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1f6cbf77e4 | %\label{dsr2}%\dim_{\Comp} \M({\rm regular},\rho_+) = \dim_{\Comp} {\S}(\rho_+) - N\ ,% | |
30348cea1a | \gamma_1 = -\gamma-1- \gamma(-2-a-b) - \delta (2-a/\delta-b/\delta-2a-2b) - 9/88(1+a+b) , | |
607ee9e5c2 | j^{mi} \equiv \frac{\delta I_{SI}}{\delta v_{m}^{i}}, \quad K^{mi} \equiv \frac{\delta I_{SI}}{\delta A_{m}^{i}} . | |
3362c78722 | [\hat{x}_{\mu}, \hat{x}_{\nu}]=\frac{i}{\kappa}
(a_{\mu}\hat{x}_{\nu}-\hat{x}_{\mu}a_{\nu}) | |
69dd0b398d | G^0_{\mu\nu} = \frac{\epsilon_{\mu\nu\lambda}p_\lambda+\delta_{\mu\nu}M+p_\mu p_\nu/M}{p^2+M^2}\;,~~D^0_{\mu\nu}(p)= \frac{\epsilon_{\mu\nu\lambda}p_\lambda}{p^2}\;, ~~{\rm and}~~\Gamma^0_{\mu\nu\lambda}=g\epsilon_{\mu\nu\lambda}\label{R} | |
4fa16f1d3f | \left\{ Q_{a},Q_{b}\right\} _{{\footnotesize PB}}=f_{abc}Q_{c}\;, | |
7a54e991b0 | Adj(Sp(2n_H)) \to Adj(U(k)) + Adj(U(n_H-k)) + 2(k,n_H-k)\label{storto} | |
50a1f82ae6 | C_0(k,q)=<k,q|\sum_{j=0}^{m-1}R^{-j}\omega^{js}|\phi>=\sum_{j=0}^{m-1}(R^j|k,q>)^+\omega^{js}|\phi>. | |
256f1e5b2a | G^{\dagger} = G, \;\;\;\;\;\; \tilde{G}^{\dagger} = \tilde{G} | |
1c7a4617a5 | I_n(x)=\int_0^xdx_n\int_0^{x_n}dx_{n-1}\cdots\int_0^{x_2}dx_1\cdot1=\frac{x^n}{n!}. | |
1de6b57b0c | N^{\mu\nu} = L^i n^{[\mu}{}_i X^{\nu]}{}' = L^* \eta^{[\mu}_1 X^{\nu]}{}'\,. | |
315201f078 | \label{al34-sol}X^{+} = {{* (2{\cal D} X^i \wedge f^{+i})} \over {* (f^{+i} \wedge f^{-i})}} = {1 \over {\cal R}} ({\cal D}_- X^i f_+^{~+i} - {\cal D}_+ X^if_-^{~+i}) | |
2d2e6105f4 | T(z)=L^{ab}:J_a(z) J_b(z): \label{introsimpleT} | |
4e721cfe49 | R(b)=\frac{4}{k-2}\;\frac{k(k-4)+k(k-2)b}{[k+2+(k-2)b]^2} | |
42c766c6dc | n!\prod_{i=1}^kd_i!{1\over (n-m)!}.\label{fac1} | |
799f625cf9 | [R,M]=-{\rm i\over2}\sum_{\rho\in\Delta_{+}}g_{|\rho|}\thinspace {|\rho|^2\cos(\rho\cdot q)\over{\sin^{2}(\rho\cdot q)}} \thinspace[{\rm e}^{2{\rm i}q\cdot\hat{H} }, \hat{s}_{\rho}]. \label{(5.9)} | |
323c465216 | \label{Eq_0407}lndet = \frac{|e\Phi|}{2\pi} \ln(ma) + R(m), | |
226979ee71 | B(0) = B_{\rm f}(0)+B_{\rm b}(0)=0,\label{BDF2abgt} | |
4d603c88f4 | H\, {\cal U} = i {\partial \over \partial t}\, {\cal U}\,, | |
16a0874240 | \Sigma_{\pm}(q_3,\left|{\bf q}\right|) = - \frac{1}{4\pi} \ln \left[ \frac{1}{2m} \left( -\frac{1}{2}\delta \pm iq_3 + \frac{\left|{\bf q}\right|^2}{2m} \right) \right] \, + \, O\left(\frac{1}{m^2}\right).\label{eq:SG1LP} | |
386bf9a37b | \frac{g^2_{YM}N}{J^3}\frac{J^4}{N^2} \frac{g^2N}{J^2} | |
7352f31651 | \label{re6a} {\cal L}_{n}(H_n)=(-T_{H_n}\bar{R}_{H_n})\phi ^n . | |
4ec40afb85 | \fbox{$\begin{array}{ll} (xy)(ax)=x(ay)x; & (xy)_{L}x_{R} = x_{L}x_{R}y_{L};\\ \\ (xa)(yx)=x(ya)x; & (yx)_{R}x_{L} = x_{L}x_{R}y_{R}; \\ \\(xax)y=x(a(xy)); & y_{R}x_{L}x_{R} = x_{L}(xy)_{R}; \\ \\y(xax)=((yx)a)x; & y_{L}x_{L}x_{R} = x_{R}(yx)_{L}. \\\end{array}$} | |
5d3a4f9d53 | {\delta_{\rho}}{\mathcal{F}_D^1}={\delta_{\rho}}{\mathcal{F}_D^0}-\alpha{Q}\oint{d}\hat{s}(\xi)\rho(\xi){\partial_{\hat{n}}}{\hat{G}_D}(\xi,\xi')+({\alpha^2}-\alpha{Q})\rho(\xi'). | |
252f1b9cea | D\sum_{I\neq J}(-1)^{p_{I}p_{J}}e_{JI}\otimes e_{IJ}D^{-1}=\sum_{I\neq J}e_{JI}\otimes e_{IJ}\label{A.7} | |
4c475e1afd | H(t) |n;t> = E_n(t) |n;t>,~~<m;t|n;t> = \delta_{mn} | |
131a79148f | \label{4eq12}E_c= \frac{4kl}{R}\left (k-\frac{r_+^2}{l^2} \right), | |
33fcdc8633 | %P\sim \exp\left(-y_0^3/M_p^2\ell\right),% | |
71feac6326 | \tilde{W}[\eta|t] = \int_{\eta_0}^{\eta(t)} \delta \eta^{' \nu}(t) \,\tilde{E}_\nu[\eta'|t],\label{Wtildeint} | |
1e8595e5e1 | \partial_z\partial_{\bar{z}}\theta=\{n(n+1)\eta^2+a\}\theta\,,\quad a\in{\cal R}\,,\quad n=0,1,2,\ldots\label{hie} | |
3bf4f3ec2d | C_{0}^{op}=g_{o}^{-2}\frac{1}{\left( 2\alpha^{^{\prime}}\right) ^{d/2}},\label{c0open}% | |
2581f66470 | \alpha^2 \partial_{x^-}^3 g_a(x)=0~; \alpha\neq 0 | |
29a6048204 | J_{gf}=\left( \begin{array}{cc}0 & J_{+}\\1 & 0\end{array}\right) \quad ;\quad \tilde{J}_{gf}=\left( \begin{array}{cc}0 & 1\\J_{+} & 0\end{array}\right) \quad ;\quad g=\left( \begin{array}{cc}-J_{+}g_{22} & -g_{21}\\g_{21} & g_{22}\end{array}\right) . | |
3b1cebad19 | Z^{int}(s)=\int {\cal D}A_\mu ~e^{-S_{bos}(A)-i\int d^3x~\varepsilon ^{\mu\nu \rho }A_\mu \partial _\nu s_\rho }~, | |
395413c90f | \hat{\Gamma}_0 \ldots \hat{\Gamma}_{(10)} = 1 | |
3b82cc7aee | {\cal L} = - V(T) \sqrt{- \det (g + F) } {\cal F}(z), \label{gauge} | |
4890fec461 | W_{\infty}^{3,D}={1\over 16}(1-\gamma)^{-1}\sum_{k=0}^{\infty}(-1)^{k}\beta^{k}(k+1)(k+3)(k+5)\sum_{m=0}^{\infty}(m^{2+k}-m^{1+k})\alpha_{m}^{-k-7}.\label{(3.20)} | |
7b00f012b2 | \label{0.1}\begin{array}{c}\left\{ M_1,M_2\right\} =aM_1M_2-M_1M_2a, \\a=\frac 12\left( r-r^{*}\right) .\end{array} | |
9bcc0bb538 | B(T)= -\; {b_2(V,T)\over [b_1(V,T)]^2}\;, | |
1be344a743 | {\cal H} = {\cal H}_a \otimes {\cal H}_A \otimes {\cal H}_c. | |
562c231867 | H r(\tau)=\frac{k}{\sqrt{1+k^2}}\mid\mbox{sn}[\frac{\tau}{\sqrt{1+k^2}},\;k]\mid. | |
78ef714591 | \delta _{\beta }\left( \alpha \right) =\alpha \wedge \beta =\alpha_{ab}\cdot \beta _{cd}\left( \chi ^{ab}\otimes \chi ^{cd}\right) Ad_{R} | |
173111e861 | \label{tens}T \sim \frac{m}{L^p} \sim \frac{1}{g \ell_s^{p+1}}. | |
3f0ef5c2d1 | (-1)^{|X||Z|}[X,[Y,Z]]+(-1)^{|X||Y|}[Y,[Z,X]]+(-1)^{|Y||Z|}[Z,[X,Y]=0,\label{jac} | |
3ab0d7b424 | {\cal M}^{uu}_{b}= \frac{ig^4}{2}[T^aT^b\otimes T_aT_b] \int \frac{d^2k}{(2\pi)^2} \left [ \frac{T(k,p_1) T^*(k,p_3)}{w_k-w_p}\right] | |
11a8eea207 | \int_C \frac{d\nu \: \nu}{i 4\sqrt{2} \sin \pi\nu} \left[ J_\nu(z_1) J_{-\nu}(z_2) + J_{-\nu}(z_1) J_\nu(z_2) \right] \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; | |
738c6bc090 | \label{v3b}\zeta_0(\nu)=-{\nu \varrho^{-2\nu} \over \pi}\int^{\infty}_{\mu} d\omega \int_{C_+}dz{2z^2 \over (z^2+\omega^2)^{\nu+1}}\breve{\Psi}(\omega;z)e^{i\epsilon z}~~~, | |
478234cbe5 | \Psi={C}\Psi^{\ast}\,. \label{Majoranacondition} | |
1a882641b7 | a_\alpha W_iZ^I_\alpha Y^{\alpha}_0 + WZ^I_\alpha Y^{\alpha}_I | |
81ba9ce4a9 | Q_A+(\gamma^{6}\gamma^{7}\gamma^{8}\gamma^{9})_A^B\tilde{Q}_B\nonumber | |
28a38729aa | %W_{(1,2\ell,1)}W_{(1,1,Z_2)}W_{(1,2\ell-1,1)}W_{(1,1,Z_2)}\label{eq:lzlz}%\sim \sum_{k=0}^{[\ell/2]-1}W_{(1,4k+2(\ell \text{ mod } 2)+1,Z_2)},% | |
1c2883bfac | V(z) = A^2 e^{-2\alpha z} - 2 A e^{-\alpha z} \ ,\ A,\alpha >0 \ . | |
3fdecf80a1 | \sigma_{\pm} = \frac{\tau\pm\sigma}{2}\ . | |
5f9a3eb396 | k^{u}(K_{uv})_{j}^{i}=\partial _{v}P_{j}^{i}+[p_{v},P]_{j}^{i},\label{eq:prepotn} | |
580036eb68 | \label{weak} \rho _0(\theta )=\frac{1}{\pi}\sqrt{\mu-\frac{1}{4}\mu ^2\theta ^2}, | |
6157449337 | C_{ABC}=\frac{1}{24 \pi} \big(A_{[A}^i\partial_BA_{C]}^i+i\frac{2}{3}f^{ijk}A_{[A}^iA_B^jA_{C]}^k \big) . \label{ccon} | |
505cc30969 | \label{E14}=g^2\left(\begin{array}{c|c}\delta_{\mu\nu}\delta_{AB}\delta^4(x-y) & 0 \\0 & \delta_{AB}\delta^4(x-y)\end{array}\right) | |
a43fa630b7 | \Psi,~~~\Phi,~~~\Pi,~~~\Xi,~~~{\cal A}_{w},~~~{\cal A}_{\theta},~~~{\cal A},~~~\Delta,~~~ \Delta^*. | |
36fbd5706b | C^{N_{P+1}}_{N_{P+1}+\dots +N_{P+K}}C^{N_{P+2}}_{N_{P+2}+\dots+N_{P+K}}\dots C^{N_{P+K-1}}_{N_{P+K-1}+N_{P+K}} | |
70a6e5b4b9 | (N_{c}+\tilde{N}_{c}, \overline{N_{c}+\tilde{N}_{c}}) (1, 1, 0, \beta_{i}), \ \ \ i=1,2,3 | |
1420ac42ed | {\cal D}=\sum_{i,j} c_{i,j}\, \frac{(g^2 N)^i}{N^{2j}}, | |
1e1bbaef2b | \label{3eq3}{\cal R}= {\cal R}_r \sin \eta, ~~~\tau= {\cal R}_r (1- \cos \eta). | |
347196fad5 | \left\{\begin{array}{rcl}\displaystyle \partial^{m}h_{m} - \frac{1}{2}\dot{h}' & = & 0 ,\vspace{2mm} \\\displaystyle \partial^{n}h_{mn} - \frac{1}{2}\partial_{m}h' & = & 0 ,\vspace{2mm} \\\displaystyle\partial^{m}\theta_{m} - \frac{1}{2}h' & = & 0 .\end{array}\right. | |
689d84bd57 | \label{gfcon}Z ({j _k}) = e ^{i Z_c ({j_k})} | |
7199fd1e3c | |\{mss\}|=\sum_{\nu=0}^k {k \choose \nu}=(1+1)^k=2^k | |
4f6c0a6083 | E_{4s}=2(Q+Q^{-1})+\xi _{1}+\xi _{1}^{-1}+\xi _{2}+\xi _{2}^{-1}\label{cba41} | |
5c01b8b718 | \label{} \stackrel{{\rm G}}{{\mathcal L}}\,:=\frac{1}{2m}\left[(D_\alpha\overline{\Psi} )D^\alpha\Psi-m^2\overline{\Psi}\Psi \right]\,. | |
7893ca4e1b | \label{pqdef}p(x) = \prod_{i=1}^{N+1} (x+\nu_{i})\, ,\quad q(x) = p(x) - e^{i\theta}/r^{N+1} = \prod_{i=1}^{N+1}(x-x_{i})\, , | |
54682978af | \tau_{cl}={\theta_0 \over 2\pi} + {4 \pi i \over g_0^2}, \label{struc} | |
3d6cbb761f | t= (H^{-2} - r^2)^{1/2} \sinh(H\tau); \quad v= (H^{-2} - r^2)^{1/2} \cosh(H\tau); \quad x=r\label{eq:3d_sta_tr} | |
39d44702c3 | \begin{array}{lll}\langle q_{i0} - Q_{i0} \rangle & = & 0 \, , \\&& \\\langle \dot{q}_{i0} \rangle & = & 0 \, , \\&& \\m\omega_{i}\omega_{j} \langle (q_{i0}-Q_{i0})(q_{j0}-Q_{j0}) \rangle & = & k_{B}T \delta_{ij} \, , \\&& \\m\langle \dot{q}_{i0}\dot{q}_{j0} \rangle & = & k_{B}T\delta_{ij} \, , \\&& \\\langle \dot{q}_{... | |
47b0771bd8 | i\frac{\partial}{\partial\tau_{2}}<m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}> =\hat{H}(m',\bar{m'})<m',\bar{m'};\tau_{2}|m,\bar{m};\tau_{1}>. \label{eq:dyn.eq.} | |
396bcb28c0 | \label{fvalue}f=\frac{1}{2}\partial_{\sigma}Z. | |
629c42b571 | {\cal L}_{eff}= {\cal L}^{(2)}+ {\cal L}^{(4)}+ {\cal L}^{(6)} + ... | |
3bd3e38c6c | {\bf E}_\pm(\xi_\pm)={\rm T}_\pm(\xi_\pm){\bf E}_0{\rm T}^{\dagger}_\pm(\xi_\pm), | |
149b313ecf | \psi'(t,x)=e^{-ie\epsilon(t,x)}\psi(t,x)\ \ ,\ \ A'_\mu(t,x)=A_\mu(t,x)+\partial_\mu\epsilon(t,x)\ , | |
e94fbdb3c3 | \delta(f(x))=\sum_{i=1}^n\frac{1}{|f^{\prime}(x_i)|}\delta(x-x_i) | |
234d3db18a | V_A=T^a\int_0^T d\tau\,\bigl[\dot x_{\mu}\varepsilon_{\mu}-2{\rm i}\psi_{\mu}\psi_{\nu}k_{\mu}\varepsilon_{\nu}\bigr]{\rm exp}[ikx({\tau})]\label{vertex} | |
742d89ddeb | \Omega : u \rightarrow\ {S_{\Omega}} \left(u \right)={{\Omega}^{-1}}u. | |
6e8fa596b4 | t_a(M) - t_a^*(M) = t_a(M)(R-R^*)t_a^*(M) | |
49ca0559c8 | -{\cal L}_{\xi} J^{\mu\nu} = \hat \xi^{\lambda} \partial_{\lambda} J^{\mu\nu}-\partial_\lambda \hat \xi^\mu J^{\lambda\nu} - \partial_\lambda \hat \xi^\nu J^{\mu\lambda}\label{grav} | |
534c7843ec | \widehat{F}_{\mu \nu} =F_{\mu \nu}+\frac{1}{4}\theta ^{\rho \sigma}\bigg( 2\left\{F_{\mu \rho},F_{\nu \sigma}\right\} -\left\{ A_{\rho},D_{\sigma}F_{\mu \nu}+\partial_{\sigma}F_{\mu \nu}\right\}\bigg) +{\cal O}\left( \theta ^{2}\right).\label{swf} | |
19c9f9219d | |q(x,\xi)|=|\chi(x,\xi)a(x,\xi)^{-1}|\geq C_K(1+|\xi|)^{-m}, | |
6f49be4f68 | \label{metric}ds_M^2 =g^{4/3}dx_{11}^2+g^{-2/3}ds_{10}^2 | |
71aea8e1bc | D(T)=\left(\begin{array}{cc}a(T)&0\\0&a(T)^{-1}\end{array}\right) \\ |a(T)|>1 | |
30a0e5b27c | S_{matter}=16\pi\int d^4x \sqrt{-g}\;e^{2(\sigma-1)\psi}\;L_{matter}. | |
3b4dd4f61d | \label{def_H}H^{(N)}(L,r,l|d,p)=\sum_{k=l+p}^N q^{-\sum_{i=d}^{p-1}m_i-e_{d-1}C^{-1}m} B^{(N)}(L,r,l+1|k). | |
7f22efe073 | \label{sigalfa}\sigma=4\pi g^2M~~ {\rm and}~~\alpha^{-1}=-\frac{\pi g^2}{2M}, | |
277fac3048 | d_{\rm el}^{-2} = \rho_{\rm pol}/A_0 \sim g^2Ta^{2-d} \, . | |
f1c1b75b1b | V(\phi) \ = \ <H_{\mu\nu}^2> \cdot (1 - [\rho^2 + \sigma^2])^2\ \sim \ \lambda ( 1 - |\phi |^2 )^2 | |
5b7bdc01b4 | \int_{\Sigma}(i(Dg)\varphi-(Ag+ig)\Lambda)\Psi_0=\int_{\Sigma}g(iD\varphi+C\Lambda- i\Lambda)\Psi_0 | |
42b805b1d8 | V_\mu^s(x,p) = {\pi}^{-4} \partial_\nu^{(x)} \int d^4q e^{2ip.q} {\bar \psi}(x+q){{\sigma_{\mu \nu}} \over {2m}} \psi(x-q). | |
3ca343759c | -\left( 2s+1\right) -\frac{\left( 2s+1\right) (e\tau )^2\mathcal{F}}3\left[s(s+1)\left( g^2-\sigma ^2\right) -1\right] \biggl \}, | |
4c5c28ef84 | \int_{0}^{t}dt^{\prime}\int d^{2}\xi^{\prime}K^{t-t^{\prime}}_{\xi\xi^{\prime}}e^{-\phi(\xi)}\left(-\frac{1}{2}(\xi^{\prime}-\xi)^{a}(\xi^{\prime}-\xi)^{b}\left(\partial_{a}\phi(\xi)\partial_{b}\phi(\xi)-\partial_{a}\partial_{b}\phi(\xi)\right)\right)\Delta^{\prime}K^{t^{\prime}}_{\xi^{\prime}\xi} | |
1e216001c4 | m\epsilon_{ln}F_{n0} - j_l = 0,\label{eq:ampere1} | |
5f71c12252 | iD_kM=[C_k,M]=B_{kl}[x^l,M]+[\hat{A}_l,M] \ . | |
4c32f7004f | {\tilde{W}}\equiv W-S_0 ~,\quad {\tilde{X}} \equiv X+S_0 ~.\label{tilde} | |
4b836d0a2c | \Delta_{\mu \nu} (p) = \frac{4\pi}{\kappa} \frac{\mu}{ p^2 (p^2+ \mu^2)} ( \mu \epsilon_{\mu \nu \lambda} p^\lambda + \delta_{\mu \nu} p^2 - p_\mu p_\nu)\label{ru5} | |
e6c3904a0c | {\mbox{Tr}}(\gamma_g)=-2~,~~~ {\mbox{Tr}}(\gamma_{R_s})={\mbox{Tr}}(\gamma_{gR_s})=0~. | |
4124513a65 | G_\alpha ^{(n)}\equiv \sum_{m=0}^n \{\tau_\alpha ^{(n-m)}, \tilde{H}^{(m)}\}+ \sum_{m=0}^{n-2} \{ \tau_\alpha ^{(n-m)}, \tilde{H}^{(m+2)}\}_{(\eta)}+\{\tau_\alpha ^{(n+1)}, \tilde{H}^{(1)}\}_{(\eta)}; \hspace {0.5 cm} n\geq2. \label{a21} |
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