harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left[B_{a}^{ij}\left(x\right),\pi_{kl}^{b}\left(y\right)\right]_{x^{0}=y^{0}}^{*}=\frac12\delta_{a}^{\;\;b}\delta_{\;\;k}^{\left[i\right.}\delta_{\;\;l}^{\left.j\right]}\delta^{3}\left({\bfx}-{\bfy}\right).
A_{n}(\varepsilon,v)={\frac{1}{\varepsilon^{\etan+1}}}\int_{0}^{\infty}\prod_{j=1}^{n}{\frac{dx_{j}}{x_{j}^{\eta+1}}}\theta(x_{j}-m_{1}/\varepsilon)\delta\left(m_{0}(a)/\varepsilon-x_{j}\right)\theta\left(1-\sum_{j=1}^{n}x_{j}\right).
C_{1}=\frac{2^{1/6}\sqrt{3}\left(\frac{3}{4}\right)^{\frac{1}{3}}}{2\left(\frac{3}{8}\right)^{\frac{5}{6}}}\frac{\Gamma\left(\frac{4}{3}\right)\left[\Gamma\left(\frac{2}{3}\right)\right]^{2}}{\Gamma\left(\frac{5}{3}\right)\left[\Gamma\left(\frac{1}{3}\right)\right]^{2}}\approx0.505468\,.
S_{2}=\intd\tau\left\{\pi_{{\bar{a}}\mu}Dx_{\bar{a}}^{\mu}-\frac12\sum^{\prime}\phi_{\bar{a}\bar{b}}(\pi_{{\bar{a}}\mu}\pi_{\bar{b}}^{\mu}+c_{\bar{a}}^{2}\delta_{\bar{a}\bar{b}})\right\},
\left(\begin{matrix}{\hat{H}^{(1)}}\\{\hat{H}^{(2)}}\\\end{matrix}\right)\to\omega\left(\begin{matrix}{\hat{H}^{(1)}}\\{\hat{H}^{(2)}}\\\end{matrix}\right),\\\\hat{\calM}\to(\omega^{-1})^{T}\hat{\calM}\omega^{-1},\\\\omega\inSL(2,{\bfR}).
{\calS}_{\Sigma}=\intd^{3}z\Biggl[{\frac{\sqrt{g}}{l_{11}^{3}}}-{\frac{i}{3!}}\epsilon^{ijk}\partial_{i}{\bfZ}^{M}\partial_{j}{\bfZ}^{N}\partial_{k}{\bfZ}^{P}C_{MNP}(X(s),\Theta(s))\Biggr],
\partial_{\bar{a}}C_{i_{1}...i_{s}}^{g}=fact({\bar{C}}_{\bar{a}})-\sum_{i=1}^{s}G_{i{\bar{a}}}(2g\!-\!2\!+\!s\!-\!1)C_{i_{1}..{\hat{i}}..i_{s}}^{g}
S_{E}={\frac{1}{2\kappa_{D}^{2}}}\intd^{D}x\sqrt{-g^{E}}\left[{\calR}_{g^{E}}-{\frac{4}{D-2}}(\partial\phi)^{2}-{\frac{1}{2\cdot(p+2)!}}e^{2a_{p}\phi}F_{p+2}^{2}-{\frac{1}{2\cdotD!}}e^{2a\phi}F_{D}^{2}\right],
H_{\mu\nu}=\frac{1}{4}g_{\mu\nu}H\qquadH\equivg^{\mu\nu}H_{\mu\nu}=\frac{1}{4}g^{-1/2}e_{n}^{\rho}\frac{\delta}{\deltae_{n}^{\rho}}\int{\calE}\sqrt{g}d^{4}x
\begin{array}{rrclcrcl}{\Omega:}&{{\calO}_{1}}&{\rightarrow}&{{\calO}_{1},}&{}&{{\calO}_{\omega_{1}}}&{\rightarrow}&{{\calO}_{\omega_{2}},}\\{}&{{\calO}_{\omega}}&{\rightarrow}&{{\calO}_{\omega},}&{}&{{\calO}_{\omega_{2}}}&{\rightarrow}&{{\calO}_{\omega_{1}}.}\\\end{array}
\int\mathrm{d}x\,J_{a}(x)\,\frac{\delta^{L}Z_{J,K,\xi}}{\deltaK_{a}(x)}\;-\;\sum_{r\geq2}\frac{1}{r!}\,\frac{\partial^{R}Z_{J,K,\xi}}{\partial\xi^{B}}\,f_{A_{1}\cdotsA_{r}}^{B}\xi^{A_{r}}\cdots\xi^{A_{1}}=0\;.
|K\rangle=\sum_{M=-\infty}^{\infty}\int_{M-{\frac{1}{2}}}^{M+{\frac{1}{2}}}dq|q\rangle\sum_{k=0}^{K-1}\varphi_{K}(q,k)b_{M}^{\dagger}(k+{\frac{1}{2}})d_{M}^{\dagger}(K-k-{\frac{1}{2}})|0\rangle_{M},
{\bf\Omega}^{\prime}(x_{1},\ldots,x_{N})=\prod_{i=1}^{N/4}f(x_{4i-1}-x_{4i})\bar{f}(x_{4i-3}-x_{4i-2})\,\,\delta\left(\sum_{i=1}^{N/4}(x_{4i-3}-x_{4i-1})\right)\,.\nonumber
(\bar{f}_{I}^{\Lambda},f_{\bar{I}}^{\Lambda})=(\bar{f}_{i}^{\Lambda}P_{I}^{i},f_{\bar{\imath}}^{\Lambda}\bar{P}_{\bar{I}}^{\bar{\imath}}),\quad\quadP_{i}^{I}\bar{P}_{\bar{\jmath}}^{\bar{J}}\eta_{I\bar{J}}=g_{i\bar{\jmath}}
{\frac{\delta}{\deltay^{\mu}(\,\vec{s}\,)}}=y^{\mu_{2}\dots\mu_{p+1}}(\,\vec{s}\,){\frac{\delta}{\delta\sigma^{\mu\mu_{2}\dots\mu_{p+1}}(\,\vec{s}\,)}}\,\qquady^{\mu_{2}\dots\mu_{p+1}}(\,\vec{s}\,)\equiv\epsilon^{m_{2}\dots\mu_{p+1}}\,\partial_{m_{2}}\,y^{\mu_{2}}\dots\partial_{m_{p+1}}\,y^{\mu_{p+1}}\.
Z_{4n}[j,J,\epsilon,\kappa]=\sum_{l,m}\int{\calD}be^{-S[\tilde{A}_{h},b]}\left[detIm\OmegaVol(\Sigma)\right]^{\frac{1}{2}}\left|\theta\left[\begin{array}{c}{\frac{l}{2}}\\{\frac{m}{2}}\\\end{array}\right](0|\Omega)\right|^{2}
\delta_{2}S=-\frac{i}{2}\intd^{4}x\delta_{2}c^{a}\int_{-L}^{L}dt\frac{\partial\bar{c}^{a}}{\partialt}+h.c.+\intd^{4}x\frac{i}{2}\delta_{2}\bar{c}^{a}M^{+}M\int_{-L}^{L}dtc^{a}+h.c.
{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}\partial_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}\partial_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\rightarrow{\hat{\Pi}}_{{\hat{\mu}}{\hat{\nu}}}{\calD}_{\hat{\imath}}{\hat{X}}^{\hat{\mu}}{\calD}_{\hat{\jmath}}{\hat{X}}^{\hat{\nu}}\,,
I=\int_{0}^{\tilde{X}}dz\,\frac{n_{1}}{c}\sqrt{(y^{\prime})^{2}+(h^{\prime})^{2}}+\int_{\tilde{X}}^{L}dz\,\frac{n_{2}}{c}\sqrt{(y^{\prime})^{2}+(h^{\prime})^{2}}
ds_{5}^{2}=-f^{-2/3}(r)\left(1-{\frac{r_{0}^{2}}{r^{2}}}\right)dt^{2}+f^{1/3}(r)\left[\left(1-{\frac{r_{0}^{2}}{r^{2}}}\right)^{-1}dr^{2}+r^{2}d\Omega_{3}^{2}\right],
{\calL}(1,\alpha-\frac{1}{2},g)=-i\epsilon_{\mu\nu\lambda}B_{\mu}^{*}(\partial_{\nu}+ia_{\nu}+iC_{\nu}+M)B_{\lambda}+\frac{g}{4}(B_{\mu}^{*}B_{\mu})^{2}-\frac{i}{8\pi(\alpha-1/2)}\epsilon_{\mu\nu\lambda}a_{\mu}\partial_{\nu}a_{\lambda}\;.
\begin{array}{l}{[S_{1}(\vec{x}),S_{2}(\vec{y})]_{P.B.}=-S_{3}(\vec{x})\delta(\vec{x}-\vec{y})\,,}\\{[S_{2}(\vec{x}),S_{3}(\vec{y})]_{P.B.}=S_{1}(\vec{x})\delta(\vec{x}-\vec{y})\,,}\\{[S_{3}(\vec{x}),S_{1}(\vec{y})]_{P.B.}=S_{2}(\vec{x})\delta(\vec{x}-\vec{y})\,.}\\\end{array}
(\mathrm{mass})_{\mathrm{n}}^{2}={\frac{2}{\alpha^{\prime}}}\left(-2+{\frac{\pi^{2}n^{2}\alpha^{\prime}}{2\beta^{2}}}\right),\quad(\mathrm{mass})_{\mathrm{w}}^{2}={\frac{2}{\alpha^{\prime}}}\left(-2+{\frac{w^{2}\beta^{2}}{2\pi^{2}\alpha^{\prime}}}\right)\quad.
\textrm{P}\frac{1}{z^{2}}=\frac{1}{2}\left[\frac{1}{(z+i\epsilon)^{2}}+\frac{1}{(z-i\epsilon)^{2}}\right]=-{\frac{\partial}{\partialz}}\left[\textrm{P}\left({\frac{1}{z}}\right)\right].
S_{SG}(\theta)=\frac{S_{0}(\theta)}{xp-p^{-1}x^{-1}}\left(\begin{array}{cccc}{xp-p^{-1}x^{-1}}&{0}&{0}&{0}\\{0}&{p-p^{-1}}&{x-x^{-1}}&{0}\\{0}&{x-x^{-1}}&{p-p^{-1}}&{0}\\{0}&{0}&{0}&{xp-p^{-1}x^{-1}}\\\end{array}\right)\,
(E^{2}-c^{2}\vec{\bfp}^{\,2})I^{(2)}\Psi=\left[EI^{(2)}-c{\bbox\sigma}\cdot{\bfp}\right]\left[EI^{(2)}+c{\bbox\sigma}\cdot{\bfp}\right]\Psi=\mu_{2}^{2}c^{4}I^{(2)}\Psi\,.
\gamma^{1}=e^{\frac{i}{2}\sigma_{3}\chi_{s}}i\sigma_{1}e^{-\frac{i}{2}\sigma_{3}\chi_{s}},\qquad\gamma^{2}=e^{\frac{i}{2}\sigma_{3}\chi_{s}}is\sigma_{2}e^{-\frac{i}{2}\sigma_{3}\chi_{s}},
C_{ij}{}^{k}\Phi_{ka_{0}\ldotsa_{n}}=\sum_{p,q=0}^{n}(-1)^{p+q-1}F_{bca_{p}}\Phi_{i}{}^{b}{}_{a_{p+1}\ldotsa_{q}}\Phi_{j}{}^{c}{}_{a_{q+1}\ldotsa_{p-1}}+(i\leftrightarrowj).
{\calR}(x)=\sum_{m=0}^{\infty}\frac{b_{m}}{m!}\Lambda^{m}(x)=\left(\frac{e^{\Lambda(x)}-1}{\Lambda(x)}\right)^{-1}\,,\qquad\Lambda_{j}^{i}(x)=x^{k}f_{kj}^{i}\,,
Q=G^{\ast}+\bar{G}^{\ast}=\theta{\frac{\delta}{\deltaX^{\ast}}}-F^{\ast}{\frac{\delta}{\delta\eta}}-\partialX{\frac{\delta}{\delta\psi}}-\bar{\partial}X{\frac{\delta}{\delta\bar{\psi}}}-(\partial\bar{\psi}+\bar{\partial}\psi){\frac{\delta}{\deltaF}}
G_{D}({\bfr^{\prime\prime}},{\bfr^{\prime}};E)=\sum_{n=0}^{\infty}\,\prod_{\alpha=1}^{n}\left[\int_{0}^{\infty}d^{D}{\bfr}_{\alpha}V({\bfr}_{\alpha})\right]\,\prod_{\beta=0}^{n}\left[G_{D}^{(0)}({\bfr}_{\beta+1},{\bfr}_{\beta};E)\right]\;.
{\calL}_{-1}^{\gammafull}=\frac{\chi_{x}\upsilon_{t}-\upsilon_{x}\chi_{t}}{\chi_{x}^{\gamma-1}-\upsilon_{x}^{\gamma-1}}-{\calH}_{-1}^{E}+\lambda\left(u-\frac{\chi_{x}^{\gamma-1}+\upsilon_{x}^{\gamma-1}}{\gamma-1}\right)+\sigma\left(\rho-\chi_{x}\upsilon_{x}\right)
\Gamma_{0}\,=\,\left(\begin{array}{cc}{{\mathbbI}}&{0}\\{0}&{-{\mathbbI}}\\\end{array}\right)\:\:\:\:\:\mathrm{and}\:\:\:\:\:{\vec{\Gamma}}\,=\,\left(\begin{array}{cc}{0}&{{\bfs}}\\{-{\bfs}}&{0}\\\end{array}\right)\;\;.
\langles_{1}\cdot\cdot\cdots_{N}|\Psi\rangle=e^{-{\frac{1}{2B}}\sum_{i}\left({\frac{\partial}{\partials_{i}}}\right)^{2}}\prod_{i<j}(e^{i2\pis_{i}/L_{x}}-e^{i2\pis_{j}/L_{x}})^{m}.
{\calZ}_{GU}={\displaystyle\int}{\calD}(A^{\mu},\pi_{\mu},\lambda)~exp~{i\displaystyle\int}d^{4}x~\left(\pi_{0}{\dot{A}_{0}}+\pi_{i}{\dot{A}_{i}}-{\widetildeH_{GU}}-\lambda\chi\right),
\widehat{d}_{-}(p)_{2}=\frac{m}{2(4\pi)^{3}}g_{\mu\nu}\beta^{\mu}\beta^{\nu}\left(1-\frac{m^{2}}{p^{2}}\right)\theta(p^{2}-m^{2})\theta\left(-p^{0}\right)\mathrm{~.}
\begin{array}{c}{\alpha^{\dagger}\hat{I}S^{(0)}=\alpha^{\dagger}\hat{I}\hat{J}\alpha^{\prime}=-\alpha^{\dagger}\Gamma_{1}\Gamma_{2}\hat{J}\alpha^{\prime}=}\\{=\alpha^{\dagger}\hat{J}\hat{I}\alpha^{\prime}=\alpha^{\dagger}\hat{J}\Gamma_{1}\Gamma_{2}\alpha^{\prime}=0}\\\end{array}
\int_{0}^{\tau}dt\cdot\omega_{\mu}(t,z,\tau)+\int_{0}^{\tau}dt\cdot\omega_{2-\mu}(t,z,\tau)=\tilde{q}^{-1}\left(\int_{0}^{-z}dt\cdot\omega_{\mu}(t,z,\tau)-\int_{0}^{-z}dt\cdot\omega_{\mu-2}(t,z,\tau)\right)
{^{(+)}\xi}^{i}=\left(-\beta\partial_{\phi}{^{(+)}\xi}^{\phi}(x^{+}),{^{(+)}\xi}^{\phi}(x^{+})\right),\qquad{^{(-)}\xi}^{i}=\left(-\beta\partial_{\phi}{^{(-)}\xi}^{\phi}(x^{-}),{^{(-)}\xi}^{\phi}(x^{-})\right)
0=\frac{\delta\Gamma}{\deltaK_{\mu}}\frac{\delta\Gamma}{\deltaA_{\mu}}+\frac{\delta\Gamma}{\delta\bar{K}}\frac{\delta\Gamma}{\delta\psi}+\frac{\delta\Gamma}{\deltaK}\frac{\delta\Gamma}{\delta\bar{\psi}}+B\frac{\delta\Gamma}{\delta\bar{c}}+\frac{\delta\Gamma}{\delta\bar{K}_{c}}+\chi\frac{\partial\Gamma}{\partial\xi}\frac{\delta\Gamma}{\deltac}
\Psi_{\kappa}(n)=\sum_{\tilde{s}=0}^{n}\frac{(-\kappa)^{\tilde{s}}}{\tilde{s}!}\biggl[\sum_{\tilde{k}=0}^{n-\tilde{s}}\frac{(-\kappa)^{\tilde{k}}}{\tilde{k}!}\binom{n}{\tilde{k}}(H_{n-\tilde{k}}+H_{\tilde{s}}-2H_{\tilde{k}})\biggr]\,.
\widetildeZ_{1}\,\frac{1}{2}\,\,\langle\,\partialA_{x}^{a}\,(\bar{c}\timesc)_{y}^{b}\,\rangle\,=\,\widetildeZ_{3}\,\frac{i}{2}\,\Big(\langle\,i(\partialD_{r}c)_{x}^{a}\,\bar{c}_{y}^{b}\,\rangle\,-\,\langle\,c_{y}^{b}\,i(\partialD_{r}\bar{c})_{x}^{a}\,\rangle\Big)\;.
T_{\mu\alpha\beta}^{V\rightarrowVV}|_{AT}=T_{\mu\alpha\beta}^{V\rightarrowAA}|_{AT}=\frac{-2i\pi^{2}}{3(2\pi)^{4}}(b-a)\left\{g_{\mu\alpha}(p-p^{\prime})_{\beta}+g_{\mu\beta}(p-p^{\prime})_{\alpha}+g_{\alpha\beta}(p-q)_{\mu}\right\}.
g_{w}(z)\equiv\frac{d^{3}}{dz^{3}}f_{w}(z)=\sum_{b}\frac{\deltaw_{b}}{\epsilonw_{b}}\left[\frac{(\alpha_{b}-\beta_{b})}{(z-\alpha_{b})(z-\beta_{b})}\right]^{2}\left[\bar{\calP}_{bc}\left(u(z)\right)-\bar{\calP}^{\prime\prime}\left(u(z)\right)\right]\,
\begin{array}{rcl}{\deltaG_{\mu}}&{=}&{\epsilon(y\partial_{\mu}c+zig[c,G_{\mu}]),}\\{\delta\phi}&{=}&{\epsilonxigc\Phi,~~\delta\phi^{\dagger}=-\epsilonxig\Phi^{\dagger}c,}\\{\deltac}&{=}&{\epsilonuigcc.}\\\end{array}
\overline{{\mathrm{\Large~v~}}}_{3}(x)={\frac{2\pi^{3/2}}{g^{2}\Lambda}}x+{\frac{1}{3}}\left(4x-{\frac{{\calE}^{2}}{x}}\right)x^{1/2}{\mit\Gamma}(1/2,x)+{\frac{2}{3}}(1-2x){\mathrme}^{-x}-{\frac{2}{3}}\,
\Phi_{1}^{(2)}=\pi^{2}\;,\;\;\Phi_{2}^{(2)}=\pi_{\mu}\psi^{\mu}\;,\;\;\Phi_{3\mu}^{(2)}=T_{\mu}=\epsilon_{\mu\nu\rho\varsigma}\pi^{\nu}\psi^{\rho}\psi^{\varsigma}+i\frac{\alpha}{2}\pi_{\mu}\;,\;\;
{\frac{3}{r_{1}^{(1+{\frac{3b}{4}})}(r_{1}^{3}-R_{1}^{3})^{(1-{\frac{3b}{4}})}}}\,\left[\left(1+{\frac{(4+3b)}{3(4-3b)}}\right)\,r_{1}^{3}-R_{1}^{3}\right]=-{\frac{V}{4}}\,({\frac{4}{3}}-b)\,e^{{\frac{3b}{2}}\,\Phi_{0}}.
\langle{\calO}_{\mu_{1}\cdots\mu_{s}}(x)~{\calO}_{\nu_{1}\cdots\nu_{s}}(0)\rangle=c_{s}\frac{1}{(|x|\mu)^{2h_{s}}}{\prod}_{\mu_{1}\cdots\mu_{s},\nu_{1}\cdots\nu_{s}}^{(s)}\left(\frac{1}{|x|^{4}}\right),
\begin{aligned}{[r]}&{{}U_{1}:}&{}&{{}\lambda(t)=t,}&{}&{{}v(t)=\frac{\mu}{t},}&{}&{{}x(t)=0,}&{}&{{}y(t)=0}\\{}&{{}U_{2}:}&{}&{{}\rho(t^{\prime})=t^{\prime},}&{}&{{}v(t^{\prime})=\mut^{\prime},}&{}&{{}x(t^{\prime})=0,}&{}&{{}u(t^{\prime})=0.}\\\end{aligned}
\psi_{SE2}(x,{\bfp};s_{1},s_{2})=-\int{\frac{(1-x)dp^{-}}{2\pi}}{\frac{\bar{u}(xP^{+},{\bfp};s_{1})\gamma^{+}\Phi(p)\Sigma_{1}(p-P)v((1-x)P^{+},-{\bfp};s_{2})}{(p-P)^{2}-m^{2}+i\epsilon}},
Y^{(l)a_{1}\cdotsa_{m}}={\frac{(-1)^{{\frac{1}{2}}(m+1)(d-m-2)}}{(l+m)(d-m-3)!}}{\frac{1}{\sqrt{h}}}\epsilon^{a_{1}\cdotsa_{2n+1}}\partial_{a_{m+1}}Y_{a_{m+2}\cdotsa_{d-2}}^{(l)}.
\left[\begin{array}{cc}{a_{0}}&{1}\\{1}&{0}\\\end{array}\right]\left[\begin{array}{cc}{a_{1}}&{1}\\{1}&{0}\\\end{array}\right]\cdots\left[\begin{array}{cc}{a_{i}}&{1}\\{1}&{0}\\\end{array}\right]=\left[\begin{array}{cc}{p_{i}}&{p_{i-1}}\\{q_{i}}&{q_{i-1}}\\\end{array}\right].
\langle\overline{{\phi}}^{2}(x)\rangle_{\beta}=\frac{1}{12\beta^{2}}(1+\eta^{2})-\eta^{2}\frac{\xi\sqrt{2}}{8\pi^{2}r^{2}}S_{1}(\zeta)+\eta^{2}\frac{1}{32\pi^{2}r^{2}}S_{2}(\zeta),
\frac{1}{u_{3}}\Big(8x_{1}x_{2}-\frac{31}{4}x_{1}x_{4}+x_{3}x_{4}-\frac{5}{4}x_{2}x_{3}\Big)+Z_{1}(\zeta)-\frac{2}{3}N_{F}\Big(\frac{x_{2}-x_{4}}{u_{3}}\Big)z_{3}=0,
2\,\sqrt{\left(\bar{z}_{+}^{(1)}\,\,\bar{z}_{+}^{(2)}\right)\Big(\begin{array}{cc}{\mathrm{e}^{V}}\\\end{array}\Big)\left(\begin{array}{c}{z_{+}^{(1)}}\\{z_{+}^{(2)}}\\\end{array}\right)\left({z}_{-}^{(1)}\,\,{z}_{-}^{(2)}\right)\Big(\begin{array}{cc}{\mathrm{e}^{-V}}\\\end{array}\Big)\left(\begin{array}{c}{\bar{z}_{-}^{(1)}}\\{\bar{z}_{-}^{(2)}}\\\end{array}\right)}
S[\tilde{A}_{(\tilde{p}+1)}]=\intd^{d}x\\sqrt{|g|}\\left[{\textstyle\frac{(-1)^{(\tilde{p}+1)}}{2\cdot(\tilde{p}+2)!}}e_{(p+1)}^{2}\tilde{F}_{(\tilde{p}+2)}^{2}+{\textstyle\frac{(-1)^{\tilde{p}}}{2\cdot(\tilde{p}+1)!}}e_{(p)}^{2}\tilde{A}_{(\tilde{p}+1)}^{2}\right]\,,
{\calB}^{[AB]}={\frac{R_{s}}{2(k-2)}}\left(\nu_{k}^{A}{\bar{\nu}}_{C}^{k}J^{BC}-\nu_{k}^{B}{\bar{\nu}}_{C}^{k}J^{AC}+{\frac{1}{2}}J^{AB}{\nu_{k}^{F}}{\bar{\nu}}_{F}^{k}\right)
[\Sigma_{\frac{1}{2}}^{\prime},\Sigma_{-\frac{1}{2}}]=\Theta^{\alpha}\Theta^{\beta}\frac{1}{2}\oint\frac{dz}{2\pii}(\partial{\hat{X}}-iQ\cdot\psi{\hat{\psi}})_{\alpha\beta}e^{-iQX}.
\left[{H(x^{\prime};n(x^{\prime})),H(x^{\prime\prime};n(x^{\prime\prime}))}\right]_{\mathrm{P.B.}}-{\frac{\partialH(x^{\prime\prime};n(x^{\prime\prime}))}{\partial\sigma(x^{\prime})}}+{\frac{\partialH(x^{\prime};n(x^{\prime}))}{\partial\sigma(x^{\prime\prime})}}=\mathrm{const.}\.
\left\{\psi_{1},\psi_{2};\alpha,a_{1},a_{2}\right\}\equiv\left(\begin{array}{cccccc}{e^{\alpha}}&{}&{}&{}&{}&{}\\{}&{e^{\alpha}}&{0}&{a_{1}}&{}&{}\\{}&{e^{\alpha}\psi_{2}}&{0}&{a_{2}}&{}&{}\\{}&{0}&{0}&{1}&{}&{}\\{}&{}&{}&{}&{1}&{\psi_{1}}\\{}&{}&{}&{}&{0}&{0}\\\end{array}\right).
\sum_{i}^{n}\!\left<\gamma_{++}\!(\xi_{1})\!\dots\!\delta_{\!f}\!\gamma_{++}\!(\xi_{i})\!\dots\!\gamma_{++}\!(\xi_{n})\right>\!+\!\frac{d\!\!-\!\!28\!\!+\!\!\lambda^{W}}{i48\pi}\!\!\int\!\!d\xi^{2}\!\,\delta\!f\!(\!\xi\!)\left<\partial_{-}^{3}\gamma_{++}\!(\xi)\gamma_{++}\!(\xi_{1})\!\dots\!\gamma_{++}\!(\xi_{n})\right>\!\!=\!\!0.
R_{\\nu}^{\mu}-{\frac{1}{2}}\delta_{\nu}^{\mu}R+2l^{2}\left[\tau_{\\nu}^{\mu}+\left(\alpha-{\frac{1}{4}}\right){\calS}_{\\nu}^{\mu}+{\frac{\beta}{3}}{\calK}_{\\nu}^{\mu}\right]={\frac{\kappa^{2}}{l}}T_{\\nu}^{\mu}\.
\int{\left\{\frac{1}{2}D_{i}\tilde{\Phi}^{a}D_{i}\tilde{\Phi}^{a}\right\}d^{3}r}=\int{\left\{\frac{1}{4}F_{ij}^{a}F_{ij}^{a}\right\}d^{3}r}=\frac{1}{2}E_{\mathrm{MP}}\,
^{c}g_{\mu\nu}^{\prime}\left(x\right)\equive_{\mu}^{\primea}\left(x\right)\widetilde{e}_{\nu}^{\primeb}\left(x\right)\eta_{ab}=U^{a}\,_{c}\left(x\right)\widetilde{U}^{b}\,_{d}\left(x\right)\eta_{ab}e_{\mu}^{c}\left(x\right)\widetilde{e}_{\nu}^{d}\left(x\right)\,.
H(\{q_{N_{s}}\})=(u_{1}-u_{2})(v_{1}-u_{2})\left[1-\frac{\mu_{1}\mu_{2}}{2(u_{1}-u_{2})}-\frac{\nu_{1}\mu_{2}}{2(v_{1}-u_{2})}\right]\prod_{s=1}^{n}(u_{s}-v_{s}-\mu_{s}\nu_{s})^{-1}
{\calL}=-\frac{1}{2}\mathrm{Tr}F^{\mu\nu}F_{\mu\nu}+\bar{q}i{\not\!\!D}q+(m\bar{q}_{R}q_{L}+\mathrm{h.c.})+\frac{i\phi}{16\pi^{2}}\mathrm{Tr}F^{\mu\nu}\tilde{F}_{\mu\nu}.
\delta_{\eta}q_{i}^{a}=\epsilon^{abc}\eta^{b}q_{i}^{c}\\,\\\delta_{\eta}p_{i}^{a}=\epsilon^{abc}\eta^{b}p_{i}^{c}\\,\\\delta_{\eta}\lambda^{a}=-\dot{\eta}^{a}+\epsilon^{abc}\eta^{b}\lambda^{c}\\\,
\begin{array}{cl}{(1,1)\times2^{4}~=(3,1)+(1,5)+(2,4)~~~~~~~}&{\mathrm{:\,massless~tensor,~Maxwell,}}\\{{}}&{{}}\\{(1,1,1)\times2^{4}=(3,1,1)+(1,1,5)+(2,1,4)~}&{\mathrm{:\,massive~tensor}\,,}\\\end{array}
\vec{n}^{\prime}\vec{\Gamma}_{\mu}^{\prime}=\vec{n}^{\prime}\Omega\vec{\Gamma}_{\mu}+\vec{n}^{\prime}\vec{\Omega}_{\mu}=\vec{n}\vec{\Gamma}_{\mu}+\vec{n}^{\prime}\vec{\Omega}_{\mu}=\vec{n}^{\prime}\vec{\Omega}_{\mu}.
\begin{array}{l}{\partialX(z)\sigma_{\vartheta}(w,\bar{w})\sim(z-w)^{-(1-\vartheta)}\tau_{\vartheta}(w,\bar{w}),}\\{\partial\bar{X}(z)\sigma_{\vartheta}(w,\bar{w})\sim(z-w)^{-\vartheta}\tau_{\vartheta}^{\prime}(w,\bar{w}),}\\\end{array}
{\calH}_{(m)}-{\calE}_{(m)}=\int_{\Sigma}\sqrt{\sigma}d^{2}zQ_{(m)}^{\mu\nu}(t)l_{\mu}p_{\nu}+\int_{C_{t}}\sqrt{\gamma}d^{2}yQ_{(m)}^{\mu\nu}(t)u_{\mu}n_{\nu}~~~,
\alpha_{L}^{1}=-i/2\left(\begin{matrix}{0}&{1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\{0}&{0}&{0}&{1}\\{0}&{0}&{-1}&{0}\\\end{matrix}\right),\;\;\alpha_{L}^{2}=-i/2\left(\begin{matrix}{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{-1}\\{-1}&{0}&{0}&{0}\\{0}&{1}&{0}&{0}\\\end{matrix}\right),\;\;\alpha_{L}^{3}=-i/2\left(\begin{matrix}{0}&{0}&{0}&{1}\\{0}&{0}&{1}&{0}\\{0}&{-1}&{0}&{0}\\{-1}&{0}&{0}&{0}\\\end{matrix}\right),
\|(t_{0},t_{1},t_{2},t_{3})\|_{c}^{2}=\int_{0}^{c}\sum_{0}^{3}\|t_{i}(s)\|^{2}+\int_{c}^{+\infty}\sum_{0}^{3}\left(\|t_{i}(s)\|^{2}-\|t_{i}(+\infty)\|^{2}\right)ds.
\langle\psi|j_{0}|\psi\rangle=\frac{1}{V}\left[1+\frac{\omega_{{\bfq}_{1}}+\omega_{{\bfq}_{2}}}{2\sqrt{\omega_{{\bfq}_{1}}\omega_{{\bfq}_{2}}}}\,\mathrm{cos}(q_{1}-q_{2})x\right],
\begin{array}{rcl}{\rho_{1}}&{=}&{{\frac{1}{\sqrt{B_{1}}}}cos{\alpha}\tilde{\rho}_{1}+{\frac{1}{\sqrt{B_{1}}}}sin{\alpha}\tilde{\rho}_{2}}\\{\rho_{2}}&{=}&{-{\frac{1}{\sqrt{B_{2}}}}sin{\alpha}\tilde{\rho}_{1}+{\frac{1}{\sqrt{B_{2}}}}cos{\alpha}\tilde{\rho}_{2}}\\\end{array}
{\calC}_{c}^{a}{\calC}_{b}^{c}={\calC}_{c}^{a}{\calC}_{b}^{d}\delta_{d}^{c}={\calC}_{c}^{a}{\calC}_{b}^{d}\varepsilon^{cp}\varepsilon_{pd}=\varepsilon^{cp}{\calC}_{c}^{a}{\calC}_{b}^{d}\varepsilon_{pd}=-\varepsilon^{cp}{\calC}_{c}^{a}{\calC}_{p}^{d}\varepsilon_{db},
U\;=\;\int\frac{d^{4}p_{1}}{(2\pi)^{4}}\cdots\int\frac{d^{4}p_{r}}{(2\pi)^{4}}\;\delta(a_{1}-p_{1}^{2})\:\cdots\:\delta(a_{r}-p_{r}^{2})\;f^{\{\alpha\}}(p_{1},\ldots,p_{k})\;\;\;\;.
\left(f\starg\right)\starh-f\star\left(g\starh\right)=\frac{1}{6}\left(\alpha^{i\ell}\partial_{\ell}\alpha^{jk}+\alpha^{j\ell}\partial_{\ell}\alpha^{ki}+\alpha^{k\ell}\partial_{\ell}\alpha^{ij}\right)\left(\partial_{i}f\partial_{j}g\partial_{k}h\right)+\mathcal{O}(\alpha^{3}).
a_{0,1}=-{\frac{1}{24}}\int_{{\calM}_{3}}h^{1/2}d^{3}x~F^{jk}F_{jk},~~a_{2,2}=-{\frac{1}{12}}\int_{{\calM}_{3}}h^{1/2}d^{3}x~F^{jk}F_{jk}.
\begin{aligned}{\mathrm{res}\left[f\left(\frac{s+1}{2}\right)\right]_{s=0}}&{{}=\frac{1}{3}\left((\lambdaS)^{2}-1\right)\,,}\\{\mathrm{res}\left[f\left(\frac{s+3}{2}\right)\right]_{s=0}}&{{}=\frac{1}{(\lambdaS)^{2}}\,.}\\\end{aligned}
\begin{array}{rcl}{\deltaG_{\mu}}&{=}&{\epsilon(\partial_{\mu}c+ig[c,G_{\mu}]),}\\{\delta\phi}&{=}&{\epsilonigc\Phi,~~\delta\phi^{\dagger}=-\epsilonig\Phi^{\dagger}c,}\\{\deltac}&{=}&{\epsilonigcc,}\\\end{array}
\mathbf{X}=\left(\begin{array}{cccc}{1}&{-1}&{i}&{i}\\{-i}&{-i}&{1}&{-1}\\{1}&{1}&{-i}&{i}\\{-i}&{i}&{-1}&{-1}\\\end{array}\right),\quad\mathbf{Y}=\left(\begin{array}{cccc}{-1}&{1}&{i}&{i}\\{-i}&{-i}&{-1}&{1}\\{-1}&{-1}&{-i}&{i}\\{-i}&{i}&{1}&{1}\\\end{array}\right).
\nu_{1}\nu_{2}q^{2}I_{\nu_{1}+1~\nu_{2}+1}^{(d+2)}-\!\left(\frac{d}{2}-\!\nu_{1}\right)\!I_{\nu_{1}\nu_{2}}^{(d)}\!+\nu_{2}I_{\nu_{1}-1~\nu_{2}+1}^{(d)}+\nu_{1}m_{1}^{2}I_{\nu_{1}+1~\nu_{2}}^{(d)}\!-\nu_{2}(q^{2}\!-m_{1}^{2})I_{\nu_{1}~\nu_{2}+1}^{(d)}\!\equiv0.
\begin{aligned}{T_{Dp-brane}}&{{}=T_{\frac{p-1}{2},1}=\frac{1}{(2\pi)^{\frac{p-1}{2}}}\,T_{F-string}^{\frac{p-1}{2}}\,T_{D-string}=\frac{1}{(2\pi)^{p}\,g_{s}\,l_{p}^{p+1}}\,,}\\{T_{NSp-brane}}&{{}=T_{1,\frac{p-1}{2}}=\frac{1}{(2\pi)^{\frac{p-1}{2}}}\,T_{F-string}\,T_{D-string}^{\frac{p-1}{2}}=\frac{1}{(2\pi)^{p}\,g_{s}^{\frac{p-1}{2}}\,l_{p}^{p+1}}\,.}\\\end{aligned}
\begin{array}{c}{Z_{1}VZ_{2}=Z_{2}VZ_{1}\quad,\qquadD_{1}V^{-1}D_{2}=D_{2}V^{-1}D_{1}\quad,}\\{\,}\\{D_{1}Z_{2}=VZ_{2}D_{1}V^{-1}+{\calP}\quad.}\\\end{array}
3.2B_{\mu\nu\alpha\beta}=R_{\\mu\\alpha}^{\rho\\sigma\}R_{\rho\nu\sigma\beta}+R_{\\mu\\beta}^{\rho\\sigma\}R_{\rho\nu\sigma\alpha}-\textstyle{\frac{1}{2}}g_{\mu\nu}R_{\alpha}^{\\rho\sigma\tau}R_{\beta\rho\sigma\tau}.
{\calL}_{1/m^{3}}^{(2+1)spin}=\frac{\alpha}{720m^{3}}\left[18F^{\nu\lambda}F_{\nu\lambda,\mu}^{~~~~\mu}+7F^{\nu\lambda,\mu}F_{\nu\lambda,\mu}-2F_{~~~\lambda}^{\nu\lambda,}F_{\nu\mu,}^{~~~\mu}\right],
J(a^{2})=-\frac{a^{2}}{2}\sum_{n=1}^{\infty}{\frac{1}{n}}\int_{0}^{\infty}dt\,e^{-na\sqrt{t+1}}=-a\sum_{n=1}^{\infty}\frac{e^{-na}}{n^{2}}\;-\;\sum_{n=1}^{\infty}\frac{e^{-na}}{n^{3}}\;.
\Phi(x,x^{\prime})=({\frac{2}{3}}\partial_{\mu}\partial_{\nu}^{\prime}-{\frac{1}{6}}\eta_{\mu\nu}\partial_{\sigma}\partial^{\sigma}-{\frac{1}{3}}\partial_{\mu}^{\prime}\partial_{\nu}^{\prime})<\phi(\vec{x},t)_{mn}\phi(\vec{x}^{\prime},t^{\prime})_{nm}>|_{t=t^{\prime}}
\langlek^{\prime}|\hat{T}_{x}^{(0)}|k\rangle=e^{i(k^{\prime}-k)x}\left[k^{0\prime}k^{0}+\vec{k}^{\prime}\cdot\vec{k}+m^{2}\right]\quad\mathrm{and}\quad\langlek^{\prime}|\hat{T}_{x}^{(1)}|k\rangle=\sigma(x)e^{i(k^{\prime}-k)x}\,.
S=\frac{1}{2}\intd^{4}x\,\left\{g_{ij}\partial_{m}f^{i}\partial^{m}f^{j}+\bar{g}^{ij}\partial_{m}\bar{f}_{i}\partial^{m}\bar{f}_{j}+2h_{~j}^{i}\partial_{m}f^{j}\partial^{m}\bar{f}_{i})-V(f)\right\}~,
\begin{matrix}{M5:}&{1}&{2}&{3}&{4}&{5}&{-}&{-}&{-}&{-}&{-}\\{M2:}&{-}&{-}&{-}&{4}&{-}&{6}&{-}&{-}&{-}&{-}\\{M2:}&{-}&{-}&{-}&{-}&{5}&{-}&{7}&{-}&{-}&{-}\\\end{matrix}
\begin{array}{rcl}{p}&{=}&{\oint_{\gamma}e^{2\phi}dC^{(0)}\,,}\\{}&{}&{}\\{q}&{=}&{\oint_{\gamma}e^{2\phi}(-C^{(0)}d|\lambda|^{2}+|\lambda|^{2}dC^{(0)})\,,}\\{}&{}&{}\\{s}&{=}&{\oint_{\gamma}e^{2\phi}d|\lambda|^{2}\,,}\\\end{array}
\hat{\lambda}(z)\gamma^{A}-\tilde{\hat{\omega}}(z)\gamma^{A}-\gamma^{A}\tilde{\hat{\omega}}(z)^{t}=\hat{\lambda}(z)\gamma^{A}-\gamma^{B}\hat{\omega}(z)_{B}^{~A}=\gamma^{B}\partial_{B}h^{A}(z)
A_{\mu\nu}=\displaystyle{\frac{\sum_{k=1}^{m}(-1)^{k-1}\left(\sum_{j=0}^{m-k}S_{1^{j}}(z)\right)(\tilde{H}^{2k-1})_{\mu\nu}}{\sqrt{\sum_{j=0}^{m}S_{1^{j}}(z)}}}\,
{\calA}\equiv{\frac{{\frac{ap(a+p)}{8}}\partial^{2}+{\frac{1}{2}}(p\alpha^{2}+q^{2}a)}{({\frac{p+a}{2}})^{2}\partial^{2}+(q+\alpha)^{2}}}\partial^{2}-{\frac{1}{2g}},
{\calL}_{s}=\frac{\alpha^{2}}2\partial_{\mu}\varphi^{\alpha}\partial^{\mu}\varphi^{\alpha}-\frac{\beta^{2}}4\partial_{\mu}\varphi^{\alpha}\partial_{v}\varphi^{\beta}\left\{\partial^{\mu}\varphi^{\beta}\partial^{v}\varphi^{\alpha}-\partial^{\mu}\varphi^{\alpha}\partial^{\nu}\varphi^{\beta}\right\}+\frac{{}_{1}}{{}^{2}}\mu\left(\varphi^{\alpha}\varphi^{\alpha}-1\right)
{\frac{(12)^{2}(34)^{2}Y}{4i}}S_{44}=-{\frac{s}{2}}\partial_{4}^{--}[(12)^{2}(34)^{2}Y]X_{2}+(12)(13)(23)[-tX_{3}+{\frac{1}{2}}(1-s-t)X_{2}]\;.
\begin{array}{c}{t_{i}e_{j}t_{i}^{-1}=q^{a_{ij}}e_{j}}\\{t_{i}f_{j}t_{i}^{-1}=q^{-a_{ij}}f_{j}}\\{{}[e_{i},f_{j}]=\delta_{ij}\frac{t_{i}-t_{i}^{-1}}{q-q^{-1}}}\\\end{array}
D(P^{\dagger})=\{F_{\psi}\inL^{2}(0,\infty):F_{\psi}\;\mathrm{abs.}\;\mathrm{cont.},\int_{0}^{\infty}\mathrm{d}q\,|{\frac{\mathrm{d}F_{\psi}}{\mathrm{d}q}}|^{2}<\infty\},
I(x^{-},x^{+})|_{x^{+}=0}=\frac{\pi}{4}|x^{-}|k_{\bot}^{2},\quad{\partial}_{+}I(x^{-},x^{+})|_{x^{+}=0}=\frac{\pi}{2}x^{-}{\delta}(x^{+})k_{\bot}^{2}-\frac{\pi}{16}x^{-}|x^{-}|k_{\bot}^{4}.
\frac{{W}_{\xi,\eta}(i)}{W_{\xi,\eta}(i-1)}=\frac{\mu_{\xi}}{\mu_{\eta}}\frac{y_{\eta}-x_{\xi}\omega^{i}}{y_{\xi}-x_{\eta}\omega^{i}},\quad\frac{\overline{{W}}_{\xi,\eta}(i)}{\overline{{W}}_{\xi,\eta}(i-1)}=\mu_{\xi}\mu_{\eta}\frac{x_{\xi}\omega-x_{\eta}\omega^{i}}{y_{\eta}-y_{\xi}\omega^{i}}
\langlep^{\prime}|J^{\mu}|p\rangle=i(p^{\mu}+p^{\prime\mu})\frac{N}{\pi}{\int_{0}}^{1}{dx}{\int_{0}}^{1-x}{dy}\frac{2m^{2}(1-x)-(m^{2}+(x+y-1)^{2}M^{2}-xyq^{2})x}{((x+y)(x+y-1)M^{2}+m^{2}-xyq^{2})^{2}},
\frac{1}{\pi}\int_{1}^{\infty}\frac{|D(\nu,\alpha)|}{\alpha+\tau}d\alpha\leq\frac{\frac{1}{\pi}\int_{1}^{\infty}\omega(\nu,\alpha)/\alphad\alpha}{1+\frac{1}{\pi}\int_{1}^{\infty}\frac{\omega(\nu,\beta)}{\beta}d\beta}<1.
\zeta^{\alpha}=\mathbb{C}^{\alpha\beta}\,\mathcal{C}\,\left(\overline{{\zeta}}_{\beta}\right)^{T}\,\quad;\quad\overline{{\zeta}}_{\beta}=\left(\zeta^{\beta}\right)^{\dagger}\gamma_{0}\quad;\quad~~~\alpha,\beta=1,\dots,2\,m
\left(\begin{array}{c}{\hat{{\bfb}}}\\{\hat{{\bfb}}^{\dagger}}\\\end{array}\right)=\Omega\left(\begin{array}{c}{\hat{{\bfa}}}\\{\hat{{\bfa}}^{\dagger}}\\\end{array}\right)+\left(\begin{array}{c}{{\bfd}}\\{{\bfd}^{*}}\\\end{array}\right),\qquad\Omega=\left(\begin{array}{cc}{\zeta}&{\eta}\\{\eta^{*}}&{\zeta^{*}}\\\end{array}\right),
\begin{aligned}{}&{{}U_{1}:}&{}&{{}\qquadx_{1}(t)=t^{d},}&{}&{{}\qquady_{1}(t)=0,}&{}&{{}\qquadu_{1}(t)=\mu,}&{}&{{}\qquadv_{1}(t)=0}\\{}&{{}U_{2}:}&{}&{{}\qquadx_{2}(t^{\prime})=t^{d},}&{}&{{}\qquady_{2}(t^{\prime})=0,}&{}&{{}\qquadu_{2}(t^{\prime})={\frac{\mu}{t^{d}}},}&{}&{{}\qquadv_{2}(t^{\prime})=0.}\\\end{aligned}
\displaystyle{\frac{m_{\pi}^{2}f_{\pi}j_{\pi}(q^{2})}{m_{\pi}^{2}-q^{2}}=\partial^{\mu}J_{\mu}(q)=\frac{q^{2}f_{\pi}j_{\pi}(q^{2})}{m_{\pi}^{2}-q^{2}}+\partial^{\mu}\widehatJ_{\mu}(q)\quad,}