harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{{\mathbfi}{\mathbfj}}{\bar{D}}^{{\mathbfk}{\mathbfl}}F_{{\mathbfk}{\mathbfl}}={\bar{D}}_{{\mathbfi}{\mathbfj}}(F^{{\mathbfk}{\mathbfl}}F_{{\mathbfk}{\mathbfl}})+\frac{3}{10}{\bar{D}}^{{\mathbfk}{\mathbfl}}(F_{({\mathbfi}{\mathbfj}}F_{{\mathbfk}{\mathbfl})})\,
\langle0|a(k^{\prime}s^{\prime})a^{*}(q,r)a(q^{\prime},r^{\prime})a^{*}(k,s)|0\rangle=\frac{2\kappa^{\prime}g(s^{\prime})}{(2\pi)^{4}\lambda}\delta_{rs^{\prime}}\delta^{4}(q-k^{\prime})\frac{2\kappag(s)}{(2\pi)^{4}\lambda}\delta_{r^{\prime}s}\delta^{4}(q^{\prime}-k)
\frac{Z_{3}}{Z_{\lambda}\lambda}\,\langle\,\partialA_{x}^{a}\,\partialA_{y}^{b}\,\rangle\,=\,\widetildeZ_{3}\,\Big((1-\frac\alpha2)\,\langle\,i(\partialD_{r}c)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\,+\,\frac\alpha2\,\langle\,c_{y}^{b}\,i(\partialD_{r}\bar{c})_{x}^{a}\,\rangle\Big)\;.
T\bigl({\calO}\bigr)=\oint_{C_{0}}dz_{2}\oint_{C_{z_{2}}}dz_{1}\,J(z_{1})V(z_{2})\quad\mathrm{with}\quad{\calO}=\oint_{C_{0}}dz\,V(z)\,\,,\,\,T=\oint_{C_{0}}dz\,J(z)\,\,,\,\,T=\mathbf{H}_{n},\mathbf{E}_{n}^{\pm},L_{n}\,\,.
((F^{s})^{I_{1}I_{2}I_{3}I_{4}})_{\mu\nu}=\sum_{\lambda=0}^{2I}\sigma_{\nu\lambda}(I_{1},I_{2},I_{3},I_{4})\sigma_{2I-\lambda,\mu}^{-1}(I_{3},I_{2},I_{1},I_{4})\,
x^{\mu}(\bar{\tau})=\frac{p^{\mu}}{m}\bar{\tau}+x^{\mu}(0),\\\\x^{\mu}(0)\stackrel{\mathrm{def}}{=}-\frac{p^{\mu}}{m}\tau_{0}-2i\{\bar{\alpha},\sigma^{\nu\mu}\beta\}_{_{\sim}}p_{\nu}
{\calL}_{Weyl}=\partial_{+}{A}_{i}(\partial_{-}{A}_{i}-\partial_{i}{A}_{-})+\frac12\left(\partial_{+}{A}_{-}\right)^{2}-\frac14\left(\partial_{i}{A}_{j}-\partial_{j}{A}_{i}\right)^{2}+{A}_{-}{j}^{-}+{A}_{i}j^{i}.
\intdyK_{\mu}^{(1)}(y;x)A^{\mu}(y)=\intdyK_{\mu}^{(1)}(y;x)A^{\mu}(y)+\frac{1}{2!}\int\intdydzK_{\mu\nu}^{(2)}(y,z;x)(A^{\mu}(y)\partial^{\nu}\theta(z)+\partial^{\mu}\theta(y)A^{\nu}(z))
L=m(\dot{x},n)-im(n_{\alpha\beta}\theta^{\alpha}\dot{\theta}^{\beta}+n_{\alpha\beta}\chi^{\alpha}\dot{\chi}^{\beta})+mb(\theta_{\alpha}\dot{\chi}^{\alpha}-\chi_{\alpha}\dot{\theta}^{\alpha})-mb\theta^{\alpha}n_{\alpha\gamma}\dot{n}^{\gamma}{}_{\beta}\chi^{\beta}+is\frac{\bar{z}\dot{z}-z\dot{\bar{z}}}{1-z\bar{z}}\;,
s(\Gamma)\equiv\int(\frac{\delta\Gamma}{\delta\varrho}\frac{\delta\Gamma}{\deltaA}+B\frac{\delta\Gamma}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\sigma}\frac{\delta\Gamma}{\deltac}+\frac{\delta\Gamma}{\delta\bar{Y}}\frac{\delta\Gamma}{\delta\psi}+\frac{\delta\Gamma}{\delta\bar{\psi}}\frac{\delta\Gamma}{\deltaY}=0
S_{\mathrm{anomalous}}={\frac{1}{T_{H}}}\int_{\Sigma}e^{-\phi}\{\varrho_{L}-{\calL}_{E}\}d^{3}r={\frac{k}{\hbar}}\int_{\Omega}\{\varrho_{L}-{\calL}_{E}\}\sqrt{g_{E}}d^{4}x.
{\calC}_{ABCD}={\textstyle\frac{1}{18}}(dxxx)^{2}\,g_{AE}\,g_{BF}\,g_{CG}\,\frac{\partial}{\partialx^{D}}C^{EFG}=-6g_{DF}\,E_{ABCE}^{F}\,x^{E}=6g_{E}\,E_{ABCD}^{E}\.
{\calL}=-{\frac{1}{4}}H_{\mu\nu}^{a}H^{a\mu\nu}+{\frac{1}{2}}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}+{\frac{1}{2}}\mu^{2}\Phi^{a}\Phi^{a}-{\frac{1}{4}}\lambda(\Phi^{a}\Phi^{a})^{2}
K_{\mathrm{m}}^{(1)}=-2{\frac{i\gamma^{0}}{h}}igh\sum_{n^{\prime}=0}^{n}\!\!{}^{\prime}(-1)^{n+n^{\prime}}(A_{0})_{{\bfm},\overline{{n^{\prime}-1}}}\psi_{\overline{{\bfm}},n^{\prime}},
R_{\ell}(r)={\frac{1}{\sqrtv}}\;{\frac{\Gamma(\ell+1+i\eta)}{\Gamma(2\ell+1)}}\;e^{-\pi\eta/2}\;(2ikr)^{\ell}\;e^{ikr}F(\ell+1+i\eta\;,\;2\ell+2\;;\;-2ikr)
a_{m}\mid0\rangle=0,\;\;\;\;\tilde{a}_{m}^{\dagger}\mid0\rangle=a_{m}^{\dagger}\mid0\rangle,\;\;\;\;\psi_{\dot{\alpha}}\mid0\rangle=0,\;\;\;\;\tilde{\psi}_{\dot{\alpha}}^{\dagger}\mid0\rangle=\psi_{\dot{\alpha}}^{\dagger}\mid0\rangle
[n_{i}^{F},n_{j}^{F}]=\frac{i\epsilon_{ijk}n_{k}^{F}}{[l(l+1)]^{1/2}},\quad\sum_{i=1}^{3}n_{i}^{F2}={\bf1},\quadl\in\{\frac{1}{2},1,\frac{3}{2}\ldots\}.
\tilde{D}_{0}^{\dagger}\tilde{D}_{0}\left(\begin{array}{c}{(d_{+})^{m}\chi_{1}^{(n)}(z,\bar{z})}\\{0}\\\end{array}\right)=\frac{4\pikm}{|\tau|}\left(\begin{array}{c}{(d_{+})^{m}\chi_{1}^{(n)}(z,\bar{z})}\\{0}\\\end{array}\right).
H=\frac{1}{2}\pi_{A}^{\mu}\pi_{A}^{\mu}+\frac{1}{4}f_{ABC}\phi_{B}^{\mu}\phi_{C}^{\nu}f_{ADE}\phi_{D}^{\mu}\phi_{E}^{\nu}-\frac{i}{2}f_{ABC}\phi_{A}^{\mu}\Lambda_{B\alpha}\Gamma_{\alpha\beta}^{\mu}\Lambda_{C\beta}
\bar{\theta}_{R}^{a}\theta_{L}^{b}|0,p^{+}\rangle_{U}\quad\longleftrightarrow\quad{\frac{1}{\sqrt{2JN^{J+2}}}}(\sum_{l=0}^{J}Tr[\chi^{a}\tilde{Z}^{l}\psi^{b}Z^{J-l}]+\sum_{l=0}^{J}Tr[\psi^{a}Z^{l}\chi^{b}\tilde{Z}^{J-l}]),
\bar{\partial}_{1}\langle{\calT}(a_{1}){\calT}(a_{2}){\calT}(a_{3})\rangle=\kappa\pi\left(\partial_{1}\delta^{(3)}(z_{1}-z_{3})\langle{\calT}(a_{1}){\calT}(a_{2})\rangle-3\partial_{1}\delta^{(3)}(z_{1}-z_{2})\langle{\calT}(a_{1}){\calT}(a_{3})\rangle\right)
{\frac{\langleWW\rangle_{dil}}{\langleW\rangle^{2}}}=\mathrm{Exp}\left[\sum_{k,A}Y_{A}^{(k)}(\theta)^{2}\frac{1}{4}\int\frac{d{\calA}}{2\pi\alpha^{\prime}}\int\frac{d{\calA}^{\prime}}{2\pi\alpha^{\prime}}G_{k}(W(\sigma,\sigma^{\prime}))_{dil}\right],
V=-\frac{n}4\frac{\tilde{\gamma}k_{T}\tilde{g}^{2}}{\tilde{\gamma}\tilde{g}^{2}-\frac1{k_{T}}}\left(m_{\bar{q}}^{2}-\frac{\frac12\tilde{\gamma}\tilde{g}^{2}-\frac1{k_{T}}}{\frac12\tilde{\gamma}\tilde{g}^{2}}m_{q_{1}}^{2}\right)^{2}-\frac{n}{\tilde{\gamma}\tilde{g}^{2}}m_{q_{1}}^{4}
{\calL}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{\kappa}{2}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+\sum_{a}\bar{\psi}_{a}[\gamma_{a}^{\mu}(i\partial_{\mu}+q_{a}A_{\mu})-m_{a}]\psi_{a}\,\,,
k\left[\left({\frac{\dot{R}}{R}}\right)^{2}+\xi_{+}^{2}R^{-{\frac{2}{3}}k^{2}}\right]^{\frac{1}{2}}-k\left[\left({\frac{\dot{R}}{R}}\right)^{2}+\xi_{-}^{2}R^{-{\frac{2}{3}}k^{2}}\right]^{\frac{1}{2}}={\kappa_{5}^{2}}\left[\lambda^{\prime}(\phi)-(3w-1)b\,\rho\right].
\sum_{k=1}^{2}\left[B_{ik}(t)\dot{B}_{jk}(t)-\dot{B}_{ik}(t)B_{jk}(t)\right]=\sum_{k=1}^{2}\left[B_{ik}^{\ast}(t)\dot{B}_{jk}^{\ast}(t)-\dot{B}_{ik}^{\ast}(t)B_{jk}^{\ast}(t)\right]=0\;.
\delta_{2}(q)=\frac{1}{2}\mathrm{arctan}\left(\frac{\mathrm{Im}\prod_{n=0}^{l-1}(q^{2}-(l-n)^{2}+2iq(l-n))}{\mathrm{Re}\prod_{n=0}^{l-1}(q^{2}-(l-n)^{2}+2iq(l-n))}\right).
{\frac{h-\sqrt{h^{2}-2(m_{q}^{2}+m_{\bar{q}}^{2})/k_{q}}}{2/k_{q}}}\leqT_{(i)}\leq{\frac{h+\sqrt{h^{2}-2(m_{q}^{2}+m_{\bar{q}}^{2})/k_{q}}}{2/k_{q}}},
\begin{array}{rl}{}&{P(1\rightarrow2)=}\\{}&{=\intdU(\theta_{2}^{a})\intdU(\theta_{1}^{a})\,<(\pi_{2})_{\theta_{2}}^{a},(A_{2})_{\theta_{2}}^{a};\eta_{\theta_{2}}|e^{-\frac{1}{2}i(t_{2}-t_{1})[(\hat{\pi}^{a})^{2}+m^{2}(\hat{A}^{a})^{2}]}|(\pi_{1})_{\theta_{1}}^{a},(A_{1})_{\theta_{1}}^{a};\eta_{\theta_{1}}>\\.}\\\end{array}
\mathrm{Tr}\mathcal{P}_{\left\{\theta_{1},\ldots,\theta_{l}\right\}}=\prod_{i=1}^{l}\mathrm{Tr}\mathcal{P}_{\left\{\theta_{i}\right\}}=\prod_{i=1}^{l}(\mathrm{Tr}P_{\left\{\theta_{i}\right\}}+\mathrm{Tr}\overline{{P}}_{\left\{\theta_{i}\right\}}).
T_{\mu\nu}^{(gd)}=\frac{1}{2\pi}\{2(g_{\mu\nu}\squareF-\nabla_{\mu}\nabla_{\nu}F)-Ug_{\mu\nu}+2V\nabla_{\mu}\phi\nabla_{\nu}\phi-g_{\mu\nu}V(\nabla\phi)^{2}\}\mathrm{,}
B={\partial}_{+}{\pi}^{+}+\sum_{i=1}^{2}{\partial}_{i}{\pi}^{i},\quad{\partial}_{-}A_{+}={\pi}^{+},\quad{\partial}_{-}A_{i}=\frac{{\pi}^{i}-\mathrm{sin}2{\theta}\;F_{+i}}{\mathrm{cos}2{\theta}}.
\frac{\partial}{\partialt^{\prime}}[\omega_{a}^{QH^{\prime}S}(t^{\prime},t)]_{Q^{S}(t,z)}=z(\bar{C}(\Psi,t^{\prime},t)_{a}^{b}-\Psi(0)\delta_{a}^{b})[\omega_{b}^{QH^{\prime}S}(t^{\prime},t)]_{Q^{S}(t,z)}
\hat{H}={\frac{1}{2}}{\vartheta}^{\frac{1}{2}}f^{-{\frac{1}{2}}}\hat{p}_{i}g^{ij}f\hat{p}_{j}f^{-{\frac{1}{2}}}{\vartheta}^{\frac{1}{2}}+{\frac{i}{2}}{\vartheta}^{-{\frac{1}{2}}}f^{\frac{1}{2}}C_{oa}^{aj}\hat{p}_{j}f^{-{\frac{1}{2}}}{\vartheta}^{\frac{1}{2}}+\lambda{\vartheta},
w=\left(\begin{array}{c}{n^{1}}\\{n^{2}}\\{0}\\{0}\\{n^{1}}\\{n^{2}}\\\end{array}\right)\,\\n^{1},n^{2}\inZ\\\\mathrm{and}\\\p=\left(\begin{array}{c}{m_{1}}\\{m_{2}}\\{-m_{1}}\\{-m_{2}}\\{m_{1}}\\{m_{2}}\\\end{array}\right)\,m_{1},m_{2}\inZ.
S={\frac{Tr_{0}^{D}}{2^{D}}}\int\sqrt{g}\;\left((1-q)\phi^{{\frac{2D}{D-2}}}+{\frac{8}{(D-2)^{2}}}[(\partial\phi)^{2}+{\frac{D-2}{4(D-1)}}\phi^{2}R]+{\calO}(\phi^{{\frac{2(D-4)}{D-2}}})\right).
\left[\left({\frac{H_{1}^{\prime}}{H_{1}}}+{\frac{H_{2}^{\prime}}{H_{2}}}\right)+{\frac{2}{r}}\right]h_{2}=-2\left({\frac{H_{1}^{\prime}}{H_{1}}}-{\frac{H_{2}^{\prime}}{H_{2}}}\right)\varphi.
j_{\mu}=\overline{{\Psi}}_{2}\gamma_{\mu}\Psi_{1}\,,\,\,\,\partial^{\mu}j_{\mu}=0,\,\,\,\,\,\left(\overline{{\Psi}}_{2}\Psi_{1}\right):=\intj_{0}d^{3}x=2m(\psi_{2},\psi_{1})\,
[\mathsf{L}_{1},\mathsf{u}_{2}]=\mathsf{u_{1}u_{2}}[\mathsf{B}_{12}^{-1}-\mathsf{I\otimesI},p(\mathsf{T}_{2})_{1}]\mathsf{u}_{1}^{-1}+i\hbar\mathsf{u_{1}u_{2}}\mathsf{(T_{2}\otimesT_{2})}\mathsf{B}_{12}^{-1}\mathsf{u}_{1}^{-1}
u=\frac{x_{13}^{2}x_{24}^{2}}{x_{12}^{2}x_{34}^{2}},\;v=\frac{x_{14}^{2}x_{23}^{2}}{x_{12}^{2}x_{34}^{2}},\;x_{ij}=x_{i}-x_{j}.\notag
{\frac{3}{\sqrt{g}}}\partial_{\mu}\left[\sqrt{g}\,\partial_{[\mu^{\prime}}A_{\mu_{1}^{\prime}\mu_{2}^{\prime}]}\,g^{\mu^{\prime}\mu}g^{\mu_{1}^{\prime}\mu_{1}}g^{\mu_{2}^{\prime}\mu_{2}}\right]-16g^{\mu_{1}^{\prime}\mu_{1}}g^{\mu_{2}^{\prime}\mu_{2}}A_{\mu_{1}^{\prime}\mu_{2}^{\prime}}=0,
\begin{array}{c}{~~e_{i}e_{j}=-\delta_{ij}+c_{ijk}\,e_{k}\,,~~~~~~~{i,j,k=1,2,\cdots,7}}\\{{}}\\{1=c_{127}=c_{163}=c_{246}=c_{347}=c_{567}=c_{253}=c_{154}~~~\mathrm{(others~zero)}\,.}\\\end{array}
S(Z)={\calD}^{-1}\left(\begin{array}{ccc}{z^{5}z^{9}-qz^{6}z^{8}}&{-q^{-1}z^{2}z^{9}+z^{3}z^{8}}&{q^{-2}z^{2}z^{6}-q^{-1}z^{3}z^{5}}\\{-qz^{4}z^{9}+q^{2}z^{6}z^{7}}&{z^{1}z^{9}-qz^{3}z^{7}}&{-q^{-1}z^{1}z^{6}+z^{3}z^{4}}\\{q^{2}z^{4}z^{8}-q^{3}z^{5}z^{7}}&{-qz^{1}z^{8}+q^{2}z^{2}z^{7}}&{z^{1}z^{5}-qz^{2}z^{4}}\\\end{array}\right).
\tilde{R}_{L}^{(n)}(t,t^{\prime};\{q\})\approx\frac{\tilde{a}_{L}^{(n)}(\{q\})+\tilde{\alpha}_{L}^{(n)}(\{q\})(\vartheta+\vartheta^{\prime})}{(z-z_{0})(z^{\prime}-z_{0})}-\frac{\tilde{b}_{L}^{(n)}(\{q\})\vartheta\vartheta^{\prime}}{(z-z_{0})(z^{\prime}-z_{0})}\Biggl[\frac{1}{z-z_{0}}-\frac{1}{z^{\prime}-z_{0}}\Biggl]\,.
S^{E}=\frac{1}{g^{2}}\int_{0}^{\beta}d\tauTr\left[\dot{Y}^{i}\dot{Y}^{i}+\theta^{T}\dot{\theta}+\frac{1}{2}\left[Y^{i},Y^{j}\right]^{2}+\theta^{T}\gamma^{j}\left[Y^{i},\theta\right]\right]
|\,V_{B}>=\mathrm{exp}\left({\frac{1}{2}}\sum_{r,s=1}^{3}\sum_{m,n=-\infty}^{\infty}\sum_{I=1}^{8}a_{mr}^{I\dagger}\bar{N}_{mn}^{rs}a_{ns}^{I\dagger}\right)|\,0>.
\langle\prod_{i=1}^{N-2+(N-k)d}{\calO}_{e}(z_{i})\rangle=\int_{{\calM}_{0,d}^{M_{N}^{k}}}\bigwedge_{i=1}^{N-2+(N-k)d}\phi_{i}^{*}(e)=\int_{{\calM}_{0,d}^{CP^{N-1}}}c_{(N,k)}\wedge(\bigwedge_{i=1}^{N-2+(N-k)d}\varphi_{i}^{*}(e))
{\cal{G}}_{1}^{(\tilde{\eta}_{o})}(x_{1},x_{4},x_{3},x_{2})=\frac{g_{*}^{2}}{C_{\tilde{O}}}\,\frac{1}{(x_{12}^{2}x_{34}^{2})^{\eta}}v^{\eta}\sum_{n,m=0}^{\infty}\frac{v^{n}(1-\frac{v}{u})^{m}}{n!m!}a_{nm}\left[-\mathrm{ln}v+b_{nm}\right],
{\calW}(G,\lambdaA,N)=\sum_{\nu\in\tilde{\Sigma}_{G}}\frac{(-1)^{\tilde{t}}}{|S_{\nu}|}e^{-\frac{n\lambdaA}{2}}\frac{(\lambdaA)^{(i+t+\tilde{t}+h)}}{i!t!\tilde{t}!h!}N^{n(2-2G)-2(t+\tilde{t}+h)-i}\left[1+{\calO}\left({\frac{1}{N}}\right)\right],
\Big\langle\psi_{0}^{0}S(z_{1})~\psi^{0}(z_{2})~S(z_{3})\Big\rangle=\Big\langleS(z_{1})~\psi^{0}(z_{2})~\psi_{0}^{0}S(z_{3})\Big\rangle=z_{12}^{-1/2}\,z_{13}^{3/8}\,z_{23}^{-1/2}~~.
E_{odd}={\frac{v}{2\sqrt{\pi}}}\int_{-\infty}^{\infty}\!\!d\tau\int_{0}^{\infty}{\frac{ds}{s^{1/2}}}\mathrm{e}^{-s(b^{2}+(v\tau)^{2})}=\frac{v}{2}\int_{-\infty}^{\infty}\frac{1}{r}\,d\tau~,
S={\frac{1}{2\kappa_{10}^{2}}}\intdx^{10}\sqrt{-\hat{G}}\,e^{-\hat{\Phi}}\left[{\calR}_{\hat{G}}+\partial_{M}\hat{\Phi}\partial^{M}\hat{\Phi}-{\frac{1}{2\cdot3!}}\hat{H}_{MNP}\hat{H}^{MNP}\right],
\int_{I}d^{3}{\bfr}\,\frac{-23}{4\pi|{\bfr}_{B}-{\bfr}|^{7}}=-\frac{23}{15}\frac{(a-\lambda)^{3}(5a^{2}+(a-\lambda)^{2})}{a(2a-\lambda)^{4}\lambda^{4}}
\gamma_{i}(\beta\sqrt{-\triangle})=\int_{0}^{\infty}\!\!{\mathrmd}t\,\,g_{i}(t)\left[\frac{2\pi}{\beta{\sqrt{-\triangle}}\,t}\,\frac{1}{{\mathrmsh}(2\pit/(\beta{\sqrt{-\triangle}}))}-\frac{1}{t^{2}}\right],
{\calL}_{(1+1)}=\frac1{2\varkappa}B\epsilon^{\mu\nu}F_{\mu\nu}+i\hbar\Psi^{*}(\partial_{t}+iA_{0})\Psi-\frac{\hbar^{2}}{2m}\left|(\partial_{x}+iA_{x})\Psi\right|^{2}-V(\Psi^{*}\Psi),
2{\calS}^{a}=2{\calS}^{AA^{\prime}}={\bar{m}}_{(i)(j)}\omega_{(i)}^{A}\pi_{(j)}^{A^{\prime}}+m_{(i)(j)}{\bar{\omega}}_{(i)}^{A^{\prime}}{\bar{\pi}}_{(j)}^{A}+a_{(i)(j)}{\bar{\pi}}_{(i)}^{A}\pi_{(j)}^{A^{\prime}}
\ddot{X}_{a}^{\mu}=g_{a}\partial_{a\rho}\left(\partial_{a}^{\mu}g_{b}\intd\tau_{b}M_{b}^{\nu\rho}G((X_{a}-X_{b})^{2})-\partial_{a}^{\nu}g_{b}\intd\tau_{b}M_{b}^{\mu\rho}G((X_{a}-X_{b})^{2})\right)\dot{X}_{a\nu}.\,
{G}(p_{1},..p_{E})=\sum_{n=1}^{L}\frac{1}{\epsilon^{n}}\sum_{a_{j}}\prod_{j=1}^{I}(\sum_{i=0}^{\infty}c_{i}m_{i}^{a_{j}})g(a_{j},n,p_{1},..p_{E})+{\calO}(\epsilon^{0})
{\calF}_{mirror}=\sum_{1\leqi,j,p\leqk}{\frac{J_{i}\cdotJ_{j}\cdotJ_{p}}{6}}t_{i}t_{j}t_{p}+\sum_{1\leqi\leqk}{\frac{c_{2}\cdotJ_{i}}{24}}t_{i}-i{\frac{\zeta(3)}{2(2\pi)^{3}}}c_{3}+\sum_{n_{1},\cdots,n_{k}}d_{\vec{n}}\,\mathrm{Li}_{3}(e^{n_{1}t_{1}+\cdots+n_{k}t_{k}}),
\left(\begin{matrix}{{\bf1}_{(0,0)}}\\{{\bf8}_{(0,0)}}\\{{\bf3}_{(4,0)}}\\{\overline{{\bf3}}_{(-4,0)}}\\{{\bf3}_{(-2,2)}}\\{\overline{{\bf3}}_{(2,2)}}\\{{\bf3}_{(-2,-2)}}\\{\overline{{\bf3}}_{(2,-2)}}\\\end{matrix}\right)\quad.
M^{zi}=\sigma_{-}(\bar{\alpha}^{i}+\alpha^{i}\bar{\alpha}_{j}^{2})-\sigma_{+}\alpha^{i}\,,\qquad\sigma_{-}\equiv\left(\begin{array}{ll}{0}&{0}\\{1}&{0}\\\end{array}\right)\,,\qquad\sigma_{+}\equiv\left(\begin{array}{ll}{0}&{1}\\{0}&{0}\\\end{array}\right)\,.
{\mathcal{M}}_{n}(\xi_{1})=\left(\!\begin{array}{cccc}{\chi_{\!n\!-\!1}\!(\xi_{1}\!)}&{0}&{0}&{0}\\{0}&{\chi_{\!n}\!(\xi_{1}\!)}&{0}&{0}\\{0}&{0}&{\chi_{\!n\!-\!1}\!(\xi_{1}\!)}&{0}\\{0}&{0}&{0}&{\chi_{\!n}\!(\xi_{1}\!)}\\\end{array}\!\right),
e=\frac{1}{\sqrt{T_{2}}}\left(\begin{matrix}{1}&{-T_{1}}\\{0}&{T_{2}}\\\end{matrix}\right)\,\qquad\tilde{e}=\frac{1}{\sqrt{t}}\left(\begin{matrix}{1}&{-\theta}\\{t\phi}&{(1-\theta\phi)t}\\\end{matrix}\right)
2\pix\int_{1}^{\infty}{\frac{\pi^{2}B_{2}(t-[t])}{(x^{2}+(2\pit)^{2})^{{\frac{3}{2}}}}}dt=2\pix\sum_{m=1}^{\infty}\int_{0}^{1}{\frac{\pi^{2}B_{2}(t)}{(x^{2}+(2\pi(t+m))^{2})^{{\frac{3}{2}}}}}dt\,
\stackrel{(0)}{S}=S_{0}^{\prime}\left[A_{\;\;\;(\sigma)}^{\alpha\beta},B_{\alpha}^{\;(\sigma)}\right]+\intd^{4}x\left(\varepsilon^{\alpha\beta\gamma\delta}A_{\alpha\beta}^{*\;\;(\sigma)}\partial_{\gamma}\eta_{\delta(\sigma)}+\eta^{*\alpha(\sigma)}\partial_{\alpha}C_{(\sigma)}\right),
\frac{1}{2}\partial_{\mu}\overline{{\theta}}^{\prime}(1-\Gamma_{11})\partial_{\nu}\theta^{\prime}=\frac{1}{2}\partial_{\mu}\overline{{\theta}}R(\Lambda)^{-1}(1-\Gamma_{11})R(\Lambda)\partial_{\nu}\theta=\frac{1}{2}\partial_{\mu}\overline{{\theta}}\partial_{\nu}\theta
Z={\calN}\,\int\prod_{x}\{\mathrm{d}v_{\mu}\,\mathrm{d}j^{\mu}\,\mathrm{d}j_{5}^{\mu}\,\mathrm{d}|h|^{2}\,\mathrm{d}\alpha\}\,e^{\mathrm{i}\int\mathrm{d}^{4}x\,{\calL}[v_{\mu},j^{\mu},j_{5}^{\mu},|h|,\alpha]},
{\frac{1}{16\piG}}\left({\frac{N(N-1)}{r^{2}}}\right)+{\frac{1}{(4\pi)^{2}\epsilon}}\left\{c_{N/2+1}\Lambda^{N/2+1}r^{N-2}\right\}={\frac{1}{16\piG_{0}}}\left({\frac{N(N-1)}{r^{2}}}\right).
e^{i\hat{\varphi}(\sigma)}=e^{i\hat{\chi}(\sigma)+i\hat{\chi}^{\dagger}(\sigma)}=e^{-\frac12[\,\hat{\chi}(\sigma),\hat{\chi}^{\dagger}(\sigma)\,]}\,e^{i\hat{\chi}^{\dagger}(\sigma)}\,e^{i\hat{\chi}(\sigma)}
Z_{R}=48+\frac{1}{4\overline{{\eta}}^{~12}}[\overline{{\theta}}_{2}^{~8}+\overline{{\theta}}_{3}^{~8}+\overline{{\theta}}_{4}^{~8}][\overline{{\theta}}_{3}^{~4}-\overline{{\theta}}_{4}^{~4}-\overline{{\theta}}_{2}^{~4}]
\nabla_{\mu}\phi_{\mu_{1}\dots\mu_{k}}=(\partial_{\mu}+\omega_{\mu})\phi_{\mu_{1}\dots\mu_{k}}-\sum_{i=1}^{k}\Gamma_{\mu_{i}\mu}^{\lambda}\phi_{\mu_{1}\dots\mu_{i-1}\lambda\mu_{i+1}\dots\mu_{k}}
{}~~~~~~~~~~~~~\times\Bigg\{\sum_{\sigma\inZ_{N}}\frac{w(x_{3},x_{13},x_{1}|d,h+\sigma)w(x_{4},x_{24},x_{2}|a,g+\sigma)}{w(x_{8},x_{58},x_{5}|e,c+\sigma)w(x_{7}/\omega,x_{67},x_{6}|f,b+\sigma)}\Bigg\}_{0}
\hat{U}=exp[\,\frac{\delta^{R}F}{\delta\Phi^{A}}\,\frac{\delta^{L}}{\delta\overline{{\Phi}}_{A}}+\frac{i}{2}\,\epsilon_{ab}\,\frac{\delta^{L}}{\Phi_{Aa}^{*}}\,\frac{(\delta^{R})^{2}F}{\delta\Phi^{A}\delta\Phi^{B}}\,\frac{\delta^{L}}{\delta\Phi_{Bb}^{*}}]
\zeta=\frac{g^{2}\mu_{1}^{2}J+g^{2}\mu_{1}(m_{1}+m_{2})I_{1}+g^{2}m_{1}m_{2}I_{0}}{M^{2}-m_{2}^{2}+g^{2}\mu_{1}^{2}J+2g^{2}\mu_{1}m_{2}I_{1}+g^{2}m_{2}^{2}I_{0}}\,.
\mathcal{A}=\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{2}}Z(\tau,\bar{\tau})\,\quadZ(\tau,\bar{\tau})=(\tau_{2})^{-24/2}\frac{1}{\eta^{24}(\tau)\bar{\eta}^{24}(\bar{\tau})}
S=\intd^{5}x\left[-\frac{1}{2}\tilde{\varphi}\left\{\partial_{\mu}\partial^{\mu}+(m\kappa)\{a,a^{\dagger}\}\right\}\tilde{\varphi}-|\kappay|^{5}V\left(|\kappay|^{-\frac{3}{2}}\tilde{\varphi}\right)\right],
\widehat{\theta}=\frac{1}{2\sqrt{2}}e^{\frac{\iota\pi}{4}}\left(\begin{array}{c}{\left(\begin{array}{c}{\theta_{N}}\\{-i\:\sigma_{2}\theta^{*M}\,C_{5NM}}\\\end{array}\right)\otimes\epsilon^{N}}\\{0_{16}}\\\end{array}\right)
\nabla_{j}=\left(\begin{array}{cc}{i\theta_{jk}^{-1}\hat{x}^{k}}&{0}\\{0}&{\partial_{j}}\\\end{array}\right)+i\left(\begin{array}{cc}{(D_{j})_{i}^{l}}&{0}\\{0}&{(\hat{D}_{j})_{\beta}^{\alpha}}\\\end{array}\right)
<ik\cdotj^{5}>=-e\int\frac{dp_{+}dp_{-}}{(2\pi)^{2}}\frac{1}{{(2p_{+}p_{-}-m^{2})}^{2}}\biggl[4{(p_{-}^{2}k_{+}A_{+}-p_{+}^{2}k_{-}A_{-})}+2m^{2}{(k_{-}A_{+}-k_{+}A_{-})}\biggr].
x=(-(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}},\,-(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}},\,(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}},\,(\frac{m_{0}m_{2}}{\Lambda_{1}^{4}\Lambda_{2}^{4}})^{\frac{1}{2}})
P_{lh}^{-}={\frac{\pi}{L}}{\sum_{n}}^{\prime}n(\alpha{_n^{\dagger}}\alpha_{n}+\beta{_n^{\dagger}}\beta_{n}+\delta{_n^{\dagger}}\delta_{n})-{\frac{\pi}{L}}Z_{L}Q_{L}+{\frac{\pi}{2L}}Z_{L}^{2}.
(d-1){\frac{2f^{2}(\epsilon)}{1-f^{2}(\epsilon)}}-{\frac{D-2}{d-2}}{\frac{2\kappaf(\epsilon)}{1-f^{2}(\epsilon)}}+{\frac{\epsilon{\widetildeL}}{2}}e^{B(\epsilon)}\left[{\widetilde\Lambda}-(d-1)\lambda\right]=0~.
\{\theta^{i},a\}=C_{aj}^{i}\theta^{j},~~\{\theta^{i},b\}=C_{bj}^{i}\theta^{j},~~\{\theta^{i},c\}=C_{cj}^{i}\theta^{j},~~\{\theta^{i},d\}=C_{dj}^{i}\theta^{j},
\left(2\piiK_{+}^{2}-\frac32k_{0}K_{+}-\frac{k^{2}k_{0}}{32\pi^{2}K_{+}}\right)-\left(2\piiK_{-}^{2}-\frac32k_{0}K_{-}-\frac{k^{2}k_{0}}{32\pi^{2}K_{-}}\right)=0
\leq\;\left\langle\prod_{b=1}^{N}\Bigg|\chi_{+}^{(b)}[Q^{0,S}]\Bigg|^{q_{b}}\Bigg|\chi_{-}^{(b)}[Q^{0,S}]\Bigg|^{2n_{b}-q_{b}}\right\rangle_{Q^{0,S}}=\left\langle\prod_{b=1}^{N}\bigg[\chi_{+}^{(b)}[Q^{0,S}]\;\chi_{-}^{(b)}[Q^{0,S}]\bigg]^{n_{b}}\right\rangle_{Q^{0,S}}\;.
\nabla\left(\begin{matrix}{V}\\{V_{AB}}\\{\bar{V}}\\{\bar{V}^{AB}}\\\end{matrix}\right)=\left(\begin{matrix}{0}&{0}&{0}&{{\frac{1}{2}}P_{CD}}\\{0}&{0}&{P_{AB}}&{{\frac{1}{2}}\bar{P}_{ABCD}}\\{0}&{{\frac{1}{2}}\bar{P}^{CD}}&{0}&{0}\\{\bar{P}^{AB}}&{{\frac{1}{2}}P^{ABCD}}&{0}&{0}\\\end{matrix}\right)\left(\begin{matrix}{V}\\{V_{CD}}\\{\bar{V}}\\{\bar{V}^{CD}}\\\end{matrix}\right)
\Phi=\tilde{\Phi},~~~\Psi~=~{\calF}~\tilde{\Psi},~(\tilde{\Psi}~=~{\calG}{\Psi}),~~~\Phi^{i}=e_{\mu}^{i}f_{a}^{\mu}\tilde{\Phi}^{a},~(\tilde{\Phi}^{a}=f_{\mu}^{a}e_{i}^{\mu}{\Phi}^{i}).
\begin{array}{lcl}{D=3:}&{[00k0]}&{\mathrm{of~SO(8)}}\\{D=4:}&{[0k0]}&{\mathrm{of~SU(4)}}\\{D=5:}&{[k]}&{\mathrm{of~SU(2)}}\\{D=6:}&{[0k]}&{\mathrm{of~USp(4)}}\\\end{array}
G_{\tilde{\lambda}}=\sum_{N_{2}}\mathrm{e}^{-\tilde{\lambda}N_{2}}\sum_{T|_{N_{2}}}1=\sum_{N_{2}}\mathrm{e}^{-(\tilde{\lambda}-\tilde{\lambda}^{\mathrm{crit}})N_{2}}\times\mathrm{subleading}(N_{2}),
X_{7}:\qquadds^{2}=dr^{2}+a(r)^{2}\sum_{a=1}^{3}(\widetilde\sigma_{a})^{2}+b(r)^{2}\sum_{a=1}^{2}(\sigma_{a}-{\frac{1}{2}}\widetilde\sigma_{a})^{2}+c(r)^{2}(\sigma_{3}-{\frac{1}{2}}\widetilde\sigma_{3})^{2}\,.
\begin{array}{ll}{\{Q_{+},\bar{Q}_{+}\}=-2(p_{0}+p_{1})\qquad}&{\qquad{}\{Q_{-},\bar{Q}_{-}\}=-2(p_{0}-p_{1})}\\{\{Q_{-},\bar{Q}_{+}\}=\,\,\,\,2Z\qquad{}}&{\qquad{}\{Q_{+},\bar{Q}_{-}\}=\,\,\,\,2\bar{Z}}\\\end{array}
h(T_{i},k_{i})\sime^{-\frac{2}{3}\pik_{i}^{2}T_{i}}\Bigl[\Theta_{3}(ik_{i}T_{i},2iT_{i})\Theta_{3}(ik_{i}T_{i},6iT_{i})+\Theta_{2}(ik_{i}T_{i},2iT_{i})\Theta_{2}(ik_{i}T_{i},6iT_{i})\Bigr]
ig\frac{1}{6}(\pi\alpha^{\prime})^{2}\int_{0}^{2\pi}d\theta~[N(0)]^{2}~\mathrm{Tr}~{\calP}\left(U[A]~\left[D^{2}D^{b}F_{ba}+D^{c}D^{b}D_{c}F_{ba}+D^{b}D^{2}F_{ba}\right]\partial_{\theta}\overline{{X}}^{a}\right);
\frac{d\Lambda_{ind}}{dt}\,=\,\frac{m^{4}}{2f^{2}}\,\frac{df}{dt}\,-\,\frac{m^{2}}{f}\,\frac{dm^{2}}{dt}\,=\,-\,\frac{1}{(4\pi)^{2}}\cdot\frac{2m^{4}}{f^{2}}\,\cdot\,\sum_{j}h_{j}^{4}\,=\,-\frac{2}{(4\pi)^{2}}\,\sum_{j}m_{j}^{4}\;.
\int_{\infty}d\Omega_{(1)}{}^{*}F_{0\Omega_{(1)}}^{I}+\int_{\infty}d\Omega_{(2)}{}^{*}F_{0\Omega_{(2)}}^{I}=\int_{\infty}d\Omega_{(3)}{}^{*}F_{0\Omega_{(3)}}^{I}.
A_{\mu}^{a}~=~\left(\begin{matrix}{V^{M}{}_{\mu}}\\{B_{\muM}}\\{A^{I}{}_{\mu}}\\\end{matrix}\right){}~~~~~~~~~~F_{\mu\nu}^{a}~=~\left(\begin{matrix}{V^{M}{}_{\mu\nu}}\\{H_{\mu\nuM}}\\{F^{I}{}_{\mu\nu}}\\\end{matrix}\right)
F_{mn}(\Box_{1},\Box_{2})X_{1}X_{2}(x)=-2\int_{-\infty}^{0}dq\,q^{m+n}\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{m+1}{\calH}_{q}X_{1}(x)\Bigr]\Bigl[\Bigl(\frac{d}{dq}\Bigr)^{n+1}{\calH}_{q}X_{2}(x)\Bigr]\quad,
\alpha_{\pm}=\frac{\theta^{2}}{2}\left(1\pm\sqrt{1-4/\theta^{2}}\right),\\\\\\frac{\lambda}{\gamma^{2}}=\frac{2\pi^{2}}{\theta^{2}}\left(1+\frac{4\alpha^{2}}{\theta^{2}}+\frac{6\alpha^{4}}{\theta^{4}}\right).
\begin{array}{l}{\partial^{(\tau,\nu,\beta)}\partial_{(\tau,\nu,\beta)}a^{(\lambda,\mu,\alpha)}-(1-\zeta_{0}^{-1})\partial^{(\lambda,\mu,\alpha)}\partial_{(\tau,\nu,\beta)}a^{(\tau,\nu,\beta)}=J^{(\lambda,\mu,\alpha)}=}\\{=-\displaystyle\frac{1}{2}\sqrt{g}\displaystyle\frac{g^{(\tau\nu\beta)(\rho\omega\gamma)}}{\partiala_{(\lambda,\mu,\alpha)}}T_{(\tau\nu\beta)(\rho\omega\gamma)},}\\\end{array}
{\calL}=-\frac{1}{4}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})^{2}+\frac{m^{2}}{2}A_{\mu}^{2}-\overline{{\psi}}(\gamma_{\mu}\partial^{\mu}+M)\psi+ie\,\overline{{\psi}}\gamma_{\mu}\partial^{\mu}\psiA_{\mu}
G^{\mathrm{ret}}(t,{\bfx})=\frac{\Theta(t)}{2\pi}\left[-\frac{\partial}{2\pir\partialr}\right]^{(D-3)/2}\left[\frac{1}{\sqrt{t^{2}-r^{2}}}\right]\,,\,\,\,\,D\,\,\mathrm{odd}.
S=-\frac{1}{g_{YM}^{2}}\intd^{p+1}x\,\frac{U^{7-p}}{d_{p}e^{2}}\left(-1+\sqrt{1-\frac{d_{p}e^{2}}{U^{7-p}}\left(\dot{U}^{2}+U^{2}\dot{\Omega}^{2}\right)}\,\,\right)
H_{b}=\frac{1}{2}\left(p_{x}^{2}+\omega_{b}^{2}x^{2}\right)=\frac{\omega_{b}}{2}[a^{+},a^{-}]_{+}=\omega_{b}\left(N_{b}+\frac{1}{2}\right),\quadN_{b}=a^{+}a^{-},
A_{+}=\frac{1}{2}\left(\begin{array}{cc}{\sigma_{-}}&{0}\\{0}&{-\sigma_{+}}\\\end{array}\right)\;,\;\;A_{-}=\frac{1}{2}\left(\begin{array}{cc}{\sigma_{+}}&{0}\\{0}&{-\sigma_{-}}\\\end{array}\right).
V_{z\ggb}\toE[1-{\frac{z^{6-p}}{Q_{p}}}(1-{\frac{3EQ_{F}}{2m_{f}z^{6-p}}})]+{\frac{J_{\perp}^{2}z^{10-2p}}{2m_{f}Q_{p}^{2}}}(1-{\frac{2EQ_{F}}{m_{f}z^{6-p}}}),
S[X,g]\=\\int_{0}^{1}d^{2}\sigma\frac{1}{2\pi}\left\{\sqrt{g}g^{\alpha\beta}\partial_{\alpha}X^{I}\partial_{\beta}X^{I}\-\iB^{IJ}\varepsilon^{\alpha\beta}\partial_{\alpha}X^{I}\partial_{\beta}X^{J}\\right\}\,
i\hbar\partial_{t}\psi=-{\frac{\hbar^{2}}{2m}}\partial_{x}^{2}\psi+i\kappa^{2}{\frac{\hbar^{2}}{m}}[(2+a)\rho\partial_{x}\psi+a\psi^{2}\partial_{x}\psi^{*}]+{\frac{\hbar^{2}\kappa^{4}}{2m}}a(2-a)\rho^{2}\psi,
{\widehat\Gamma}_{\Lambda_{1}\dots\Lambda_{n}}^{(n)}(k_{1}\dotsk_{n})=2^{1-n}P_{\Lambda_{1}\alpha_{1}}(k_{1}^{0})\dotsP_{\Lambda_{n}\alpha_{n}}(k_{n}^{0})\Gamma_{\alpha_{1}\dots\alpha_{n}}^{(n)}(k_{1}\dotsk_{n})\,,
e^{2i\varphi}={\frac{p_{(2)}+ip_{(3)}}{p_{(2)}-ip_{(3)}}}={\frac{({\bar{\alpha}}^{A}{\bar{\pi}}_{A})(\beta^{A^{\prime}}\pi_{A^{\prime}})}{({\bar{\beta}}^{B}{\bar{\pi}}_{B})(\alpha^{B^{\prime}}\pi_{B^{\prime}})}},
\overline{X}_{i}\Big\{f,g\Big\}-\Big\{\overline{X}_{i}f,g\Big\}-\Big\{f,\overline{X}_{i}g\Big\}=-F_{i\phantom{jl}}^{\phantom{i}jl}\Big(\overline{X}_{j}f\Big)\Big(\overline{X}_{l}g\Big)
\sum_{a,b}\eta_{ab}\vartheta\left[\begin{array}{c}{a}\\{b}\\\end{array}\right]\vartheta\left[\begin{array}{c}{a}\\{b+v_{5}}\\\end{array}\right]\prod_{r=3}^{4}\vartheta\left[\begin{array}{c}{a+\frac{1}{2}}\\{b+v_{r}}\\\end{array}\right]=0
A^{-}=-{\frac{i}{\sqrt{L}m}}\sum_{n=1}^{\infty}{\frac{p^{-}(n)}{\sqrt{p_{-}(n)}}}\left(C(n)e^{-ip(n)x}-C^{*}(n)e^{ip(n)x}\right)-{\frac{1}{Lm^{2}}}Q_{+}
G({\bf{x}}_{b},{\bf{x}}_{a};E)=\frac1{\sqrt{r_{b}r_{a}}}\sum_{l_{D}}G(r_{b},r_{a};E_{D},l_{D})\sum_{k=-l_{D}}^{l_{D}}Y_{l_{D},k,q}(\theta_{b},\varphi_{b})Y_{l_{D},k,q}^{\star}(\theta_{a},\varphi_{a}),