# Gauge symmetries of a theory

1. Mar 13, 2010

### ubugnu

1. The problem statement, all variables and given/known data

I want to derive Gauge symmetries of the following gauge theory:

$$S=\int\;dt L=\int d^4 x \;\epsilon^{\mu\nu\rho\sigma} B_{\mu\nu\;IJ} F_{\mu\nu}^{\;\;IJ}$$

Where $$B$$ is an antisymmetric tensor of rank two and $$F$$ is the curvature of a connection $$A$$ i.e: $$F=dA+A\wedge A$$, $$\mu,\nu...$$ are space-time indices and $$I,J...$$ are Lie Algebra indices (internal indices) I would like to find its symmetries.

2. Relevant equations

I rewrite the Lagrangian by splitting time and space indices $$\{\mu,\nu...=0..3\}\equiv \{0; i,j,...=1..3\}$$ I find:

$$L = \int d^3 x\;(P^i_{\;IJ}\dot{A}_i+B_i^{\,IJ}\Pi^i_{\,IJ}+A_0^{\;IJ}\Pi_{IJ})$$

Where $$\dot{A}_i = \partial_0 A_i$$, $$P^i_{\;IJ} = 2\epsilon^{ijk}B_{jk\,IJ}$$ is hence the conjugate momentum of $$A_i^{\,IJ}$$

$$B_i^{\,IJ}$$ and $$A_0^{\;IJ}$$ being Lagrange multipliers we obtain respectively two primary and two secondary constraints:

$$\Phi_{IJ} = P^0_{\;IJ} \approx0$$

$$\Phi_{\;\;IJ}^{\mu\nu} = P^{\mu\nu}_{\;\;IJ} \approx0$$

$$\Pi^i_{\,IJ} = 2\epsilon^{ijk}F_{jk\,IJ} \approx0$$

$$\Pi_{IJ}=(D_i P^i)_{IJ} \approx0$$

Where $$P^0_{\;IJ}$$ are the conjugate momentums of $$A_0^{\,IJ}$$ and $$P^{\mu\nu}_{\;\;IJ}$$ those of $$B_{\mu\nu}^{\;\;IJ}$$. Making these constraints constant in time produces no further constraints.

Whiche gives us a general constraint:

$$\Phi = \int d^3 x \;(\epsilon^{IJ}P^0_{\,IJ}+\epsilon_{\mu\nu}^{IJ}\;P^{\mu\nu}_{\;\;IJ}+\eta^{IJ}\Pi_{IJ}+\eta_i^{IJ}\Pi^i_{\;IJ})$$

3. The attempt at a solution

Each quantity $$F$$ have thus a Gauge transformation $$\delta F = \{F,\Phi\}$$ where $$\{...\}$$ denotes the Poisson bracket.

Knowing that this theory have the following Gauge symmetry:

$$\delta A = D\omega$$

$$\delta B = [B,\omega]$$

Where $$\omega$$ is a 0-form, I would like to retrieve these transformations using the relation below. (where $$\Phi$$ is considered as the generator of the Gauge symmetry) but my problem is that I don't know how to proceed, I already did this with a Yang-Mills theory and it worked... but for this theory it seems to le intractable! Someone to guide me?