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\int | S \rangle \star Q | S \rangle =\langle S | c_0 \left(\alpha' p^2 +\frac{1}{2} M^2 \right)| S \rangle\label{eq:kinetic} | 32742287b1.png | none |
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\xi _ \alpha \bar \xi ^ { \dot \beta } = q _ { \alpha ( \beta + 2 ) } \bar \xi ^ { \dot \beta } \xi _ \alpha \ , | 1b538a5ad1.png | tokenized |
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G_{MN}=\left( \begin{array}{cc}f(x)& 0 \\ 0& h_{ij}(x,y)\end{array}\right) | 6a61540090.png | none |
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R ^ { 2 } ~ = ~ x ^ { 2 } + y ^ { 2 } + \left( { \frac { p _ { 0 } } { m } } x ^ { - } - { \frac { m } { 4 p _ { 0 } } } x ^ { + } \right) ^ { 2 } . | 54142ffed9.png | normalized |
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\operatorname { e x p } ( i ( q _ { i j } + q _ { j k } + q _ { k i } ) ) = \operatorname { e x p } ( i 4 \pi \mu ) , | 5cd3fdac09.png | normalized |
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\label{triple}\partial_a^3{\cal F}_{gauge}=\frac{C_2}{C_1^2\Lambda}\Big(\frac{dw_1}{dz}\Big)^{-3}\frac{\alpha^2}{(1-z)z^{2\alpha+3}}, | 17b3be79d0.png | none |
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\pi _ 0 ' = \frac { \Gamma ( \pi _ 0 - V _ 3 \pi _ 3 ) } { 1 - \pi _ 0 + \Gamma ( \pi _ 0 - V _ 3 \pi _ 3 ) } | 4c7f0d2f22.png | tokenized |
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\alpha ^ a _ q = \int ^ 1 _ 0 d \tau f _ q ( \tau ) X ^ a ( \tau ) . | 448195d420.png | tokenized |
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V(\varphi)={1\over2}m^2\varphi^2+u_4\varphi^4+u_6\varphi^6+\cdots, | 4bd46d0320.png | none |
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\label{eqren}\frac{d m^2(t)}{dt}=\frac{2\alpha^2 m^2(t)}{Q^2}+\Delta_{m^2}m^2(t) | d7847d2ec7.png | none |
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( \phi _ c , \dot { \phi } _ c , \psi _ c , \dot { \psi } _ c ) | _ { t = 0 } = ( 4 \Phi _ { \rm c r } , 0 , 0 , 0 ) \; . | 78ac9a3d67.png | tokenized |
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{ B } _ i = \frac { i } { 2 } \, \epsilon _ { i j k } \, { \stackrel { \circ } { R } } _ { j k 0 l } \, \alpha ^ l | 2e7a8bcefa.png | tokenized |
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Q ( \tau ) = g ( \tau ) K g ^ { - 1 } ( \tau ) . | 289fe055a1.png | normalized |
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\label{glA.6} A^o_{\mu} = z_A A^\mu + z_V V^\mu \; \; \; , | 33285d1467.png | none |
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B_{\mu \nu} \rightarrow B^{\prime}_{\mu \nu} = B_{\mu \nu} + \partial_\mu f_\nu- \partial_\nu f_\mu ~.\label{14} | 8bb343b16e.png | none |
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( { \rm g s } ) \equiv \left ( n _ { r } = 1 , l = 0 \right ) \; , | 1d1be66653.png | tokenized |
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{\Gamma}=\frac{1}{l+\frac{1}{2}}(-\vec{\sigma}\vec{L}^R+\frac{1}{2}).\label{28} | 63bad8b91d.png | none |
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h _ { c m } \ = \ - \sum _ { i = 1 } ^ { N - 1 } \partial _ i ^ 2 + \frac { 1 } { N } ( \sum _ { i = 1 } ^ { N - 1 } \partial _ i ) ^ 2 + 2 \sum _ { i = 1 } ^ { N - 1 } y _ i \partial _ i | 38cf898574.png | tokenized |
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\eta ^ { 6 } + \alpha \zeta ( \zeta ^ { 1 0 } + 1 1 \zeta ^ { 5 } - 1 ) = 0 | 3212c32dd7.png | normalized |
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\varepsilon \partial _ { \mu } j ^ { \mu } = \delta _ { \varepsilon } { \cal { L } } = - \varepsilon \partial _ { \mu } \kappa ^ { \mu } ~ . | 547c77f9aa.png | normalized |
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E ^ { i n j } = M / \sqrt { 1 + \beta ^ { 2 } / ( n + \mu _ { i } + 1 ) ^ { 2 } } , | 6e6333ce66.png | normalized |
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{ \bar { \varepsilon } _ { d } } = \varepsilon _ { d } ^ { T } \ \ C _ { d } , | 2876ef9887.png | normalized |
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\mathcal { L } = \mathcal { L } _ { g r a v } + W \, \delta ( x ^ 5 ) \, \mathcal { L } _ { b r a n e } \, , | 549095d291.png | tokenized |
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g _ { s } = g _ { s } ^ { \prime } \frac { l _ { s } ^ { p } } { R _ { 1 0 - p } ^ { \prime } \cdots R _ { 9 } ^ { \prime } } . | 36b3f2696f.png | normalized |
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M = { \frac { r _ { + } } { 1 + a ^ { 2 } } } \ , \qquad Q ^ { 2 } = { \frac { ( r _ { + } ) ^ { 2 } } { 1 + a ^ { 2 } } } \ . | 19d2688d72.png | normalized |
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( q ^ { \hat N } ) ^ + = q ^ { - \hat N } \ \ ( q ^ + = q ^ { - 1 } ) . | 5006525233.png | tokenized |
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g _ { m n } = \partial _ m Z ^ { \underline M } \partial _ n Z ^ { \underline N } E ^ { \underline { a } } _ { \underline N } \eta _ { \underline { a } \underline { b } } E ^ { \underline { b } } _ { \underline M } , | 2c2d75b392.png | tokenized |
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s ^ { 1 1 } = - s ^ { 2 2 } = 1 ~ , ~ ~ ~ ~ s ^ { 1 2 } = s ^ { 2 1 } = 0 ~ , | 44d5f72638.png | tokenized |
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U _ { \mathcal { N \cap M } , \mathcal { M } } ( a ) : = \operatorname { e x p } ( \frac { i a } { 2 \pi } ( \operatorname { l n } \Delta _ { N \cap M } - \operatorname { l n } \Delta _ { M } ) ) | 2300a95d9e.png | normalized |
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T_{\mu \nu} = {1 \over 2} \left\{ \partial_\mu \phi(t,x),\partial_\nu \phi(t,x) \right\}, | 7c10915fb6.png | none |
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g_{f}(x)={g \over (1+g { x \pi \sqrt{3} \over 9})} | 1ac3ec2309.png | none |
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H^2+\frac{c^2k}{R^2}=\frac{8\pi G\rho}{3}+\frac{c^2\Lambda}{3}, | 3a7bbdeb4a.png | none |
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x_0=x_1x_2\ ;\qquad \zeta=(x_1/x_2)^{24}\ . \label{defx0zeta} | 669dc09728.png | none |
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\stackrel { \circ } { \Psi } ( y ) = ( y , \stackrel { \circ } { p } ) ^ { - E _ { o } - \frac 1 2 } \zeta ^ { + } | 46b6161fad.png | normalized |
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X ^ \eta + X ^ \xi + ( X ^ \chi + v X ^ a - a X ^ v ) _ { , \chi } = 0 . | b6cbd7781a.png | tokenized |
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F^{X\psi}=-\frac{1}{2}(\psi _{\mu}\partial X_{\mu}+iQ_{\mu}\partial \psi _{\mu})\label{11} | 35fd39dfa9.png | none |
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\Delta ^ i _ { m n } = \epsilon ( m ) \delta _ { m , n } + \frac { 2 \pi i \alpha ' } { n } H ^ i _ { - m + n } , | 6024874a8d.png | tokenized |
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( b _ 0 + \tilde { b } _ 0 ) \int _ { \pi } ^ { \infty } d s ' \, e ^ { - s ' ( L _ 0 + \tilde { L } _ 0 ) } | \mathcal { B } \rangle . | 75e6efbb4d.png | tokenized |
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ds^2 = g_{\mu \nu} \, dx^{\mu} \, dx^{\nu} \, . \label{eq:(1.1)} | 4e7be08592.png | none |
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m _ { 3 / 2 } \sim \frac { \Lambda ^ 2 } { M _ p } | 10b0539133.png | tokenized |
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\psi _ { \alpha } ( x ) = \int \frac { d p _ { \alpha } ^ { 1 } } { \sqrt { 4 \pi } p _ { \alpha } ^ { 0 } } | 7546ef671d.png | normalized |
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m_\mu'=m_\mu+{\bf w}_\mu\cdot{\bf k} -\frac{1}{2}{\bf w}_\mu\cdot{\bf w}_\nu n_\nu, | 7521ba723b.png | none |
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\varepsilon = Ts-p+\mu \rho \label{eq:inten}\ . | 18b625cb6f.png | none |
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a _ { i } a _ { j } = - \delta _ { i j } + \epsilon _ { i j k } a _ { k } . | 450509aaa1.png | normalized |
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d s ^ 2 = \sum _ { j = 1 } ^ n \frac { d \zeta _ j d \bar { \zeta } _ j } { ( 1 + | \zeta _ j | ^ 2 ) ^ 2 } , | 279849125b.png | tokenized |
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=2\left( \delta _{\mu \nu }\alpha _\alpha +\delta _{\alpha \nu}\alpha _\mu +\delta _{\mu \alpha }\alpha _\nu \right). \label{25} | 47ac2b29c8.png | none |
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\mathcal { L } ^ { R _ { 1 } , R _ { 2 } } [ \alpha , \beta ] ( t ) = c l \tilde { \mathcal { L } } ^ { R _ { 1 } , R _ { 2 } } [ \frac { \kappa } { \alpha } , \frac { \kappa } { \beta } ] \left( l \right) | 10e950bd18.png | normalized |
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\delta T = \Lambda r \qquad \delta r ={1\over 2}( \Lambda T^* + \Lambda^* T) | 16d705cac2.png | none |
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- \frac { \ddot { a } ( \tau _ 0 ) } { a _ 0 } \left [ \frac { 2 } { a _ 0 ^ { 2 } } + \frac { \kappa } { 2 } \right ] = 0 \, . | 56f757d1bb.png | tokenized |
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A'^{(1)}_{\rm cl} (y)= \tanh (y-y_1), \label{eq:BPSsol_cubic} | 1dfc6c0102.png | none |
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\frac { \partial ^ R { \cal S } ' } { \partial \xi ^ A } = S _ A ^ 0 + O ( k _ A + 1 ) | 1be4980602.png | tokenized |
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\tilde{q}_1+\tilde{q}_2+\cdots+\tilde{q}_{r+1}=0. \label{sumzero} | 1b3516fc31.png | none |
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\beta ^ { 2 } = c \lambda r L ^ { 2 } ( r ^ { 2 } - L ^ { 2 } ) ^ { - n } \left ( \int _ { r _ { b } } ^ { r } \frac { ( s ^ { 2 } - L ^ { 2 } ) ^ { n } } { s ^ { 2 } } d s \right ) . | 6f4f872b61.png | tokenized |
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e ^ { i Z ( \lambda _ { j } ) } = - ( - 1 ) ^ { \delta } . | 426c855270.png | tokenized |
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z ^ { \mu } _ { \tau } ( \tau , \vec \sigma ) = ( \sqrt { { g \over { \gamma } } } l ^ { \mu } + g _ { \tau { \check r } } \gamma ^ { { \check r } { \check s } } z ^ { \mu } _ { \check s } ) ( \tau , \vec \sigma ) , | 10b91acb0a.png | tokenized |
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A _ i ^ U = U ^ { \dagger } \left ( A _ i + \frac { 1 } { g } \, \partial _ i \right ) U . | 6c17c2f58a.png | tokenized |
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\label{2.17}\bigl\{T_a^{(1)},\,A^{(2)}\bigr\}_{(y)}=-\,G_a^{(1)}\,, | 4110e2cc54.png | none |
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p ( z ) = \frac { 1 - \mu _ { 1 } } { z } + \frac { 1 - \mu _ { 2 } } { z - 1 } + \frac { 1 - \mu _ { 3 } } { z - z _ { 3 } } - \frac { 1 } { z - z _ { A } } | eb74871637.png | normalized |
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Z = \sum _ { i = 1 } ^ { n } a _ { i } ^ { + } a _ { i } ^ { - } + b _ { i } ^ { + } b _ { i } ^ { - } | 2502dca0b3.png | normalized |
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\int_{0}^{1}dxh_{weak}^{4}(x)=\frac{2}{A^{2}}\label{4.21} | 356a45c8bb.png | none |
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{ \cal G } _ d = \cases { Z _ 2 , & i f D = 4 k + 2 \cr S O ( 2 ) , & i f D = 4 k \cr } | 2b0830ad87.png | tokenized |
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~ y _ { c } = \frac { \operatorname { l n } ( \Sigma ) } { \Delta _ { - } } , | 51a63f549f.png | normalized |
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S _ { V / A } = S _ { L R } \mp \frac { 1 } { 2 \pi } \int d ^ { 2 } z \; \frac { 2 } { 1 \pm B ( x ) } J _ { L } \bar { J } _ { R } . | 525b0bf89a.png | normalized |
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e^{\varphi_{SW}}= {e^{-\varphi/2}\over 2\pi|u^2-\Lambda^4|}.\label{comepuo} | f5b40ccac0.png | none |
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{ \frac { 1 } { 2 } } F _ { \mu \nu } D ^ { \mu \nu } = d ^ { \mu } F _ { \mu \nu } v ^ { \nu } + m ^ { \mu } { F * } _ { \mu \nu } v ^ { \nu } | 776b30627c.png | normalized |
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a_{ii} = 1. \label{4.12} | 707eb8e1d2.png | none |
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S_{\rm eff}^{\rm CS}=-i \frac{N_f}{2}\frac{1}{4\pi}\frac{m}{|m|}\int d^3x\epsilon^{\mu\nu\rho}A_\mu \partial_\nu A_\rho\label{indquad} | ff6acfb354.png | none |
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|j,-\theta \rangle _{out}=K_j(\theta )|j,\theta \rangle _{in} \label{kI} | 6825f8eab2.png | none |
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\epsilon ^{\omega \mu \nu \rho }D_{\mu }F_{\nu \rho }=D_{\mu }\;^{*}F^{\mu\omega }=\pm D_{\mu }F^{\mu \omega }=0. | 4329dcc6e3.png | none |
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\psi _ { \pm } \partial _ { \pm 2 } \psi _ { \pm } = 0 , | 476055177c.png | normalized |
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F _ { \mu \nu \rho } = \partial _ { \mu } B _ { \nu \rho } + \partial _ { \rho } B _ { \mu \nu } + \partial _ { \nu } B _ { \rho \mu } \equiv \partial _ { \left[ \mu \right. } B _ { \left. \nu \rho \right] } . | edb9dec39e.png | normalized |
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r _ + ^ 2 = ( 1 - a ^ 2 ) \frac { Q ^ 2 e ^ { 2 a \phi _ 0 } } { \beta ^ 2 } . | 68d95ef677.png | tokenized |
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\Omega_2(R_1)\Omega_1(0)=-\Omega_1(R_2)\Omega_2(0).\label{toron} | 7460285587.png | none |
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\hat { I } _ 4 = d Q _ 3 , \qquad \delta Q _ 3 = d Q _ 2 ^ 1 , | 746448c9cd.png | tokenized |
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\partial_+r(x_+,x_-)|_{x_-=0}=0~~.\label{2.4} | 5843e4b4f3.png | none |
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{ \cal { E } } \sim - \frac { e B } { 4 \pi } | m | + \frac { 1 } { 2 4 \pi } \frac { ( e B ) ^ 2 } { | m | } + \frac { B ^ 2 } { 2 } . | 2de96f3597.png | tokenized |
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d s ^ { 2 } = - \rho ^ { 2 } d \omega ^ { 2 } + d \rho ^ { 2 } + d y ^ { 2 } + d z ^ { 2 } , | 71f65af18b.png | normalized |
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\mu=\frac{\rho\wedge\bar{\rho}}{(det\,Im\,\Omega)^{13}} | 7e023628ff.png | none |
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(x)[-1/2g_{0}f^{\alpha \beta \gamma }c^{\beta }c^{\gamma }](x)-\xi ^{\alpha }(x) | 1fc307e4d3.png | none |
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h _ { \mu \nu } ( x , y ) = \sum _ { n = 0 } ^ { \infty } h _ { \mu \nu } ^ { ( n ) } ( x ) \Psi ^ { ( n ) } ( y ) | 768b4f68cc.png | tokenized |
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\psi ( x ) = \varphi ( x ) + \int d x ^ { ' } G [ x , x ^ { ' } ] V ( x ^ { \prime } ) \psi ( x ^ { ' } ) , | b0e0f92059.png | normalized |
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\label{f4.3} \langle q',t=0\mid q,0 \rangle = \delta (q'-q)+A\delta(q'+q), | 599afbe3c8.png | none |
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(T(R)u_\chi)(\zeta,\stackrel {o} {p};\lambda) = e^{i\lambda\Theta(R)}u_\chi(\zeta,\stackrel {o} {p};\lambda). | 619b6b4cbb.png | none |
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\lim _ { \nu \to 0 ^ \pm } F = \pm \frac { 1 } { 2 } \, . | 15f310385c.png | tokenized |
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\int d x \, e ^ { 2 n x } \exp \left ( - \frac { 1 } { 2 } \, e ^ { 2 x - z } \right ) = 2 ^ { n - 1 } \Gamma \left ( n \right ) \, e ^ { n z } ~ , | 351a38694b.png | tokenized |
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J^B_\mu=i(W\partial_\mu W^*-W^*\partial_\mu W)\label{J^B} | e07a20434c.png | none |
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p _ { 2 n } = e ^ { - \bar { n } } \frac { ( \bar { n } ) ^ { n } } { n ! } , \qquad \bar { n } = \frac 1 2 \mathrm { t r } \, \beta ^ { + } \beta . | 6e7093d63b.png | normalized |
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c = \frac { 3 v _ { s o u n d } } { \pi T L } \frac { \partial S } { \partial T } = \frac { 3 j _ { e f f } } { j _ { e f f } + 1 } \; ; \; \; \; \; j _ { e f f } = \frac { N - 1 } { 4 } | 49f7ef5e7f.png | normalized |
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H _ { i j } \tilde { n } ^ i _ R \tilde { n } ^ j _ S = \delta _ { R S } , \hspace * { 5 m m } H _ { i j } x ' ^ i \tilde { n } ^ j _ R = 0 , | 6614245403.png | tokenized |
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( \xi ^ \alpha ) ^ * = \left ( \lambda \epsilon ^ \alpha \right ) ^ * = \lambda ^ * ( \epsilon ^ \alpha ) ^ * = - \lambda \bar { \epsilon } ^ { \dot { \alpha } } \stackrel { ! } { = } \bar { \xi } ^ { \dot { \alpha } } , | 2bedad24ed.png | tokenized |
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\mu _ { \ell } = \left\{ \begin{array} { l l } { 1 } & { \mathrm { i f ~ \ell = 0 ~ } } \\ { 2 ( 2 \ell + 1 ) } & { \mathrm { i f ~ \ell \neq ~ 0 ~ } } \\ \end{array} \right. \nonumber | 72b7f5b9ac.png | normalized |
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M^{(\pm)}_2(0)=\sum_{\ell=1}^{6}c_{\ell}=0, | 203b072867.png | none |
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C = \frac { 9 \alpha } { Z _ { h o r } ^ 2 } ( \Pi _ { \phi } - \Pi _ { \psi } ) + q _ I X ^ I _ { h o r } . | 2380fede2e.png | tokenized |
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div({\bf j}_{\tt cond}+{\bf j}_{\tt disp})=0 | 3b71dff3f0.png | none |
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W ^ \prime _ { \rm e f f } = \frac { \det T \cdot f ( Z ) } { \Lambda ^ { 2 N _ f - 3 } } . | 7b771c25ba.png | tokenized |
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\partial _ { + } J _ { - } ^ { B } \, = \, - \, { \frac { 1 } { 2 \pi } } \, ( \, \partial _ { - } A _ { + } \, - \, \partial _ { + } A _ { - } ) | 3a6a97f76e.png | normalized |
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f ^ \prime ( x ) + f ^ \prime ( x + a ) + f ^ 2 ( x ) - f ^ 2 ( x + a ) = k . | 157c556761.png | tokenized |
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\hbox{Tr}(\phi_{\{l_1}\cdots\phi_{l_{p-4}\}}F_{\alpha\beta}F^{\alpha \beta}F_{\dot\alpha \dot\beta}F^{\dot\alpha\dot\beta})-\hbox{traces}\label{trac5} | 657f91d975.png | none |
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\lambda = - \frac { \mathbf { P } _ - . \mathbf { W } _ - } { \| \mathbf { P } _ - \| ^ 2 } = \frac { \mathbf { P } _ - . \mathbf { J } _ - } { \| \mathbf { P } _ - \| } , | 6dd99e5d80.png | tokenized |
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{ \rm C . F . } ( 2 ) = \ { \rm C . F . } ( 2 ) _ A + \ { \rm C . F . } ( 2 ) _ B | 13f9caf780.png | tokenized |
Subsets and Splits