harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|\frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta}-\frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime}|^{2-n}=\sum_{N=0}^{N_{0}}\sum_{\alpha=0}^{N}\frac{1}{\lambda_{N}}Y_{\alpha}^{N}(\frac{a_{{\eta}^{\prime}}}{a_{B}}{\eta})Y_{\alpha}^{N}(\frac{a_{B}}{a_{{\eta}^{\prime}}}{\eta}^{\prime})
I=\frac{1}{4\pi\alpha^{\prime}}\intd^{2}z\sqrt{h}\,\left[(G_{\mu\nu}h^{ab}+B_{\mu\nu}\frac{\varepsilon^{ab}}{\sqrt{h}})\partial_{a}x^{\mu}\partial^{a}x^{\nu}+\alpha^{\prime}R^{(2)}\Phi(x)\right]
H_{k^{\prime}k}^{\prime}\equiv\langlek^{\prime}|H^{\prime}|k\rangle=\left((1+A)|k|+\mu+B-{\frac{g^{2}}{6}}\sum_{\mathbf{l}}{\frac{\delta_{k,l_{1}+l_{2}+l_{3}}}{\sum_{i=1}^{3}\omega_{l_{i}}-\omega_{k}}}\right)\delta_{k^{\prime},k},
\Gamma_{\mu\nu}=P_{\mu\nu}\,(p^{2}+\Pi_{1})+Q_{\mu\nu}\,(p^{2}+\Pi_{2})+\epsilon_{\mu\nu\lambda}\,p_{\lambda}\,(\kappa+\Pi_{\mathrm{odd}})+\frac{1}{\xi}\,p_{\mu}p_{\nu},
\beta(n,s)=\frac{-i}{2\pi}\int\!d\alpha\frac{e^{-in\alpha}}{\sqrt{4\omega_{n}\omega_{s}}}\left\{i\omega_{n}f_{s}^{+\ast}(t^{\prime}(0,\alpha))e^{-is\alpha^{\prime}(0,\alpha)}-\partial_{t}\left\{f_{s}^{+\ast}(t^{\prime}(t,\alpha))e^{-is\alpha^{\prime}(t,\alpha)}\right\}\left|_{t=0}\right.\right\}
\Psi[\xi,t)=exp\biggl[i\intd^{2}x\biggl(\xi({\bfx})\rho({\bfx},t)+\int^{t}d\tauJ_{i}({\bfx},\tau)\epsilon_{ij}{\frac{\partial_{j}}{\nabla^{2}}}\rho({\bfx},\tau)\biggr)\biggl]
\frac{\left|e\right|A_{0}^{\left(1\right)}}{c\hbar}=\frac{f_{1}\left(u^{1}\right)}{u^{0}},\;\,A_{1}^{\left(1\right)}=0,\;\frac{\left|e\right|A_{2}^{\left(1\right)}}{c\hbar}=f_{2}\left(u^{1}\right),\;\,A_{3}^{\left(1\right)}=0\,,
ds^{2}=\left(-\partial_{a}\partial_{b}w\right)dx^{a}dx^{b}=m^{2}d\xi^{2}+\frac{1}{2}\frac{1}{(4\pi)^{\frac{D}{2}}}{m^{2}}^{\frac{D}{2}-2}\Gamma(2-\frac{D}{2})(dm^{2})^{2}
U_{[1/2]}^{c}\left(\vert{\bfp},\,\uparrow>^{+}\pmi\,\vert{\bfp},\,\uparrow>^{-}\right)=\mpi\,\left(\vert{\bfp},\,\uparrow>^{+}\pmi\,\vert{\bfp},\,\uparrow>^{-}\right)\quad.
j_{\sigma\sigma^{\prime}}^{\mu}({\vec{p}},{\vec{q}})=\sum_{\sigma_{p}=-1/2}^{1/2}j_{\sigma\sigma_{p}}^{\mu}({\vec{q}}(-){\vec{p}},{\vec{p}})\,\,D_{\sigma_{p}\sigma^{\prime}}^{1/2}\left\{V^{-1}(\Lambda_{p},q)\right\}\quad.
\begin{array}{ccc}{A_{\bar{\mu}}=\left(\begin{array}{cc}{0}&{c_{\mu}^{*}}\\{c_{\mu}}&{0}\\\end{array}\right),}&{X_{p+1}=\left(\begin{array}{cc}{0}&{d^{*}}\\{d}&{0}\\\end{array}\right),}&{X_{9}=\left(\begin{array}{cc}{0}&{w^{*}}\\{w}&{0}\\\end{array}\right)}\\\end{array}
\begin{array}{l}{K_{(\tau,\nu,\beta)}^{(\lambda,\mu,\alpha)}(r^{\lambda\mu},r^{\tau\nu})=<e_{r^{\lambda\mu}}^{(\lambda,\mu,\alpha)},e_{(\tau,\nu,\beta)}^{r^{\tau\nu}}>,}\\{K_{(\lambda,\mu,\alpha)}^{(\tau,\nu,\beta)}(r^{\tau\nu},r^{\lambda\mu})=(K^{T})_{(\lambda,\mu,\alpha)}^{(\tau,\nu,\beta)}=<e_{r^{\tau\nu}}^{(\tau,\nu,\beta)},e_{(\lambda,\mu,\alpha)}^{r^{\lambda\mu}}>,}\\\end{array}
C_{\Omega}=2^{10/3}3^{-1}\pi^{13/6}\kappa^{-5/3}N^{1/2}m^{1/2}r_{+}\frac{15r_{+}^{4}+3l^{2}r_{+}^{2}-2l^{4}}{3r_{+}^{4}+4l^{2}r_{+}^{2}+l^{4}}\frac{3r_{+}^{2}+l^{2}}{3r_{+}^{2}-l^{2}}.
V_{0}+V_{v^{2}}\rightarrow-\frac{4\alpha_{S}}{3r}\left[1-\frac{1}{2}\;\left(\bfv_{\mathrm{1}}\cdotv_{\mathrm{2}}+v_{\mathrm{1}}\cdot\hat{r}\;v_{\mathrm{2}}\cdot\hat{r}\right)\right].
y^{2}=\prod_{a=1}^{n_{f}/2+1}(x-\phi_{a})^{2}\prod_{k=1,k\neq(n_{c}-n_{f}/2+1)/2}^{n_{c}-n_{f}/2}\phi_{k}^{2}-4\Lambda^{2n_{c}-n_{f}}\prod_{i=1}(x+m_{i}),
S_{1}({\bf{k}})=1+\sum_{{\bf{q}},i}\frac{{\bar{n}}_{{\bf{k}}-{\bf{q}}}}{({\tilde{\omega}}_{i}(-{\bf{q}})+\frac{{\bf{k.q}}}{m}-\epsilon_{{\bf{q}}})^{2}}g_{i}^{2}(-{\bf{q}})+\sum_{{\bf{q}},i}\frac{1-{\bar{n}}_{{\bf{k}}+{\bf{q}}}}{({\tilde{\omega}}_{i}(-{\bf{q}})+\frac{{\bf{k.q}}}{m}+\epsilon_{{\bf{q}}})^{2}}g_{i}^{2}(-{\bf{q}})
\begin{array}{ll}{{\calL}_{K}=}&{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-(1-g^{2}\zeta\eta^{2})\tilde{G}_{\mu}\tilde{G}^{\mu}+2(m+gh\zeta\eta^{2})A_{\mu}\tilde{G}^{\mu}+}\\{}&{-2h\zeta\eta^{2}\Sigma\partial_{\mu}A^{\mu}+h^{2}\zeta\eta^{2}A_{\mu}A^{\mu}+\zeta\eta^{2}\partial_{\mu}\Sigma\partial^{\mu}\Sigma,}\\\end{array}
M\!\inGl(2,{\bfC})\longmapsto{\widehat{\calA}}(M)=\left(\begin{array}{cc}{I-|1\rangle\langle1|}&{0}\\{}&{}\\{0}&{I-|2\rangle\langle2|}\\\end{array}\right)+\left(\begin{array}{rr}{M_{11}|1\rangle\langle1|}&{M_{12}|1\rangle\langle2|}\\{}&{}\\{M_{21}|2\rangle\langle1|}&{M_{22}|2\rangle\langle2|}\\\end{array}\right).
\begin{array}{llrl}{MW(i)}&{\qquad\qquad}&{q_{i}}&{=\int{\mathrm{tr}}F_{0i}}\\{M2(ij)}&{\qquad\qquad}&{m_{ij}}&{=\int{\mathrm{tr}}F_{ij}}\\{M2(i\natural)}&{\qquad\qquad}&{m_{i\natural}}&{=\intT_{0i}}\\{M5(jklm\natural)}&{\qquad\qquad}&{f_{i}}&{=\int{\mathrm{tr}}\epsilon_{ijklm}F_{jk}F_{lm}\quad,}\\\end{array}
\langlex|\eta,n\rangle=\frac{1}{\sqrt{2L}}\left(\begin{array}{c}{1}\\{-\mathrm{i}\eta\,\mathrm{sgn}(k_{n})}\\\end{array}\right)\mathrm{e}^{\mathrm{i}k_{n}x},\k_{n}=\frac{2\pi}{L}(n+1/2).
\Pi(\tau)=\frac{3}{8}\frac{\zeta^{2}}{\tau^{4}\,\pi^{2}}\,\mathrm{e}^{-\zeta^{2}/2}\left[K_{0}\left(\frac{\zeta^{2}}{2}\right)+K_{1}\left(\frac{\zeta^{2}}{2}\right)\right]\;,
Z={\calN}\int\prod_{x}\left\{\mathrm{d}v_{\mu}\mathrm{d}j_{5}^{\mu}\,\mathrm{d}\theta\right\}\,e^{\mathrm{i}\,\int\mathrm{d}^{2}x\,{\calL}[v_{\mu},j_{5}^{\mu},\theta]},
\begin{array}{l}{R(\theta-\theta^{\prime})[K^{-}(\theta)\otimes1]R(\theta+\theta^{\prime})[K^{-}(\theta^{\prime})\otimes1]=}\\{{[}K^{-}(\theta^{\prime})\otimes1]R(\theta+\theta^{\prime})[K^{-}(\theta)\otimes1]R(\theta-\theta^{\prime})}\\\end{array}
+i\hbar(V_{AB}^{CE}V_{E}^{D}(-1)^{\varepsilon_{C}}-(C\leftrightarrowD)(-1)^{\varepsilon_{C}\varepsilon_{D}})-[V_{AB}^{CD},\Omega](-1)^{\varepsilon_{C}+\varepsilon_{D}}-\frac{1}{2}(i\hbar)^{2}V_{AB}^{EF}V_{FE}^{CD}.
S=\intd\tau\left({\bfP}_{m}(\dot{X}^{m}-\bar{\theta}\Gamma^{m}\dot{\theta})+{\frac{1}{2}}V({\bfP}^{2}+Z^{2})-Z\bar{\theta}\Gamma^{11}\dot{\theta}+\bar{\chi}_{1}d_{2}\right)\.
\begin{array}{rcl}{R_{\mu\nu}+\frac{1}{2}\partial_{\mu}\varphi\partial_{\nu}\varphi+\frac{1}{12}{\calV}_{0}e^{-\varphi}g_{\mu\nu}}&{=}&{0\,,}\\{}&{}&{}\\{\nabla^{2}\varphi+\frac{1}{2}{\calV}_{0}e^{-\varphi}}&{=}&{0\,.}\\\end{array}
\begin{aligned}{J^{1}}&{{}=e^{12}+e^{34},}&{\qquadJ^{\prime}{}^{1}}&{{}=e^{56}+e^{78},}\\{J^{2}}&{{}=e^{14}+e^{23},}&{\qquadJ^{\prime}{}^{2}}&{{}=e^{58}+e^{67},}\\{J^{3}}&{{}=e^{13}+e^{42},}&{\qquadJ^{\prime}{}^{3}}&{{}=e^{57}+e^{86}.}\\\end{aligned}
F_{m}=\sum_{n=1}^{\infty}\;f_{mm_{2}\ldotsm_{n}}^{(n)}\phi_{m_{2}}\ldots\phi_{m_{n}},\;\;\;\;\;\;V_{m}^{\prime}=\sum_{n=1}^{\infty}\;v_{mm_{2}\ldotsm_{n}}^{(n)}\phi_{m_{2}}\ldots\phi_{m_{n}}
V(\alpha^{*},\alpha)=\frac{g}{3!}\left(V_{abc}\alpha^{a}\alpha^{b}\alpha^{c}+3\alpha_{a}^{*}V_{\;\;bc}^{a}\alpha^{b}\alpha^{c}+3\alpha_{a}^{*}\alpha_{b}^{*}V_{\quadc}^{ab}\alpha^{c}+\alpha_{a}^{*}\alpha_{b}^{*}\alpha_{c}^{*}V^{abc}\right)
L=\int\,dx^{-}\;\Bigl[{\dot{\varphi}}\varphi^{\prime}+{\frac{1}{2}}({\dot{A}}_{-}-{A_{+}^{\prime}})^{2}-({\frac{e}{\sqrt{\pi}}})(A_{+}\varphi^{\prime}-A_{-}{\dot{\varphi}})\Bigr]+({\frac{e}{\sqrt{\pi}}}){\dot{\omega}}{h(\tau)}\nonumber
\tilde{H}={\frac{1}{4}}\eta_{2}\sum_{\alpha=1}^{n-1}\partial_{\mu}{\bfe}_{\alpha}\cdot\partial_{\mu}{\bfe}_{\alpha}+\left({\frac{1}{2}}\eta_{1}-{\frac{1}{4}}\eta_{2}\right)\partial_{\mu}{\bfe}_{n}\cdot\partial_{\mu}{\bfe}_{n}.
G_{\pm}^{(2)}(\rho,u_{\pm})=\mp\frac{g(d)}{\rho^{\frac{d-3}{2}}L}\sum_{n=1}^{\infty}{}cos(\frac{n\pi{}u_{\pm}}{L})((\frac{n\pi}{L})^{2}+m^{2})^{\frac{d-3}{4}}K_{\frac{d-3}{2}}(\rho(m^{2}+(\frac{n\pi}{L})^{2})^{\frac{1}{2}})
J_{0}=i\{\pi\phi-\pi^{*}\phi^{*}-\fracg{2e}\varepsilon^{ij}\partial_{i}[\phi^{*}(D_{j}\phi)-\phi(D_{j}\phi)^{*}]+i\frac{g^{2}}{2e}\partial_{i}(\left|\phi\right|^{2}F^{0i})\}
I_{n,k,d}(z_{1},z_{2},\dots;\epsilon)=\frac{I_{n-1,k,d}(z_{2},z_{3},\dots,z_{n};\epsilon)-I_{n-1,k,d}(z_{1},z_{3},\dots,z_{n};\epsilon)}{z_{1}^{2}-z_{2}^{2}}\,.
\tilde{G}_{d-m}^{N,k,d+f}(n;\sigma_{m}\cup(f)):=\sum_{j=0}^{(k-N)f}G_{d-m}^{N,k,d}(n-j;\sigma_{m})-\sum_{j=0}^{(k-N)f}G_{d-m}^{N,k,d}(1+(k-N)(d+f)-j;\sigma_{m}).
A_{+}^{1}\,,A_{+}^{2}\,,A_{+}^{3}\,,A_{1}^{1}\,,A_{1}^{2}\,,A_{1}^{3}\,,A_{2}^{1}\,,A_{2}^{2}\,,A_{2}^{3}\,,A_{-}^{1}\,,A_{-}^{2}\,,A_{-}^{3}
\begin{aligned}{V_{A}(\chi)}&{{}=+\frac{2}{9\sqrt{3}}\epsilone^{-2\chi}(e^{\chi}-1)^{3/2}}\\{V_{B}(\chi)}&{{}=-\frac{2}{9\sqrt{3}}\epsilone^{-2\chi}(e^{\chi}-1)^{3/2}}\\\end{aligned}
a_{{\bf{k}}}({\bf{q}})=\frac{1}{\sqrt{n_{{\bf{k}}-{\bf{q}}/2}}}c_{{\bf{k}}-{\bf{q}}/2}^{\dagger}(\frac{n^{\beta}({\bf{k}}-{\bf{q}}/2)}{\langleN\rangle})^{\frac{1}{2}}e^{i\mathrm{~}\theta({\bf{k}},{\bf{q}})}c_{{\bf{k}}+{\bf{q}}/2}
\varphi^{\prime}\left(r\right)=-\frac{e_{0}^{2}}{4\pi}\,\frac{Q_{0}}{r^{2}}-\frac{e_{0}^{2}}{4\pi}\,\frac{1}{r^{2}}\sum_{\kappa=1}^{\infty}\kappa\int_{r}^{\infty}\mathrm{d}r^{\prime}\left[\varrho_{\kappa}\left(r^{\prime}\right)+\varrho_{-\kappa}\left(r^{\prime}\right)\right]\,,
P=\left(\begin{array}{ccccc}{0}&{0}&{\cdots}&{0}&{P^{0n}}\\{0}&{0}&{\cdots}&{P^{1(n-1)}}&{P^{1n}}\\{\vdots}&{\vdots}&{{}}&{\vdots}&{\vdots}\\{0}&{P^{(n-1)1}}&{\cdots}&{P^{(n-1)(n-1)}}&{P^{(n-1)n}}\\{P^{n0}}&{P^{n1}}&{\cdots}&{P^{n(n-1)}}&{P^{nn}}\\\end{array}\right).
S_{\mathrm{eff}}[A,g]~=~\intdx~\left(\frac{N}{2\gamma}~\mathrm{Tr}\Big|\nablag(x)+i[A(x),g(x)]\Big|^{2}-\sum_{R}\left(\lambda_{R}\mathrm{Tr}~g^{R}(x)\right)\right)\;.
\frac{d^{D}\mathbf{x}\,d^{D}\mathbf{p}}{\Bigl[1+\betap^{2}\Bigr]^{D-1}\Bigl[1+(\beta+\beta^{\prime})p^{2}\Bigr]^{1-\frac{\beta^{\prime}}{2(\beta+\beta^{\prime})}}}
\frac{1}{M_{p}^{2}p^{2}}\left(1+\langleT_{CFT}\left(p\right)T_{CFT}\left(-p\right)\rangle\frac{1}{M_{p}^{2}p^{2}}\right)+\frac{1}{M_{w}^{8}}\langleT_{CFT}\left(p\right)T_{CFT}\left(-p\right)\rangle,
\frac{1}{2}D^{\mu}\partial_{[\mu}G_{\nu\alpha]}{}^{\nu^{\prime}\alpha^{\prime}}(z,w)=-\delta(z,w)(g_{\nu}{}^{\nu^{\prime}}g_{\alpha}{}^{\alpha^{\prime}}-g_{\alpha}{}^{\nu^{\prime}}g_{\nu}{}^{\alpha^{\prime}})+\partial^{\nu^{\prime}}\Lambda_{\nu\alpha}{}^{\alpha^{\prime}}-\partial^{\alpha^{\prime}}\Lambda_{\nu\alpha}{}^{\nu^{\prime}},
\,_{1}\langle\,_{2}\langle\,_{3^{\prime}}\langleV|Y_{a_{1}}\rangle_{1}|Y_{a_{2}}\rangle_{2}|I\rangle_{3^{\prime}}\rangle_{3}=f_{a_{1}a_{2}}^{b}c_{0}|Y_{b}\rangle_{3}+Q|\Lambda_{a_{1}a_{2}}\rangle_{3}.
\theta_{g_{1}g_{2}h_{1}h_{2}}\left(\begin{array}{cc}{\tau_{1}}&{\epsilon}\\{\epsilon}&{\tau_{2}}\\\end{array}\right)=\sum_{n=0}^{infty}\frac{2^{2n}}{(2n)!}\frac{d^{n}}{(d\tau_{1})^{n}}\theta_{g_{1}h_{1}}(\tau_{1})\frac{d^{n}}{(d\tau_{2})^{n}}\theta_{g_{2}h_{2}}(\tau_{2})\epsilon^{2n}
\langle\,p_{f}\,|\,q_{i}\,\rangle=\int\frac{dP_{1}}{2\pi\hbar}\,dQ_{N}\,\langle\,p_{f}\,|\,Q_{N}\,\rangle\,e^{iP_{1}Q_{N}/\hbar}\langle\,P_{1}\,|\,q_{i}\,\rangle\,.
T(z)=\frac{1}{k-\frac{3}{2}}\Bigl(g_{ab}J^{a}J^{b}+\varepsilon_{\alpha\beta}j^{\alpha}j^{\beta}\Bigr)(z)\,,\,\,\mathrm{~with~}\,\,\varepsilon_{\alpha\beta}\varepsilon^{\beta\gamma}=\delta_{\alpha}^{\gamma}
S_{H}=\intd^{3}x\left[-\frac{1}{2}\left(D_{\mu}Q^{A}D^{\mu}Q^{A}\right)-W_{H}(Q^{A})+\frac{\kappa}{2}\epsilon^{\mu\nu\rho}\partial_{\mu}A_{\nu}^{a}A_{\rho}^{a}\right].
(\tilde{X}^{I};\tilde{F}_{I})|_{\mathrm{1-loop}}=(1,T^{a}\eta_{ab}T^{b},iT^{a};iS,iST^{a}\eta_{ab}T^{b}+2ih(T^{a})-iT^{a}{\frac{\partialh}{\partialT^{a}}},-ST^{a}+{\frac{\partialh}{\partialT^{a}}})
\mu_{\mathrm{H}}^{\prime}\\stackrel{x_{\mathrm{H}}\rightarrow0}{\longrightarrow}\,frac{1}{2}\frac{1}{1+\gamma^{2}}\,\\\mathrm{i.e.}\\(xN^{\prime})_{\mathrm{H}}\\stackrel{x_{\mathrm{H}}\rightarrow0}{\longrightarrow}\,frac{\gamma^{2}}{1+\gamma^{2}}\.
\hat{x}_{12}=x_{12}+{\frac{2i}{(12)}}[(1^{-}2)\theta_{1}^{+}\bar{\theta}_{1}^{+}-(12^{-})\theta_{2}^{+}\bar{\theta}_{2}^{+}+\theta_{1}^{+}\bar{\theta}_{2}^{+}+\theta_{2}^{+}\bar{\theta}_{1}^{+}]\.
Q_{\perp}^{-}(T)=\frac{2^{3/4}\pi\mathrm{i}}{L}\sum_{|n^{\perp}|\leT}\int_{0}^{\infty}dk\frac{n^{\perp}}{\sqrt{k}}\left[a_{ij}^{\dagger}(k,n^{\perp})b_{ij}(k,n^{\perp})-b_{ij}^{\dagger}(k,n^{\perp})a_{ij}(k,n^{\perp})\right]\,,
\begin{array}{rcl}{G(x_{1},x_{2},x_{3})}&{=}&{\displaystyle\intd^{4}x_{4}<0|T\left(\varphi(x_{1})\varphi(x_{2})\varphi(x_{3})(\varphi\star\varphi\star\varphi)(x_{4})\right)|0>,}\\\end{array}
S_{E}={\frac{\pi^{3}R^{8}}{4\kappa_{10}^{2}}}\int\!dz\,d^{4}x\,z^{-3}\left[(\partial_{z}\phi)^{2}+(\partial_{i}\phi)^{2}+{\frac{m^{2}R^{2}}{z^{2}}}\phi^{2}\right]\,.
\hat{\epsilon}^{\hat{\alpha}_{1}\hat{\alpha}_{2}\hat{\alpha}_{3}\hat{\mu}_{1}\ldots\hat{\mu}_{4}\hat{\nu}_{1}\ldots\hat{\nu}_{4}}\left(\partial\hat{C}+{\textstyle\frac{1}{3}}\partial\hat{L}\right)_{\hat{\mu}_{1}\ldots\hat{\mu}_{4}}\left(\partial\hat{C}+{\textstyle\frac{1}{3}}\partial\hat{L}\right)_{\hat{\nu}_{1}\ldots\hat{\nu}_{4}}=0\,.
S=\fraci{4\pi\alpha^{\prime}}\int\left\{g_{i\bar{\jmath}}(\partialu^{i}\bar{\partial}u^{\bar{\jmath}}+\bar{\partial}u^{i}\partialu^{\bar{\jmath}})-iB_{i\bar{\jmath}}(\partialu^{i}\bar{\partial}u^{\bar{\jmath}}-\bar{\partial}u^{i}\partialu^{\bar{\jmath}})\right\}\,d^{2}z,
H_{c}=\int{d^{3}x}\left({-\frac{1}{2}\Pi_{i}^{a}\Pi^{ai}+\Pi_{i}^{a}\partial^{i}A^{a0}+\frac{1}{4}F_{ij}^{a}F^{aij}-gf^{abc}\Pi^{ai}A^{b0}A_{i}^{c}}+A_{0}^{a}J^{0}\right).
\begin{array}{l}{c_{1}^{[\alpha]_{k}|[\beta]_{l}}=0,\quadk\neql,\quadk+l\neqn,}\\{c_{1}^{[\alpha]_{k}|[\beta]_{n-k}}=c_{1}^{(k)}\varepsilon^{[\alpha]_{k}[\beta]_{n-k}},\quadk,n-k\ge4,\quad2k\neqn,}\\{\varepsilon(c_{1}^{(k)})=n\,(mod\,\,2),\quadc_{1}^{(k)}=-c_{1}^{(n-k)}.}\\\end{array}
\sum_{l_{2}=1}^{*}{\frac{(-1)^{l_{2}}l_{2}^{2}}{(l_{1}^{2}-l_{2}^{2})^{3}}}=(-1)^{l_{1}}\left({\frac{1}{32l_{1}^{4}}}+{\frac{\pi^{2}}{96l_{1}^{2}}}\right).
\begin{aligned}{\mathcal{A}_{r}(\sigma^{1},\sigma^{2},\tau)=}&{{}\sum\mathcal{A}_{r}^{A}(\tau)Y_{A}(\sigma^{1},\sigma^{2})}\\{\Pi^{r}(\sigma^{1},\sigma^{2},\tau)=}&{{}\sum\sqrt{W}\Pi^{r,A}(\tau)Y_{A}(\sigma^{1},\sigma^{2})}\\\end{aligned}
\partial_{\mu}q^{u}\to{\mathcalD}_{\mu}q^{u}=\partial_{\mu}q^{u}-k_{I}^{u}A_{\mu}^{I}\,\qquad\partial_{\mu}z^{i}\to{\mathcalD}_{\mu}z^{i}=\partial_{\mu}z^{i}-k_{I}^{i}A_{\mu}^{I}\.
I_{rel}=\frac{-i\lambda\pi^{\frac{3}{2}}}{6(2\pi)^{3}}\int_{0}^{\infty}\frac{d\alpha}{\alpha^{3/2}}e^{-\frac{\tilde{p}^{2}}{4\alpha}-\alpham^{2}}=\frac{-i\lambdae^{-\tilde{p}m}}{24\pi\tilde{p}}\,.
f_{\mu\nu}(x):=\partial_{\mu}(n^{A}(x){\calA}_{\nu}^{A}(x))-\partial_{\nu}(n^{A}(x){\calA}_{\mu}^{A}(x))+{\frac{i}{g}}{\bfn}(x)\cdot[\partial_{\mu}{\bfn}(x),\partial_{\nu}{\bfn}(x)].
{\vec{v}}^{\prime}=\gamma^{-1}\vec{v}+\gamma^{-2}\vec{V}\left[\displaystyle\frac{\vec{V}\vec{v}}{c^{2}\left(\gamma+\sqrt{\gamma^{2}+\displaystyle\frac{\vec{V}^{2}}{c^{2}}}\right)}-\frac{\vec{\sigma}\vec{v}}{c^{2}}\gamma_{0}^{-2}-1\right],
\begin{array}{c}{a_{2}^{+}a_{1}^{+}=q\a_{1}^{+}a_{2}^{+}\\\a_{2}a_{1}=q^{-1}\a_{1}a_{2}}\\{a_{2}a_{1}^{+}=a_{1}^{+}a_{2}\\\a_{1}a_{2}^{+}=a_{2}^{+}a_{1}}\\\end{array}
f\starg\=f\mathrm{e}^{\frac{i\hbar}{2}\sum_{j=1}^{N}(\stackrel{\leftarrow}{\partial}_{x_{j}}\stackrel{\rightarrow}{\partial}_{p_{j}}-\stackrel{\leftarrow}{\partial}_{p_{j}}\stackrel{\rightarrow}{\partial}_{x_{j}})}g
{\calM}=\left(\begin{array}{ccc}{G^{-1}}&{G^{-1}(B+C)}&{G^{-1}A}\\{(-B+C)G^{-1}}&{(G-B+C)G^{-1}(G+B+C)}&{(G-B+C)G^{-1}A}\\{A^{T}G^{-1}}&{A^{T}G^{-1}(G+B+C)}&{I_{16}+A^{T}G^{-1}A}\\\end{array}\right).
R_{k}(\xi)_{ac,bd}=\delta_{a+b,c+d}\cdot\frac{\vartheta_{1}(\frac{\mu-\xi-a+b}{k},\frac{\tau}{k})\,\vartheta_{1}^{\prime}(0,\frac{\tau}{k})}{k\,\vartheta_{1}(\frac{\mu-a+c}{k},\frac{\tau}{k})\,\vartheta_{1}(\frac{\xi-b+c}{k},\frac{\tau}{k})}
\begin{array}{l}{\displaystylea(r)=\frac{\pi^{2}}{2\nu}r\frac{d}{dr}G_{\rhom_{_W}}(r)+\frac{1}{\nu}f^{a}(r),}\\{\displaystyleh(r)=-\frac{\pi^{2}}{2\nu}G_{\rhom_{_H}}(r)+\frac{1}{\nu}f^{h}(r),}\\\end{array}
\begin{array}{ccl}{\left(t_{51}^{(9)}\right)_{BiCj}}&{=}&{\frac{1}{\theta}\left(\sigma_{1}\delta_{1B}\delta_{1C}\delta_{ij}+\sigma_{2}\delta_{2B}\delta_{2C}\delta_{ij}+\sigma_{3}\delta_{3B}\delta_{3C}\delta_{ij}+\sigma_{4}\delta_{4B}\delta_{4C}\delta_{ij}\right)}\\{}&{=}&{\frac{1}{\theta}\left(\displaystyle\sum_{A=1}^{4}\sigma_{A}\delta_{AB}\delta_{AC}\delta_{ij}\right)}\\\end{array}
\hat{Q}^{+}(\xi_{\alpha})=\xi_{\alpha}^{2}\frac{\partial}{\partial\xi_{\alpha}}-J_{\alpha}\xi_{\alpha},\\\hat{Q}^{-}(\xi_{\alpha})=-\frac{\partial}{\partial\xi_{\alpha}},\\\hat{Q}^{3}(\xi_{\alpha})=\xi_{\alpha}\frac{\partial}{\partial\xi_{\alpha}}-\frac{J_{\alpha}}{2}.
\langle\overline{{\phi}}^{2}(x)\rangle_{\beta}=\frac{1}{4\pi^{2}r^{2}}\sum_{n=1}^{\infty}\sum_{l=0}^{\infty}(2l+1)Q_{\nu_{l}-1/2}(z_{n}-\eta^{2}\lambda_{n}^{2})\,
{\calH}_{\mu\nu}=c_{1}F_{\mu\nu}+c_{2}\tilde{F}_{\mu\nu}+{\frac{c_{3}}{2}}\epsilon_{\mu\nu\alpha\beta}F_{\alpha\beta}+{\frac{c_{4}}{2}}\epsilon_{\mu\nu\alpha\beta}\tilde{F}_{\alpha\beta}\,;
|\mathrm{D}p_{\psi},\eta\rangle_{\mathrm{NS}}^{(0)\,\,T}=\left(\widehatC{\widehat{\gamma}}^{6}\dots{\widehat{\gamma}}^{5+s}\frac{1+\mathrm{i}\eta\widehat\gamma}{1+\mathrm{i}\eta}\right)_{LM}|L\rangle|\widetildeM\rangle
{\bfb}^{-}(t,{\bfr})=\int_{0}^{\Lambda}d\omega\sum_{l,m}\lbrace\alpha_{lm}(\omega)\,e^{-i\omegat}\,{\bfY}_{lm}(\theta,\phi)\,K_{j}(\omegar)+\beta_{lm}(\omega)\,e^{i\omegat}\,\bigl[{\bfY}_{lm}(\theta,\phi)\,K_{j}(\omegar)\,\bigr]^{*}\rbrace.
{\itF}_{0}^{a}=0;\quad{\itF}_{n}^{a}=-\frac{1}{a}\varepsilon_{mnc}D_{m}\phi^{a}H_{c}^{(ext)};\quad{\itF}^{a}=\frac{1}{2a}\varepsilon_{mnc}F_{mn}^{a}H_{c}^{(ext)}.
A=\sum_{j=0}^{l-m}\frac{(-1)^{j+m^{-}}(l+m)!}{(m^{-}+j)!(l^{+}-j)!}\:\frac{l^{-}!(1-x)^{l-m-j}}{(l-m-j)!}\:\frac{l^{+}!(1+x)^{j}}{j!}.
\left(\begin{matrix}{F_{11}^{(2)}}\\{F_{3}^{(2)}}\\\end{matrix}\right)\rightarrow\left(\begin{matrix}{a}&{b}\\{c}&{d}\\\end{matrix}\right)\left(\begin{matrix}{F_{11}^{(2)}}\\{F_{3}^{(2)}}\\\end{matrix}\right)\,.
\overline{{{\cal{A}}_{i}^{\gamma}}}({\bf{r}})={\delta}_{{i}\,{\gamma}}\,{\xi}_{A}(r)+\frac{r_{i}\,r_{\gamma}}{r^{2}}\,{\xi}_{B}(r)+{\epsilon}_{i{\gamma}n}\frac{r_{n}}{r}\,{\xi}_{C}(r)
\begin{array}{l}{Y_{1,1}=\beta^{2}\,\Y_{2,2}=-\beta^{2}/4\,\Y_{2,4}=\beta^{2}/4\,\}\\{Y_{3,3}=\beta^{2}/6-\beta^{4}/12\,\Y_{3,5}=-7\beta^{2}/8+\beta^{4}/4\,}\\{Y_{3,7}=5\beta^{2}/8\.}\\\end{array}
C_{i,j}^{-1}(x,y)=\left(\begin{array}{ccc}{a^{-1}(\vec{x},\vec{y})}&{p(\vec{x},y^{\bot})}&{0}\\{-p(\vec{y},x^{\bot})}&{0}&{-c^{-1}(y^{\bot},x^{\bot})}\\{0}&{c^{-1}(x^{\bot},y^{\bot})}&{0}\\\end{array}\right),
\begin{array}{l}{\alpha(K_{\alpha}^{})\,\Phi(\alpha(q),v)\,A_{k}^{\alpha}(q,u)}\\{\mathrm{}-\,{\textstyle\frac{1}{2}}\,N_{\theta\alpha,\alpha}^{}\,\Phi(\alpha(q),u+v)\,\Bigl(A_{k}^{\alpha+\theta\alpha}(q,-u)\,-\,A_{k}^{\alpha+\theta\alpha}(q,v)\Bigr)}\\{\mathrm{}-\,{\displaystyle\sum_{j=1}^{r}}\,\alpha_{j}^{}\,A_{j}^{\alpha}(q,u)A_{k}^{\alpha}(q,v)~=~0~.}\\\end{array}
\delta{\calL}_{total}=\omega\left[J^{\mu,a}(D_{\mu}c^{a})-i\overline{{\eta}}^{a}(-{\frac{1}{2}}f^{abc}c^{b}c^{c})+iF^{a}\eta^{a}+H^{\mu}{\frac{\partial{\calL}_{total}}{\partialn^{\mu}}}\right]
{\calP}(t)=-E+q^{2}(\partial{\cal\psi})^{2}-\frac14\int_{0}^{1}d\rho(e^{\frac{1-\rho}{8}{\partial}^{2}}(-q^{2}{\partial}^{2}+1)\partial_{t}\psi)(e^{-\frac{1-\rho}{8}{\partial}^{2}}\partial_{t}\psi).
H_{c}=\intd^{2}x\left(-\frac{1}{2}F_{i0}F^{i0}+\frac{1}{4}F^{ij}F_{ij}+\pi_{i}\partial^{i}A_{0}-\frac{\theta}{2}\varepsilon^{ij}A_{0}\partial_{i}A_{j}+A_{0}J^{0}\right).
\bar{\calX}_{,i}^{\mu}\bar{\calX}_{,j}^{\nu}{\calG}_{\mu\nu}=\bar{g}_{ij},\;\bar{\calX}_{,i}^{\mu}\bar{\eta}_{A}^{\nu}{\calG}_{\mu\nu}=0,\;\bar{\eta}_{A}^{\mu}\bar{\eta}_{B}^{\nu}{\calG}_{\mu\nu}=g_{AB}
\hat{J}_{3}=\hat{p}\,\\hat{J}_{+}=\hat{W}\sqrt{\lambda^{2}-\left(\hat{p}+1/2\right)^{2}}\,\\hat{J}_{-}=\sqrt{\lambda^{2}-\left(\hat{p}+1/2\right)^{2}}{\hat{W}}^{\dagger}\.
\{(r-s)z^{12}z_{1}^{r-s}-2(z_{1}^{r-s}z_{2}+\cdots+z_{1}z_{2}^{r-s})\}=z_{12}\{(r-s)z_{1}^{r-s}+\sum_{\ell=1}^{r-s-1}2(r-s-\ell)z_{1}^{r-s-\ell}z_{2}^{\ell}\}\,.
{\bf\hat{V}}(\tau)=:e^{-\gamma\left(\hat{q}+\frac{\hat{p}\tau}{2\pi}\right)}\left(1+m_{\alpha}^{2}\int_{0}^{2\pi}\int_{0}^{2\pi}dy\,d\bar{y}\,\,G_{\alpha}(z,\bar{z};\,y,\bar{y})\,e^{2\gamma\left(\hat{q}+\frac{\hat{p}(y+\bar{y})}{4\pi}\right)}\right):\,,
H_{f}=\frac{\pi_{f}^{2}}{2R^{3}}+\frac{R}{2}(\nabla\phi_{f})^{2}+R^{3}\Bigl(\frac{m^{2}}{2}+\frac{3\lambda}{2}\phi_{c}^{2}\Bigr)\phi_{f}^{2}+\frac{\lambdaR^{3}}{4}\phi_{f}^{4}.
u_{\Omega,A}^{(\sigma)}=\int_{-\infty}^{\infty}\!dk\sum_{M,N}\left(\alpha_{\Omega}^{k\{M,N\}(\sigma)}U_{k}+\beta_{\Omega}^{k\{M,N\}(\sigma)}U_{k}^{*}\right)\Psi_{\{M,N\}}\cdot\phi_{A}\,
S_{Sieg}=\intd^{4}x[G^{a}G^{a}]_{D}=\intd^{4}x\;Tr\left\{-\frac{1}{2}(D_{\mu}C^{a})^{2}-\frac{1}{2}\bar{\chi}{}^{a}\hat{D}\chi^{a}-\frac{1}{2}(v^{a})_{\mu}^{2}\right\},
\Pi_{i}^{a}\left({\bfx}\right)\left|\Omega\right\rangle=\overline{\psi}\left({\bfy}\right)U({\bfy},{\bf0})\psi\left({\bf0}\right)\Pi_{i}^{a}\left({\bfx}\right)\left|0\right\rangle+gT^{a}\int_{\bf0}^{\bfy}{dz_{i}}\delta\left({{\bfx}-{\bfz}}\right)\left|\Omega\right\rangle.
(\psi^{n},\phi^{n})=\int\cdots\int(\sum_{\mu_{i}}\bar{\psi}_{\mu_{1},\cdots,\mu_{n}}^{n}(p_{1},\cdots,p_{n}))(\sum_{\nu_{i}}\phi_{\nu_{1},\cdots,\nu_{n}}^{n}(p_{1},\ldots,p_{n}))dp_{1}\ldotsdp_{n}
{\calL}_{2}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}\theta^{\alpha\beta}F_{\alpha\mu}F_{\beta\nu}F^{\mu\nu}+\frac{1}{8}\theta^{\alpha\beta}F_{\alpha\beta}F_{\mu\nu}F^{\mu\nu},
{(z^{1})}^{2}+{\frac{((p\cdotq)-2p^{1}q_{1})}{(p^{0}q_{1}-p^{2}p^{3})}}z^{1}-{\frac{(p^{1}q_{0}+q_{1}q_{2})}{(p^{0}q_{1}-p^{2}p^{3})}}=0\,
I=\intd^{4}x\epsilon^{\mu\nu\alpha\beta}\epsilon_{abcd}\bigg(c_{1}{^+}G_{\mu\nu}^{ab}{^+}G_{\alpha\beta}^{cd}+c_{2}{^-}G_{\mu\nu}^{ab}{^-}G_{\alpha\beta}^{cd}+c_{3}{^+}F_{\mu\nu}^{ab}{^+}G_{\alpha\beta}^{cd}+c_{4}{^-}F_{\mu\nu}^{ab}{^-}G_{\alpha\beta}^{cd}\bigg).
\int_{S^{2}}c_{1}=-\int_{S^{2}}\frac{{\calF}}{2\pi}=-\frac{1}{2\pi}[\int_{H^{+}}dA^{+}+\int_{H^{-}}dA^{-}]=-\frac{1}{2\pi}\int_{S^{1}}A^{+}-A^{-}
\sum_{n=-\infty}^{\infty}e^{-\alpha_{1}\pi^{2}T^{2}(2n+1)^{2}}\,e^{-\alpha_{2}\pi^{2}T^{2}(2n+1+2l)^{2}}=e^{-4\pi^{2}l^{2}T^{2}\alpha_{2}}\,\Theta_{2}\left(4\piilT^{2}\alpha_{2}|4\piiT^{2}(\alpha_{1}+\alpha_{2})\right)
\hat{r}_{c}(p_{0})=\frac{i}{2\pi}\,p_{0}^{\omega+1}\,\int_{-\infty}^{+\infty}\!\!dk_{0}\,\frac{\hat{d}(k_{0})}{(k_{0}-i0)^{\omega+1}(p_{0}-k_{0}+i0)}\,,
(3,1):\quad\left(\begin{array}{c}{u_{L}^{i}}\\{d_{L}^{i}}\\{b_{L}^{i}}\\\end{array}\right),~~~~\quad(1,\bar{3}):\quad\left(\begin{array}{c}{\hat{u}_{R}^{i}}\\{\hat{d}_{R}^{i}}\\{\hat{b}_{R}^{i}}\\\end{array}\right)
G(p)_{\mu\nu}=-i\frac{(-p^{2})\Pi_{e}}{\Pi_{e}^{2}p^{2}-\Pi_{o}^{2}M^{2}}\left[{\calP}_{\mu\nu}-iM\frac{\Pi_{o}}{\Pi_{e}}\epsilon_{\mu\nu\rho}\frac{p^{\rho}}{p^{2}}\right]
\frac{1}{V}<H^{\theta}>=\frac{1}{8\pi}[\frac{4}{3}\Lambda^{3}+\frac{1}{3}m^{3}-\frac{2}{3}(\lambda\eta^{2})^{\frac{3}{2}}+\lambda\eta^{2}\Lambda]-\frac{1}{4}\frac{g^{2}}{\pi^{2}}m^{2}
{\calL}_{\mathrm{rel}}=\frac{1}{2}\,(g_{\mathrm{rel}})_{\mu\nu}\dot{z}^{\mu}\dot{z}^{\mu}-{\calU}_{\mathrm{rel}}=\frac{1}{2}\left(1+\frac{1}{r}\right)\dot{\bfr}^{2}+\frac{1}{2}\left(\frac{1}{1+{1}/{r}}\right)(\dot{\chi}+{\bfw}({\bfr})\cdot\dot{\bfr})^{2}-\frac{1}{2}\left(\frac{a^{2}}{1+1/r}\right).
R^{\iota_{1}}=\sum_{i=1}^{N}qe_{ii}\otimese_{ii}+\sum_{i,j=1\atopi\neqj}^{N}b_{ij}e_{ii}\otimese_{jj}+(q-q^{-1})\sum_{i,j=1\atopi>j}^{N}e_{ij}\otimese_{ji}
\{\Psi_{2}^{2},\Psi_{3}^{2}\}\stackrel{\Sigma}{=}0,~~~~\{\Psi_{2}^{2},\Psi_{2}^{4}\}\stackrel{\Sigma}{=}0,~~~~\{\Psi_{2}^{3},\Psi_{3}^{1}\}\stackrel{\Sigma}{=}0
|\phi_{n}\rangle_{gh}^{x}=\hat{\Gamma}^{n}|0\rangle_{gh}^{x}=\hat{Q}\hat{\Phi}\hat{\Gamma}^{n-1}|0\rangle_{gh}^{x}=\hat{Q}^{\dagger}\hat{\Phi}^{\prime}\hat{\Gamma}^{n-1}|0\rangle_{gh}^{x}
{\frac{P^{\mu}}{\alpha}}=\left({\frac{m}{\alpha}}+{\frac{\kappa_{2}^{2}}{2}}\right)X^{\mu}{}^{\prime}-{\frac{\kappa_{2}\kappa_{2}{}^{\prime}}{\kappa_{1}}}\,\eta_{1}^{\mu}-\Big[\left({\frac{\kappa_{2}^{\prime}}{\kappa_{1}}}\right)^{\prime}-\kappa_{1}\kappa_{2}\Big]\eta^{\mu}{}_{2}\,,
X_{a}^{i}=\left(\begin{array}{cc}{X_{a,11}^{i}}&{X_{a,12}^{i}\nonumber}\\{X_{a,21}^{i}}&{X_{a,22}^{i}}\\\end{array}\right)=\left(\begin{array}{cc}{C_{a}^{i}+{{\calO}(t)}}&{{{\calO}(t^{\gamma})}\nonumber}\\{{{\calO}(t^{\gamma})}}&{{\frac{T_{a}^{i}}{s-\mu_{a}}}}\\\end{array}\right).
\{\Phi^{\alpha}(x),\Phi^{\beta}(y)\}=\omega^{\alpha\beta}(x,y)=\left(\begin{array}{cccccc}{0}&{0}&{1}&{0}&{0}&{0}\\{0}&{0}&{0}&{\delta^{ij}}&{0}&{0}\\{-1}&{0}&{0}&{0}&{0}&{0}\\{0}&{-\delta^{ij}}&{0}&{0}&{0}&{0}\\{0}&{0}&{0}&{0}&{0}&{\epsilon^{ijk}}\\{0}&{0}&{0}&{0}&{-\epsilon^{ijk}}&{0}\\\end{array}\right)\delta(x-y).
\dot{\epsilon}_{\delta\sigma}=\epsilon_{\delta\sigma}\left[3{\frac{\dot{a}}{a}}(1+w)+\left\{2{\frac{\dot{G}_{5}}{G_{5}}}-{\frac{6+8\epsilon}{1+\epsilon}}{\frac{\dot{c}}{c}}\right\}(1+\epsilon_{\delta\sigma})+{\frac{9Cc^{6}}{32\pi^{2}G_{5}^{2}\sigmaa_{0}^{4}\varrho_{m}}}{\frac{\dot{c}}{c}}\right].