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Contour Integral for the Partition Function of $\mathcal{N}=2$ Topologically Twisted on $\mathbb{CP}^2$ and Physical Fluxes

Lorenzo Ruggeri

Abstract

We compute the contour integral for the partition function of an $\mathcal{N}=2$ $SU(2)$ topologically twisted theory on $\mathbb{CP}^2$, dimensionally reducing from an $\mathcal{N}=1$ theory on $S^5$. Earlier works presented the partition function as a sum over three equivariant fluxes, one for each toric divisor of $\mathbb{CP}^2$. Our result depends only on a single physical flux, assigned to the non-trivial two-cycle of the manifold. The reduced summation over fluxes is compensated by a contour of integration, arising from a different solution of the BPS equations, which captures more poles in each topological sector. As our observable involves a position-dependent Yang-Mills coupling, we compute new equivariant invariants of $\mathbb{CP}^2$, which reduce to Donaldson invariants in the non-equivariant limit. Stability conditions of gauge bundles over $\mathbb{CP}^2$ appear intrinsically via the dimensional reduction.

Contour Integral for the Partition Function of $\mathcal{N}=2$ Topologically Twisted on $\mathbb{CP}^2$ and Physical Fluxes

Abstract

We compute the contour integral for the partition function of an topologically twisted theory on , dimensionally reducing from an theory on . Earlier works presented the partition function as a sum over three equivariant fluxes, one for each toric divisor of . Our result depends only on a single physical flux, assigned to the non-trivial two-cycle of the manifold. The reduced summation over fluxes is compensated by a contour of integration, arising from a different solution of the BPS equations, which captures more poles in each topological sector. As our observable involves a position-dependent Yang-Mills coupling, we compute new equivariant invariants of , which reduce to Donaldson invariants in the non-equivariant limit. Stability conditions of gauge bundles over appear intrinsically via the dimensional reduction.

Paper Structure

This paper contains 27 sections, 73 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of Partition Function on Quasi-Toric Four-Manifolds
  3. Five-Dimensional Gauge Theory
  4. Partition Function.
  5. Dimensional Reduction
  6. Finite Quotients of $M$.
  7. Classical Part.
  8. One-Loop Part.
  9. Instanton Part.
  10. Partition Function on the Base.
  11. Analytic Structure of the Partition Function on $\mathbb{CP}^2$
  12. Partition Function on $\mathbb{CP}^2$
  13. Geometry.
  14. Product Over Roots and Sum Over Fluxes.
  15. Partition Function.
  16. ...and 12 more sections

Figures (4)

  • Figure 1: Zeroes distribution of the one-loop determinant for $\mathfrak{m}_1=3$. Simple zeroes are in the blue region and double zeroes are in the red region. The remaining points are regular points.
  • Figure 2: Poles distribution of the instanton partition function for $\mathfrak{m}_1=3$. Simple poles are in the green region and triple poles are in the yellow region. The remaining points are regular points.
  • Figure 3: Poles and zeroes distribution of the full partition function for $\mathfrak{m}_1=3$. Simple poles are in the green region and simple zeroes are the blue points. The remaining points are regular points.
  • Figure 4: Division of poles (green regions), zeroes (blue points) and regular points (black points) of the full partition function into six different regions. After taking into account all cancellations, the residue sums receives contribution from the interior of region A with a -2 factor and from the border of region A with a -1 factor.