Scheme dependence of instanton counting in ALE spaces

TL;DR

This paper compares two different methods of instanton counting in ALE spaces, showing they generally produce different results and proposing relations between these methods, with tests in specific gauge theories.

Contribution

It highlights the scheme dependence in instanton counting for ALE spaces and establishes relations between different counting schemes, supported by explicit calculations.

Findings

Different schemes yield different partition functions.

Proposed relations between the two schemes.

Validated relations through explicit examples.

Abstract

There have been two distinct schemes studied in the literature for instanton counting in A_{p-1} asymptotically locally Euclidean (ALE) spaces. We point out that the two schemes---namely the counting of orbifolded instantons and instanton counting in the resolved space---lead in general to different results for partition functions. We illustrate this observation in the case of N=2 U(N) gauge theory with 2N flavors on the A_{p-1} ALE space. We propose simple relations between the instanton partition functions given by the two schemes and test them by explicit calculations.