[Paper Review] Localization of a supersymmetric gauge theory in the presence of a surface defect

Joël Lamy-Poirier|arXiv (Cornell University)|Dec 1, 2014

Black Holes and Theoretical Physics12 references11 citations

TL;DR

This paper computes the partition function of $\mathcal{N}=2$ super-Yang-Mills theory on $S^4$ coupled to a gauged linear sigma model (GLSM) surface defect on a $S^2$ subspace using supersymmetric localization. The result is a modified partition function on $S^4$ with a shifted GLSM contribution and a deformed instanton partition function, arising from the coupling of 4d vector multiplets to chiral multiplets on the defect via transverse derivatives, preserving $\mathcal{N}=(2,2)$ supersymmetry.

ABSTRACT

We use supersymmetric localization to compute the partition function of N=2 super-Yang-Mills on S^4 in the presence of a gauged linear sigma model surface defect on a S^2 subspace. The result takes the form of a standard partition function on S^4, with a modified instanton partition function and an additional insertion corresponding to a shifted version of the gauged linear sigma model partition function.

Motivation & Objective

• To compute the partition function of a 4d $\mathcal{N}=2$ supersymmetric gauge theory on $S^4$ coupled to a 2d GLSM surface defect on a $S^2$ subspace.
• To extend supersymmetric localization techniques to theories with surface defects, particularly in the context of $\mathcal{N}=(2,2)$ supersymmetry on the defect.
• To provide a framework for computing physical observables such as Wilson loops and verifying AGT duality in the presence of defects.
• To derive the one-loop determinant and instanton partition function for chiral multiplets coupled to 4d vector multiplets via transverse derivatives.
• To generalize the result to include matter multiplets and vector multiplets on the defect, setting the stage for further computations.

Proposed method

• Uses supersymmetric localization on $S^4$ with a $S^2$ surface defect, deforming the action to localize the path integral onto the Coulomb branch.
• Restricts the 4d vector multiplet to the $S^2$ defect, producing a tower of 2d fields including a vector multiplet and chiral multiplets from transverse derivatives.
• Treats the 4d gauge fields as background fields for the 2d chiral multiplets, coupling them via a transversally elliptic Dirac operator $D_{10}$.
• Applies the equivariant index theorem to compute the one-loop determinant of the chiral multiplet, using the $\mathcal{Q}$-cohomology structure.
• Derives the one-loop determinant as $Z_{1-\text{loop}} = \prod_{w\in R} \frac{\Gamma(\omega\cdot\Lambda_N - irM)}{\Gamma(1 - \omega\cdot\Lambda_S + irM)}$, where $\Lambda_N, \Lambda_S$ are background gauge connections at the poles.
• Constructs the full partition function as a product of 4d and 2d Coulomb branch integrals, with a modified instanton partition function and a shifted GLSM contribution.

Experimental results

Research questions

• RQ1 How does the presence of a surface defect on $S^2 \subset S^4$ modify the partition function of a 4d $\mathcal{N}=2$ gauge theory?
• RQ2 What is the structure of the one-loop determinant for chiral multiplets coupled to 4d vector multiplets via transverse derivatives in the defect setup?
• RQ3 How is the instanton partition function deformed when the $\Omega$-background includes a 2d defect on $\mathbb{R}^2$?
• RQ4 Can supersymmetric localization be consistently applied to a 4d-2d coupled system with preserved $\mathcal{N}=(2,2)$ supersymmetry on the defect?
• RQ5 What is the role of the equivariant index in computing the one-loop determinant for chiral multiplets in this defect configuration?

Key findings

• The partition function takes the form of a 4d $S^4$ partition function with a modified instanton partition function and an additional insertion from a shifted GLSM partition function.
• The one-loop determinant for chiral multiplets is derived as $\prod_{w\in R} \frac{\Gamma(\omega\cdot\Lambda_N - irM)}{\Gamma(1 - \omega\cdot\Lambda_S + irM)}$, where $\Lambda_N, \Lambda_S$ are background gauge connections at the north and south poles.
• The coupling between the 4d vector multiplet and the 2d chiral multiplet arises from the restriction of the 4d gauge field's transverse derivatives, forming a tower of 2d multiplets.
• The resulting theory preserves $\mathcal{N}=(2,2)$ supersymmetry on the $S^2$ defect, breaking half the original 4d supercharges.
• The instanton partition function is a modified version of Nekrasov's, incorporating a 2d defect in the $\Omega$-background, though its full derivation is left for future work.
• The method generalizes to include matter multiplets and vector multiplets on the defect, enabling further computations of physical observables in the defect setup.