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\left[ U_{n}^{-},U_{n}^{+}\right] =\left[ V_{n}^{-},V_{n}^{+}\right]=\left[ X_{n}^{-},X_{n}^{+}\right] =\left[ Y_{n}^{-},Y_{n}^{+}\right] =1 | expression | |
\label{4.16}\xi^\mu \Gamma_\mu=\frac 14 \tilde{\gamma}^t \tilde{\gamma}^i B_{,i}=\tilde{\Gamma}_t~~~, | expression | |
\tilde{a}_A = \frac{d-2}{24} + \frac{D}{48}, | expression | |
\label{Eq:Ek-q}\frac{1}{2\mu}P^{2}=\frac{1}{2\mu}\tilde p^{2}+\tilde H^{(q)}_I(\tilde x,\tilde p), | expression | |
\label{10}\left.\frac{\partial}{\partial r}\left(\frac{\Delta}{r^2}\right)\right|_{r=R}=0. | expression | |
\mu_{p} = N \sqrt{16 \pi G_{D}} \, \tau_{p}. | expression | |
{\mathcal G}_{l+\nu} (r'',r';E) =\sum_{n=0}^{\infty}{\mathcal G}^{(n)}_{l+\nu} (r'',r';E) \; ,\label{eq:reduced_GreenF_perturb_expansion_radial} | expression | |
\Delta [ {\bf z} ]= \left\{ \prod_t \det \left[{1 \over m} (1 - \dot{\bf z}^2)^{1/2}( \delta^{hk}-{\dot{z}}^h {\dot{z}}^k ) \right]\right\}^{-\frac{1}{2}} \> . | expression | |
T_2 = c_{++} J^+_n J^+_n + c_{+0} J^+_n J^0_n + c_{00} J^0_n J^0_n + c_{0-} J^0_n J^-_n + c_{--} J^-_n J^-_n + | expression | |
\label{dconifold}\sum_{i=1}^4 z_i^2 =%-2\det_{i,j} z_{ij} =\varepsilon^2\ , | expression | |
U(\vec x) = {\mathrm{e}}^{\mathrm{i} f_2(r) \hat\cdot\vec\tau}\label{eq:dib} | expression | |
H_{Susy} = p_{tA} p_{tA} + \frac{1}{2}\ f_{ABC} f_{AB'C'} q_{sB} q_{tC} q_{sB'} q_{tC'} + iq_{tC} f_{ABC} \gamma_{\alpha\beta}^t \Theta_{\alpha A} \Theta_{\beta B}\ , | expression | |
\label{C17}Z[K_{\alpha,\epsilon}]=e^{-W_1[K_{\alpha,\epsilon}]}=e^{-W_1^{BW}(\alpha,\epsilon)}~~~. | expression | |
u_{l}(r) \stackrel{(r \rightarrow 0 )}{\sim} C^{(+)}_{l}\,\sqrt{r}\,r^{ s_{l}}\; .\label{eq:ISP_BC_u} | expression | |
\label{eq:g2 spectrum tw1}\begin{array}{ll}1) & h=3/8, \quad a=3/80 , \\2) & h=7/8, \quad a=7/16 , \\3) & h=3/8+x, \quad a=3/80 . \end{array} | expression | |
S^{int}_{\phi\sigma\sigma} = \int d^3 x \; \lambda_\phi \phi(x) \sigma^2(x) | expression | |
Z={\sin\left(\left( J+1/2\right) hT\right)\over\sin\left( hT/2\right)}\ ,\label{genmitsukekka} | expression | |
\label{gquant}{2 \over \alpha^\prime} g_{ab} \in Z\ . | expression | |
\partial \bar\partial X(z,\bar z) = 0. \label{xeom} | expression | |
\delta \beta_{0z}=\partial_0\lambda_z -\partial_z\lambda_0. | expression | |
\psi =\left( \begin{array}{c}ik \\ E-m\end{array}\right) e^{ikx} | expression | |
\Gamma_S^{Q'Q}(-i\Omega_m, \stackrel{\rightharpoonup}{p})=\frac{g_{Q'Q}}{2\Delta(-i\Omega_m, \stackrel{\rightharpoonup}{p})}, | expression | |
{\cal U}(r)=\sqrt{m^2+\frac{J^2}{r^2}}+ U(r). \label{effective} | expression | |
\label{a3}[\hat{S}_+,\hat{S}_-]=\displaystyle\frac{\partial G(S_3)}{\partial S_3}\vert_{S_3\to \hat{S}_3}\stackrel{def.}{=}[[\hat{S_3}]]~,~~~~[\hat{S}_3,\hat{S}_{\pm}]=\pm \hat{S}_{\pm}~, | expression | |
\label{pap.laro}\frac{8\pi}{3}\rho_{eff}=(\frac{2}{Q})^{\frac{1}{2}}\Big{(}(2+\frac{E}{\alpha^{4}})^{2} -1 \Big{)} (1-\alpha^{4}) ^{\frac{5}{2}} | expression | |
\label{e12a}\left( \hat{{\cal P}}_{\nu} \gamma^{\nu} - m \right)S^c(x,x')=-\delta(x-x'). | expression | |
ds^2 = B(y)\left\{ \left[ \eta_{\mu\nu} + {h_{\mu\nu}\over B(y)}\right]dx^\mu dx^\nu +dy^2\right\}\,.\label{eq=metrich} | expression | |
f=f(\varepsilon ^{mnpq} S^{\alpha}_{\ mn}S^{\alpha}_{\ pq}) \label {11} | expression | |
U=U_0(\mbox{\boldmath$r$})= exp\{ {i F(r) \frac{\mbox{\boldmath$\tau$}\cdot \mbox{\boldmath$r$}}{r}}\} \label{eq:ansatz} | expression | |
\label{8b}\gamma_{\bar\psi\psi} = 2 \gamma - 2 Res, | expression | |
{\cal L}_{\sigma}=\frac{\partial_{\mu}W\partial^{\mu}W^{*}}{(1+|W|^{2})^{2}}\label{ldw} | expression | |
d\dot{x}^{\mu}=-\frac{q}{mc}F^{\mu}_{\,\,\,\nu}d{x}^{\nu} | expression | |
\label{Cubicaction} S=\frac{1}{3}\int \Phi\star\Phi\star \Phi \ , | expression | |
\label{1.80} (D_{1}+i\sigma_{1}D_{2})\phi=0 \;\; and \;\;(D_{1}+i\sigma_{2}D_{2})\chi=0 | expression | |
S=\Lambda^3 f(X)=\Lambda^3 {\widetilde f} ({\widetilde X})\simeq \Lambda^3{\widetilde X}~. | expression | |
[e_q(ix^\dagger(1-q)z)]^*=E_q(-ix(1-q)z),\label{estar=E} | expression | |
X (x^\mu, t) = P \exp \Big\{ - \int_{x_{0}}^x (\alpha_+ A_+ dy^++ \alpha_- A_- dy^-)\Big\}, | expression | |
Z_\Psi=\int dZ \exp i\int_{\tau_1}^{\tau_2}d\tau [P_A\dot Q^A-H+\{\Psi,\Omega\}] | expression | |
\label{(20)}\frac 12\Delta _{(ab)}V_{\quad c}^b\frac{\partial f^c\left(Q\right) }{\partial Q^d}=\overline{V}_{cd},\qquad for\;every\;real\;Q^a\neq0. | expression | |
\label {simple_case_intertw}T_{w}=T_{i_{1}}\circ T_{i_{2}}\circ\cdots\circ T_{i_{l} }. | expression | |
R=\frac{d(d-1)k}{a^2},%R= \frac{2k}{a^2},\label{2.7} | expression | |
{\widetilde \Lambda}\equiv -\gamma\lambda~, | expression | |
\Psi_j = \sqrt{| \lambda_j |} \, \psi_j~. | expression | |
u_1^{\dot{a}}Q^{\dot{a}+}u_2^a Q^{b-}\mid B\rangle=u_1^{\dot{a}}\gamma^I_{\dot{a}b}u_2^b\oint (\partial X^I+M^{IJ}\bar{\partial} X^J)\mid B\rangle | expression | |
\frac{2i\varepsilon \delta \theta (1-\theta )}{\theta ^{3}[\theta k_{0}^{2}-\frac{1}{\theta }|\mathbf{k}|^{2}+i\varepsilon ](k^{2}+i\varepsilon )} | expression | |
\label{a17}\frac{\partial F^{AB}(\alpha,\eta)}{\partial \alpha}=\check{W}F^{AB}(\alpha,\eta)\;. | expression | |
P=\pm\ {\lambda^2 -|k|^2\over \lambda^2 +|k|^2} \qquad Q =\pm\ {2\over \lambda^2 +|k|^2}\qquad R=\mp\ {2\lambda\over \lambda^2 +|k|^2}. | expression | |
|B, p=0\rangle=\int [dx] \exp\left({i\over 8\pi}F_{\mu\nu}{\int_{0}}^{2\pi} d\sigma ~x^{\mu}(\sigma)\cdot\partial_{\sigma}x^{\nu} \right) |x, p=-\bar{p}=0\rangle | expression | |
a*_gb(y) =\exp \left(\frac{i\hbar }2\Lambda ^{ij}\frac \partial {\partial y^i}\frac \partial{\partial z^j}\right) a(y)b(z)|_{z=y}, \label{wlp} | expression | |
Z(\vartheta )=\lim _{N\, \rightarrow \, \infty }Z_{N}(\vartheta )\: . | expression | |
\label{realdec}P_i^\pm=P_i\mp {\rm i} J_{vi}\,, | expression | |
A_F = P_{12} x_{12} \partial_1 , \ \ \ \ [A_F, H_F^{(0)} ] = 0. | expression | |
\vec{v}_\varepsilon = \frac{\vec{S}}{u} , | expression | |
{\bf\tau}^5=\chi_1d\eta +\chi_{-1}d\xi +(a+v\chi_{-1}e^{2\xi})dv- | expression | |
R^{ab}({\alpha})=(D^2)^{ab}+g\overleftarrow{D}_\mu^{ac}f^{cbd}\alpha^d_\mu. | expression | |
{-i\mu^2 \over 8 H_5 \pi V} (2 \pi)^3 \delta^3(\vec{k}+\vec{k}') \int dt |t| \int\int du du' e^{i(u-u'){k_y\over H_5}} e^{-i({\rm cosh} u+{\rm cosh} u') mt} e^{-\epsilon |t|},\label{eq:rateq} | expression | |
{\overline A}_{\mu}(x,\beta) = -\frac{i}{g} e^{i\theta_{a}T^{a}} \left[ {\overline\sigma}_{\mu\nu}\frac{\rho^{2}(x-x_{0})_{\mu}} {(x-x_{0})^{2}[(x-x_{0})^{2}+\rho^{2}]} \right] e^{-i\theta_{a}T^{a}} \, , \label{explicitsinginst} | expression | |
\left\{ S_\mu , p_\nu \right\}^{\ast} \,=\, \epsilon_{\mu \nu \gamma}p^\gamma\quad. | expression | |
y^2 = (x^2 - \Lambda^4) (x-u), \qquad u= \langle \hbox{\rm Tr} \Phi^2 \rangle , | expression | |
{{cdt}\over{dx}}={{\hat{c}d\hat{t}}\over{d\hat{x}}},\ \ \ \ \ \ \ {\hbar\over{dEdt}}={\hat{\hbar}\over{d\hat{E}d\hat{t}}},\ \ \ \ \ \ \ {{G_4dE}\over{c^4dx}}={{\hat{G}_4d\hat{E}}\over{\hat{c}^4d\hat{x}}},\ \ \ \ \ \ \ {e^2\over{dEdx}}={\hat{e}^2\over{d\hat{E}d\hat{x}}},\label{dmlssrts} | expression | |
\label{27}{\theta}^2B-\frac{dU}{d{\sigma}^2}=0, | expression | |
{ d^2 \tilde \psi_\infty \over d {\rho^*}^2 } - \left [ {3 \over 4} - \mu \right ] { \rho^2 \over R^4} \tilde \psi_\infty =0 .\label{eq-boundary} | expression | |
\{A,B\} = {\partial A\over\partial y} {\partial B\over\partial z} -{\partial A\over\partial z}{\partial B\over\partial y}\,. \label{eq:poisson} | expression | |
\{Q_{a}\,,\,Q_{b}\}=-2\,i\,P^{\mu }(\gamma _{\mu })_{ab}\,,\label{algscalgau} | expression | |
D_a\Phi = \partial_a \Phi + i[A_a,\Phi] \, . | expression | |
\overline{\nabla }_{\alpha }(\overline{\omega })=\overline{(\nabla _{\alpha}\omega ).} | expression | |
\delta T^{*bc}=\partial_a T^{*[ab]c}.\label{anti2} | expression | |
\hat Z_a(\beta )=\exp \bigg( a{\beta\over 2\xi}\bigg)Z_a(\beta )\ ,\label{sgzfredef} | expression | |
f = C\eta(z_i) - 3h(z_i)^2,~~g = h(z_i) [C \eta(z_i) - 2h(z_i)^2]\label{fgspec} | expression | |
\sigma^*=s_og^4+g^4x^2{s_1\over \epsilon}+g^6s_2+...\label{sigma*} | expression | |
A'=-a\beta \tanh(ay)\left(2+\frac{1}{\cosh^2(ay)}\right)\,\,\,\,,\,\,\,\,\,\,A^{''}=-\frac{3a^2\beta}{\cosh^4(ay)} | expression | |
-{1\over 2}\Bigl[ (\eta^{\mu_1\mu_2} \eta^{\nu_1\nu_2} - \eta^{\mu_1\nu_2} \eta^{\mu_2\nu_1}) (\eta^{\mu_3\mu_4} \eta^{\nu_3\nu_4} - \eta^{\mu_3\nu_4} \eta^{\mu_4\nu_3}) + (2\leftrightarrow 3), (2\leftrightarrow 4) \Bigr] \nonumber | expression | |
[W] = \sum_{n=1}^{N} [J^{(n)}] \label{eq:11} | expression | |
\overline{g}=\sqrt{g^{2}+g^{\prime 2}} = \frac{1}{2\sqrt{\omega}}\sqrt{\tilde{g}^{2}+\tilde{g}^{\prime 2}} \label{VII34} | expression | |
Z_{\hat \Lambda}( - \frac 1 \tau) = Z_{\hat \Lambda} (\tau) . | expression | |
T_{a_1 \ldots a_r}^{b_1 \ldots b_s}T_{c_1 \ldots c_m}^{a_j} = T_{a_1 \ldots a_{j-1} c_1 \ldots c_m a_{j+1} \ldots a_r}^{b_1 \ldots b_s} | expression | |
\begin{array}{lll}SO(10) &\begin{array}{l}\\\end{array}SO(9) & SO(8) \\\phi _{(XY)} & \left\{\begin{array}{l}\phi _{(IJ)}+ \\\phi _{I\,or\,J}+\phi\end{array}\right. & \left\{\begin{array}{l}(\phi _{(ij)}+\phi _{i\,or\,j}+\phi ^{\prime }) \\+(\phi _{i\,or\,j}+\phi ^{\prime \prime })+\phi\end{array}\right. \\=54 & =44+9+... | expression | |
[\Phi_{m}, \Phi_{n}] = (m - n) \Phi_{m + n}~, | expression | |
\chi(p^2=0,\rho)={M_P^{2-D}\over 2\pi(D-2)} T(p)\ln\left({\rho^2\over b^2} \right)~, | expression | |
\sup_{x_{1},x_{2},\nu}\left|\left(G_{I}(\nu)\right)_{x_{1},x_{2}}\right|< c_{2}.\label{S1242} | expression | |
\label{AA50}g^{\alpha\beta} = \delta^{\alpha\beta} - \frac {q^\alpha q^\beta}{R^2}. | expression | |
\frac{1}{2}\widehat{\chi }_{p}\left( \tau \right) \left( \widehat{\chi }_{p}\left( \frac{\tau }{2}\right) -\widehat{\chi }_{p}\left( \frac{\tau +1}{2}\right) \right) | expression | |
\label{eq:r0ed2}r_{min} ={\textstyle \frac{1}{2}}\biggl \{ \pm \frac{a}{\sqrt{1 -a^{2}}}\Sigma+\sqrt{(M \mp\frac{a \Sigma}{\sqrt{1-a^{2}}})^{2}-\Sigma^{2}} \biggr \} \, . | expression | |
Q^{\pm}_{1}=\int {dz\over 2\pi i} q_{1}^{\pm}(z) | expression | |
\frac{\delta^n\langle\phi_2|\phi_1\rangle_J}{\delta J(x_1)\ldots \delta J(x_n)}= i^n \langle \phi_2|T(\phi(x_1)\ldots\phi(x_n)) |\phi_1 \rangle\label{eq:10}, | expression | |
{\widehat M}_P\sim \Lambda\sim {r\over\alpha^\prime}~. | expression | |
t({\cal QH}_l({\bf CP}^n))=t(\rho_l(T{\bf CP}^n))\oplus t({\tau^l}).\label{decc} | expression | |
\label{n7132}F_{\chi_0}(z)\equiv z^{-\Delta_0}e^{d^\dagger\bar D(-z)e^\dagger}\chi(\chi_0)\,. | expression | |
H = dB + \frac{\alpha'}{4\pi}(\omega_Y-\omega_T), | expression | |
\widehat{\phi *\phi}=\widehat{\phi} \quad \longrightarrow \quad \widehat{\phi}^2=\widehat{\phi} \label{eq:OE} | expression | |
\begin{array}{ll}\langle \alpha \mid & =\langle \alpha \mid h\rangle \langle h\mid \\ \mid \beta \rangle & =\mid h\rangle \langle h\mid \beta \rangle\end{array} | expression | |
\label{SUSY}\left(D_0 y^i\gamma^i + {\mu\over 6} \sum_{i=1}^3 y^i\gamma^i\gamma_{123} +{i\over 2}[y^i,y^j]\gamma_{ij}\right)\epsilon(t)=0 | expression | |
C(x+\Delta )-e^{-ih}D^{-1} C(x-\Delta ) D=2\pi i \delta(x)Y,\label{e} | expression | |
\left( {m \over a}\right)^2 > -\biggl({d-1 \over 2}\biggr)^2, | expression | |
p_\mu=e^{-1}(\dot{x}_\mu+i\lambda\xi_\mu),\label{p2} | expression | |
=- (2 \pi \sqrt{\alpha'})^2\,\left(\frac{1}{2} + \frac{1}{(2 \pi\sqrt{\alpha'})^2} \tilde{b} \right) + (2 \pi \sqrt{\alpha'})^2 = (2\pi \sqrt{\alpha'})^2 \,\left(\frac{1}{2} - \frac{1}{(2 \pi\sqrt{\alpha'})^2} \tilde{b} \right)\label{for968} | expression | |
2r \frac{\partial_0 \beta}{\beta} = 0\label{a3} | expression | |
-ng(\theta)-(1+\cos\theta)\dot{g}(\theta)+\tilde{\lambda} f(\theta) -\frac{1}{2}g(\theta)=0 %(98) | expression | |
\label{PiTad}\begin{array}{lll}\Pi_{tad}^{\mu\nu ,\,\alpha\beta}\left(k\right)\equiv\displaystyle{\frac {1}{4}}&\left(\eta^{\mu\alpha}\tilde\Gamma^{\nu\beta}+ \eta^{\mu\beta} \tilde\Gamma^{\nu\alpha}+ \eta^{\nu\alpha}\tilde\Gamma^{\mu\beta}+ \eta^{\nu\beta}\tilde\Gamma^{\mu\alpha}\right)\\&-\eta^{\mu\nu}\Delta^{\alpha\... | expression | |
\label{m2} {k/2} \int_M \left(A^3_\varphi - \bar{A}^3_\varphi\right)\delta^2(\underline{x})d\varphi d^2x, | expression |
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