Abstract
The low energy effective couplings of a four-dimensional mathcal{N} = 2 supersymmetric gauge theory to topological invariants of the background gravitational field are described by two functions A and B. These two functions play an important role in the study of topo- logically twisted four-dimensional mathcal{N} = 2 supersymmetric gauge theories and in the computation of central charges of mathcal{N} = 2 superconformal theories. In this paper, we compute A and B from the partition function in the Ω-background for SU(2) gauge theories. Our results not only confirm the predicted expressions of the effective gravitational couplings, but also give the previously undetermined overall multiplicative factors. We also analyze A and B for the SU(N ) super-Yang-Mills theory, and confirm all the previous predictions.
Highlights
Low energy effective theory is described in terms of r abelian vector multiplets
The low energy effective couplings of a four-dimensional N = 2 supersymmetric gauge theory to topological invariants of the background gravitational field are described by two functions A and B
It is remarkable that F can be solved exactly, and the solution is elegantly encoded in the Seiberg-Witten geometry
Summary
It is worth emphasizing that the masses mf appearing in (2.15), (2.16), (2.22), (2.23) differ from the masses mf in the original paper [52, 53] by a constant shift of ε+ [69,70,71], mf = mf + ε+ This shift is due to the fact that the scalars in a hypermultiplet become spinors in the Donaldson-Witten twist, and the Dirac complex is the Dolbeault complex twisted by the square-root of the canonical bundle of the four-manifold. This shift can often be ignored in many applications of the Ω-background, since it will not modify the dynamics of the theory on flat space where ε+ = 0. The simplest but most important example is the SU(2) super-Yang-Mills theory
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