R}})}2^{l+1}(r_{0}\omega)^{-l-\frac{1}{2}}\left(\frac{\delta_{5}}{\delta_{1}}\frac{T_{L}}{T_{R}}\right)^{\frac{1}{4}},
u={\frac{1}{2}}\sum_{n}\int{\frac{d^{d}k}{(2\pi)^{d}}}\int_{0}^{\infty}{\frac{dt}{t}}t^{-1/2}e^{-t(k^{2}+n^{2}\pi^{2}/a^{2})}{\frac{1}{\Gamma(-{\frac{1}{2}})}},
{\calL}_{WZ}^{(1)}\,=\,-\,{\frac{1}{2}}\,(\,\dot{\theta}^{2}\,+\,3\,{\theta^{\prime}}^{2}\,)\,-\,\lambda\,(\,\dot{\theta}\,+\,\theta^{\prime}\,)\,-\,{\frac{1}{2}}\,\lambda^{2}\;\;,
\left[p^{\mu}\partial_{\mu}+\vec{Q}\cdot\vec{F}_{\mu\nu}p^{\mu}\partial_{p}^{\nu}+p^{\mu}\vec{Q}\times\vec{A}_{\mu}\cdot{\frac{\partial}{\partial\vec{Q}}}\right]f(x,p,\vec{Q})=C(x,p,\vec{Q})+S(x,p,\vec{Q})
M=\left(\begin{matrix}{0}&{E_{0}^{-}}&{-E_{0}^{+}}\\{E_{0}^{+}(V^{\prime}(Y)+U^{\prime}(Y)Y^{+}Y^{-})}&{-\Omega_{0}+E_{0}^{+}U(Y)Y^{-}}&{E_{0}^{+}U(Y)Y^{+}}\\{-E_{0}^{-}(V^{\prime}(Y)+U^{\prime}(Y)Y^{+}Y^{-})}&{-E_{0}^{-}U(Y)Y^{-}}&{\Omega_{0}-E_{0}^{-}U(Y)Y^{+}}\\\end{matrix}\right)\;.
\Gamma_{a,s}=\tilde{\Gamma}_{s}\Gamma_{a,s}^{(o)}(l_{1s})\tilde{\Gamma}_{s}^{-1},\qquad\Gamma_{b,s}=\tilde{\Gamma}_{s}\Gamma_{b,s}^{(o)}(l_{2s})\tilde{\Gamma}_{s}^{-1}=\Gamma_{a,s}({1/2}-l_{2s})\Gamma_{s}\qquad
C_{nl}(\delta_{1},\delta_{2})=\frac{2(-\alpha)^{\frac{3}{2}}}{\left(n+\frac{\delta_{1}+\delta_{2}}{2}\right)^{2}}\frac{1}{\Gamma(2l+\delta_{1}+\delta_{2}+2)}\sqrt{\frac{\Gamma(n+l+\delta_{1}+\delta_{2}+1)}{(n-l-1)!}}
Z_{g=2}=\sum_{e,f,g}C(e,f,g)\int\prod_{i=1}^{6}\frac{1}{dv_{pr}^{2}}T^{-5}\prod_{k<l}^{6}\overline{{a_{kl}}}^{-3}a_{kl}^{-2}\overline{{\calO}}_{f}^{2}\overline{{\calO}}_{g}^{2}[{\calP}^{X}+{\calP}_{e}^{gh}]{\calO}_{e},
\hat{\hat{\Omega}}_{\hat{\hat{a}}}{}^{\hat{\hat{b}}\hat{\hat{c}}}\;=\;\hat{\hat{\omega}}_{\hat{\hat{a}}}{}^{\hat{\hat{b}}\hat{\hat{c}}}\;+\;\hat{\hat{K}}_{\hat{\hat{a}}}{}^{\hat{\hat{b}}\hat{\hat{c}}}\;.
F(T,\Omega)=-c_{p}V_{p}N^{2}\lambda^{-\frac{p-3}{p-5}}T^{\frac{2(7-p)}{5-p}}(1+x)^{2(6-p)/(5-p)}\left(1+\frac{x}{x_{c}}\right)^{-2\frac{7-p}{5-p}}
S_{\mathrm{D7}}=-T_{D7}\intd^{8}\xi\\Bigl\{e^{-\phi}\sqrt{|\mathrm{det}(g+(2\pi\alpha^{\prime}){\calF})|}+\frac{(2\pi\alpha^{\prime})}{8!}\epsilon^{i_{1}\dotsi_{8}}{\calG}_{i_{1}\dotsi_{8}}^{(8)}\Bigr\}\,,
S_{\mathrm{gf}}=-\mathrm{tr}_{K}\,[S_{a},\tilde{a}_{n}^{\prime}]^{2}-\rho^{4}\mathrm{tr}_{K}\,[S_{a},\tilde{\chi}_{b}^{\perp}]^{2}-{\frac{1}{2}}\mathrm{tr}_{K}\left(\rho^{-4}[\hat{a}_{n}^{\prime},\hat{a}_{m}^{\prime}]^{2}+2[\hat{\chi}_{a},\hat{a}_{n}^{\prime}]^{2}+\rho^{4}[\hat{\chi}_{a},\hat{\chi}_{b}]^{2}\right)\,.
|z\rangle\equivU(z)|0\rangle=e^{-\frac{1}{2}|z|^{2}}e^{za^{\dagger}}|0\rangle=e^{-\frac{1}{2}|z|^{2}}\sum_{n=0}^{\infty}\frac{z^{n}}{\sqrt{n!}}|n\rangle
g_{v,2}(x)=\frac{k}{2}C_{1}^{2}({\hat{h_{v}}}+4\Omega_{sph}^{2})^{-1}\bigg[\{(5+4k^{2})-9k^{2}sn^{2}(x)\}sn(x)cn^{2}(x)dn(x)\bigg].
V_{1}(r)=d^{2}\mathrm{e}^{-r^{2}/w^{2}}+2d\mathrm{e}^{-r^{2}/2w^{2}}\qquad\mathrm{and}\qquadV_{2}(r)=C\mathrm{e}^{-r^{2}/2w^{2}}\.
\langle\phi_{i_{1}}...\phi_{i_{s}}\rangle_{W}^{g=0}=res(\frac{\phi_{i_{1}}...\phi_{i_{s}}}{\partialW})\equiv\frac{1}{2\pii}\oint_{\gamma}dX\frac{\phi_{i_{1}}(X)...\phi_{i_{s}}(X)}{\partialW}
\begin{array}{rcl}{{\tilde{S}}[X^{\mu},b_{i}]}&{=}&{-T_{M2}l\intd^{2}\xi\sqrt{|D_{i}X^{\mu}D_{j}X^{\nu}g_{\mu\nu}|}}\\{}&{}&{}\\{}&{}&{-{\textstyle\frac{T_{M2}l}{2}}\intd^{2}\xi\epsilon^{ij}D_{i}X^{\mu}D_{j}X^{\nu}B_{\mu\nu}\,.}\\\end{array}
\begin{aligned}{F_{r_{1}r_{2}}}&{{}\rightarrowR_{r_{1}r_{2}s_{1}s_{2}}\,,}\\{\chi}&{{}\rightarrow\psi_{s_{1}s_{2}}\,,}\\{D_{r}\chi}&{{}\rightarrowD_{r}\psi_{s_{1}s_{2}}\,,}\\\end{aligned}
S=\frac{1}{2}\intd\tau\left[\frac{1}{e}\left(\dot{x}^{\mu}-\frac{i}{2}\chi\xi^{\mu}\right)^{2}+em^{2}+im\chi\xi_{D+1}-i\left(\xi_{\mu}\dot{\xi}^{\mu}-\xi_{D+1}\dot{\xi}_{D+1}\right)\right].
\{\mathbf{S}_{Q},\mathbf{S}_{Q}\}=-2\mathbf{Z},\qquad\{\mathbf{S}_{Q},\mathbf{\bar{Q}}_{\mu}\}=-i\mathbf{P}_{\mu},\qquad\{\mathbf{\bar{Q}}_{\mu},\mathbf{\bar{Q}}_{\nu}\}=-2\delta_{\mu\nu}\mathbf{\bar{Z}}.
I_{non-planar}^{T=0}=\frac{g}{2}\int\frac{d^{2}p}{(2\pi)^{2}}\frac{d^{2}k}{(2\pi)^{2}}\frac{e^{i\tilde{p}k}}{e^{\beta(\frac{k^{2}}{2}-\mu)}-1}=\frac{g}{8\pi^{2}\theta^{2}}\frac{1}{e^{-\frac{\mu}{T}}-1}\,.
\left(\begin{array}{c}{\sigma^{i}[\omega_{0}^{+0i}\tilde{\epsilon}_{2}-iF^{+0i}\tilde{\epsilon}^{1}]}\\{\sigma^{i}[-\omega_{0}^{+0i}\tilde{\epsilon}^{1}-iF^{+0i}\tilde{\epsilon}_{2}]}\\\end{array}\right)=0\,,
\begin{array}{rcl}{<P_{\mu},X^{\nu}>}&{=}&{i\delta_{\mu}^{\nu}}\\{<\Lambda^{\mu}{}_{\nu},M^{\alpha\beta}>}&{=}&{i\left(g^{\alpha\mu}\delta_{\nu}^{\beta}-g^{\beta\mu}\delta_{\nu}^{\alpha}\right)}\\{<\Lambda^{\mu}{}_{\nu},1>}&{=}&{\delta_{\nu}^{\mu}}\\\end{array}
R_{\mu\nu\lambda}^{\kappa}=\Gamma_{{\nu\lambda},\mu}^{\kappa}-\Gamma_{{\mu\lambda},\nu}^{\kappa}+\Gamma_{\mu\sigma}^{\kappa}\Gamma_{\nu\lambda}^{\sigma}-\Gamma_{\nu\sigma}^{\kappa}\Gamma_{\mu\lambda}^{\sigma}
{J_{5}^{\mu}}=-\sum_{r=0,2,...}{\tilde{\overline{\Psi}}}_{L_{r}}\gamma^{\mu}\tilde{\Psi}_{L_{r}}+\sum_{s=1,3,...}{\tilde{\overline{\Phi}}}_{L_{s}}\sigma^{3}\gamma^{\mu}\tilde{\Phi}_{L_{s}}.\nonumber
\begin{array}{rcl}{\kappa(z)\kappa(-z)}&{=}&{\displaystyle\frac{h(z+w)h(-z+w)}{h^{2}(w)},}\\{\kappa(z)\kappa(-z-nw)}&{=}&{\displaystyle\frac{h(-z)h(z+nw)}{h^{2}(w)}.}\\\end{array}
{\calA}_{\mu\nu}=-{\calA}_{\nu\mu}=-\frac{\mathrm{i}}2\langle\psi_{\varepsilon}^{+}(\sigma_{\mu\nu}[\mathrm{e}^{-\varepsilon\cdot\stackrel{\rightarrow}{D}},\gamma\cdot\stackrel{\rightarrow}{D}]\mathrm{e}
[-{\frac{d^{2}}{dv^{2}}}+{\frac{1}{\psi}}{\frac{d^{2}\psi}{dv^{2}}}+n^{2}e^{(\rho^{2}-\rho_{0}^{2})/\rho_{0}}({\frac{\rho}{\rho_{0}}})^{{\frac{D-1}{D-2}}}]\varphi(v)=m^{2}\varphi(v)~.
\widehat{{\calM}}^{(B=0)}=\hat{V}^{(B=0)}(x,t)(\hat{V}^{B=0)})^{T}(x,{\frac{1}{t}})=\left(\begin{array}{cc}{{\calM}(\omega)}&{0}\\{0}&{{\calM}^{-1}(\omega)}\\\end{array}\right)
\left({\frac{\dot{a}_{0}}{a_{0}}}\right)^{2}={\frac{32\pi^{2}G_{5}^{2}\sigma}{9c^{4}\sqrt{I}}}\varrho+{\frac{32\pi^{2}G_{5}^{2}\sigma}{9c^{6}}}\left({\frac{1}{2}}{\frac{\dot{\Phi}^{2}}{c^{2}}}+V\right)+{\frac{4\piG_{5}}{3c^{2}}}\left(\Lambda+{\frac{4\piG_{5}}{3c^{4}}}\sigma^{2}\right)+{\frac{{\calC}c^{2}}{a_{0}^{4}}}-{\frac{kc^{2}}{a_{0}^{2}}},
\langleT_{\eta}^{\eta}\rangle^{C}=\frac1{2C}\phi_{c}^{^{\prime}}\phi_{c}^{^{\prime}}+2\xi\frac{C^{^{\prime}}}{C^{2}}\phi_{c}\phi_{c}^{^{\prime}}+\frac{3\xi}C(\frac{C^{^{\prime}}}C+k)\phi_{c}^{2}+\frac{m^{2}}2\phi_{c}^{2}-2\kappa\phi_{c}^{4}+\frac32\lambda^{2}\phi_{c}^{4}
{\calV}=e\left(\gamma^{-}{A}_{-}+\gamma^{i}{A}_{i}\right)\left[1+\frac{e}{2}\gamma^{+}\frac{1}{i\partial_{-}}\ast{\calW}_{-1}[\widehat{a}]\left(\gamma^{-}{A}_{-}+\gamma^{j}{A}_{j}\right)\right].
F=F^{(0)}-\frac3{\betamc^{2}}\left[\left(\frac58-\frac{3A}2+B\right)\fracN{\beta}+A\left(F^{(0)}+\beta\frac{\partialF^{(0)}}{\partial\beta}\right)+BV\frac{\partialF^{(0)}}{\partialV}\right].
P^{+}=\sum_{N>0}\left({\frac{N\pi}{L}}\right)a{_N^{\dagger}}a_{N}+\sum_{n>0}\left({\frac{n\pi}{L}}\right)\left[b{_n^{\dagger}}b_{n}+d{_n^{\dagger}}d_{n}\right]-{\frac{\pi}{2L}}z_{R}^{2}\;,
\left\langle\frac{1}{2N}\mathrm{Tr}\,(\vec{\sigma}(\vec{A}\times\vec{A})\vec{\sigma}(\vec{A}\times\vec{A}))\right\rangle=\left\langle\frac{6}{N}\mathrm{Tr}\,(A_{1}A_{2}A_{1}A_{2}-A_{1}^{2}A_{2}^{2})\right\rangle=-6\left(\frac{a^{2}}{4}\right)^{2}
\deltaT_{\beta_{1}...\beta_{d}}^{\alpha_{1}...\alpha_{u}}\to\deltaT_{\beta_{1}...\beta_{d}}^{\alpha_{1}...\alpha_{u}}+\xi^{\mu}\partial_{\mu}T_{\beta_{1}...\beta_{d}}^{\alpha_{1}...\alpha_{u}}-\partial_{\mu_{i}}\xi^{\alpha_{i}}T_{\beta_{1}...\beta_{d}}^{\alpha_{1}...\mu_{i}...\alpha_{u}}+\partial_{\beta_{j}}\xi^{\nu_{j}}T_{\beta_{1}...\nu_{j}...\beta_{d}}^{\alpha_{1}...\alpha_{u}}.
\theta_{\mu\nu}~=~g_{ij}(\varphi)\,\partial_{\mu}\varphi^{i}\,\partial_{\nu}\varphi^{j}\,-\,{\textstyle{\frac{1}{2}}}\,\eta_{\mu\nu}\,\eta^{\kappa\lambda}\,g_{ij}(\varphi)\,\partial_{\kappa}\varphi^{i}\,\partial_{\lambda}\varphi^{j}~~,
\left(\begin{matrix}{\Gamma^{\phi_{P}^{0}11}(p)}&{\Gamma^{\phi_{P}^{0}12}(p)}\\{\Gamma^{\phi_{P}^{0}21}(p)}&{\Gamma^{\phi_{P}^{0}22}(p)}\\\end{matrix}\right)=\frac{-i\sum_{Q}m_{Q}^{2}}{[(p^{2})^{2}+\varepsilon^{2}][(k+h)^{2}+s^{2}-r^{2}]}\left(\begin{matrix}{(p^{2}-i\varepsilon)(k+h+is)}&{-ip^{2}r}\\{-ip^{2}r}&{-(p^{2}+i\varepsilon)(k+h-is)}\\\end{matrix}\right)
G_{\mu\nu}(q)=\frac{-i}{q^{2}+i\epsilon}\left[g_{\mu\nu}-\frac{q\cdotN^{\ast}(q_{\mu}N_{\nu}+q_{\nu}N_{\mu})}{q\cdotNq\cdotN^{\ast}+i\epsilon}+\frac{N^{2}(q\cdotN^{\ast})^{2}q_{\mu}q_{\nu}}{(q\cdotNq\cdotN^{\ast}+i\epsilon)^{2}}\right].
\left(\begin{matrix}{\Theta_{Q}(X)}\\{\Theta_{S}(X)}\\\end{matrix}\right)=\left(\begin{matrix}{{\calK}_{Q}{}^{+}(X)}&{{\calK}_{Q}{}^{-}(X)}\\{0}&{{\calK}_{S}{}^{-}(X)}\\\end{matrix}\right)\left(\begin{matrix}{\theta_{+}}\\{\theta_{-}}\\\end{matrix}\right)
\widetilde{\Omega}_{i}(A^{\mu},\pi_{\mu},\phi,\pi_{\phi};\Phi^{j})=\Omega_{i}+\sum_{n=1}^{\infty}\widetilde{\Omega}_{i}^{(n)}=0,~~~~~~\Omega_{i}^{(n)}\sim(\Phi^{j})^{n}.
F_{\mu\nu}^{(0)}:=i\,(k_{\mu}\zeta_{\nu}-k_{\nu}\zeta_{\mu})e^{i\,k\cdotx}+i\,({k^{\prime}}_{\mu}{\zeta^{\prime}}_{\nu}-{k^{\prime}}_{\nu}{\zeta^{\prime}}_{\mu})e^{i\,k^{\prime}\cdotx}
{\frac{\partial}{\partialu}}[p-p_{i}p^{i}/r_{1}^{2}-\gamma^{2}(u)/r_{1}^{2}r_{5}^{2}]\ll[p-p_{i}p^{i}/r_{1}^{2}-\gamma^{2}/r_{1}^{2}r_{5}^{2}]^{3/2},
\left.\sum_{i=1,2}\frac{\partial^{2}{W}}{\partial\varphi_{i}^{2}}\right|_{\{\varphi\}=\{\varphi_{\mathrm{vac}}(z=+\infty)\}}=\left.\sum_{i=1,2}\frac{\partial^{2}{W}}{\partial\varphi_{i}^{2}}\right|_{\{\varphi\}=\{\varphi_{\mathrm{vac}}(z=-\infty)\}}\,.
\frac{G^{\prime}(\frac{1}{2}+\frac{1}{\sqrt{2}}-i{\beta},\frac{1}{2}-\frac{1}{\sqrt{2}}-i{\beta};\frac{iz_{0}}{{\hbar}c})}{G(\frac{1}{2}+\frac{1}{\sqrt{2}}-i{\beta},\frac{1}{2}-\frac{1}{\sqrt{2}}-i{\beta};\frac{iz_{0}}{{\hbar}c})}=-\frac{i}{2{\hbar}c}-(\frac{1}{2}+i{\beta})\frac{1}{z_{0}}.
ds^{2}\rightarrow\ell_{s}^{2}\bigg\{\frac{1}{\sqrt{\lambda}}U^{\frac{7-p}{2}}(dx^{2})_{d}+\sqrt{\lambda}U^{\frac{p-7}{2}}dU^{2}+\sqrt{\lambda}U^{\frac{p-3}{2}}d\Omega_{8-p}^{2}\bigg\},
Z=N\int\prod_{\mu}\prod_{\nu}\prod_{\lambda}{\calD}a_{\mu}{\calD}B_{\nu}^{*}{\calD}B_{\lambda}{\calD}c{\calD}\bar{c}\mathrm{exp}\left(-\int_{0}^{\beta}d\tau\intd^{2}x{\calL}\right)\;,
\beta_{2}\!\left(\frac{gN}{2}\right)~=~2\beta_{1}^{2}\!\left(\frac{gN}{2}\right)~-~8~+~\frac{1}{2}\int_{0}^{gN}dy\left[y\beta_{1}\!\left(\frac{y}{2}\right)\beta_{1}^{\prime\prime}\!\left(\frac{y}{2}\right)~-~\frac{2\Omega_{2}(y)}{y^{2}}\right]
{\frac{(g_{m}\zeta)^{2}}{2}}\left(V^{4}+V^{*4}\right)={\frac{(g_{m}\zeta)^{2}}{2}}\mathrm{e}^{-{\frac{8\pi}{g^{2}}}\Lambda}:\left(V^{4}+V^{*4}\right):~.
\gamma^{0}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\gamma^{1}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right),\gamma^{5}=-i\gamma^{0}\gamma^{1}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right)
\sum_{\nu\in\Sigma(G,n,i)}1/|S_{\nu}|=\sum_{p_{1},\ldots,p_{i}\inT_{n}}\sum_{s_{1},t_{1},\ldots,s_{G},t_{G}\inS_{n}}\left[\frac{1}{n!}\delta(p_{1}\cdotsp_{i}\prod_{j=1}^{G}s_{j}t_{j}s_{j}^{-1}t_{j}^{-1})\right].
{Z_{3}}\,\langle\,\partialA_{x}^{a}\,\partialA_{y}^{b}\,\rangle\,=\,{Z_{\lambda}\,\lambda}\,\Big\{\,\delta^{ab}\delta_{xy}\,-\,i\widetildeZ_{1}\,\frac{1}{2}\,\langle\,(\partialA\timesc)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\,+\,\frac{Z_{\lambda}\,\lambda}{Z_{3}}\,\frac{{\widetildeZ}_{1}^{2}}{4}\,\langle\,\big((\bar{c}\timesc)\timesc\big)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\,\Big\}\;.
g^{2}\left(\begin{array}{c|c}{-\frac{1}{\bar{D}^{2}}\delta_{\sigma}\nu+\frac{1}{\bar{D}^{2}}\bar{D}_{\sigma}\frac{1}{E}\bar{D}_{\nu}\frac{1}{\bar{D}^{2}}}&{-i\frac{1}{\bar{D}^{2}}\bar{D}_{\sigma}\frac{1}{E}}\\{i\frac{1}{E}\bar{D}_{\nu}\frac{1}{\bar{D}^{2}}}&{\frac{1}{E}-1}\\\end{array}\right)
m\left[{\dot{\phi}}_{c}^{(i)}(\beta){\ddot{\phi}}_{c}^{(i)}(\beta)-{\dot{\phi}}_{c}^{(i)}(-\beta){\ddot{\phi}}_{c}^{(i)}(-\beta)\right]=-2E_{0}^{(i)}\int_{-\beta}^{\beta}({\dot{\phi}}_{c}^{(i)})^{2}d\tau
\alpha=2\frac{m_{V}^{2}}{m_{H}^{2}},~~~\mu=\frac{2\chi\Lambda}{m_{H}^{2}},~~~\nu=\chi\frac{m_{H}^{2}}{2\lambda},~~~\mu_{\mathrm{ph}}=\frac{16\pi\Lambda_{\mathrm{ph}}}{M_{P}^{2}m_{H}^{2}}.
tr(\prod_{i=1}^{2m+1}\Gamma_{a_{i}^{\prime}})=tr(\prod_{i=1}^{2m}\Gamma_{a_{i}^{\prime}}\Gamma_{a^{\prime\prime}})=tr(\prod_{i=1}^{2m+1}\Gamma_{a_{i}^{\prime}}\Gamma_{a^{\prime\prime}b^{\prime\prime}})=tr(\prod_{i=1}^{2m}\Gamma_{a_{i}^{\prime}}\Gamma_{a^{\prime\prime}}\Gamma_{b^{\prime\prime}}\Gamma_{c^{\prime\prime}})=0
\sum_{k=1}^{\infty}f_{k}=1,\qquad\sum_{k=1}^{\infty}{\frac{f_{k}}{k}}={\frac{\pi^{2}}{6}},\qquad\sum_{k=1}^{\infty}h_{k}(1)=1,\qquad\sum_{k=1}^{\infty}{\frac{h_{k}(1)}{k}}=0.
\bigtriangleup^{(3/2)}\psi_{\mu}=-(\gamma^{\alpha}\nabla_{\alpha})^{2}\psi_{\mu}=\left[\left(-\nabla^{\alpha}\nabla_{\alpha}+\frac14R\right)\delta_{\mu}^{\nu}-\frac12R_{~\mu\rho\sigma}^{\nu}\gamma^{\sigma}\gamma^{\rho}\right]\psi_{\nu}~~~,
\gamma_{\theta}\gamma_{S_{1}}\gamma_{\theta}^{-1}=\gamma_{S_{2}}~,~~~\gamma_{\theta}\gamma_{S_{2}}\gamma_{\theta}^{-1}=\gamma_{S_{3}}~,~~~\gamma_{\theta}\gamma_{S_{3}}\gamma_{\theta}^{-1}=\gamma_{S_{1}}~.
\begin{array}{rll}{<S_{i}^{+}S_{j}^{-}>=<S_{i}^{-}S_{j}^{+}>}&{\stackrel{N\rightarrow\infty}{=}}&{(-\Lambda)^{-j+i}(q^{3}+2+q^{-3})/(q^{2}+1+q^{-2})\;,}\\{<S_{i}^{z}S_{j}^{z}>}&{\stackrel{N\rightarrow\infty}{=}}&{(-\Lambda)^{-j+i}(q^{2}+2+q^{-2})/(q^{2}+1+q^{-2})\;.}\\\end{array}
I_{2n}^{B}=\intd^{2}w_{1}\cdotsd^{2}w_{2n}\prod_{1\leqi\not=j\leqn\atop2n\geqi\nej>n}(|w_{i}-w_{j}|^{2})^{\hat{\beta}^{2}}\prod_{1\leqi\leqn<j\leq2n}(|w_{i}-w_{j}|^{2})^{-\hat{\beta}^{2}}.
S=\int_{\tau^{\prime}}^{\tau^{\prime\prime}}d\tau(\dot{x^{\mu}}p_{\mu}+\frac{i}{2}(\dot{\theta^{\mu}}\theta_{\mu}+\dot{\theta_{5}}\theta_{5})-N{\calH}-M{\calQ}_{0})-\frac{i}{2}\theta(\tau^{\prime\prime})\cdot\theta(\tau^{\prime})-[B]_{\tau^{\prime}}^{\tau^{\prime\prime}},
\overline{{F}}_{\sigma}^{(n)}(\zeta)^{kk_{1}\cdotsk_{2n-1}k}=\prod_{a\inA}\oint_{C_{a}}\frac{dw_{a}}{2\pi\sqrt{-1}w_{a}}\Psi_{\sigma}^{(n)}(w|\zeta)Q^{(n)}(w|\zeta)^{kk_{1}\cdotsk_{2n-1}k}.
\left[\sigma_{-}^{(n)}(\widehat{x}),\sigma_{+}^{(n)}(\widehat{x})\right]=-\sigma_{3}^{(n)},\qquad\left[\sigma_{\pm}^{(n)}(\widehat{x}),-\sigma_{3}^{(n)}\right]=\pm2\sigma_{\pm}^{(n)}(\widehat{x}).
V^{\mu}(\vec{t})=\frac1{2\pi}\sum_{k,n=0,\pm1,\cdots}e^{i(k-n)\theta}\left[v_{\uparrow\downarrow}^{\mu}\bar{\psi}_{-n}^{\uparrow}(t)\psi_{k}^{\downarrow}(t)+v_{\downarrow\uparrow}^{\mu}\bar{\psi}_{-n}^{\downarrow}(t)\psi_{k}^{\uparrow}(t)\right]
S_{F,n}=\intd^{3}x\,\left[{\bar{\chi}}_{n}^{(1)}(\not\!d+\omega_{n}+e\varphi)\chi_{n}^{(1)}\,+\,{\bar{\chi}}_{n}^{(2)}(x)(\not\!d-\omega_{n}-e\varphi)\chi_{n}^{(2)}(x)\right]\;.
\Psi_{1}\equiv\pi_{a}\pi^{a}-m^{2}+{\frac{2eF_{ab}\pi^{a}}{\pi^{2}}}\Theta_{2}^{b}\approx0~~;~~\Psi_{2}\equiv\pi.S-{\frac{mj}{2}}+{\frac{ejF_{ab}\pi^{a}}{2m\pi^{2}}}\Theta_{2}^{b}\approx0,
T_{0}(z^{\prime})T_{1}(z)=\frac{\frac{c_{1}}{2}I_{1}(z)+\frac{c_{2}}{2}I_{0}}{(z^{\prime}-z)^{4}}+\frac{T(z)+2T_{1}(z)}{(z^{\prime}-z)^{2}}+\frac{\partial_{z}T_{1}(z)}{z^{\prime}-z}+\cdots\:\:\:.
\frac{p_{r}^{2}}{g_{rr}}+\frac{p_{\theta}^{2}}{g_{\theta\theta}}+\frac{-{g^{\prime}}_{tt}}{-\calD}\left(p_{\phi}+\frac{g_{t\phi}+\Omega_{0}g_{\phi\phi}}{{g^{\prime}}_{tt}}E\right)^{2}=\left(\frac{1}{-{g^{\prime}}_{tt}}E^{2}-\mu^{2}\right),
\begin{array}{rcl}{}&{}&{G=4\partialC+6F^{1\,m}B_{m}\,,}\\{}&{}&{}\\{}&{}&{H_{m}=3\partialB_{m}+3\epsilon_{mnp}F^{1\,n}A^{2\,p}\,,}\\{}&{}&{}\\{}&{}&{F^{2\,m}=2\partialA^{2\,m}\,,}\\\end{array}