harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{A}\frac{\partial\phi}{\partial\xi^{A}}\partial_{i}\partial^{i}\xi^{A}+\sum_{A,B}\frac{\partial^{2}\phi}{\partial\xi^{A}\partial\xi^{B}}\partial_{i}\xi^{A}\partial^{i}\xi^{B}=i(\partial_{i}A^{i})\phi+2iA^{i}\sum_{A}\frac{\partial\phi}{\partial\xi^{A}}\partial_{i}\xi^{A}+A_{i}A^{i}\phi\;\;.
u_{t}^{i}(x)=\{u^{i}(x),H\}=\hat{A}_{ij}\frac{\partialH}{\partialu^{j}(x)}=(g^{ij}\partial_{k}\partial_{j}h+b_{k}^{ij}\partial_{j}h)u_{x}^{k}=v_{k}^{i}(u)u_{x}^{k}
\Psi_{\alpha}(\vec{n}a,t)=\frac{1}{\sqrt{V}}\sum_{k_{1},k_{2},k_{3}=-\frac{N}{2}}^{\frac{N}{2}-1}\tilde{\psi}_{\alpha}(\frac{\pi\vec{k}}{L},t)e^{i\frac{\pi}{L}\vec{k}\cdot\vec{n}a},
exp\bigg(-\int_{X}\tilde{L}_{\omega+}^{**}\bigg)=\int{\calD}R^{+}exp\Bigg(-\int_{X}\epsilon^{\mu\nu\rho\sigma}\big(\tau^{+}R_{\mu\nu}^{+ab}R_{ab\rho\sigma}^{+}+iR_{\mu\nu}^{+\\ab}G_{\rho\sigmaab}^{+}\big)\Bigg).
I_{W}^{\ast}=\frac{L}{2}\int_{\bfR}dt\int_{\Sigma}\eta_{ab}\{-^{(2)}A^{(+)a}\wedge^{\mathrm{~}(2)}\dot{A}^{(+)b}+^{(2)}A^{(-)a}\wedge^{\mathrm{~}(2)}\dot{A}^{(-)b}\}.
\sigma^{2}=\left(\frac1{2m}e_{\mu\nu\alpha\beta}p_{\alpha\beta}\right)^{2}=\frac1{m^{2}}\left(\frac{1}{2}J_{\mu\nu}^{2}p^{2}-J_{\mu\sigma}J_{\nu\sigma}p_{\mu}p_{\nu}\right).
\langle0\vert(a_{n})^{r_{n}}\cdot...\cdot(a_{1})^{r_{1}}(a_{1}^{\dagger})^{r_{1}}\cdot...\cdot(a_{n}^{\dagger})^{r_{n}}\vert0\rangle\quad=\prod_{i=1}^{n}[r_{i}]_{q}!
T\sum_{n=-\infty}^{\infty}f(p_{3})=\frac{1}{2}\int_{-\infty}^{\infty}\frac{dz}{2\pi}[f(z)+f(-z)]+\int_{-\infty+i\epsilon}^{\infty+i\epsilon}\frac{dz}{2\pi}\left(\frac{f(z)}{e^{-i\betaz+\beta\mu}-1}+\frac{f(-z)}{e^{-i\betaz-\beta\mu}-1}\right)\;,
{\calS}_{\mathrm{probe}}=-M_{1}\intd^{2}\sigma\left[1-\frac{Q_{1}}{r^{2}}+\frac{2V}{(4\pi^{2}\alpha^{\prime})^{2}}\frac{Q_{1}}{r^{2}}-\frac{2V}{(4\pi^{2}\alpha^{\prime})^{2}}\frac{Q_{1}}{r^{2}}\right]=-M_{1}\intd^{2}\sigma\left[1-\frac{Q_{1}}{r^{2}}\right]
f(\lambda)=\left\langle\frac{\theta(|\hat{\bf{q}}^{\prime\prime}+\lambda\hat{\bf{e}}|)-1}{(\hat{\bf{q}}^{\prime\prime}+\lambda\hat{\bf{e}})^{2}}\right\rangle_{\hat{\bf{q}}^{\prime\prime}}=\frac{\Omega_{D-1}}{\Omega_{D}}\int_{-\lambda/2}^{1}dx\frac{(1-x^{2})^{\frac{D-3}{2}}}{1+2\lambdax+\lambda^{2}}\;.
{\calF}_{N}\star_{N}{\calG}_{N}({\calW})={\calF}_{N}\,{\calG}_{N}({\calW})+\sum_{m=1}^{N}\frac{(N-m)!}{N!\,m!}\,{\calF}_{N}({\calW})\,\,\vdots\underbrace{(\partial^{^{\!\!\!\!\!\leftarrow}}\,K\,\partial^{^{\!\!\!\!\rightarrow}})\cdots(\partial^{^{\!\!\!\!\!\leftarrow}}\,K\,\partial^{^{\!\!\!\!\rightarrow}})}_{m\,factors}\,\vdots\,{\calG}_{N}({\calW})\,.
{\frac{g}{\gamma}}={\frac{\int_{0}^{t_{f}}{\frac{dt}{{\calR}_{g}(t)}}\sqrt{h({\calR}_{g})-{\frac{\dot{\calR}_{g}(t)^{2}}{h({\calR}_{g})}}}}{\int_{0}^{t_{f}}{\frac{dt}{{\calR}(t)}}{\frac{d\tau}{dt}}}}\,.
(G^{-1})^{i\bar{j}}=\left(\begin{matrix}{L^{2}+4b^{2}C/(3|H|^{2})}&{-2bC/(3H)}&{-\frac{2}{3}bC}\\{-2bC/{3\bar{H}}}&{\frac{1}{3}{C}}&{\frac{1}{3}{HC}}\\{-\frac{2}{3}{bC}}&{\frac{1}{3}{\bar{H}C}}&{\frac{1}{3}C(|H|^{2}+C)}\\\end{matrix}\right),
\beta(B_{12}+F_{12})m_{(r,s;0)}=\beta(B+F)m_{(r,s;0)}=\beta(r-s)m_{(r,s;0)}+\sum_{\ell=1}^{\lfloor(r-s-1)/2\rfloor}2\beta(r-s-2\ell)m_{(r-\ell,s+\ell;0)}\,.
\delta_{h}{S}=\int_{t_{1}}^{t_{2}}dt\{[{L}_{q^{i}}]\delta_{h}q^{i}+[\frac{d}{dt}{H}+\frac{\partial}{\partialt}{L}]\deltat+\frac{d}{dt}(\frac{\partial{L}}{\partial\dot{q}^{j}}\delta_{h}q^{j}-{H}\deltat)\}.
V_{abc}(k^{a},k^{b},k^{c})=\frac{2g}{\sqrt{LV}}\bigg[\frac{\delta_{bc}}{p_{a}}(k_{a}^{b}p_{a}-k_{a}^{a}p_{b})+\frac{\delta_{ac}}{p_{b}}(k_{b}^{b}p_{a}-k_{b}^{a}p_{b})-\frac{\delta_{ab}}{p_{c}}(k_{c}^{b}p_{a}-k_{c}^{a}p_{b})\bigg].
{\calL}_{\mathrm{bulk}}+{\calL}_{\mathrm{source}}\sim\intd^{n}\xi\;\Bigl[{\frac{1}{g_{I}^{2}}}(\partial_{\xi}m_{K})^{2}+{\frac{1}{g_{I}}}\sum_{s}f_{s}(m_{K})\delta(\xi-\xi_{s})\Bigr].
P\equiv\int_{\phi(-\infty)=\phi_{f}}^{\phi(\infty)=\phi_{f}}\!\![D\phi]_{FP}\,e^{-\frac{1}{\hbar}\int_{-\infty}^{\infty}\!\!d\sigma\sqrt{2V[\phi]\int\!d^{3}x\dot{\phi}^{2}}},
\betaJ_{ij}=-\frac{1}{4}A_{ij},\quade^{A_{ij}}=\frac{(a_{i}-a_{j})(b_{i}-b_{j})(a_{i}-b_{j})(b_{i}-a_{j})}{(a_{i}+a_{j})(b_{i}+b_{j})(a_{i}+b_{j})(b_{i}+a_{j})},
{\hat{\vartheta}}{\binom{u}{v}}={\hat{h}}{\binom{u}{v}}+{\binom{G_{2}^{u}(u,v)}{G_{2}^{v}(u,v)}}+{\binom{G_{3}^{u}(u,v)}{G_{3}^{v}(u,v)}}+\cdot\cdot\cdot
\widetilde{S}=\left(\begin{array}{ccc}{2S_{a,b}}&{S_{a,f}}&{S_{a,f}}\\{S_{f,b}}&{\frac{1}{2}}&{-\frac{1}{2}}\\{S_{f,b}}&{-\frac{1}{2}}&{\frac{1}{2}}\\\end{array}\right),
K_{n}^{(N)}=-\sum_{n^{\prime}=1}^{n}(-1)^{n+n^{\prime}}\sum_{k=1}^{N}{\frac{(-igh)^{k}}{k!}}{\calA}_{n^{\prime}}^{k}K_{n^{\prime}}^{(N-k)}+{\frac{(igh)^{N}}{2^{N}N!}}(-1)^{n}{\calA}_{0}^{N}K_{0}^{(0)},
\begin{array}{l}{ab=q^{-1}ba,\;\;\;\;\;\;\;ac=q^{-1}ca,}\\{bc=cb,\;\;\;\;\;\;\;dc=qcd,}\\{db=qbd,\;\;\;\;\;\;\;da-ad=(q-q^{-1})bc,}\\{det_{q}T=ad-q^{-1}bc=1,}\\\end{array}
\lbrack\frac{5}{6(D+1)}\eta_{(\mu\nu}^{T}\epsilon_{\lambda)\beta}\alpha_{-1}^{\mu\nu\lambda}\alpha_{-2}^{\beta}+\epsilon_{\mu\nu}(\alpha_{-2}^{\mu}\alpha_{-3}^{\nu}-\frac{1}{2}\alpha_{-1}^{\mu}\alpha_{-4}^{\nu})]\left\vert0,k\right\rangle,\mathrm{~}
\tilde{\ell}_{2n+1}(\lambda):=\left(\begin{array}{cc}{-(\partial_{1}+2v)(\rho_{n}(\lambda)/2-R_{n})}&{e^{2w}\rho_{n}(\lambda)}\\{e^{-2w}(\lambda\rho_{n}(\lambda)-\partial_{1}(\partial_{1}+2v)(\rho_{n}(\lambda)/2-R_{n}))}&{(\partial_{1}+2v)(\rho_{n}(\lambda)/2-R_{n})}\\\end{array}\right).
M^{2}(t)=\frac{1}{2}v_{\mathrm{R}}^{2}-\frac{1}{2}\hat{m}^{2}a^{2}(t)\left(1+\frac{\lambda_{\mathrm{R}}}{16\pi^{2}}\bar{c}\right)+O(\lambda_{\mathrm{R}}^{2})\,,
D_{2}\eta_{1}^{\alpha_{1}}=\pi^{\alpha_{1}}+\frac12C_{\;\;\beta_{1}\gamma_{1}}^{\alpha_{1}}\eta_{1}^{\beta_{1}}\eta_{2}^{\gamma_{1}}+\frac12C_{\;\;\beta_{1}\gamma_{3}}^{\alpha_{1}}\left(\eta_{1}^{\beta_{1}}\eta_{2}^{\gamma_{3}}+\eta_{2}^{\beta_{1}}\eta_{1}^{\gamma_{3}}\right),
{\calW}_{\mu}=\intd^{2}x\,\Big(\varepsilon_{\mu\nu}n^{\nu}\bar{c}^{a}\,\frac{\delta}{\delta\phi^{a}}-A_{\mu}^{a}\,\frac{\delta}{\deltac^{a}}+\partial_{\mu}\bar{c}^{a}\,\frac{\delta}{\deltab^{a}}\Big)\quad
\tilde{\Psi}^{-}=i\frac1r(2k+1)^{-1/2}\left(\begin{array}{c}{(k-m+1/2)^{1/2}Y_{k}^{m-1/2}(\vartheta,\varphi)A(r)}\\{-(k+m+1/2)Y_{k}^{m+1/2}(\vartheta,\varphi)A(r)}\\\end{array}\right)
\eta(\alpha)u^{\beta(\alpha)}(-\vec{p})=-(-1)^{\alpha}\gamma^{2}\gamma^{4}\gamma^{5}u^{\alpha\,*}(\vec{p})\,,\qquade_{\mu}^{-\ell*}(-\vec{q})=-R_{\mu\nu}e_{\nu}^{\ell}(\vec{q})\,,
Z(i\lambda\,;R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)=\pi^{\zeta/2}\,e^{\mp\frac{i\pi}{4}\eta}\,\lambda^{-\zeta/2}\,\widetilde{Z}(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)\,.
S=\intd^{2}\sigma\sqrt{g}\left(\frac{1}{4\pi\alpha^{\prime}}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}g_{\mu\nu}(X)+\frac{i}{2\pi}\epsilon^{\alpha\beta}B_{\mu\nu}(X)\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}-\frac{1}{4\pi}\phi(X)R\right)
I_{n}=\int_{D}d^{2}\xiR^{n}(\xi_{i})=\int_{D}d^{2}\xi\Biggl(\epsilon_{kl}\frac{\partial}{\partial\xi_{k}}\left(p_{m}(\xi_{i})\frac{\partialx_{m}(\xi_{i})}{\partial\xi_{l}}\right)\Biggr)^{n}.
\bar{S}=S+\intd^{D}x\left(\varphi_{a}^{*(m)}C_{m}^{a}+\left(\varepsilon_{mnp}C_{na}^{*(m)}+\bar{\varphi}_{a}^{(p)}\right)p_{p}^{a}-\left(p_{ma}^{*(m)}+\bar{C}_{ma}^{(m)}-\tilde{\varphi}_{a}\right)l^{a}\right)\,.
N_{Q5}(-i\Omega_{m},\stackrel{\rightharpoonup}{p})=-\frac{d_{Q}(R)}{2}\int\frac{d^{3}l}{(2\pi)^{3}}T\sum_{n=-\infty}^{\infty}\mathrm{tr}[i\gamma_{5}S_{Q}(-i\omega_{n}+\mu_{Q},\stackrel{\rightharpoonup}{l})i\gamma_{5}S_{Q}(-i\omega_{n}-i\Omega_{m}+\mu_{Q},\stackrel{\rightharpoonup}{l}+\stackrel{\rightharpoonup}{p})].
\gamma_{m}^{\mu\nu}=\left(\begin{array}{cc}{~1}&{~0}\\{~0}&{~1}\\\end{array}\right),~\left(\begin{array}{cc}{~0}&{-1}\\{-1}&{~0}\\\end{array}\right),~\left(\begin{array}{cc}{~1}&{~0}\\{~0}&{-1}\\\end{array}\right).
\int_{-\infty}^{\infty}\left(\frac{1}{2}\phi^{2}+U(\phi)\right)dy\rightarrow\int_{-\infty}^{\infty}\left(\frac{-3\xi^{2}}{8\kappa^{2}}+\frac{3\xi}{2\kappa^{2}}\delta(y)\right)dy\;.
\psi(2)_{b=d=0}=i\int\frac{d^{4}x_{1}d^{4}p^{\prime}}{(2\pi)^{4}}e^{-ip_{\mu}^{\prime}x_{2}^{\mu}}\;\frac{{p_{\nu}}^{\prime}\gamma^{\nu}}{p_{\tau}^{\prime}p^{\tau\,\prime}-i\epsilon}e^{ip_{\eta}^{\prime}x_{1}^{\eta}}N_{\sigma}\gamma^{\sigma}\psi(1).
\Sigma+{\frac{1}{2x_{0}^{+}}}=\frac{Ke^{2\Phi}}{\sqrt{\left(1-{\frac{N}{12}}e^{2\Phi}\right)}}=\frac{K}{\sqrt{-\lambda^{2}x_{0}^{+}x^{-}}\sqrt{-\lambda^{2}x_{0}^{+}x^{--}{\frac{N}{12}}}}\;.
Z=\beta^{N/2}\Gamma((N-1)/2)^{-1}{\LARGE\int_{0}^{\infty}}d\sigma{\LARGE\int_{-\infty}^{\infty}}d\sigma_{1}(4\beta^{2}\sigma^{2}+(2\beta\sigma_{1}+v)^{2}+m^{2})^{M}\sigma^{N-2}exp[-\beta(\sigma^{2}+\sigma_{1}^{2})]
S=\intd^{d+1}x\sqrt{-g}\Big\{\frac{R}{2\kappa^{2}}-\Lambda_{b}+\lambda^{\prime}\big(\alphaR^{2}+\betaR_{AB}R^{AB}+\gammaR_{ABCD}R^{ABCD}\big)\Big\}+S_{B}+S_{m}
\langle\partialX^{i}(z_{1})\partialX^{j}(z_{2})\rangle=\delta^{ij}\frac{1}{2}\left(\frac{\sqrt{z_{1}}}{\sqrt{z_{2}}}+\frac{\sqrt{z_{2}}}{\sqrt{z_{1}}}\right)\frac{1}{(z_{1}-z_{2})^{2}}.\quad(i,j=4,5,6,7)
S_{{\calM}_{6}}=T_{5}\int_{{\calM}_{6}}d^{6}y\big[-\sqrt{-det(g_{mn}+{}^{*}{H}_{mnp}v^{p})}+\sqrt{-g_{6}}{\frac{1}{4}}v_{l}{~}^{*}H^{lmn}H_{mnp}v^{p}\big],
\mathrm{Index}=\int_{M_{4}}\frac{\mathrm{ch}(\Omega^{2,0}\oplus\Omega^{1,1}\,\ominus\,\Omega^{2,0}\otimes\Omega^{1,1}\,\oplus\,\Omega^{4,0})\,\mathrm{td}(TM_{4}\otimes{\bfC})}{\mathrm{e}(TM_{4})}\,=5\chi-7\tau\,,
(D_{\mu}{\mit\Phi})^{a}=\partial_{\mu}{\mit\Phi}^{a}+gf^{aib}(\bar{A}+\tilde{A})_{\mu}^{i}{\mit\Phi}^{b}=(\bar{D}_{\mu}{\mit\Phi})^{a}+gf^{aib}\tilde{A}_{\mu}^{i}{\mit\Phi}^{b}
\hat{\hat{K}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\;=\;{\textstyle\frac{1}{2}}\left(\hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{c}}\hat{\hat{b}}}\;+\;\hat{\hat{T}}_{\hat{\hat{b}}\hat{\hat{c}}\hat{\hat{a}}}\;-\;\hat{\hat{T}}_{\hat{\hat{a}}\hat{\hat{b}}\hat{\hat{c}}}\right)\;,
|\mathrm{D}p_{\psi},\eta\rangle_{\mathrm{R}}^{(0)\,\,T}=\left(\bar{C}\bar{\gamma}^{0}...\bar{\gamma}^{r}\frac{1+\mathrm{i}\eta\bar{\gamma}}{1+\mathrm{i}\eta}\right)_{AB}|A\rangle|\widetildeB\rangle
J_{2}^{NLS}=\left(\begin{array}{cc}{D+\varepsilon^{2}\left\{\begin{array}{c}{\eta^{-1}\,D^{3}+D^{3}\,\eta^{-1}}\\{-\frac{1}{2}\left[(\eta^{-1})_{xx}\,D+D\,(\eta^{-1})_{xx}\right]}\\\end{array}\right\}}&{\frac{1}{2}\,D\,\upsilon}\\{\frac{1}{2}\,\upsilonD}&{\frac{1}{2}(\eta\,D+D\,\eta)}\\\end{array}\right)
\tilde{\Theta}(\bar{z})=M\Theta(z),\quad\tilde{\psi}^{\mu}(\bar{z})=\left(\frac{g+B}{g-B}\right)^{\mu}{_\nu}\psi^{\nu}(\bar{z}),\quad\tilde{\psi}^{i}(\bar{z})=-\psi^{i}(\bar{z}),
g^{\mu\alpha}g^{\nu\beta}g^{\rho\gamma}h_{\mu\nu\rho}h_{\alpha\beta\gamma}=\tilde{g}^{\mu\alpha}\tilde{g}^{\nu\beta}\tilde{g}^{\rho\gamma}\tilde{h}_{\mu\nu\rho}\tilde{h}_{\alpha\beta\gamma}\,\,.
T_{\mu\nu}[g]=T_{\mu\nu}[\hat{g}]+{\frac{1}{48\pi}}\left(-4\hat{\nabla}_{\mu}\hat{\nabla}_{\nu}\sigma+4\partial_{\mu}\sigma\partial_{\nu}\sigma+g_{\mu\nu}(4\hat{\Box}\sigma-2(\hat{\nabla}\sigma)^{2})\right).
B_{4}(\triangle)={\itTr}\Biggl(\frac{1}{180}\left(R_{\mu\nu\sigma\lambda}^{2}-R_{\mu\nu}^{2}\right)+\frac{1}{2}\left(Z+\frac{R}{6}\right)+\frac{1}{12}Y_{\mu\nu}Y^{\mu\nu}+\frac{1}{6}\Box\left(\frac{R}{5}+Z\right)\Biggr)
U^{\prime}(w^{\prime})=\prod_{X^{\prime}\in{\itG}}e^{-m^{\prime}\sum_{i^{\prime}\inJ_{X^{\prime}}}t_{i^{\prime}}^{\prime}-\lambda^{\prime}\sum_{\stackrel{i^{\prime}<j^{\prime}}{\left\{i^{\prime},j^{\prime}\right\}\inJ_{X^{\prime}}}}t_{i^{\prime}}^{\prime}t_{j^{\prime}}^{\prime}{\bf1}_{(w(t_{i^{\prime}}^{\prime})=w(t_{j^{\prime}}^{\prime}))}}\times
f_{m}(k)\=\-\frac{2\pikR}{4i}\left[J_{m}(kR)H_{m-\beta+1}^{(1)}-J_{m+1}(kR)H_{m-\beta}^{(1)}(kR)+\frac{\beta}{kR}J_{m}(kR)H_{m-\beta}^{(1)}(kR)\right]\.
AII_{T}=\int{\frac{\prod_{i=1}^{6}d^{2}A_{i}}{{T}^{5}\prod_{i<j}^{6}|A_{ij}|^{2}}}\,{\frac{d^{2}Z_{4}}{|Y(Z_{4})|^{2}}}\,{\frac{1}{|Y(Z_{1}^{0})Y(Z_{2}^{0})Y(Z_{3}^{0})|^{2}}}.
\left\{\varphi_{1},\varphi_{2}\right\}=\frac12\int(\left\langle{\calH}X_{1},X_{2}\right\rangle-\left\langle{\calH}X_{2},X_{1}\right\rangle)dx+\left\langle\left\langleB\partialX_{1},X_{2}\right\rangle\right\rangle,
m_{*}^{2}=m^{2}+{\frac{\lambda}{2}}\int_{-\infty}^{\infty}{\frac{dp}{2\pi}}\;{\frac{\sqrt{p^{2}+m_{*}^{2}}}{\mathrm{exp}\big(\beta\sqrt{p^{2}+m_{*}^{2}}\big)-1}}\;{\frac{1}{p^{2}+m_{*}^{2}}}\;,
\langle\omega_{1},\ldots,\omega_{n};\overline{\mathrm{out}}|\omega;\overline{\mathrm{in}}\rangle=\left(\frac{\sqrt{2\pi}}{\overline{\mu}}\right)^{n-1}\frac{\Gamma(1+i\omega)}{\Gamma(2-n+i\omega)}2\pi\delta(\omega_{1}+\ldots+\omega_{n}-\omega).
Z(m^{2},\lambda)|_{m^{2}=1}=1-{\frac{3}{32}}\,\lambda+{\frac{105}{2048}}\,{\lambda}^{2}-{\frac{3465}{65536}}\,{\lambda}^{3}+{\frac{675675}{8388608}}\,{\lambda}^{4}-{\frac{43648605}{268435456}}\,{\lambda}^{5}+O({\lambda}^{6})
\left[\bar{\partial}^{\Psi},L_{1}^{\Psi}\right]:\!k^{\nu}\bar{\partial}^{\Psi}X^{\nu}\!:=:\!\left(R_{\nu\mu}-C_{\nu}\!^{\sigma\rho}C_{\mu\sigma\rho}+iC_{\eta\nu\mu}\!^{;\eta}\right)k^{\nu}\bar{\partial}^{\Psi}\!X^{\mu}\!:+O(\epsilon^{4}).
\begin{array}{ccc}{\Phi(s,u)\,,\,\Phi(s,v)}&{\rightarrow}&{w(s)}\\{\Phi(s,u-v)\,,\,\Phi(s,v-u)\,,\,\zeta(s)}&{\rightarrow}&{-\,{\displaystyle\frac{w^{\prime}(s)}{w(s)}}}\\{\begin{array}{c}{\zeta(u)\,,\,\zeta(v)\,,\,\zeta(u-v)\,,\,\zeta(v-u)}\\{\zeta(s+u)\,,\,\zeta(s+t+u)}\\\end{array}}&{\rightarrow}&{0\rule{0mm}{7mm}}\\\end{array}
\hat{g}_{\hat{\mu}\hat{\nu}}=\left(\begin{array}{cc}{g_{\mu\nu}+g_{ab}\,V_{\mu}^{\,\,a}\,V_{\nu}^{\,\,b}}&{g_{ab}\,V_{\mu}^{\,\,b}}\\{g_{ab}\,V_{\nu}^{\,\,b}}&{g_{ab}}\\\end{array}\right)\,,
\tilde{W}(t)(\alpha^{*},\alpha)=\frac{g}{3!}\left(W_{abc}\alpha^{a}\alpha^{b}\alpha^{c}+3\alpha_{a}^{*}W_{\;\;bc}^{a}\alpha^{b}\alpha^{c}+3\alpha_{a}^{*}\alpha_{b}^{*}W_{\quadc}^{ab}\alpha^{c}+\alpha_{a}^{*}\alpha_{b}^{*}\alpha_{c}^{*}W^{abc}\right)
\frac{d\lambda_{4}}{d\Lambda}=-\frac{\Lambda\,T^{2}}{2\pi}\,g\left(\frac{\Lambda}{T}\right)\left[{\frac{15\,\lambda_{{6}}+24\,\rho_{{0}}\lambda_{{8}}}{{\Lambda}^{2}+4\,\rho_{{0}}\lambda_{{4}}}}-{\frac{\left(6\,\lambda_{{4}}+12\,\rho_{{0}}\lambda_{{6}}\right)^{2}}{2\,\left({\Lambda}^{2}+4\,\rho_{{0}}\lambda_{{4}}\right)^{2}}}\right]+3\lambda_{6}\frac{d\rho_{0}}{d\Lambda}\;,
\frac{1}{2}\mathrm{Tr}{\calA}_{\mu\nu}{\calA}^{\mu\nu}=\frac{1}{2}m^{2}\left[-n_{\mu}n^{\mu}\left(A_{1\nu}A_{1}^{\nu}+A_{2\nu}A_{2}^{\nu}\right)+n_{\mu}n_{\nu}\left(A_{1}^{\mu}A_{1}^{\nu}+A_{2}^{\mu}A_{2}^{\nu}\right)\right].
\bar{M}={\frac{\delta^{2}}{2\delta^{2}-\gamma}}{\frac{1}{2\ell\kappa_{2}^{2}}}\Phi_{H}^{2-{\frac{\gamma}{\delta^{2}}}},\\\\\\\\\bar{T}={\frac{1}{4\pi\ell}}\Phi_{H}^{1-{\frac{\gamma}{\delta^{2}}}},\\\\\\\\\bar{S}={\frac{2\pi}{\kappa_{2}^{2}}}\Phi_{H},
\chi=\frac{C_{F}g^{2}}{8\pi}\Biggl(1-\frac{g^{2}}{8\pi^{2}}\biggl[\frac{11C_{A}-2n_{f}}{3}\Biggl(\frac{1}{n-4}+ln\;\frac{m^{2}}{4\pi\nu^{2}}\Biggr)+\Phi+O(g_{0}^{2};n-4)\biggr]\Biggr),
S=\intdt_{1}dt_{2}\Big[\partial_{i}\phi^{\dagger}\partial_{i}\phi+\frac{1}{4\theta^{2}}t_{i}t_{i}\phi^{\dagger}\phi+\frac{i}{2\theta}\big\{\phi^{\dagger}\epsilon_{ij}t_{i}\partial_{j}\phi-\phi\epsilon_{ij}t_{i}\partial_{j}\phi^{\dagger}\big\}+m^{2}\phi^{\dagger}\phi\Big]
3\left({\frac{\dot{a(t)}}{a(t)}}\right)^{2}+{\frac{3\alpha^{2}}{8(\zeta+2)}}[y^{3}-(1-4\zeta){\frac{1}{y}}+{\frac{2\zeta}{a^{2}}}(y^{2}-3)]={\frac{8\piG\rho_{0}}{a^{3(\omega+1)}}}
\mathbb{K}\left(p,q\right)=\left[\begin{array}{cc}{\mathcal{K}_{11}\left(p,q\right)}&{\mathcal{K}_{12}\left(p,q\right)}\\{\mathcal{K}_{21}\left(p,q\right)}&{\mathcal{K}_{22}\left(p,q\right)}\\\end{array}\right].
{\calS}_{\mathrm{probe}}=-M_{1}\intd^{2}\sigma\left[\frac{R_{9}}{\sqrt{2\alpha^{\prime}}}-\frac{R_{9}^{2}}{2\alpha^{\prime}}\frac{Q_{1}}{r^{2}}+\frac{2V}{(4\pi^{2}\alpha^{\prime})^{2}}\frac{Q_{1}}{r^{2}}\right].
\Pi_{\mu\rho,\nu\sigma}^{\mathrm{{non-minimal}}}=+~\frac{\kappa^{2}(-p^{2})^{\frac{3}{2}}}{512}\left(P_{\mu\rho,\nu\sigma}^{(2)}+2(1+16\lambda)^{2}Q_{\mu\rho,\nu\sigma}\right).
V=\frac{NT_{0}}{L}\sqrt{(1+\frac{r^{2}}{{\tilde{L}}^{2}})(1+\frac{4L^{2}r^{4}}{N^{2}-1})}-\frac{2NT_{0}}{\sqrt{N^{2}-1}}\frac{r^{3}}{{\tilde{L}}}.
<p_{2},q_{2}|p_{1},q_{1}>=e^{-\frac{m\omega}{4\hbar}\left[(q_{2}-q_{1})^{2}+\left(\frac{p_{2}-p_{1}}{m\omega}\right)^{2}-\frac{2i}{m\omega}(p_{2}+p_{1})(q_{2}-q_{1})\right]}\\\,\\\<p,q|p,q>=1\\\.
x_{12,\mu}x_{34,\nu}\langleW_{\mu}^{I_{1}I_{2}...I_{r}I_{r+1}}(x_{2})W_{\nu}^{J_{1}J_{2}...J_{r}J_{r+1}}(x_{4})\rangle\cong(r+1)(r+1)!(1-(\frac{v}{u})^{2})P_{Y_{(r,1)}^{(\mathrm{sym})}}^{I_{1}...I_{r+1},J_{1}...J_{r+1}}(x_{24}^{2})^{-2(r+1)}
T_{MN}=\frac{1}{2}{\scrL}G_{MN}-\frac{\partial{\scrL}}{\partialG^{MN}}\,\quadT_{\mu\nu}^{0}=\frac{1}{2}L^{0}g_{\mu\nu}^{0}-\frac{\partialL^{0}}{\partialg_{0}^{\mu\nu}}\,\quadT_{\mu\nu}^{L}=\frac{1}{2}L^{L}g_{\mu\nu}^{L}-\frac{\partialL^{L}}{\partialg_{L}^{\mu\nu}}\.
\intd^{4}z\left(P(x,z)\:P(z,y)\:\sum_{k=1}^{K_{zy}}\frac{\partial{\cal{L}}(\lambda^{zy})}{\partial\lambda_{k}^{zy}}\:F_{k}^{yz}\:-\:\sum_{k=1}^{K_{xz}}\frac{\partial{\cal{L}}(\lambda^{xz})}{\partial\lambda_{k}^{xz}}\:F_{k}^{xz}\:P(x,z)\:P(z,y)\right)\;=\;0\;.
\begin{matrix}{\psi_{\alphaA}\rightarrow\psi_{\alpha\dot{A}},}&{\psi^{\dot{\alpha}A}\rightarrow\psi^{\dot{\alpha}\dot{A}},}\\{\lambda_{i\alphaA}\rightarrow\lambda_{i\alpha\dot{A}},}&{\lambda^{i\dot{\alpha}A}\rightarrow\lambda^{i\dot{\alpha}\dot{A}},}\\{\epsilon_{AB}\rightarrow\epsilon_{\dot{A}\dot{B}},}&{\epsilon^{AB}\rightarrow-\epsilon^{\dot{A}\dot{B}},}\\\end{matrix}
S_{s}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-G^{s}}\left[e^{-2\phi}\left\{{\calR}_{G^{s}}+4(\partial\phi)^{2}\right\}-{\frac{1}{2\cdot2!}}e^{2(a-{\frac{D-4}{D-2}})\phi}F_{2}^{2}\right].
\begin{array}{l}{u_{+}^{2}=u_{+};\\u_{+}u_{-}=0;\\v_{+}^{(n)}u_{+}=u_{-}v_{+}^{(n)}}\\{u_{+}v_{+}^{(n)}=v_{+}^{(n)}u_{-}=0;\\v_{+}^{(\ell)}v_{-}^{(m)}=-\delta_{\ellm}u_{0};\\v_{+}^{(\ell)}v_{+}^{(m)}=\epsilon_{\ellmn}v_{-}^{(n)}.}\\\end{array}
\tilde{T}_{a}\Psi^{(0)}=\left(\begin{array}{c}{e^{i\chi(a;s^{k})}\phi^{(0)}(s^{i}+a^{i})}\\{e^{-i\chi(a;s^{k})}\phi^{\ast(0)}(s^{i}+a^{i})}\\{A_{k}^{(0)}(s^{i}+a^{i})-\frac{1}{q}\partial_{k}\chi(a;s^{i})}\\\end{array}\right),
\chi_{1,2}\equiv2^{-{\frac{\alpha}{2}}}e^{\sqrt\alpha\phi_{0}}\left(e^{{\frac{K_{\mathrm{matt}}}{2}}}|W|\right)|_{T_{1,2}}=e^{\sqrt\alpha\phi_{0}}\sqrt{-V_{0}(\tau_{1,2})/(3-\alpha)}.
{}~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(\omegau_{2}^{\prime}u_{3},\sigma+j_{2}+j_{3})w(qu_{2}^{\prime\prime},\sigma)s(\sigma,j_{1})}{w(q^{-1}u_{2}^{\prime},\sigma+j_{2})w(u_{2}^{\prime\prime}u_{3},\sigma+i_{3})}\Bigg\}_{0}.
\widehat{\Omega}_{ij}=\left(\begin{matrix}{1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\{0}&{0}&{P_{\mu\nu,\mu^{\prime}\nu^{\prime}}}&{0}\\{0}&{0}&{0}&{-i\delta_{ab}\gamma^{\sigma}\nabla_{\sigma}}\\\end{matrix}\right)
\prod_{n=1}^{L}b(w_{n})\prod_{i=1}^{V-L_{0}}Z(z_{0i})\prod_{i=1}^{L_{1}}W(z_{0i}^{\prime})=\sum_{\alpha}O_{\alpha}^{\prime}(z_{0})\epsilon^{k_{\alpha}}\prod_{n=1}^{L}(w_{n}-z_{0})^{j_{\alpha,n}}.
\begin{array}{ll}{\deltaA_{0}=i\bar{C},}&{\deltaA_{i}=i\frac{\partial_{i}\partial_{0}}{\nabla^{2}}\bar{C},}\\{\deltaC=-A_{0}+\frac{\partial_{i}\partial_{0}}{\nabla^{2}}A_{i}+\frac{g_{0}}{\nabla^{2}}\bar{\psi}\gamma_{0}\psi,}&{\delta\bar{C}=0,}\\{\delta\psi=[\frac{g_{0}}{\nabla^{2}}\partial_{0}\bar{C}]\psi,}&{\delta\bar{\psi}=\bar{\psi}\frac{g_{0}}{\nabla^{2}}\partial_{0}\bar{C}.}\\\end{array}
{\cal{L}}_{\Lambda}=\frac{1}{2}[(\partial\eta)^{2}-2\mu^{2}\eta^{2}]-\sqrt{\lambda}\mu\eta^{3}-\frac{\lambda}{4}\eta^{4}-\frac{1}{2}\delta\mu^{2}\left(\eta^{2}+\frac{2\mu}{\sqrt{\lambda}}\eta\right)+O(\hbar^{2}).
S=S_{0}^{L}\left[A_{\mu}^{a}\right]+\intd^{4}x\left(A_{a}^{*\mu}\left(\partial_{\mu}C^{a}+C_{\mu}^{a}\right)+C_{a}^{*}\eta^{a}-C_{a}^{*\mu}\partial_{\mu}\eta^{a}\right),
\psi_{+2\dot{q}}^{1-}=\frac{i}{32}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{+p}^{-2i},\\psi_{-2q}^{2+}=\frac{i}{32}\gamma_{q\dot{p}}^{i}\Omega_{-\dot{p}}^{+2i},\\nabla_{+p}\psi_{-2q}^{2+}=\nabla_{-\dot{p}}\psi_{+2\dot{q}}^{1-}=0,
K^{F}{}_{F}=\left(\begin{matrix}{-(c\partial)_{3}}&{-2\kappa[y_{h}(u\partial)_{2}+u(\partialy_{h})]}&{0}&{0}\\{-2(u\partial)_{\frac{3}{2}}}&{-(c\partial)_{2}}&{0}&{0}\\{0}&{0}&{-(c\partial)_{-1}}&{-2\kappa(1-\alpha)y_{h}(u\partial)_{-1}}\\{0}&{0}&{-2(u\partial)_{-\frac{1}{2}}}&{-(c\partial)_{-2}}\\\end{matrix}\right)\nonumber\,
\begin{array}{rclcrclcrclcrclcrcl}{\!\!\!f_{1}\!\!\!}&{=}&{\!\!\!-5!\,i\,\rho\!\!\!}&{;}&{\!\!\!f_{2}\!\!\!}&{=}&{\!\!\!-20\sqrt{2}\,a_{7}\!\!\!}&{;}&{\!\!\!\!f_{3}\!\!\!}&{=}&{\!\!\!-5!\,i\,\rho\!\!\!}&{;}&{\!\!\!f_{4}\!\!\!}&{=}&{\!\!\!-\frac{2}{\sqrt{2}}\,i\,a_{7}\,e^{\frac{\phi}{2}}\!\!\!}&{;}&{\!\!\!f_{5}\!\!\!}&{=}&{\!\!\!-30\,i\,\sqrt{2}\,a_{7}}\\\end{array}
(-i)S(p)={\frac{p^{2}+b^{2}-m^{2}+2\left(b\cdotp+mb_{\mu}\gamma^{\mu}\right)\gamma_{5}}{\left(p^{2}+b^{2}-m^{2}+i\varepsilon\right)^{2}-4\left[(b\cdotp)^{2}-m^{2}b^{2}\right]}}\left(\gamma^{\nu}p_{\nu}+m+b_{\nu}\gamma^{\nu}\gamma_{5}\right)\.
Tr\left(\frac{F}{2\pi}\right)^{4}=Tr\left(\frac{R}{2\pi}\right)^{4}+2(T\cdotT)^{2}-2(e\cdotR\cdote)^{2}+8(T\cdotR\cdote)(e\cdotT)+4(e\cdotR^{3}\cdote-T\cdotR^{2}\cdotT),
I=\intdtd^{2}{\bfr}(i\psi^{\ast}\partial_{t}\psi-A_{0}^{e}\psi^{\ast}\psi-{\frac{1}{2}}kr^{2}\psi^{\ast}\psi-{\frac{1}{2}}|(D_{1}^{e}-iD_{2}^{e})\psi|^{2})
{\psi}_{0}^{\prime}(x)=e^{i{t}^{a}{\omega}_{a}(x)}{\psi}_{0}(x),\\\\overline{{\psi}}_{0}^{\prime}(x)=\overline{{\psi}}_{0}(x)e^{-i{t}^{a}{\omega}_{a}(x)}.
\Phi(\tau,\theta,\overline{{\theta}})=Z(\tau)+\theta^{a}\chi_{a}(\tau)+\frac{i}{2}\overline{{\theta}}\theta\dot{Z}(\tau)+\theta\thetaF(\tau)-\frac{i}{4}\theta\theta\overline{{\theta}}_{a}\dot{\chi}^{a}-\frac{1}{16}\overline{{\theta}}\overline{{\theta}}\theta\theta\ddot{Z}(\tau)
\langle\langle\widehat\chi\rangle\rangle={\varepsilon^{\alpha}}_{\mu\beta}\left({\calF}_{+\alpha}\partial^{\mu}{\calF}_{+}^{\beta}+{\calF}_{-\alpha}\partial^{\mu}{\calF}_{-}^{\beta}\right)+m\left({\calF}_{+\gamma}{\calF}_{+}^{\gamma}-{\calF}_{-\gamma}{\calF}_{-}^{\gamma}\right).
\int_{0}^{\infty}dp\,e^{ip|\vec{x}|}\left({\frac{p}{\kappa}}\right)^{i\kappax_{0}}=\kappa\left(-i\kappa|\vec{x}|\right)^{-1-i\kappax^{0}}\Gamma\left(1+i\kappax^{0}\right)
{\calZ}=\int{\calD}A^{\mu}{\calD}B^{\mu\nu}{\calD}\theta{\calD}Q^{i}{\calD}\Phi^{i}{\calD}\Phi^{ij}{\calD}Q^{0}\delta(Q^{0})\delta(\Gamma_{\beta})det\mid\{\widetilde{\varphi}_{\alpha},\Gamma_{\beta}\}\mide^{iS_{T}},
\phi_{\dot{f}}^{\alpha}\to\tilde{U}_{\dot{f}}^{\alpha}v\,,\quadA_{\mu}\toi\tilde{U}\partial_{\mu}\tilde{U}^{\dagger}\,,\quad\tilde{U}_{\dot{f}}^{\alpha}=\frac{x_{\dot{f}}^{\alpha}}{\sqrt{x^{2}}}\,,
LL^{\dagger}=\left(\begin{array}{cc}{-\partial_{r}^{2}-{\frac{\partial_{r}-a^{\prime}}{r}}+{\frac{(a+j-1/2)^{2}}{r^{2}}}+f^{2}}&{2\partial_{r}f}\\{2\partial_{r}f}&{-\partial_{r}^{2}-{\frac{\partial_{r}}{r}}+{\frac{(j+1/2)^{2}}{r^{2}}}+f^{2}}\\\end{array}\right)
\sigma^{1}=\left(\begin{array}{cc}{0}&{1}\\{1}&{0}\\\end{array}\right),\qquad\sigma^{2}=\left(\begin{array}{cc}{0}&{-i}\\{i}&{0}\\\end{array}\right),\qquad\sigma^{3}=\left(\begin{array}{cc}{1}&{0}\\{0}&{-1}\\\end{array}\right).
\delta\theta^{\mu\nu}=-\theta^{\mu\alpha}\epsilon_{\alpha\beta}\theta^{\beta\nu}-\frac{1}{4}\theta^{\mu\alpha}\epsilon_{\alpha\gamma}\theta^{\gamma\rho}\epsilon_{\rho\beta}\theta^{\beta\nu}=-\theta^{\mu\alpha}\delta\theta_{\alpha\beta}^{-1}\theta^{\beta\nu}+O(\epsilon^{2}).
\mu_{12}=-{\frac{1}{6\pi^{2}}}{\frac{\pi^{D/2}}{2^{D}}}\left[-{\frac{(D-3)(D-1)}{D}}\Gamma\left(2-{\frac{D}{2}}\right)\zeta(2-D)+{\frac{\pi^{3/2-D}}{2}}(D+1)\Gamma\left({\frac{D+1}{2}}\right)\zeta(D+1)\right]\,{\frac{\hbarL^{D-1}}{a^{D}}}.
\begin{array}{rcl}{\chi_{1,r(k+1)+k}^{(p,kp+p-1)}(q)}&{=}&{\displaystyle\sum_{n_{1}\geq\cdots\geqn_{k}\geq0}\frac{q^{n_{1}^{2}+\cdots+n_{k}^{2}+r(n_{1}+\cdots+n_{k})}}{(q)_{n_{1}-n_{2}}\cdots(q)_{n_{k-1}-n_{k}}(q)_{2n_{k}+r}}F_{p-1;r}^{(2n_{k}+r)}(q).}\\\end{array}
\begin{array}{cc}{\bar{Q}_{i}=Q^{i\dagger}\gamma^{0}=(Q^{j})^{t}\bar{\calE}_{ji}~~~~}&{~~~~\bar{\tilde{Q}}_{i}=\tilde{Q}^{i\dagger}\tilde{\gamma}^{0}=(\tilde{Q}^{j})^{t}\bar{\calE}_{ji}}\\\end{array}
\begin{aligned}{\beta}&{{}=a_{1}-\left(a_{2,1}+\frac14a_{2,2}+\frac16a_{2,3}\right)R_{0},}\\{{m_{0}}^{2}}&{{}=\frac{\beta}{3a_{2,1}+a_{2,2}+a_{2,3}},}\\\end{aligned}\qquad\begin{aligned}{\gamma=\frac{a_{2,3}}{2\beta},}\\{{m_{2}}^{2}=-\frac{2\beta}{a_{2,2}+4a_{2,3}},}\\\end{aligned}
\begin{array}{rl}{\psi_{\pm}(x)=}&{\frac{1}{\sqrt{L}}\,e^{\pm\frac{i\pi}{L}x}\,e^{i\frac{\pi}{2}\etap_{\mp}}\,e^{\pmi\lambdaq_{\pm}}\,e^{\frac{2i\pi\lambda}{L}p_{\pm}x}}\\{}&{\times\prod_{n=1}^{\infty}\,e^{\pm\frac{i\lambda}{\sqrt{n}}a_{\pm,n}^{\dagger}e^{\pm\frac{2i\pi}{L}nx}}\prod_{n=1}^{\infty}\,e^{\pm\frac{i\lambda}{\sqrt{n}}a_{\pm,n}e^{\mp\frac{2i\pi}{L}nx}}\,}\\\end{array}