BROKERS_20231227_0501_26页_563kb

math
\beta_{\mu} = g_{\mu\nu} \partial^{\mu} \phi + \partial_{\mu} \phi g^{\mu\nu}

```math
A^{\mu} A_{\mu} = A^{\mu} g_{\mu\nu} A^{\nu}

R^{\mu}_{\nu\rho\sigma} = \partial_{\lambda} \left( \Gamma^{\mu}_{\rho\sigma} + \Gamma^{\mu}_{\lambda\rho} - \Gamma^{\lambda}_{\rho\sigma} \right) \text{stuff}

\psi_{ABCD} \psi_{ABCD} \text{stuff}

\mathcal{N} \psi = \text{some equation with } \psi^+ \text{ and } \psi^-

\eta^{\mu\nu} = \text{Minkowski metric}

\epsilon^{0123} q_{[\mu} p_{\nu]} \text{stuff}

D_\mu = \partial_\mu + i e A_\mu

\partial_{\mu} \phi = 0 \text{ for boundary}

A^\mu = A_0^\mu \exp(i e \int^{\mu'}/e A_0^\nu dx_{\nu})

\delta g^{\mu\nu} = \text{various terms}

g_{\mu\nu} = \eta_{\mu\nu} + \delta g_{\mu\nu}

\Gamma^{\mu}_{\nu\rho} = \frac{1}{2} g^{\mu\sigma} (\partial_{\nu} g_{\sigma\rho} + \partial_{\rho} g_{\nu\sigma} - \partial_{\sigma} g_{\nu\rho})

\rho_{\text{em}} = -e \text{ stuff}

-K \frac{\partial \bar{L}}{\partial \partial_{\mu} \Phi} \partial^{\mu} \Phi + \text{variation}

\mathcal{F}_{\mu\nu} = \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}

\mathcal{D}_\mu \Phi = \partial_\mu \Phi + i e A_\mu \Phi

\bar{\psi} \gamma^\mu \partial_\mu \psi = \text{matter coupling}

S_{total} = S_{gravity} + S_{gauge} + S_{matter}

\epsilon^{ijkl} \bar{\psi}_{ij} \partial_k \partial_l \text{something}

\Delta_{\mu\nu} \text{tensor decomposition}

u^\mu = \left( \frac{1}{\sqrt{-g^{00}}}, \text{stuff} \right)

g_{\mu\nu} g^{\nu\lambda} = \delta_{\mu}^{\lambda}

\text{equation set involving } \Gamma, g, T

[Y_{\mu\nu}]^{ab} \approx \int d^4x \, e^{\alpha \phi_0} g^{\mu\nu} Y_{ab}

\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 g^{\mu\nu} \text{ if } \mu \neq \nu \text{ anti-commution}

\det(-\Box) \text{ zeta function}

\text{Glueball mass formula}

\text{for } j \text{ integer}, \omega = \frac{2\pi j}{T}

H_{\mu} = \delta_{\mu}^0 \phi'+ this + \partial_\mu \phi' + g_{\mu\nu} stuff

\Delta_\mu^\nu = \delta_\mu^\nu + \text{some tensor}

\text{improved energy-momentum tensor}

T^{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta S}{\delta \partial_\mu g_{\nu\rho}} \sqrt{-g}

\Delta = -L + \Delta^{\text{matter}}

\mathcal{D} \phi = 0

\gamma^{\mu} \gamma^{\nu} + \gamma^{\nu} \gamma^{\mu} = 2 g^{\mu\nu} \text{ when } \mu \neq \nu

\sigma^{\mu\nu} = i [ \gamma^\mu, \gamma^\nu ]

[ M^{\mu\nu}, M^{\rho\sigma} ] = \text{commutator}

g_{\mu\nu} = \eta_{\mu\nu} for Minkowski

R_{\mu\nu\rho}^{\sigma} = \partial_{\lambda} \Gamma_{\rho\sigma}^{\mu} + \text{connection terms}

W^{\alpha}{}_{\mu\nu\sigma} \text{ candidate for curvature}

\theta^{\mu} = \delta^{\mu}_{\kappa'} \text{stuff}

\epsilon^{0123} p_{[\mu} q_{\nu]} \phi \text{ class}

\sigma^{\mu\nu} = \begin{pmatrix} 0 & \sigma^{\mu\nu} \\ \sigma^{\mu\nu} & 0 \end{pmatrix}

\text{directed polymer saddle point equation}

\frac{\partial S}{\delta \phi} = 0

\rho = \int \frac{d^3p}{(2\pi)^3} n(p) f(p) \text{ or something}

\theta^{\mu}{}_{\nu} \theta_{\mu}{}_{\rho} g^{\nu\rho} \text{ tensor}

\bar{\psi} \gamma^\mu \psi \equiv \frac{\partial S_{matter}}{\partial \partial_\mu \phi}

\int \mathcal{L} \sqrt{-g} d^4x = \text{action}

\Gamma^{\mu}_{\nu\rho} = \frac{1}{2} g^{\mu\sigma} (\partial_\nu g_{\sigma\rho} + \partial_\rho g_{\nu\sigma} - \partial_\sigma g_{\nu\rho})

\Gamma_{\mu\nu\rho} = \text{ Christoffel symbol from metric}

G_{\mu\nu} = 8\pi G_N e^{\alpha \phi} (\eta_{\mu\nu} - \text{terms})

\text{effective field theory matter terms}

\Delta_{ABCD} \psi^{ABCD} \text{ glueball ansatz}

\mathbb{P}_{\mu\nu} = \frac{1}{2}(g_{\mu\nu} - \eta_{\mu\nu})

\phi \sim \int \psi \sqrt{g} d^2 x \text{ Chern-Simons}

\Delta S = S_0 + \int L_{\text{source}} d^4x

\Gamma^{\mu} = \partial_{\lambda} \mathbf{A}}^{\mu\lambda} + \text{some modification}

\text{Variation of action with metric}

\delta S = \frac{1}{2} \int \sqrt{-g} \delta g_{\mu\nu} ( -2 G^{\mu\nu} + \Lambda g^{\mu\nu} ) d^4x + \text{other terms}

\text{Sudbery's generalized Fierz identity}

\gamma^I \gamma^J + \gamma^J \gamma^I = 2\eta^{IJ} for orthogonal group

\Gamma^\mu = \{ \alpha_{\mu}, \beta_{\mu} \}

\epsilon^{...\mu\nu...} \text{ via Levi-Civita symbol}

\Delta^\mu = K^\mu + \nabla^\mu

\Gamma^{\mu}{}_{\nu\rho} = c^{\mu}{}_{\nu\rho} + \text{connection}

\text{gravitational and gauge anomalies}

\bar{\epsilon} \gamma^\mu D_\mu \epsilon \sim J

\psi_{ABCD} \text{ mass dimension for scalars}

\Delta_{\mu\nu} \sim \text{propagator}

\rho_{em} = -e^2 \int \text{some loop}

\Delta S = \int \frac{d^4p}{(2\pi)^4} \log(p^2 + m^2)