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Suppose I have the action (Latin indices run from 1 to 4)

$S = S_{QED}(\psi,\bar{\psi},A_\mu) + g \int \epsilon^{abcd} Du_a \wedge Du_b \wedge Du_c \wedge D u_d$

where

$D = d + ieA \wedge$ is the exterior covariant derivative with 1-form photon field $A$

$u_a$ is a 0-form excitation field that underlies also an $U(1)$ symmetry (therefore, exterior covariant derivative applies on it)

$\epsilon^{abcd}$ is 4-dimensional Levi-Civita permutation symbol

$g$ is a coupling constant and $e$ elementary charge

$S_{QED}$ is well-known ordinary quantum electrodynamics action dependent on fermion fields $\psi$.

The second term is a topological term that does not affect the form of the energy-momentum tensor (not dependent on metric tensor). But because the photon field is contained in this term, the equation for current conservation is modified by Noether's theorem. The $u_a$ excitation fields do not transfer energy or momentum when it is considered as a classical field. A classical solution would be $u_a = 0$ for all $a \in \{1,\dots,4\}$; hence I can assume that the exciton field $u_a$ has a pure quantum nature.

Question:

How can I compute the partition function (integration over all exciton fields); I have not Gaussian integrals in exciton fields in general? This additional term must yield a topological invariant such that I could treat it nonperturbatively (i.e. without Feynman diagram expansion). Any ideas?

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