# Gauging non-compact lie groups

## Main Question or Discussion Point

I know that gauging a lie-goup with a kinetic term of the form:

\Tr{F^{\mu \nu} F_{\mu \nu} }

Is not allowed for a non-compact lie group because it does not lead to a positive definite Hamiltonian. I was wondering if anyone knew of a general way to gauge non-compac lie groups. I know there must be a way since the Lorentz group can be gauged to give the Einstein Hilbert action.

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I have been reading a little about this and it seems that people can gauge non-compact lie groups with a kinetic term of the form:

Q_{\alpha \beta} F^{\alpha}_{\mu \nu} F^{\beta}_{\eta \rho} \epsilon^{\mu \nu \eta \rho}

with a specific choice of constant matrix Q. My question now is, 1) what mathematically motivates the choice above and 2) what conditions on the matrix Q is there to obtain a non-topological term *and* a positive definite hamiltonian?