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Instantons on Quivers and Orientifolds

Francesco Fucito, Jose F. Morales, Rubik Poghossian

TL;DR

This work develops a Nekrasov-style localization framework for computing the prepotential of gauge theories descended from N=4 SYM via quiver projections and mass deformations, covering product gauge groups with bifundamental matter and extending to orientifold setups with SO/Sp groups. It formulates the D(-1)/D3 system in BRST language, derives the moduli space tangent space under orbifold and orientifold projections, and obtains the centered instanton partition function by summing over fixed points labeled by Young diagrams. The resulting quiver prepotentials are explicitly constructed for two-node cases and generalized to Z_p orbifolds, with matter contributions shown to enter off-diagonally and to reproduce Seiberg-Witten curves in the absence of gravitational corrections. The orientifold analysis further yields SO/Sp prepotentials, including fractional instantons in Sp theories, and demonstrates consistency with known SW structures while providing a streamlined determinant expression for the fixed-point contribution.

Abstract

We compute the prepotential for gauge theories descending from ${\cal N}=4$ SYM via quiver projections and mass deformations. This accounts for gauge theories with product gauge groups and bifundamental matter. The case of massive orientifold gauge theories with gauge group SO/Sp is also described. In the case with no gravitational corrections the results are shown to be in agreement with Seiberg-Witten analysis and previous results in the literature.

Instantons on Quivers and Orientifolds

TL;DR

This work develops a Nekrasov-style localization framework for computing the prepotential of gauge theories descended from N=4 SYM via quiver projections and mass deformations, covering product gauge groups with bifundamental matter and extending to orientifold setups with SO/Sp groups. It formulates the D(-1)/D3 system in BRST language, derives the moduli space tangent space under orbifold and orientifold projections, and obtains the centered instanton partition function by summing over fixed points labeled by Young diagrams. The resulting quiver prepotentials are explicitly constructed for two-node cases and generalized to Z_p orbifolds, with matter contributions shown to enter off-diagonally and to reproduce Seiberg-Witten curves in the absence of gravitational corrections. The orientifold analysis further yields SO/Sp prepotentials, including fractional instantons in Sp theories, and demonstrates consistency with known SW structures while providing a streamlined determinant expression for the fixed-point contribution.

Abstract

We compute the prepotential for gauge theories descending from SYM via quiver projections and mass deformations. This accounts for gauge theories with product gauge groups and bifundamental matter. The case of massive orientifold gauge theories with gauge group SO/Sp is also described. In the case with no gravitational corrections the results are shown to be in agreement with Seiberg-Witten analysis and previous results in the literature.

Paper Structure

This paper contains 11 sections, 71 equations, 2 figures.

Table of Contents

  1. Introduction
  2. D-instantons
  3. ADHM manifold
  4. D-instanton partition
  5. Quiver gauge theories
  6. Generalities
  7. The quiver prepotential
  8. Orientifolds
  9. ${\cal N}=(2,2)$ gauge theory
  10. $SU(2)\times SU(2)$ with one bifundamental
  11. $SU(2)\times SU(2)$ with two bifundamentals

Figures (2)

  • Figure 1: Two generic Young diagrams denoted by the indices $\alpha, \beta$ in the main text. In the figures the diagram $Y_\alpha$ where the box "s" belongs to is always displayed with solid lines. The hook starts on box "s" a run horizontally till the end of diagram $Y_\alpha$ and vertically till the top end of $Y_\beta$. The two tableaux are depicted on top of each other so that in the picture in the left side the solid diagram $Y_\alpha$ contain both solid and dashed boxes.
  • Figure 2: D(-1)-D3 branes in the presence of an orientifold plane. We have also drawn a D(-1)-D(-1) open string. See the text for a more detailed explanation of the figure.