Derivative Expansion and the Effective Action for the Abelian Chern-Simons Theory at Higher Orders

TL;DR

This paper systematically analyzes higher order derivative corrections to the parity-violating effective action in Abelian Chern-Simons theory in 2+1 dimensions, revealing summation structures and gauge invariance properties.

Contribution

It introduces a method to sum higher order derivative and field corrections in the effective action, providing explicit calculations and gauge invariance proofs.

Findings

Higher order derivatives can be summed to all orders in principle.

Effective actions can be expressed in terms of the static limit's leading order.

Gauge invariance is maintained at all orders.

Abstract

We study systematically the higher order corrections to the parity violating part of the effective action for the Abelian Chern-Simons theory in 2+1 dimensions, using the method of derivative expansion. We explicitly calculate the parity violating parts of the quadratic, cubic and the quartic terms (in fields) of the effective action. We show that each of these actions can be summed, in principle, to all orders in the derivatives. However, such a structure is complicated and not very useful. On the other hand, at every order in the powers of the derivatives, we show that the effective action can also be summed to all orders in the fields. The resulting actions can be expressed in terms of the leading order effective action in the static limit. We prove gauge invariance, both large and small of the resulting effective actions. Various other features of the theory are also brought out.