harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle\bar{W}^{4}|W^{4}|W^{4}|\bar{W}^{4}\rangle=\frac{1}{x_{13}^{8}x_{42}^{8}}\left[a_{2}+\frac{b_{4}}{s}+\frac{c_{1}}{s^{2}}+\frac{b_{1}}{s^{3}}+\frac{a_{1}}{s^{4}}\right]\;,
\left\langle{a(x,x^{5})a(y,y^{5})}\right\rangle=\int_{k5}{\frac{i}{k^{2}-(k^{5})^{2}}}\,e^{-ik\cdot(x-y)}(e^{ik^{5}(x^{5}-y^{5})}+Pe^{ik^{5}(x^{5}+y^{5})})\,
H\,=\,\frac{1}{R}\,\sum_{n=1}^{\infty}\sum_{j=1}^{D-2}\,\omega_{n}\,a_{n}^{j+}\,a_{n}^{j}+\frac{D-2}{2}\,\frac{1}{R}\,\sum_{n=1}^{\infty}\,\omega_{n}\,{.}
I(z)=i(-1)^{\alpha}\int{\frac{d^{n}k}{(2\pi)^{n}}}\,{\frac{e^{i(\bar{\theta}+z)\cdotk}}{(k^{2}+M^{2})^{\alpha}}}=i(-1)^{\alpha}{\frac{\pi^{\frac{n}{2}}}{(2\pi)^{n}}}\,{\frac{1}{\Gamma(\alpha)(M^{2})^{\alpha-{\frac{n}{2}}}}}\,2\left({\frac{|\bar{\theta}+z|M}{2}}\right)^{\alpha-{\frac{n}{2}}}\,K_{\alpha-{\frac{n}{2}}}(|\bar{\theta}+z|M)
\intdt\left\{-\Lambda\frac{dV_{\Lambda}}{d\Lambda}+\frac{1}{2}\Lambda\frac{dZ_{\Lambda}}{d\Lambda}{\dot{\phi}}^{2}\right\}=\frac{\Lambda^{3}}{4\pi^{2}}\intdt\left\{\sqrt{A_{\Lambda}}+\frac{1}{2}\dot{\phi}^{2}\left[\frac{B_{\Lambda}}{\sqrt{A_{\Lambda}}}-\left(\frac{C_{\Lambda}}{\sqrt{A_{\Lambda}}}\right)^{\prime}\right]\right\}.
\sum_{j}(\delta_{j}^{(2)}+\delta_{j}^{(3)})-\sum_{j}(\delta_{j}^{(1)}+\delta_{j}^{(4)})=\sum_{j}(\delta_{j}^{(2)}-\delta_{j}^{(1)})+\sum_{j}(\delta_{j}^{(3)}-\delta_{j}^{(4)})=0
\Omega^{3}=-i\Big(\mathrm{d}z_{+}^{M}\wedge\mathrm{d}\bar{z}_{+}^{M}+\mathrm{d}z_{-}^{M}\wedge\mathrm{d}\bar{z}_{-}^{M}\Big)\,\quad\Omega^{+}=\mathrm{d}z_{+}^{M}\wedge\mathrm{d}z_{-}^{M}\,\quad\Omega^{-}=(\Omega^{+})^{*}\,
V(u)=E\left[1-\frac{m_{p}\tilde{f}^{2}}{2E}\left(\frac{u}{R}\right)^{7-p}\left(1-\frac{\tilde{f}}{{\calA}^{2}}\right)\right]+\frac{L^{2}\tilde{f}^{3}}{2m_{p}u^{2}{\calA}^{2}}.
{\calL}_{cov}^{pzm}=\partial_{i}a_{-}\left(\partial_{i}a_{+}-\partial_{+}a_{i}\right)+\frac12\left(\partial_{+}a_{-}\right)^{2}-\frac14\left(\partial_{i}a_{j}-\partial_{j}a_{i}\right)^{2}+a_{\mu}J_{pzm}^{\mu}+\Lambda_{pzm}(\partial_{+}a_{-}-\partial_{i}a_{i})
\left(\begin{matrix}{x_{+}^{\prime}}\\{x_{-}^{\prime}}\\{1}\\\end{matrix}\right)=\delta\left(\begin{matrix}{x_{+}}\\{x_{-}}\\{1}\\\end{matrix}\right)=\left(\begin{matrix}{\sigma\omega}&{0}&{u_{+}}\\{0}&{\sigma\omega^{-1}}&{u_{-}}\\{0}&{0}&{1}\\\end{matrix}\right)\otimes\left(\begin{matrix}{x_{+}}\\{x_{-}}\\{1}\\\end{matrix}\right),
U\;=\;{\frac{4}{3}}{\frac{k^{2}r(k^{2})}{\left[d(k^{2})-p(k^{2})\right]n(k^{2})^{2}}}-{\frac{1}{3}}-{\frac{2}{3}}{\frac{d(k^{2})-p(k^{2})}{n(k^{2})^{2}}}\;,
S_{\mathrm{bosonic}}=\frac{1}{2\kappa_{11}^{2}}\intd^{11}x(-g)^{1/2}\bigl(R-\frac{1}{2}|F_{4}^{M}|^{2}\bigr)-\frac{1}{6}\intA_{3}^{M}\wedgeF_{4}^{M}\wedgeF_{4}^{M},
\hat{W}^{(8)}:=\frac{1}{2\tilde{m}-1}\Big(\tilde{m}\,M^{(8)}-W^{(8)}\Big)\quad\mathrm{and}\quad\hat{W}^{(5)}:=\frac{1}{2\tilde{n}-1}\Big(\tilde{n}\,M^{(5)}-W^{(5)}\Big)
\frac{dW}{dt}=\Phi_{E}\sigma(1-\beta)=\Phi_{E}\frac{8\pi\alpha^{2}}{3m^{2}}\frac{m^{2}}{W^{2}}\approx\Phi_{E}\frac{\alpha^{2}}{m_{Pl}^{2}}\frac{4\pi}{3}
\Omega(x,y;x^{\prime},y^{\prime})=\intd^{d-1}z{\frac{\delta}{\delta\phi^{*a}(z)}}\sigma^{*}(x,y){\frac{\delta}{\delta\phi^{a}(z)}}\sigma(x^{\prime},y^{\prime})=\delta^{d-1}(y-y^{\prime})\sigma(x^{\prime},x)
S^{BH}={\frac{\pi}{6}}\sum_{i=1}^{N}\int_{\Sigma}d\sigma\left[2\langle\hat{\phi}_{i}^{2}\rangle+{\frac{1}{m_{d,i}}}\langle\hat{\bar{\psi}}_{i}\hat{\psi}_{i}\rangle-2\langle\hat{V}_{i}^{2}\rangle\right]~~~.
X-\frac{1}{8}Z\big|_{R\rightarrowW}=\frac{3}{2}\,\int\!\mathrm{d}^{16}\,\theta\,\left(\theta{\calC}\Gamma^{mnr}\theta\,\theta{\calC}\Gamma_{r}{}^{pq}\theta\,R_{mnpq}\right)^{4}\,
\lambda^{2}B_{n}^{4}\sum_{N=0}^{\infty}\sum_{\alpha=0}^{N}\frac{(2a)^{8-3n}{\pi}^{\frac{n}{2}}\Gamma(\frac{8-3n}{2})\Gamma(N-\frac{8-4n}{2})}{\Gamma(N+\frac{8-2n}{2})\Gamma(\frac{4n-8}{2})}\int_{C}d^{n}{\eta}d^{n}{\eta^{\prime}}Y_{\alpha}^{N}(\eta)Y_{\alpha}^{N}(\eta^{\prime})
Im\Pi_{i}(p_{i}^{2}-m_{i}^{2})FT\frac{\delta^{4}}{\deltaJ^{4}}\begin{array}{c}{lim}\\{\varepsilon\rightarrow0}\\\end{array}<<[i\delta\alpha\intd^{4}x\phi\partial_{0}^{2}\phi]>>\mid_{_{J=0}}
\begin{array}{lllllll}{d\backslashofdf}&{}&{\Psi}&{}&{A_{\mu}}&{}&{}\\{3}&{}&{2}&{-}&{2}&{=}&{0}\\{4}&{}&{4}&{-}&{3}&{=}&{1}\\{6}&{}&{8}&{-}&{5}&{=}&{3}\\{10}&{}&{16}&{-}&{9}&{=}&{7}\\\end{array}
\eta_{\underline{{ab}}}\,V_{m}^{\underline{{b}}}\,\eta^{i\ell_{1}}\,\epsilon^{m\ell_{2}\dots\ell_{d}}\,\epsilon_{\ell_{1}\dots\ell_{d}}\,+\,2\,(d!)\,a_{1}\,\eta_{\underline{{ab}}}\,\Pi_{j}^{\underline{{b}}}\,h^{ij}=0
B_{\bar{\mu}\bar{\nu}}=\left(\begin{matrix}{B_{\mu\nu}+{\frac{1}{2}}(A_{\mu}^{m}B_{m\nu}-B_{\mun}A_{\nu}^{n})}&{B_{\mun}+A_{\mu}^{m}B_{mn}}\\{B_{m\nu}+B_{mn}A_{\nu}^{n}}&{B_{mn}}\\\end{matrix}\right).
S=\Gamma[Q]-\frac12\int_{\partial\calM}Tr[q^{-1}\partial_{+}q\,q^{-1}\partial_{-}q+\,2\,\partial_{+}qq^{-1}k_{-}-\,2\,q^{-1}\partial_{-}qk_{+}]\,\theta\,dt\,d\phi\quad.
|M_{[\alpha,\psi_{\alpha}]}^{i}|\leqA_{[\alpha,\psi_{\alpha}][\alpha,\psi_{\alpha}]}^{i}\;\;\;\;\;\mathrm{and}\;\;\;\;\;M_{[\alpha,\psi_{\alpha}]}^{i}=A_{[\alpha,\psi_{\alpha}][\alpha,\psi_{\alpha}]}^{i}\;\;\;\;\mathrm{mod}\;\;2\;\;\;\;.
H_{R}=\sqrt{m_{1}^{2}+V_{1}({\vec{R}}_{12}^{2})+{\vec{\kappa}}_{1}^{2}}+\sqrt{m_{2}^{2}+V_{2}({\vec{R}}_{12}^{2})+{\vec{\kappa}}_{2}^{2}}+U({\vec{R}}_{12}^{2})^{\prime}
{\calL}_{eff}^{pzm}=\frac{1}{2L}\pi\partial_{+}a_{-}-{\calH}_{D}^{pzm}=\frac{1}{2L}\pi\partial_{+}a_{-}+\frac{1}{2L}\pi{\calG}_{(\perp)}[0]\ast\partial_{i}j^{i}+a_{-}j^{-}-\frac12j^{i}{\calG}_{(\perp)}[0]\astj^{i}\;,
d{}^{*}G_{3}+d\tau\wedge\left[i\mathrm{e}^{\phi}{}^{*}G_{3}+{}^{*}H_{3}\right]-i{\tilde{F}}_{5}\wedgeG_{3}=-2i\kappa^{2}\left[\frac{\delta{\cal{L}}_{b}}{\deltaB_{2}}-\tau\frac{\delta{\cal{L}}_{b}}{\deltaC_{2}}\right]
\left(\begin{matrix}{\alpha_{2}}&{\beta_{2}}\\{\gamma_{2}}&{\delta_{2}}\\\end{matrix}\right)\left(\begin{matrix}{\alpha_{1}}&{\beta_{1}}\\{\gamma_{1}}&{\delta_{1}}\\\end{matrix}\right)=\left(\begin{matrix}{\alpha_{2}\alpha_{1}+\beta_{2}\gamma_{1}}&{\alpha_{2}\beta_{1}+\beta_{2}\delta_{1}}\\{\gamma_{2}\alpha_{1}+\delta_{2}\gamma_{1}}&{\gamma_{2}\beta_{1}+\delta_{2}\delta_{1}}\\\end{matrix}\right).
\deltax^{\mu}=\varepsilon^{\mu}+\varepsilon^{\mu}{}_{\nu}x^{\nu}=\varepsilon^{\mu}+\frac{1}{2}\varepsilon^{\alpha\beta}(\delta_{\alpha}^{\mu}\eta_{\nu\beta}-\delta_{\beta}^{\mu}\eta_{\nu\alpha})x^{\nu}\,,
\varepsilon\rightarrow\varepsilon^{\prime}=D(\alpha,\beta,\gamma)\varepsilonD^{T}(\alpha,\beta,\gamma)=\varepsilon+\left(\begin{array}{cccc}{0}&{(\gammab-\betac)}&{(\alphac-\gammaa)}&{(\betaa-\alphab)}\\{-(\gammab-\betac)}&{0}&{0}&{0}\\{-(\alphac-\gammaa)}&{0}&{0}&{0}\\{-(\betaa-\alphab)}&{0}&{0}&{0}\\\end{array}\right)
N=(1,0):\quadW_{(\alpha_{1}\ldots\alpha_{n})}=\phi_{(\alpha_{1}\ldots\alpha_{n})}+\theta_{i}^{\beta}\psi_{(\beta\alpha_{1}\ldots\alpha_{n})}^{i}+\theta^{\betai}\theta_{i}^{\gamma}F_{(\beta\gamma\alpha_{1}\ldots\alpha_{n})}+\mathrm{\small~d.~t.~}
\eta_{1}(v-\frac{m-1}{2})\xi_{1}(v-\frac{m-2}{2})\cdots\xi_{m-1}(v)\eta_{m-1}(v+\frac{1}{2})=(x^{2r};x^{2r})_{\infty}^{3(m-1)}c_{m}\eta_{m}(v)
\sum_{k_{1}+\cdots+k_{n}=k}=\int{\frac{d^{4}k_{1}}{(2\pi)^{4}}}\cdots\int{\frac{d^{4}k_{n}}{(2\pi)^{4}}}(2\pi)^{4}\delta^{(4)}(k-k_{1}-\cdots-k_{n}).
{sdet}~e_{M}^{~A}={sdet}^{-1}~e_{A}^{~M}=det^{-1}[e_{a}^{~m}-e_{a}^{~\mu}(e_{\mu}^{~\alpha})^{-1}e_{\alpha}^{~m}]~det~e_{\alpha}^{~\mu},
\int\omega_{i}\bar{\omega}_{j}=\sum_{k=1}^{g}\left(\oint_{a_{k}}\omega_{i}\oint_{b_{k}}\bar{\omega}_{j}-\oint_{b_{k}}\omega_{i}\oint_{a_{k}}\bar{\omega}_{j}\right)=-2i\left(\Omega_{2}\right)_{ij}
\hat{K}_{ij}^{-1}=\left(\begin{matrix}{{\frac{1}{f}}g_{\rho\mu}}&{0}&{0}&{0}\\{0}&{0}&{2/C^{\prime}}&{0}\\{0}&{2/C^{\prime}}&{\left({\frac{4}{C}}-{\frac{4Z}{{C^{\prime}}^{2}}}\right)}&{0}\\{0}&{0}&{0}&{-{\frac{2}{C}}P_{\rho\sigma,\mu\nu}}\\\end{matrix}\right)\,
\left(e_{j}\right)_{\ell\ell^{\prime}}~=~\delta(\ell_{j},\ell_{j+2}){}~{\frac{\left(v_{\ell_{j+1}}{}~v_{\ell_{j+1}^{\prime}}\right)^{1/2}}{v_{\ell_{j}}}}~\prod_{k\neqj+1}\delta(\ell_{j},\ell_{j}^{\prime})\.
\tilde{o}^{A}=\textrm{e}^{v}o^{A},\,\,\,\,\tilde{\iota}^{A}=\textrm{e}^{-v}\iota^{A},\,\,\,\,\tilde{\rho}^{\mu}=\textrm{e}^{-(v+\bar{v})}\rho^{\mu}
DBC^{-1}=\left(\begin{array}{cccccccc}{0}&{0}&{1}&{0}&{\cdots}&{\cdots}&{\cdots}&{0}\\{0}&{0}&{0}&{1}&{0}&{\cdots}&{\cdots}&{0}\\{0}&{0}&{0}&{0}&{1}&{0}&{\cdots}&{0}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}&{\cdots}\\{0}&{0}&{0}&{0}&{0}&{\cdots}&{0}&{1}\\\end{array}\right)
\begin{array}{rcl}{x^{i}x^{j}}&{=}&{qx^{j}x^{i}~,~~~~~\mathrm{for}~i<j~~\mathrm{and}~~i\not=-j~,}\\{x^{i}x^{-i}}&{=}&{x^{-i}x^{i}+\lambdaq^{i-3/2}L_{i-1}}\\{}&{=}&{q^{-2}x^{-i}x^{i}+\lambdaq^{i-3/2}L_{i}~,~~~~~\mathrm{for}~i>0~,}\\\end{array}
{\frac{1}{16}}\omega^{2}(3+4\omega)^{2}+\omega^{2}(\omega+1)^{2}\left({\frac{z_{0}^{2}}{l^{2}}}\right)^{2}+{\frac{\omega^{2}}{2}}(3+4\omega)(\omega+1){\frac{z_{0}^{2}}{l^{2}}}>0\,.
{\calG}_{(1)}\left(\begin{matrix}{\sigma_{1}}&{0}\\{0}&{\sigma_{1}}\\\end{matrix}\right){\calG}_{(1)}^{T}=\left(\begin{matrix}{{\calI}_{(1)}}&{0}\\{0}&{-4}\\\end{matrix}\right)\;\qquad{\calG}_{(2)}\left(\begin{matrix}{\sigma_{1}}&{0}\\{0}&{\sigma_{1}}\\\end{matrix}\right){\calG}_{(2)}^{T}={\calI}_{(2)}\,
{\frac{\partial^{2}g_{ab}}{\partial\eta^{2}}}=-{\frac{1}{2}}\;{g^{a^{\prime}b^{\prime}}}\;\;{\frac{\partialg_{aa^{\prime}}}{\partial\eta}}\;{\frac{\partialg_{b^{\prime}b}}{\partial\eta}}+2\;{}^{(5)}R_{\etaa\etab}.
\lambda_{1}=\sqrt{\zeta\left((\mathbf{p}_{1})^{\dagger}\mathbf{p}_{1}+(\mathbf{p}_{1}^{\prime})^{\dagger}\mathbf{p}_{1}^{\prime}\right)}\,,\quad\lambda_{2}=\sqrt{\zeta\left((\mathbf{p}_{2})^{\dagger}\mathbf{p}_{2}+(\mathbf{p}_{2}^{\prime})^{\dagger}\mathbf{p}_{2}^{\prime}\right)}\,,
\psi^{\prime}=\left(\begin{array}{c}{\psi_{1}^{\prime}}\\{\psi_{2}^{\prime}}\\{\psi_{0}^{\prime}}\\\end{array}\right)\equiv\left(\begin{array}{c}{z_{1}^{\prime}}\\{z_{2}^{\prime}}\\{\theta^{\prime}}\\\end{array}\right)\,.
Q_{A}(\alpha,\bar{\alpha})=\sum_{K=0}^{M}\frac{P!}{(P-K)!}\left(1+\sum_{i}^{N-1}|\alpha_{i}|^{2}\right)^{-K}A_{\nu_{1}\ldots\nu_{K}}^{\mu_{1}\ldots\mu_{K}}\alpha_{\mu_{1}}\ldots\alpha_{\mu_{K}}\bar{\alpha}_{\nu_{1}}\ldots\bar{\alpha}_{\nu_{K}}\:,
m_{1}=m_{2}=m_{3}=m_{4}={\frac{C}{\sqrt{1-4C^{2}}}}\,\qquadm_{5}=0\qquadZ_{\infty}^{5}=-Z_{\infty}^{6}=Z_{\infty}^{7}=-Z_{\infty}^{8}=C\,
\zeta_{\pm}^{1}(z)=N_{1}\left(\begin{array}{c}{z^{\frac{M}{2}}(1-z)^{-\frac{M}{2}}F(1,-1,\frac{1}{2}+M;z)}\\{\pm\frac{i}{\frac{1}{2}+M}z^{\frac{1+M}{2}}(1-z)^{\frac{M}{2}}F(-\frac{1}{2}+M,\frac{3}{2}+M,\frac{3}{2}+M;z)}\\\end{array}\right)
\Phi=U_{3}^{-1}({\bfr})\left(\begin{matrix}{t_{4}-{\frac{1}{2y}}}&{0}&{0}&{0}\\{0}&{t_{2}}&{0}&{0}\\{0}&{0}&{t_{2}}&{0}\\{0}&{0}&{0}&{t_{1}+{\frac{1}{2y}}}\\\end{matrix}\right)U_{3}({\bfr})+O(b/y^{2})
d_{\hat{D}}l^{2}=<\Psi_{0}|\hat{D^{\prime}}(\Psi)^{+}\hat{D^{\prime}}(\Psi)|\Psi_{0}>=R^{2}(|D_{00}^{\prime}(\Psi)|^{2}+\sum_{i=1}^{N}|D_{0i}^{\prime}(\Psi)|^{2})d\theta^{2}.
R_{11}=-\frac{1}{g^{2}}\left(\frac{f^{\prime\prime}}{f}-\frac{f^{\prime}}{f}\frac{g^{\prime}}{g}\right)-\frac{2}{g^{2}}\left(\frac{h^{\prime\prime}}{h}-\frac{h^{\prime}}{h}\frac{g^{\prime}}{g}\right)
\epsilon_{RPA}({\bf{q}},\omega)=1+\frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{{\bar{n}}_{{\bf{k}}+{\bf{q}}/2}-{\bar{n}}_{{\bf{k}}-{\bf{q}}/2}}{\omega-\frac{{\bf{k.q}}}{m}}
S=\intd^{3}x\{\frac{-1}{2g^{2}}Tr(G_{ij})^{2}-Tr[D_{i},\psi]^{2}+\frac{\lambda}{8}[2Tr(\psi)^{2}+v^{2})]^{2}-\frac{m^{2}}{g^{2}}Tr(\tilde{A}_{i})^{2}\}
(C_{q+1}^{2})_{\mu\nu}=g_{\mathrm{D}q}^{\rho_{1}\sigma_{1}}\cdotsg_{\mathrm{D}q}^{\rho_{q}\sigma_{q}}C_{\rho_{1}\dots\rho_{q}\mu}C_{\sigma_{1}\dots\sigma_{q}\nu}\,\quadC_{q+1}^{2}=g_{\mathrm{D}q}^{\mu\nu}(C_{q+1}^{2})_{\mu\nu}\.
\begin{array}{c}{s\left(\lambda,\mu\right)=-\frac12\frac{t\left(\lambda+\mu\right)}{\left(1-\lambda^{2}\right)\left(1-\mu^{2}\right)},}\\{a\left(\lambda,\mu\right)=\frac12\fract{\lambda-\mu}\left(\frac{\lambda^{2}}{1-\lambda^{2}}+\frac{\mu^{2}}{1-\mu^{2}}\right).}\\\end{array}
\mathcal{L}_{m_{3/2}}=\frac{1}{2}\sum_{n}\frac{n+\omega}{R}\bar{\psi}_{\mu}^{(n)}\gamma^{\mu\nu}\psi_{\nu}^{(n)}+\frac{W}{\piR}\sum_{n,m}\bar{\psi}_{\mu}^{(n)}\gamma^{\mu\nu}\psi_{\nu}^{(m)}\,,
V(a)=-\frac{\kappa_{(4)}^{2}\rho_{0}}{6}a^{-3\gamma+2}-\frac{\kappa_{(4)}^{2}\rho_{0}^{2}}{12\lambda}a^{-6\gamma+2}-\frac{\sigma_{0}^{2}}{6}a^{-4}-\frac{\Lambda_{(4)}}{6}a^{2}-\frac{{\calU}_{0}}{\lambda\kappa_{(4)}^{2}}a^{-2}+V_{0},
|\psi|^{2}:=\psi_{1}\psi_{1}^{\dagger}+\psi_{2}\psi_{2}^{\dagger}\qquad\mathrm{and}\qquad|\psi^{\dagger}|^{2}:=\psi_{1}^{\dagger}\psi_{1}+\psi_{2}^{\dagger}\psi_{2}
\bar{\calR}^{\frac{1}{2}\,z\,\omega_{1}\,,\,\frac{1}{2}\,z\,\omega_{2}}\,=\,c\left(\begin{array}{cccc}{t_{1}}&{0}&{0}&{0}\\{0}&{s^{-1}}&{w}&{0}\\{0}&{0}&{st_{1}t_{2}^{\,-1}}&{0}\\{0}&{0}&{0}&{-t_{2}^{\,-1}}\\\end{array}\right)\,,
\zeta(2\alpha)=\frac{\pi^{\alpha}}{\Gamma(\alpha)}\left\{\frac{1}{2\alpha(2\alpha-1)}+{\calK}\right\},\,\,\,\,\,\,\,\,{\calK}=\int_{1}^{\infty}dt\,\left(t^{-1/2-\alpha}+t^{\alpha-1}\right)\sum_{k=1}^{\infty}e^{-k^{2}\pit}
\chi_{F}^{(3)}\left(e_{i},e_{j}\right)-\chi_{S}^{(3)}\left(e_{i},e_{j}\right)={\frac{1}{48\pii}}\sum_{k=1}^{3h-3}{\frac{\partialS_{cl}^{(h)}}{\partial{z_{k}}}}\oint_{\calC}\Omega^{k+1-h_{0}}\left[e_{j},e_{i}\right].
{\calL}_{nonrel}={\frac{\kappa}{2}}\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}A_{\rho}+i\psi^{*}D_{t}\psi-{\frac{1}{2\mu}}|D_{\mu}\psi|^{2}+{\frac{1}{2\mu\kappa}}(|\psi|^{2})^{2}.
K^{*}\left(\left.b\begin{array}{c}{a}\\{a^{\prime}}\\\end{array}\right|z\right)=\delta_{aa^{\prime}}\sum_{d}\frac{G_{d}}{G_{a}}W\left(\left.\begin{array}{cc}{d}&{a}\\{a}&{b}\\\end{array}\right|2z\right)K\left(\left.d\begin{array}{c}{a}\\{a}\\\end{array}\right|-\frac{n}{2}-z\right).
G^{(2,2,2,2)}(1\vert2\vert3\vert4)=\langle\mathrm{Tr}\left(\tilde{q}^{+}(1)\right)^{2}\mathrm{Tr}\left(q^{+}(2)\right)^{2}\mathrm{Tr}\left(\tilde{q}^{+}(3)\right)^{2}\mathrm{Tr}\left(q^{+}(4)\right)^{2}\rangle
\{r^{\mu},r^{\nu}\}=-{\frac{S^{\mu\nu}}{m^{2}c^{2}}}=-{\frac{j}{m^{3}c^{3}}}\epsilon^{\mu\nu\lambda}P_{\lambda};~~\{r^{\mu},P^{\nu}\}=g^{\mu\nu};~~\{P^{\mu},P^{\nu}\}=0.
\bar{M}_{bdfhi}^{a}=f_{ce}^{a}f_{dg}^{c}\left(f_{hf}^{e}f_{bi}^{g}-f_{if}^{e}f_{bh}^{g}+3f_{bf}^{e}f_{hi}^{g}\right)+f_{cf}^{e}\left(f_{bg}^{c}\left(f_{he}^{a}f_{di}^{g}-f_{ie}^{a}f_{dh}^{g}\right)+3f_{be}^{a}f_{dg}^{c}f_{hi}^{g}\right)\,,
\tilde{A}_{1}(p)=-2g^{2}\delta_{b}^{a}\delta_{J}^{I}\int\frac{d^{4}k}{(2\pi)^{4}}d^{2}\thetad^{2}{\overline{\theta}}\,\frac{1}{k^{2}[(p-k)^{2}+m^{2}]}\left\{\Phi_{aI}^{\dagger}(p,\theta,{\overline{\theta}})\Phi^{bJ}(-p,\theta,{\overline{\theta}})\right\}\,.
\bar{\calM}_{\alpha\beta}^{\zeta}=\left(\begin{matrix}{-i\alpha}&{\gamma\partial_{t}+{\textstyle{\frac{1}{2}}}\dot{\gamma}+i\alpha}\\{-\gamma\partial_{t}-{\textstyle{\frac{1}{2}}}\dot{\gamma}+i\alpha}&{-i\alpha}\\\end{matrix}\right)
\cdot\sqrt{\frac{(n-l-1)!}{(n-l^{\prime}-1)!}\frac{\Gamma(n+l+\delta_{1}+\delta_{2}+1)}{\Gamma(n+l^{\prime}+\delta_{1}+\delta_{2}+1)}}\frac{1}{\Gamma(l-l^{\prime}+1)\Gamma(l^{\prime}-l+1)}.
\begin{array}{ccccc}{\mathrm{IIA:}}&{\mathrm{(NS+,NS+)}}&{\mathrm{(R+,R~-~)}}&{\mathrm{(NS+,R~-~)}}&{\mathrm{(R+,NS+)}}\\{\mathrm{IIB:}}&{\mathrm{(NS+,NS+)}}&{\mathrm{(R+,R+)}}&{\mathrm{(NS+,R+)}}&{\mathrm{(R+,NS+)}}\\\end{array}
<T_{uu}>\mid_{v\rightarrow\infty}=-t_{uu}(u)=\frac{N\hbar}{12\pi}(\frac{1}{16}F^{\prime}(r)^{2}-\frac{1}{8}F(r)F^{\prime\prime}(r))\mid_{u=v_{0}-2\intF^{-1}\,dr}.
2iH_{A}^{\\C}\sigma_{C\dot{B}}^{\\\m}+2iH_{\dot{B}}^{\\\dot{C}}\sigma_{A\dot{C}}^{\\\m}-2iH_{A\dot{B}}^{\\\m}-\nabla_{A}H_{\dot{B}}^{\m}-\nabla_{\dot{B}}H_{A}^{\m}=0
\langle\partial_{\mu}j_{R}^{\mu}\rangle=-\frac{\partial\Gamma}{\partial\alpha}\Big|_{\alpha=0}=-\frac{1}{32\pi}\mathrm{Im}\frac{\partial}{\partial\alpha}\intd^{4}xd^{2}\theta_{1}d^{2}\theta_{2}F\Big(e^{2i\alpha}\Phi(e^{-4i\alpha\gamma_{5}}\theta)\Big)\Big|_{\alpha=0}.
Z(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle^{A})=\biggl(\,\prod_{k=0}^{N}|det(A)|^{(-1)^{k}}\biggr)\,Z(R(S),\langle\cdot\,,\cdot\rangle_{H},\langle\cdot\,,\cdot\rangle)
{F}_{\mathrm{ren}}^{\infty}=-\frac1{8\pi^{2}}\int\!{\mathrmd}^{3}x\,{g}^{1/2}{\mathrmtr}\Big\{R_{ij}\gamma_{1}(-\triangle)R^{ij}+R\gamma_{2}(-\triangle)R+{\mathrm{O}}[\Re^{3}]\Big\}.
\left.\left|\left(1_{m}\right)^{\rho\tilde{\rho}};0;\beta_{T}\right\rangle\!\right\rangle=\left.\left|1,\tilde{1};m,\widetilde{m};\beta_{T}\right\rangle\!\right\rangle_{\alpha}^{\rho\widetilde{\rho}}\left.\left|0\left(\beta_{T}\right)\right\rangle\!\right\rangle_{\beta},
N\simeq\left(\frac{M_{s}}{M_{P}}\right)^{2}\frac{(M_{s}^{3}\,A_{T})^{3/2}}{(2\pi)^{7/2}g_{s}^{3/2}\,V_{6}^{1/6}\,F}\frac{\Psi_{*}}{M_{s}}\,.
E\left(\beta\right)=\frac{d}{d\beta}\left(\betaF\left(\beta\right)\right)=\frac{\pi^{2}a}{6\beta^{2}}-\frac{1}{2\beta}-\frac{4\pi^{2}a}{\beta^{2}}\sum_{p,\,n=1}^{\infty}\,p\,\,e^{-4\pi^{2}n\,p\,a/\beta},
f(y,\bar{y})\starg(y,\bar{y})=f(y,\bar{y})e^{-i(\stackrel{\,_{\leftarrow}}{\partial}{\!}^{\alpha}\!\!\!\stackrel{\,_{\rightarrow}}{\partial}{\!}_{\alpha}+\stackrel{\,_{\leftarrow}}{\partial}{\!}^{\dot{\alpha}}\!\!\!\stackrel{\,_{\rightarrow}}{\partial}{\!}_{\dot{\alpha}})}g(y,\bar{y}),
G_{0}^{(2)}(p\,,\dot{-}p^{\prime})=\frac{i}{2}\mu(-p_{0})\mu(p_{0})\frac{\delta^{(4)}(p\dot{+}(\dot{-}p^{\prime}))+\delta^{(4)}((\dot{-}p^{\prime})\dot{+}p)}{{\mathcalC}_{\kappa}(p)-m^{2}}~.
\nu_{n}^{2}=\left(\frac{32V_{0}}{b^{2}}\right)^{\frac{1}{2}}\left[n+\frac{1}{2}+\frac{1}{4}\left(\sqrt{8V_{0}a^{2}+1}-\sqrt{8V_{0}a^{2}}\right)\right],~~~~n=0,1,2,\dots
{\calL}=\frac{1}{2}F_{+-}^{a}F_{+-}^{a}+\sum_{i=1,2}\left(F_{+i}^{a}F_{-i}^{a}+\frac{\eta^{2}}{2}F_{+i}^{a}F_{+i}^{a}\right)-\frac{1}{2}F_{12}^{a}F_{12}^{a}-\rho_{m}^{a}A_{+}^{a},
V_{\calN}^{-}(q)=\frac{C_{1}^{2}}{2}q^{4}+C_{1}C_{2}q^{3}+\left(\frac{C_{2}^{2}}{2}+C_{1}C_{3}\right)q^{2}+(C_{2}C_{3}-{\calN}C_{1})q.
4\Lambda^{2n_{c}-n_{f}}\prod_{i=1}^{n_{f}}(\Lambda\omega^{2\ell-1}+\delta_{\ell}+m_{i})=4\Lambda^{2n_{c}}(\omega^{2\ell-1})^{n_{f}}\left[1+\sum_{i=1}^{n_{f}}\frac{\delta_{\ell}+m_{i}}{\Lambda\omega^{2\ell-1}}\right].
e_{1}=\left|\begin{array}{c}{\overrightarrow{e_{1}}}\\{0}\\\end{array}\right|,\quade_{2}=\left|\begin{array}{c}{\overrightarrow{e_{2}}}\\{0}\\\end{array}\right|,\quade_{3}=\left|\begin{array}{c}{\overrightarrow{e_{3}}}\\{0}\\\end{array}\right|,\quade_{4}=\left|\begin{array}{c}{0}\\{1}\\\end{array}\right|.
\Gamma(-\Box_{1},-\Box_{2},-\Box_{3})=\int_{0}^{\infty}{\frac{d{m_{1}}^{2}\,d{m_{2}}^{2}\,d{m_{3}}^{2}\,\rho(m_{1},m_{2},m_{3})}{({m_{1}}^{2}-\Box_{1})({m_{2}}^{2}-\Box_{2})({m_{3}}^{2}-\Box_{3})}},
D_{{\vec{c}},p-1}=D_{{\vec{c}},0}e^{-i2\pi\left(q\midc\mid+N_{\Phi}c_{\gamma}\right)/\ell}e^{\frac{i\pi\tau}{\omega\kappa^{2}}\left(p-1\right)^{2}}e^{i\tau\betaN_{A}\left(p-1\right)/{\omega\kappa^{2}}}e^{-i\alphaN_{A}\left(p-1\right)/{\omega\kappa^{2}}}
\sum_{l=1}^{N}\frac{w_{pR(r)}(n-l)}{w_{qr}(l-m)w_{pq}(k-l)}=R_{pqr}^{\prime}\frac{w_{pR(q)}(n-m)w_{R^{-1}(q)r}(k-n)}{w_{pr}(k-m)}
V_{(j,m)}^{(L)}(z,\bar{z})=\sum_{k,l,n,p=0}^{L}A_{klnp}\phi^{-\delta_{n,0}}\partial^{n}\phi(z)(\partial\phi(z))^{k}\partial^{l}\gamma(z)(\partial\gamma(z))^{p}\gamma^{j-m-p-1}\timesc.c.~e^{2j\phi/\alpha_{+}}
\rho(Z_{1},Z_{2},T)=\rho_{c}+{\frac{1}{20\cdot2^{1/3}\,\alpha(h)^{2}}}\left[{\frac{\Phi(Z_{1},T)-\Phi(Z_{2},T)}{Z_{1}-Z_{2}}}\right]\epsilon^{1/3}+{\calO}(\epsilon^{2/3})\.
\sum_{m=0}^{\infty}\sum_{j=1}^{23}{\frac{-\Gamma({\frac{j}{24}}+m)(-)^{m}\prod_{j\neqk=1}^{23}\Gamma({\frac{k-j}{24}}-m)}{m!(m+{\frac{j}{24}})\Gamma(-m-{\frac{j}{24}}+1)\prod_{k=-2}^{8}\Gamma({\frac{k}{8}}-m-{\frac{j}{24}})\prod_{l=10}^{1}\Gamma({\frac{l}{12}}-m-{\frac{j}{24}})}}(-z)^{-m-{\frac{j}{24}}}.
S_{GF}\,=\,-\,\intd^{4}x\,\delta_{1}\,\delta_{2}\,{\frac{1}{2}}Tr\,\Big(\,A_{\mu}A^{\mu}\,+\,({\overline{c}}\,-\,{\overline{\rho}}\,)\Box^{-1}\partial_{\mu}\Psi^{\mu}\,-\,c\,({\overline{c}}\,+{\overline{\rho}}\,)\,\Big)\,\,,
\left\langle\hat{\varphi}_{s}|\frac{d\hat{\varphi}_{m}}{dt}\right\rangle=\frac1{\left({\cal\vec{E}}_{m}-{\cal\vec{E}}_{s}\right)}\left\langle\hat{\varphi}_{s}\left|\frac{dH_{1}}{dt}\right|\hat{\varphi}_{m}\right\rangle
\left|2\Gamma_{B}(m^{2}-m_{n}^{2}-m_{n^{\prime}}^{2})-\Gamma_{D}m_{n}^{2}-\Gamma_{D}^{y}m_{n^{\prime}}^{2}\right|\ll\Gamma_{D}m_{n}^{2}+\Gamma_{D}^{y}m_{n^{\prime}}^{2}.
\frac{1}{\kappa}\Omega_{WZ}^{\rho}(g)=-\frac{1}{2}\int_{0}^{2\pi}dx\,\langle(g^{-1}dg)\stackrel{\wedge}{,}(g^{-1}dg)^{\prime}\rangle-\frac{1}{2}\langle(g^{-1}dg)(0)\stackrel{\wedge}{,}dM{\scriptstyle\,}M^{-1}\rangle+\rho(M)
<\dot{\omega}\left({q}\right)\omega\left({f}_{1}\right)\cdots\omega\left({f}_{n}\right)>~=~-\int{d^{2}k[k,q]\left(\left(q-k\right)^{2}-{k}^{2}\right)}<~\psi\left(k\right)\psi\left(q-k\right)\omega\left(f_{1}\right)\cdots\omega\left(f_{n}\right)~>
{\frac{i}{2}}\sqrt{R}\varepsilon_{ul}-{\frac{i}{4}}\sqrt{\lambda}(R^{\prime}-R\varphi^{\prime})\varepsilon_{ul}+{\frac{1}{4}}{\bfP}^{i}\gamma^{i}\varepsilon_{lu}+{\frac{1}{12}}(-{\bfQ}^{ij}\mp2i{\bfP}_{ij})\gamma^{ij}\varepsilon_{lu}=0,
ds^{2}=-\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)dt^{2}+\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},
\alpha_{n}^{\left(\pm\right)0^{\prime}}=\frac{p_{2}\cdot\hat{\alpha}_{n}^{\pm}}{m_{2}},\quad\alpha_{n}^{\left(\pm\right)I}=\hat{\alpha}_{n}^{\left(\pm\right)I}+\frac{p_{2}^{I}}{m_{2}}\frac{p_{2}\cdot\hat{\alpha}_{n}^{\pm}}{E_{2}^{\prime}+m_{2}}\,\,.
G_{polar}(k)=-\frac{ie^{2}}{4\pi^{2}}\gamma^{0}\otimes\gamma^{0}\frac{(k^{1})^{2}}{(k^{2}-e^{2}/\pi+i\epsilon)(k^{2}+i\epsilon)}\longrightarrow-\frac{i}{4\pi}\gamma^{0}\otimes\gamma^{0}\frac{(k^{1})^{2}}{(k^{2}-e^{2}/\pi+i\epsilon)}\;,
\prod_{u=1}^{N}dv_{u}^{2}dv_{u}^{2*}=d\xi_{1}^{2}d\xi_{1}^{2*}\cdot2^{N-1}(\xi_{2}^{2})^{2N-3}\,d\xi_{2}^{2}\,d^{2N-3}\hat{\Omega}_{2}\.
P_{n}\left(x\right)=\frac{1}{\sqrt{h_{n}h_{n+1}}}\left(\begin{array}{llll}{\gamma_{0}}&{\gamma_{1}}&{...}&{\gamma_{n}}\\{\gamma_{1}}&{\gamma_{2}}&{...}&{\gamma_{n+1}}\\{...}&{...}&{...}&{...}\\{\gamma_{n-1}}&{\gamma_{n}}&{...}&{\gamma_{2n-1}}\\{1}&{x}&{...}&{x^{n}}\\\end{array}\right),
\mathcal{L}=\frac{1}{4}\partial^{\mu}\vec{\phi}\cdot\partial_{\mu}\vec{\phi}+\theta_{1}\frac{1}{4}((\partial^{\mu}\vec{\phi}\cdot\partial^{\nu}\vec{\phi})(\partial_{\mu}\vec{\phi}\cdot\partial_{\nu}\vec{\phi})-(\partial^{\mu}\vec{\phi}\cdot\partial_{\mu}\vec{\phi})^{2})+\theta_{2}\mathcal{L}_{V},
\partial_{\mu}J{^{5,\mu}_{3}}=\partial_{+}J{_3^{R}}-\partial_{-}J{_3^{L}}={\frac{g}{\pi}}(\partial_{-}A_{3}-\partial_{+}v)={\frac{g}{2\pi}}\epsilon^{\mu\nu}F_{\mu\nu}^{3}\;,
L_{B}=-{\frac{1}{2}}\left(\,\partial_{\mu}\phi\,\right)^{2}+{\frac{m}{4!}}\epsilon^{\mu\nu\rho\sigma}\,\phi\,\partial_{\,[\,\mu}A_{\nu\rho\sigma\,]}-{\frac{1}{2\cdot4!}}F^{2}{}_{\mu\nu\rho\sigma}(A)
\Box\,h_{\,\,\bar{a}}^{g}\,h_{\,\,\bar{b}}^{h}\,\sigma_{ab}^{m}\,\sigma_{gh}^{n}\,G_{mn}={\textstyle{\frac{1}{4}}}\epsilon_{abce}\epsilon_{fghk}\,\delta^{ch}\,h_{\,\,\bar{a}}^{f}\,h_{\,\,\bar{b}}^{g}\,\,F^{++ek}
\left(\tilde{y}T(v)\right)_{\beta}=\mathrm{Tr}\left(\tilde{G}_{1}(\beta_{1})...\tilde{G}_{N}(\beta_{N})\right),\quad\mathrm{with}\quad\left(\tilde{G}_{i}(\beta)\right)_{\mu\nu}=\sum_{\alpha}R_{(}\mu,\alpha;\nu,\beta)\tilde{g}_{i}(\alpha),
F(\bar{y},z)=\sum_{s}\oint\frac{\bar{du}}{2\pii}\frac{1}{(\bar{y}-\bar{u})^{2}}\oint\frac{dv}{2\pii}\frac{1}{(v-z)}\frac{\xi_{s}^{\prime}(v)}{\xi_{s}(v)(1-\bar{\xi}_{s}(u)\xi_{s}(v))}
\chi_{i}=\frac{E^{d/2}}{J_{i}^{d-2}}=d^{d/2}2^{4-3d}\left(\frac{V_{p}V(S^{d-1})}{16\piG}\right)^{-(d-4)/2}h^{-(d-2)^{2}/2}\left(\frac{r_{0}}{l_{i}}\right)^{d-2}
{\calL}_{bulk}=r_{c}M^{3}\left\{\frac{1}{2}(\partial\psi)^{2}-V(\psi)+\sum_{\vec{n}}\frac{a_{\vec{n}}(\psi)}{M^{|n|}}(\partial\psi)^{n_{1}}...(\partial^{N}\psi)^{n_{N}}\right\},
0=ga_{0}+a_{0}^{3}+2a_{0}\Sigma_{2}+2\Sigma_{2}a_{0}+\sum_{n>0}{\frac{1}{n}}\left(a_{n}^{\dagger}a_{0}a_{n}+a_{n}a_{0}a_{n}^{\dagger}-a_{0}\right)+6\Sigma_{3}\;.
{\calS}\equiv\left(\begin{array}{cc}{M}&{0}\\{0}&{\tilde{M}}\\\end{array}\right)\quad,\quad{\calK}\equiv\left(\begin{array}{cc}{0}&{K}\\{K^{\epsilon}}&{0}\\\end{array}\right)\quad,\quad{\calK}\longmapsto{\calK}^{\prime}={\calS}{\calK}{\calS}^{-1}\quad,
I_{0}(\Omega)=\int\frac{d^{2}p}{(2\pi)^{2}}\frac{1}{p^{2}+\frac{g^{2}}{\pi}\left(p_{0}^{2}V_{(1)}^{s}+p_{1}^{2}V_{(0)}^{s}\right)+8\pi\Omega^{2}}
W\equiv\left|\begin{matrix}{f_{1}}&{f_{2}}&{\cdots}&{f_{n}}\\{Df_{1}}&{Df_{2}}&{\cdots}&{Df_{n}}\\{\vdots}&{\vdots}&{}&{\vdots}\\{D^{n-1}f_{1}}&{D^{n-1}f_{2}}&{\cdots}&{D^{n-1}f_{n}}\\\end{matrix}\right|\quad.