harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\gamma^{0}=\left(\begin{array}{clcr}{\sigma_{3}}&{\;\;\;0}\\{0}&{-\sigma_{3}}\\\end{array}\right)\;\;\;\gamma^{1}=\left(\begin{array}{clcr}{i\sigma_{1}}&{\;\;\;0}\\{0}&{-i\sigma_{1}}\\\end{array}\right)\;\;\;\gamma^{2}=\left(\begin{array}{clcr}{i\sigma_{2}}&{\;\;\;0}\\{0}&{-i\sigma_{2}}\\\end{array}\right)\;\;\;.
\psi_{\tau\alpha}=-{\frac{2}{3}}k^{1/2}a^{3}\delta_{\tau\tau_{2}}\epsilon_{\tau_{1}\tau_{3}}(\gamma_{5}\gamma_{\mu})_{\alpha\alpha_{1}}(C\gamma_{\mu})_{\alpha_{2}\alpha_{3}}(\lambda_{\tau_{1}\alpha_{1}}\lambda_{\tau_{2}\alpha_{2}}\lambda_{\tau_{3}\alpha_{3}}).
L\left(X_{\kappa},\Theta_{\kappa},A_{\kappa}\right)=L\left(X,\Theta,A\right)-\frac12L_{MN}\left[\Omega_{g}^{MN}-\Omega^{MN}\left(\Theta\right)\right]-\frac12\left(A_{\kappa}^{ij}-A^{ij}\right)X_{i}\cdotX_{j}.
A_{\mu}^{a}\left(x\right)=\sum_{\vec{k}}\left(\sum_{\sigma=1,2}A_{\vec{k},\sigma}^{a}f_{k,\mu}^{\sigma}\left(x\right)+A_{\vec{k}}^{L,a}f_{k,L,\mu}\left(x\right)+B_{\vec{k}}^{a}f_{k,S,\mu}\left(x\right)\right)+h.c.,
\begin{array}{lcr}{{(\bf\frac{{Q^{+}}_{-{\frac{1}{k}}})^{r}}{[r]!}}\vert\Omega_{s}^{+}\rangle,{\bf\frac{{(Q^{-}}_{-{\frac{1}{k}}})^{r}}{[r]!}}\vert\Omega_{-s}^{-}\rangle,}\\{\bf{Q^{\mp}}_{-{\frac{1}{k}}}\vert\Omega_{s}^{\pm}\rangle=0,\bf{Q^{-}}_{-{\frac{1}{k}}}=[\bf{Q^{+}}_{-{\frac{1}{k}}}]^{+}}\\{\bf{Q^{-}}_{1-{\frac{1}{k}}}=[\bf{Q^{+}}_{1-{\frac{1}{k}}}]^{+},}\\\end{array}
\gamma_{m}^{\mathrm{var}}(\alpha)=\frac{3}{2}\frac{\alpha}{\pi}-\frac{9}{32}\pi^{2}\left(\frac{\alpha}{\pi}\right)^{3}+\frac{27}{32}\pi^{2}\left(\frac{\alpha}{\pi}\right)^{4}-\frac{243}{128}\pi^{2}\left(1-\frac{\pi^{2}}{20}\right)\left(\frac{\alpha}{\pi}\right)^{5}+{\calO}(\alpha^{6}),
\beta={\frac{1}{T_{H}}}={\frac{\pir_{+}^{3}\sqrt{(r_{+}^{4}+q_{1})(r_{+}^{4}+q_{2})}}{3r_{+}^{8}+2r_{+}^{6}+r_{+}^{4}(q_{1}+q_{2})-q_{1}q_{2}}}.
(\hat{r}^{2}-1)\frac{d^{2}T_{\hat{E}\hat{N}}}{d\hat{r}^{2}}+(3\hat{r}-\frac{1}{\hat{r}})\frac{dT_{\hat{E}\hat{N}}}{d\hat{r}}+(\frac{\hat{E}^{2}}{\hat{r}^{2}-1}-\frac{\hat{N}^{2}}{\hat{r}^{2}}+4\lambda^{2}+1)T_{\hat{E}\hat{N}}=0.
E_{stat}\left[\bbox{\phi}\right]=\Lambda\int_{{\mathbbR}^{3}}d{\bfx}\;\frac{1}{2}\left(\partial_{i}\bbox{\phi}\right)^{2}+\frac{g_{1}}{8}\left(\partial_{i}\bbox{\phi}\times\partial_{j}\bbox{\phi}\right)^{2}+\frac{g_{2}}{8}\left(\partial_{i}\bbox{\phi}\right)^{2}\left(\partial_{j}\bbox{\phi}\right)^{2}\,.
j_{\mu}=\frac{1}{V}\sum_{{\bfk},{\bfk}^{\prime}}\frac{k_{\mu}+k_{\mu}^{\prime}}{2\sqrt{\omega_{{\bfk}}\omega_{{\bfk}^{\prime}}}}\,e^{i(k-k^{\prime})x}\,a_{{\bfk}}^{\dagger}a_{{\bfk}^{\prime}}.
\left[x\prod_{a=1}^{n_{c}}(x-\phi_{a}^{2})+2\Lambdam_{1}\cdotsm_{n_{f}}\right]^{2}-4\Lambda^{2}\prod_{i=1}^{n_{f}}(x+m_{i}^{2})=-4\Lambda^{2}x\prod_{a=1}^{n_{c}}(x-\alpha_{a})^{2}.
2\omega+\frac{m_{1}^{2}}{2p_{1}}-\frac{m_{2}^{2}}{2p_{2}}-\frac{m_{3}^{2}}{2p_{3}}=\deltaE_{D}^{(4)}+\frac{\xi}{M_{D}}\left(p_{2}p_{3}+(p_{2}+p_{3})\omega+\omega^{2}\right)
\Psi^{(0)}(\rho)=C_{m}\rho^{m+1}\left(\begin{array}{l}{e^{im\phi}K_{1}\left(\frac{\rho}{2m+2}\right)}\\{e^{i(m+1)\phi}K_{0}\left(\frac{\rho}{2m+2}\right)}\\\end{array}\right)e^{i\frac{2\pi}{L}kz}
\widehat{R}_{i\:\nu\rho\sigma}^{\mu}=\kappa_{i\:\rho}^{\mu}\kappa_{\nu\sigma}^{i}-\kappa_{i\:\sigma}^{\mu}\kappa_{\nu\rho}^{i}+h_{i\:\nu}^{\alpha}h_{i\:\rho}^{\beta}h_{i\:\sigma}^{\gamma}R_{\:\:\alpha\beta\gamma}^{\mu}
{\hat{A}}_{\hat{\imath}}=|{\hat{k}}|^{-2}{\hat{k}}_{\hat{\mu}}\left(\partial_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}-\frac{m}{2}\right.\left.l_{p}^{2}\,{\hat{b}}_{\hat{\imath}}{\hat{h}}^{\hat{\mu}}\right)\,.
\vert1\rangle=\epsilon^{i_{1}\cdotsi_{k-2}}\epsilon^{j_{1}\cdotsj_{k-2}}(J_{A_{1}B_{1}}\nu_{i_{1}}^{A_{1}}\nu_{j_{1}}^{B_{1}})\cdots(J_{A_{k-2}B_{k-2}}\nu_{i_{k-2}}^{A_{k-2}}\nu_{j_{k-2}}^{B_{k-2}})\vert\Phi\rangle\,.
\begin{array}{c}{E_{1,2}^{2}=p_{\bot}^{2}+m^{2}+\frac{g^{2}}4\left(\tau_{1}+\tau_{2}\right)\pmg\sqrt{\tau_{1}p_{1}^{2}+\tau_{2}p^{2}+\frac{\left(H_{z}^{3}\right)^{2}}4}}\\{E_{3}^{2}=p_{\bot}^{2}+m^{2},\\\left(p_{\bot}^{2}=p_{1}^{2}+p_{2}^{2}\right).}\\\end{array}
\mu^{2}-(q_{1}\pmq_{2})^{2}=\mu^{2}[1-(q_{1}^{+}\pmq_{2}^{+})({\frac{1}{q_{1}^{+}}}\pm{\frac{1}{q_{2}^{+}}})]=\mp\mu^{2}{\frac{(q_{1}^{+2}+q_{2}^{+2}\pmq_{1}^{+}q_{2}^{+})}{q_{1}^{+}q_{2}^{+}}},
\begin{array}{lll}{}&{[x^{\alpha},x^{\mu}]=0\,,}&{[\pi_{\alpha},p_{\mu}]=-i\hbar\partial_{\mu}A_{\alpha}^{\nu}\circp_{\nu}\,,}\\{}&{[x^{\mu},\pi_{\alpha}]=-i\hbarA_{\alpha}^{\mu}\,,}&{[x^{\alpha},p_{\mu}]=0\,.}\\\end{array}
<A_{2}|z>=\left(\frac{k\tau_{2}}{2\pi^{2}|\tau|^{2}}\right)^{1/4}\,e^{-\frac{1}{2}|z|^{2}}\,e^{\frac{k}{4i\pi\tau}A_{2}^{2}}\,e^{\frac{1}{2}\frac{i\bar{\tau}}{-i\tau}z^{2}+\frac{zA_{2}}{\tau}\sqrt{\frac{k\tau_{2}}{\pi}}}\\\.
\Gamma_{a}^{(11)}=\left(\begin{matrix}{0}&{\Gamma_{b}^{(9)}}\\{-\Gamma_{b}^{(9)}}&{0}\\\end{matrix}\right)\,,\quad\Gamma_{0}^{(11)}=\left(\begin{matrix}{0}&{{\bf1}_{2}}\\{{\bf1}_{2}}&{0}\\\end{matrix}\right)\,,\quad\Gamma_{10}^{(11)}=\left(\begin{matrix}{{\bf1}_{2}}&{0}\\{0}&{-{\bf1}_{2}}\\\end{matrix}\right)\,,
2{\frac{a}{a_{0}}}\left({\frac{\rho_{\psi}-\rho_{\mathrm{r}}}{2\rho_{\psi}+\rho_{\mathrm{m}}}}\right)=1-B^{2}={\frac{a_{\mathrm{min}}a_{\mathrm{max}}}{{a_{0}}^{2}}}.
S_{i}=-\frac{k}{2\pi}\int_{\calM}d^{2}x\left[\frac{1}{2y_{(i)}^{2}}\partial_{\mu}y_{(i)}\partial^{\mu}y_{(i)}+\frac{1}{4y_{(i)}^{2}}f_{(i)\mu\nu}f_{(i)}^{\mu\nu}\right]+(SU(2)\mathrm{~WZW~with~level~}k)_{i}
\frac{\kappa_{j}-ik_{j}}{\kappa_{j}+ik_{j}}^{2}e^{-2ik_{j}L}=\prod_{\nu=1}^{M}\frac{\lambda_{\nu}-k_{j}-ic/2}{\lambda_{\nu}-k_{j}+ic/2}\cdot\frac{\lambda_{\nu}+k_{j}-ic/2}{\lambda_{\nu}+k_{j}+ic/2}.
\frac1{[(k+k_{i})^{2}-m^{2}]}=\frac1{(k^{2}-m^{2})}+\sum_{j=1}^{N}\frac{\left(-1\right)^{j}\left(k_{i}^{2}+2k_{i}\cdotk\right)^{j}}{\left(k^{2}-m^{2}\right)^{j+1}}+\frac{\left(-1\right)^{N+1}\left(k_{i}^{2}+2k_{i}\cdotk\right)^{N+1}}{\left(k^{2}-m^{2}\right)^{N+1}\left[\left(k+k_{i}\right)^{2}-m^{2}\right]},
h_{\ell}^{\prime\prime}(it,r)=-2t\xi_{\ell}(tr)h_{\ell}^{\prime}(it,r)+\left[\sigma(r)-2t^{2}\left.\frac{d\xi_{\ell}(\tau)}{d\tau}\right|_{\tau=tr}\right]h_{\ell}(it,r)\,.
\left(T_{\lambda\mu\nu}^{AVV}(\left(k_{3}-k_{1}\right),\left(k_{1}-k_{2}\right))\right)_{phys}=T_{\lambda\mu\nu}^{AVV}(\left(k_{3}-k_{1}\right),\left(k_{1}-k_{2}\right))-T_{\lambda\mu\nu}^{AVV}\left(0\right),
\left[\tilde{H},\tilde{G}_{a}^{(2)}\right]=-f_{\;\;ab}^{c}A_{0}^{b}\left(\tilde{G}_{c}^{(2)}+2gA_{c}^{i}G_{i}^{(2)}\right)+2g\left(\pi_{a}^{i}+4gA_{aj}\pi^{ij}\right)G_{i}^{(2)},
\deltax^{\mu}=\imath\psi^{\mu}\epsilon\;,\;\;\deltae=\imath\chi\epsilon\;\;,\delta\psi^{\mu}=\frac{z^{\mu}}{2e}\epsilon\;\;,\delta\psi^{2n+1}=\frac{m}{2}\epsilon,\;\;\delta\chi=\dot{\epsilon}\;,\;\;\delta\kappa=0\;,
Z^{(\mathrm{cov)}}(G,p,t_{l}^{(A)})=\sum_{\{r_{l}^{(A)}\}}Z^{(\mathrm{cov)}}(G,p,r_{l}^{(A)})\prod_{A=1}^{p}\prod_{l=1}^{\infty}\left[t_{l}^{(A)}\right]^{r_{l}^{(A)}}~.
\sum_{N_{r}^{\prime}}\sum_{p=1,2}\int_{C_{p}^{(r)}}\chi_{N_{r}^{\prime}}(t)dt\left[E_{p,N_{r}}^{(-2)}(Y_{p,N_{r}^{\prime}}(t))-\partial_{q_{N_{r}}}Y_{p,N_{r}^{\prime}}(t)+(-1)^{e(N_{r}^{\prime})e(N_{r})}\partial_{q_{N_{r}^{\prime}}}Y_{p,N_{r}}(t)\right]=0
{\gamma\cdotD\Lambda={e_{{\hat{L}}}}^{M}\gamma^{\hat{L}}\left(\partial_{M}+\frac{1}{4}\omega_{M}^{\hat{M}\hat{N}}\gamma_{\hat{M}\hat{N}}\right)\Lambda=(\rho\gamma^{\hat{5}}\partial_{5}+\rho\gamma^{\hat{\mu}}\partial_{\mu}-2\gamma^{\hat{5}})\Lambda\;,}
{\calQ}_{4}\longrightarrowx_{1}x_{2}(x_{1}^{2}+x_{2}^{2})\left[(A_{1}^{2}+A_{2}^{2})x_{1}x_{2}+A_{1}A_{2}(x_{1}^{2}+x_{2}^{2})\right]\,\,\,.
Q^{\dagger}Qh_{\mu\nu}-\Box^{(4)}\tilde{h}_{\mu\nu}=\sum_{\sigma=\pm}\left(\frac{3}{4}U_{B}h_{\mu\nu}^{\sigma}+2\kappa_{5}^{2}\sqrta\Sigma_{\mu\nu}^{\sigma}\right)\delta(u-u_{\sigma})
<H|T_{\nu}^{\mu}(r)|H>=\left(\begin{array}{cc}{T_{\alpha}^{\alpha}-\Omega^{-1}(r)H(r)}&{0}\\{0}&{\Omega^{-1}H(r)\}\\\end{array}\right)+\Omega^{-1}(\frac{-\pi}{24a^{2}}+\frac{1}{192\pim^{2}})\left(\begin{array}{cc}{1}&{0}\\{0}&{-1\}\\\end{array}\right).
T_{-\dot{p}+2}^{+q}=\frac12\gamma_{q\dot{q}}^{i}\Omega_{-\dot{p}}^{+2i}\psi_{+2\dot{q}}^{1-}+\frac{\chi}{2}\gamma_{q\dot{p}}^{i}\Omega_{+2}^{+2i},\T_{-\dot{p}-2}^{+q}=\frac{\chi}{2}\gamma_{q\dot{p}}^{i}\Omega_{-2}^{+2i},
\left(\begin{array}{cccccc}{0}&{0}&{0}&{1}&{0}&{-1}\\{0}&{0}&{-1}&{0}&{1}&{0}\\{0}&{-1}&{0}&{0}&{0}&{1}\\{1}&{0}&{0}&{0}&{-1}&{0}\\{0}&{1}&{0}&{-1}&{0}&{0}\\{-1}&{0}&{1}&{0}&{0}&{0}\\\end{array}\right)~=\left(\begin{array}{ccc}{0}&{1}&{-1}\\{-1}&{0}&{1}\\{1}&{-1}&{0}\\\end{array}\right)\otimes\left(\begin{array}{cc}{0}&{1}\\{-1}&{0}\\\end{array}\right)~.
V_{-n}(\kappa,\rho;\tau)=\left(\frac{m}{\alpha}\right)^{n}\sum_{l=0}^{n}C_{n}^{l}\,e^{-2(n-2l)\kappa\tau}\,\,\prod_{j=0}^{l-1}\frac{1}{4\kappa^{2}+(n-2j)^{2}}\,\,\delta[2\rho-i(n-2l)],
S_{\Lambda}=\intd^{2}xdu\;\left({\frac{i}{2}}\lambda_{+}^{a}\partial_{--}\lambda_{+}^{a}+i\sigma_{-}^{--a}\partial^{++}\lambda_{+}^{a}+{\frac{1}{4}}g^{-aA^{\prime}}\partial^{++}g_{A^{\prime}}^{-a}\right)\;.
\begin{array}{c}{\pi_{\dot{q}}^{1-}=e^{+2}\psi_{+2\dot{q}}^{1-}+e^{-2}\psi_{-2\dot{q}}^{1-}+e_{q}^{+}h_{+q\dot{q}}^{1-}+e_{q}^{-}\chi_{q\dot{q}}^{1},}\\{\pi_{\dot{q}}^{2+}=e^{+2}\psi_{+2\dot{q}}^{2+}+e^{-2}\psi_{-2\dot{q}}^{2+}+e_{q}^{+}\chi_{q\dot{q}}^{2}+e_{q}^{-}h_{-q\dot{q}}^{2+}}\\\end{array}
\delta{\calL}^{[\frac{1}{2}]}=[\kappa\\bar{\kappa}]{\calP}_{+}\left[\begin{array}{c}{-\sqrt{-g}v^{i}\tilde{\calH}^{j}\gamma_{ij}\bar{P}}\\{-\sqrt{-g}v^{i}\bar{\tilde{\calH}}^{j}\gamma_{ij}P}\\\end{array}\right]
S_{E}\,=\,M_{0}^{2}\,\int_{0}^{T}\!\!dt\,\int_{0}^{R}\!\!dr\,\left[1\,+\,\frac{1}{2}\,{\bfu}\,(1\,+\,\alpha\,r_{s}^{2}\,\partial^{2})\,\partial^{2}\,{\bfu}\,\right]\,{.}
S_{D2f}=\frac{1}{D-1}\sum_{i\neqj}^{D}\frac{1}{N^{2}}\mathrm{Tr}\,M_{i}^{2}\mathrm{Tr}\,M_{j}^{2}\\\\\\\\mathrm{and}\\\\\\\S_{D4f}=\frac{1}{D-1}\sum_{i\neqj}^{D}\frac{1}{N^{2}}\mathrm{Tr}\,M_{i}^{4}\mathrm{Tr}\,M_{j}^{4}
S=\frac{1}{2\pi\alpha^{\prime}}\intd\sigma^{+}d\sigma^{-}\left(\partial_{-}X^{\mu}\partial_{+}X_{\mu}+\frac{i}{2}\psi_{+}^{\mu}\partial_{-}\psi_{+\mu}+\frac{i}{2}\psi_{-}^{\mu}\partial_{+}\psi_{-\mu}\right),
\sum_{\lambda=\pm}\epsilon_{\mu}(k,\lambda)\epsilon_{\nu}^{*}(k,\lambda)+\frac{1}{2}\left(e_{\mu}^{\left(+\right)}(k)e_{\nu}^{\left(-\right)}(k)+e_{\mu}^{\left(-\right)}(k)e_{\nu}^{\left(+\right)}(k)\right)=-g_{\mu\nu}
{\calE}=\int_{-\infty}^{+\infty}\mathrm{d}z\left[\partial_{z}\bar{\phi}\,\partial_{z}\phi+\bar{F}F\right]\equiv\int_{-\infty}^{+\infty}\mathrm{d}z\left\{\left[e^{-i\eta}\,\partial_{z}{\calW}+\mathrm{H.c.}\right]+\left|\,\partial_{z}\phi+e^{i\eta}\,F\,\right|^{2}\right\}\,,
\begin{array}{rcl}{H}&{\equiv}&{\frac{i}{2}({\bar{\partial}}-\partial)J_{0}=-\frac{1}{4}\left(du-zdv\right)\wedged{\bar{z}}\wedged{\bar{v}}-\frac14\left(d{\bar{u}}-\bar{z}d\bar{v}\right)\wedgedz\wedgedv}\\{}&{=}&{\frac14\left(\bar{\alpha}\wedge\bar{\beta}\wedge\gamma+\alpha\wedge\beta\wedge\bar{\gamma}\right)\,.}\\\end{array}
E_{non-disp.}=(\varepsilon-1)^{2}\Bigl(-\frac{23}{128}\frac{V}{\pi^{2}\lambda^{4}}+\frac{23}{384}\frac{S}{\pi^{2}\lambda^{3}}-\frac{23}{384}\frac{1}{\pi\lambda}+\frac{23}{1536}\frac{1}{\pia}\Bigr).
2\pi{\rho}_{a,j}(\beta)=\sigma_{j}^{(\infty)}*(\rho_{a,L}(\beta)+\rho_{a,R}(\beta))-\sum_{b=1}^{N-1}\sum_{l=1}^{\infty}A_{jl}^{(\infty)}*K_{ab}^{(N)}*\widetilde\rho_{b,l}(\beta)
\gamma_{i}^{*}=\gamma_{1}^{*}+(\beta_{21}^{-})^{*}+(\beta_{32}^{-})^{*}+\cdots+(\beta_{i,i-1}^{-})^{*}=\mu_{1}^{*}+\mu_{2}^{*}+\cdots+\mu_{i}^{*}.
A(0)\simeq-~ie^{2}(4|eH|)\int\frac{d^{4}\hat{q}}{(2\pi)^{4}}\frac{\mathrm{e}^{-\hat{q}_{\perp}^{2}}}{\hat{q}^{2}}\frac{A(q_{\parallel})}{2|eH|\hat{q}_{\parallel}^{2}+m^{2}},
{\cal{\pi}}_{0i}^{NC}(p_{0}=0;T=0)=\frac{1}{\pi\kappa}\epsilon_{0ij}p^{j}\left[\frac{2|\kappa|}{3}\frac{\frac{1}{2}\frac{m_{\sigma}}{M}}{(1+\frac{m_{\sigma}}{M})^{2}}+\frac{\tilde{p}}{4}(m^{2}-\frac{3}{4}\xi\kappa^{2})\right]
I=\frac1{2\pi}\intd^{2}x\sqrt{\pmg}\Big[R\tilde{\chi}(\Phi)+4\omegae^{-2\phi}{\nabla\phi}^{2}+V(\Phi)-\frac{\kappa}4{\nablaZ}^{2}-\frac12\sum_{i=1}^{N}(\nablaf_{i})^{2}\Big]-\frac1{\pi}\intd\Sigma\sqrt{\pmh}K\tilde{\chi},
\tau_{j}^{1}\;=\;\Bigl(\prod_{i<j}\sigma_{i}^{z}\Bigr)\,\sigma_{j}^{x}\;,\;\;\;\;\;\;\;\;\;\tau_{j}^{2}\;=\;\Bigl(\prod_{i<j}\sigma_{i}^{z}\Bigr)\,\sigma_{j}^{y}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(j=1\ldotsL)
[{\calD}\mu]={\calD}\xi{\calD}\pi_{i}{\calD}A^{\mu}{\calD}\rho{\calD}\pi_{\rho}\prod_{j}\{\delta[\Gamma_{j}(A^{0}+\xi,A^{i},\pi_{i},\rho)]\}\mathrm{det}\mid\{\widetilde{\Omega}_{i},\Gamma_{j}\}\mid.
\hat{\phi}_{(I)}(x)=\int_{(\infty)}\frac{d^{3}\vec{k}}{(2\pi)^{3}2\omega(\vec{k}\,)}\left[a(\vec{k}\,)e^{-ik\cdotx}+b^{\dagger}(\vec{k}\,)e^{ik\cdotx}\right]\,
\left({\frac{f^{\prime}}{f}}+{\frac{6}{r}}\right)h_{2}=-{\frac{4}{3}}\left({\frac{f_{1}^{\prime}}{f_{1}}}+{\frac{f_{5}^{\prime}}{f_{5}}}-2{\frac{f_{K}^{\prime}}{f_{K}}}\right)\delta\lambda+4\left({\frac{f_{1}^{\prime}}{f_{1}}}-{\frac{f_{5}^{\prime}}{f_{5}}}\right)\delta\nu.
g_{ij}=\left[\begin{array}{lll}{1}&{0}&{0}\\{0}&{g_{2}}&{0}\\{0}&{0}&{g_{3}}\\\end{array}\right]\mathrm{~and~}h_{ab}=\left[\begin{array}{ll}{h_{4}}&{0}\\{0}&{h_{5}}\\\end{array}\right]
\tilde{\calZ}_{\{n_{E}^{a},n_{M}^{a}\}}=\sqrt{2}\,Q^{a}\,a_{a}^{*}\,,\qquad\left|\tilde{\calZ}_{\{n_{E}^{a},n_{M}^{a}\}}\right|=\left|{\calZ}_{\{-n_{E}^{a},n_{M}^{a}\}}\right|\,,
S(\varphi,b)=\intd^{3}x\left(b_{\mu}^{\dagger}\partial^{\mu}\varphi+b_{\mu}\partial^{\mu}\varphi^{\dagger}-b_{\mu}^{\dagger}b^{\mu}+m^{2}\varphi\varphi^{\dagger}\;+V(\left|\varphi\right|)\right)\;.
\begin{aligned}{\Psi^{(1)}}&{{}=\frac{1}{2}J^{2}\wedgeJ^{2}+\frac{1}{2}J^{3}\wedgeJ^{3}-\frac{1}{2}J^{1}\wedgeJ^{1},}\\{\Psi^{(2)}}&{{}=\frac{1}{2}J^{3}\wedgeJ^{3}+\frac{1}{2}J^{1}\wedgeJ^{1}-\frac{1}{2}J^{2}\wedgeJ^{2},}\\{\Psi^{(3)}}&{{}=\frac{1}{2}J^{1}\wedgeJ^{1}+\frac{1}{2}J^{2}\wedgeJ^{2}-\frac{1}{2}J^{3}\wedgeJ^{3}.}\\\end{aligned}
{\frac{e^{2}}{2}}{\frac{\Gamma(\lambda)}{4{\pi}^{\lambda+1}}}\int_{0}^{T}d\tau_{a}\,d\theta_{a}\int_{0}^{T}d\tau_{b}\,d\theta_{b}{\frac{DX_{a}\cdotD{X_{b}}}{{({(X_{a}-X_{b})}^{2})}^{\lambda}}}\quad
\mathbf{x}=\sum_{\mu=1}^{8}\tilde{\sigma}_{\mu}x^{\mu}=\left(\begin{array}{cc}{z_{2}}&{z_{1}}\\{-\bar{z}_{1}}&{\bar{z}_{2}}\\\end{array}\right)\,,\quad\mathbf{x}^{\prime}=\sum_{\mu=1}^{8}\tilde{\sigma}_{\mu}^{\prime}x^{\mu}=\left(\begin{array}{cc}{z_{4}}&{z_{3}}\\{-\bar{z}_{3}}&{\bar{z}_{4}}\\\end{array}\right)\,.
\{\psi,\chi\}_{_{\sim}}\stackrel{\mathrm{def}}{=}\{\psi,\stackrel{\sim}{\chi}\},\\\\\stackrel{\sim}{\left(\begin{array}{c}{\chi_{l}}\\{\chi_{r}}\\\end{array}\right)}\stackrel{\mathrm{def}}{=}\left(\begin{array}{c}{\chi_{r}}\\{\chi_{l}}\\\end{array}\right)
d_{n}=\sum_{i=1}^{n}{\left(-1\right)^{i}\left(\sum_{\begin{array}{c}{0<k_{1},k_{2},\ldots,k_{i}\leqn}\\{k_{1}+k_{2}+\ldots+k_{i}=n}\\\end{array}}\frac{1}{\left[k_{1}\right]!\left[k_{2}\right]!\ldots\left[k_{i}\right]!}\right)}
m_{G}^{2}\;=\;-2e^{2}\int_{{\mathcalR}}\frac{d^{2}p}{(2\pi)^{2}}\,\frac{\partial}{\partialp_{\mu}}\left[\frac{\Lambda^{2}}{p^{2}(p^{2}+\Lambda^{2})}p_{\mu}\right]\;,
{\calR}_{{\calA}}{}^{{\calB}}(z;g)=\left(\begin{array}{cl}{R_{A}^{~B}(z;g)}&{i(\Sigma^{j}(z;g)\tilde{\gamma}_{A}L(z;g))^{\beta}}\\{0}&{L_{\alpha}^{~\beta}(z;g)U_{i}^{~j}(z;g)}\\\end{array}\right)
\begin{array}{l}{\left\{e^{-1}n_{\lambda}\cdot\Gamma~,~~e^{-1}n_{y}\cdot\Gamma~,~~e^{-1}n_{A}\cdot\Gamma~,~~e^{-1}n_{\psi}\cdot\Gamma~,~~e^{-1}n_{\varphi}\cdot\Gamma~\right\}\,,}\\\end{array}
\partial_{M}\left(\sqrt{-G}G^{MN}\partial_{N}\phi\right)+\frac{3-p}{4}(e^{{\frac{(p-3)}{4}}\phi}\sqrt{-{\hat{g}}}\sum_{i}q_{i}\delta^{9-p}(y-y_{i})-e^{{\frac{(3-p)}{2}}\phi}\sqrt{-{g}}\frac{|F_{p+2}|^{2}}{(p+2)!})=0\,,
\Sigma=m+\frac{4g^{2}}{\beta}\sum_{n=-\infty}^{\infty}\int^{\Lambda}\frac{d^{d}{\bfq}}{(2\pi)^{d}}\frac{\Sigma}{{\bfq}^{2}+(\rho-\bar{\omega}_{n})^{2}+\Sigma^{2}},
\left|{\bfp}\right|^{v^{\prime}v^{\prime\prime}}=\vert{\bfF}\vert\alpha_{0v}\int\tilde{\psi}_{v^{\prime}}^{\ast}\tilde{\psi}_{v^{\prime\prime}}dx+\vert{\bfF}\vert\alpha_{v}^{1}\intx\tilde{\psi}_{v^{\prime}}^{\ast}\tilde{\psi}_{v^{\prime\prime}}dx~.
\lambda^{2}(-\frac{1}{2}\frac{\partial^{2}}{\partialX^{2}}+\frac{iQ_{B}}{2}\frac{\partial}{\partialY}+\frac{1}{2}\frac{\partial^{2}}{\partialY^{2}}+e^{\sqrt{\frac{2}{|\kappa|}}(X+Y)}){\calT}=\frac{N}{24(x^{+})^{2}}{\calT}
{\calL}_{b}(\frac{1}{2},\alpha,g)=\bar{\psi}[\sigma_{\mu}(\partial_{\mu}+ia_{\mu}+iC_{\mu})+M+ib]\psi-\frac{i}{8\pi\alpha}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}+\frac{1}{g}b^{2}\;.
S_{\mathrm{boson}}=\intd^{4}x\sqrt{-g}\left\{-\frac{1}{4}(F_{\mu\nu}^{a})^{2}+\frac{1}{2}\vertD_{\mu}\Phi^{a}\vert^{2}-\frac{\lambda}{2}\left(\vert\Phi^{a}\vert^{2}-v^{2}\right)^{2}\right\},
\begin{array}{rcl}{<r|\hat{U}^{-1}(\theta_{0})|A_{\bar{z}}(z,\bar{z})>}&{=}&{<r|A_{\bar{z}}(z,\bar{z})>\\\,}\\{}&{}&{}\\{<r|\hat{U}^{-1}(k_{1},k_{2})|A_{\bar{z}}(z,\bar{z})>}&{=}&{<r|A_{\bar{z}}(z,\bar{z})>\\\,}\\\end{array}
\begin{array}{ll}{{\bfi.}}&{A_{1}(\hat{S}_{3})=[[\hat{S}_{3}]]~,~~~A_{2}(\hat{S}_{3})=2i([[\hat{S}_{3}]]-[[\hat{S}_{3}-1]])~,~~~~A_{4}(\hat{S}_{3})=i~,}\\{}&{A_{3}(\hat{S}_{3})=2i([[\hat{S}_{3}]]-[[\hat{S}_{3}+1]])~,~~~~A_{5}=4(2[[\hat{S}_{3}]]-[[\hat{S}_{3}-1]]-[[\hat{S}_{3}+1]])~.}\\\end{array}
|p_{(L_{1},\cdots,L_{k})}|^{2}-2|p_{(L_{1},\cdots,L_{k},L_{k+1})}|^{2}+|p_{(L_{1},\cdots,L_{k},L_{k+1},L_{k+2})}|^{2}=-\zeta_{(L_{k+1})-(L_{k+2})},
Z_{MN}=\left(\begin{array}{lllllll}{0}&{-w_{9}}&{-k_{9}}&{0}&{0}&{\cdots}&{0}\\{w_{9}}&{0}&{0}&{0}&{0}&{\cdots}&{0}\\{k_{9}}&{0}&{0}&{p_{0}}&{p_{1}}&{\cdots}&{p_{8}}\\{0}&{0}&{-p_{0}}&{0}&{0}&{\cdots}&{0}\\{0}&{0}&{-p_{1}}&{0}&{0}&{\cdots}&{0}\\{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\vdots}&{\ddots}&{0}\\{0}&{0}&{-p_{8}}&{0}&{0}&{0}&{0}\\\end{array}\right)
a=(4\lambda^{2}+\mathcal{E}\beta^{2})^{1/4}\sqrt{\frac{2t}{\beta}}-\frac{k\beta^{2}}{2}(4\lambda^{2}+\mathcal{E}\beta^{2})^{-1/4}\left(\frac{2t}{\beta}\right)^{3/2}+O(t^{5/2}),
W=\intd^{4}x{\calL}(x)=\intd^{4}x\left[-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\frac{1}{2}}M^{2}V_{\mu}^{2}+\bar{\Psi}(i\not{\!\partial}-m)\Psi-V_{\mu}J^{\mu}\right]~,
\begin{array}{l}{\lbrackq^{-},p^{+}]=i\,,}\\{\left[\alpha_{n}^{-}\,,\alpha_{m}^{+}\right]=n\,\delta_{n+m,0}\,\,\,,}\\{\left[s_{n}\,,s_{m}\right]=\left(\frac{k}{2}-1\right)\,n\,\delta_{n+m,0}\,\,\,.}\\\end{array}
det(\widetilde{T}_{k(A)}^{*(A)}\widetilde{T}_{k(A)})=\,det(Q_{k}(A_{k}^{*}A_{k})^{-1}Q_{k})\,det(P_{k-1}(A_{k-1}^{*}A_{k-1})P_{k-1})\,det(\widetilde{T}_{k}^{*}\widetilde{T}_{k}).
\langlej|\psi(b_{1})\psi(b_{2})|j\rangle_{\partial}=\frac{jC}{\pi^{2}}\,\mathrm{Im}\int\frac{d\vec{k}}{(2\pi)^{3}}\,\frac{1}{2\omega_{k}}e^{i\omega_{k}(t_{1}+t_{2})}\left\lbracee^{-i\vec{k}\cdot(\vec{x_{1}}-\vec{x_{2}})}+c.c.\right\rbrace\.\,
\begin{array}{l}{w=\left(1+\frac{\alpha}{r_{H}^{7-p}}\right)t\mp\frac{\sqrt{\alpha\left(r_{H}^{7-p}+\alpha\right)}}{r_{H}^{7-p}}x^{10},}\\{y=\mp\frac{\alpha}{r_{H}^{7-p}}t+\frac{\sqrt{\alpha\left(r_{H}^{7-p}+\alpha\right)}}{r_{H}^{7-p}}x^{10},}\\\end{array}
f(\nu)=-ie^{2}\int\frac{d^{n}k}{{(2\pi)}^{n}}\,p_{\alpha}D_{F}^{\alpha\beta}(k)p_{\beta}\int_{0}^{\nu}d\nu^{\prime}\,\int_{0}^{\nu^{\prime}}d\nu^{\prime\prime}\,e^{i\nu^{\prime\prime}(k.p)}\,\,.
h^{\alpha\dot{\alpha}}=-\frac{2}{3}{\bar{D}}^{3\dot{\alpha}{\mathbfi}}L_{\mathbfi}^{\alpha}{\quad\mathrm{{and}}\quad}{\bar{h}}^{\alpha\dot{\alpha}}=-\frac{2}{3}D^{3\alpha{\mathbfi}}{\bar{L}}_{\mathbfi}^{\dot{\alpha}}\.
\langleA_{i}^{a}(x)A_{j}^{b}(y)\rangle=\frac{\delta_{ab}}{N^{2}-1}\sum_{a,u}\sum_{J=1}^{[N/2]}(-2)Tr(Q_{J}^{2})A_{i}(x;a,u;J)A_{j}(y;a,u;J)
2\left(D_{x}^{\dagger}D_{x}\right)^{-1}(z,z^{\prime})=(\sigma_{0}+i\sigma_{3})e^{-\phi(z)}K_{+}(z,z^{\prime};x)e^{-\phi(z^{\prime})}+(\sigma_{0}-i\sigma_{3})e^{\phi(z)}K_{-}(z,z^{\prime};x)e^{\phi(z^{\prime})},
\partial_{k}E_{k}=\partial_{k}(M_{kj}\ddot{u}_{j})-c^{2}\partial_{k}(M_{kj}\partial^{2}u_{j})-\partial_{k}J_{k}-\partial_{k}{\frac{\partial_{k}\dot{n}}{\partial^{2}}}-\nu^{2}\partial_{k}(M_{kj}\partial^{4}u_{j}),\nonumber
\left[\begin{array}{l}{m}\\{n}\\\end{array}\right]_{q}=\frac{(q;q)_{m}}{(q;q)_{n}(q;q)_{m-n}},~~(p;q)_{m}=\prod_{k=0}^{m-1}(1-pq^{k}),~(p;q)_{0}=1.
\nu_{0}^{TE}\simeq\frac{{n_{2}}\,R\,\Omega}{c}\,\left\{1-k_{1}^{TE}\,\left(\frac{{n_{2}}\,R\,\Omega}{c}\right)^{-2/3}+k_{2}^{TE}\,\left(\frac{{n_{2}}\,R\,\Omega}{c}\right)^{-4/3}+{\calO}\left(\frac{R\,\Omega}{c}\right)^{-2}\right\},
(dG)_{11\bar{I}\bar{J}\bar{K}\bar{L}}=-4\sqrt{2}\pi\left(\frac{\kappa}{4\pi}\right)^{2/3}(J^{(1)}\delta(x^{11})+J^{(2)}\delta(x^{11}-\pi\rho))_{\bar{I}\bar{J}\bar{K}\bar{L}},
i\dot{\Psi}(t,x)\pm\frac{1}{\kappa^{2}}\int^{x}dy\dot{\rho}(t,y)\Psi(t,x)=-\frac{1}{2m}\Psi^{\prime\prime}(t,x)\pm\frac{1}{\kappa^{2}}j(t,x)\Psi(t,x)
\mathcal{A}_{N}^{cl}(p_{1},...,p_{N})=C_{0}^{cl}\mathcal{N}_{t}^{N}\int\frac{1}{dV_{abc}}\prod_{i=1}^{N}\left[d^{2}z_{i}\left|V_{i}^{\prime}(0)\right|^{\frac{\alpha^{\prime}}{2}p_{i}^{2}-2}\right]\prod_{\substack{i,j=1\,i<j}}^{N}\left|z_{i}-z_{j}\right|^{\alpha^{\prime}p_{i}\cdotp_{j}}.
{\calA}_{M}=g_{\mathrm{st}}^{M-2}\intdz_{4}d\overline{{z}}_{4}\cdotsdz_{M}d\overline{{z}}_{M}\langleV_{-{\frac{1}{2}}}(0,0)V_{-{\frac{1}{2}}}(1,1)V_{-1}(\infty,\infty)V_{0}(z_{4},\overline{{z}}_{4})\cdotsV_{0}(z_{M},\overline{{z}}_{M})\rangle~,
\tilde{Z}^{(1)}(q_{N_{r}};l_{1r},l_{2r})=\frac{k_{r}^{\sigma_{r}l_{1r}}H_{(1)}(q_{N_{r}})[1+(-1)^{l_{2r}}k_{r}]^{2l_{1r}}}{k_{r}^{(3-2l_{1_{r}})/2}[1+(-1)^{l_{2r}}k_{r}^{1/2}]^{4l_{1r}}}Z^{(1)}(k_{r};l_{1r},l_{2r})
S_{FP}=\intd^{4}x\biggl[-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\bar{\psi}}(i\gamma^{\mu}D_{\mu}-m)\psi-{\frac{1}{2\xi}}(\partial_{\mu}A^{\mu})^{2}+i{\bar{C}}\partial_{\mu}\partial^{\mu}C\biggr]\,,
\begin{array}{rl}{{\calL}=}&{-\frac{\mu^{2}}{4}F_{\mu\nu}F^{\mu\nu}+\frac{\kappa}{2}\varepsilon^{\mu\nu\lambda}F_{\mu\nu}A_{\lambda}}\\{}&{+g(\phi^{3})\varepsilon^{\alpha\beta}\phi^{\alpha}D_{0}\phi^{\beta}-\frac{1}{2m}|D_{i}\phi^{a}|^{2}-U(\phi^{3})+\lambda(1-|\phi^{a}|^{2}),}\\\end{array}
\frac{1}{q^{2}}={\frac{2\pia}{e^{2}}}\int_{0}^{\infty}\mathrm{d}r\,\mathrm{e}^{{\frac{1}{2}}B}\left(P^{2}(r)+{\frac{a^{2}e^{2}}{\kappa^{2}}}\mathrm{e}^{B}\right)~.
A_{4F}=\int_{0}^{1}dxx^{k^{(3)}k^{(4)}-\frac{1}{4}Q^{2}-1}(1-x)^{k^{(2)}k^{(3)}-\frac{1}{4}Q^{2}-1}K(\zeta^{(r)},k^{(r)},x),
S_{R^{4}}=(\alpha^{\prime})^{-1}\left[a\zeta(3)\tau_{2}^{3/2}+b\tau_{2}^{-1/2}+ce^{2\pii\tau}+\cdots\right]R^{4}\equiv(\alpha^{\prime})^{-1}f(\tau,\bar{\tau})R^{4},
(\deltag^{(1)},\deltag^{(2)})=\int\!\sqrt{g}\;\deltag_{\mu\nu}^{(1)}\left(g^{\mu\alpha}g^{\nu\beta}+g^{\mu\beta}g^{\nu\alpha}+Cg^{\mu\nu}g^{\alpha\beta}\right)\deltag_{\alpha\beta}^{(2)},
\Theta=e^{-i\frac{\pi}{4}}\left(\begin{array}{c}{e^{i\frac{\phi}{2}}\frac{1+i\gamma_{4}}{2}\theta_{N}\otimes\epsilon^{N}+e^{-i\frac{\phi}{2}}\frac{1-i\gamma_{4}}{2}\theta_{c}^{N}\otimes\epsilon_{cN}}\\{-e^{-i\frac{\phi}{2}}\frac{1-i\gamma_{4}}{2}\theta_{N}\otimes\epsilon^{N}+e^{i\frac{\phi}{2}}\frac{1+i\gamma_{4}}{2}\theta_{c}^{N}\otimes\epsilon_{cN}}\\\end{array}\right)
{\overline{C}}_{R}^{\mu\nu}(p)=\left[2\epsilon^{\mu\nu}{}_{\sigma\rho}F_{-}^{\sigma\rho}(p)-L^{2}p^{2}{\overline{C}}_{R}^{\mu\nu}(p)\right]\int{\frac{d^{2}k}{(2\pi)^{2}}}~{\frac{1}{k^{2}+p^{2}}}~.
Z_{\mathrm{gr}}[\varphi,\mu,\bar{\mu}]=\int{\calD}T{\calD}\bar{T}\,\,e^{-\intd^{2}z\,\,T\mu-\intd^{2}z\,\,\bar{T}\bar{\mu}+W[\mu]+W[\bar{\mu}]+K[\varphi,T,\bar{T},\mu,\bar{\mu}]}.
{\Phi}=\frac{1}{2}\left(\begin{array}{c}{-A_{Ij}E_{\nu}^{j}}\\{D_{ij}E_{\nu}^{j}}\\{E_{\nu}^{i}}\\\end{array}\right)={\frac{1}{2}}\left(\begin{array}{ccc}{{\calE}_{I}^{M}}&{0}&{0}\\{0}&{T_{i}^{\mu}}&{0}\\{0}&{0}&{T_{i}^{\mu}}\\\end{array}\right){\hat{\Phi}}
ds^{2}=\alpha^{\prime}\tilde{F}^{1/2}\left[\frac{u^{4}}{R^{4}}(-\tilde{f}d\tilde{x}_{0}^{2}+d\tilde{x}_{1}^{2})+\frac{1}{\tilde{G}}(d\tilde{x}_{2}^{2}+d\tilde{x}_{3}^{2})+\tilde{f}^{-1}du^{2}+u^{2}d\Omega_{5}^{2}\right],
f(x,p)\starg(x,p)=\sum_{s=0}^{\infty}\sum_{i=0}^{s}{\frac{\theta^{s}}{s!}}(-)^{i}c_{s}^{i}(\partial_{x}^{i}\partial_{p}^{s-i}f)(\partial_{x}^{s-i}\partial_{p}^{i}g),
B_{ij}=\left(\begin{array}{cccc}{0}&{1/2}&{0}&{-1/2}\\{-1/2}&{0}&{1/2}&{0}\\{0}&{-1/2}&{0}&{1/2}\\{1/2}&{0}&{-1/2}&{0}\\\end{array}\right)~,~~~G_{ij}=\left(\begin{array}{cccc}{2}&{-1}&{0}&{1}\\{-1}&{2}&{-1}&{0}\\{0}&{-1}&{2}&{-1}\\{1}&{0}&{-1}&{2}\\\end{array}\right)~.
\Psi_{\kappa\lambda}(t,r)=\frac{1}{\sqrt{2\pi}}\frac{\kappa!(2\lambda)^{\kappa+1/2}}{\sqrt{(2\kappa)!}}\left[\frac{i}{(\lambda+it+ir)^{\kappa+1}}-\frac{i}{(\lambda+it-ir)^{\kappa+1}}\right].
T_{0}^{\mu\nu}=-\left(\frac{g_{str}\alpha^{\prime}}{2}\right)^{2}\prod_{u}\frac{\delta(\sqrt{\alpha^{\prime}}k_{u})}{\sqrt{\tau_{2}}}\left(\frac{\eta^{\mu\nu}}{2\pi\alpha^{\prime}}\prod_{o}\frac{\delta(\sqrt{\alpha^{\prime}}k_{o})}{\sqrt{\tau_{2}}}X_{1,1}+\frac{k_{\mu}k_{\nu}}{4}2^{-(d+1)}X_{1,0}\right)