[Paper Review] Interfaces and Quantum Algebras, II: Cigar Partition Function

Mykola Dedushenko, Nikita Nekrasov|arXiv (Cornell University)|Jun 28, 2023

Black Holes and Theoretical Physics4 citations

TL;DR

This paper investigates the supersymmetric cigar partition function in 3D $σ=2$ gauge theories, using curved supergravity backgrounds to compute exact path integrals via localization. It establishes a physical realization of quantum K-theory and elliptic cohomology, linking the half-index to vertex functions, stable envelopes, and wall-crossing phenomena through Janus interfaces and $\mathcal{Q}$-exact deformations.

ABSTRACT

The supersymmetric cigar (half-)index or cigar partition function of 3d $\mathcal{N}=2$ gauge theories contains a wealth of information. Physically, it captures the spectrum of BPS states, the non-perturbative corrections to various partition functions, the effective twisted superpotential and the data of supersymmetric vacua. Mathematically, it defines the K-theoretic Vertex counting vortices/quasimaps, and connects to quantum K-theory, as well as elliptic cohomology and stable envelopes. We explore these topics from the physics standpoint, systematically developing the foundations and explaining various mathematical properties using the quantum field theory machinery.

Motivation & Objective

• To establish a physical framework for quantum K-theory and elliptic cohomology in 3D $\mathcal{N}=2$ gauge theories using the supersymmetric cigar background.
• To derive the half-index (cigar partition function) as a generating function for BPS spectra, twisted superpotentials, and non-perturbative corrections.
• To connect mathematical structures like stable envelopes and vertex functions to physical interfaces and $\mathcal{Q}$-exact deformations in supersymmetric field theory.
• To unify the description of Coulomb and Higgs branch localizations via $\mathcal{Q}$-exact terms, including moment map and superpotential deformations.
• To explore wall-crossing phenomena via Janus interfaces and their relation to quantum algebras such as Yangians acting on the Hilbert space of BPS states.

Proposed method

• Utilizes a curved $S^1 \times S^2_b$ background with squashing and Gomis-Lee deformation to preserve a single supercharge and enable exact path integral computation.
• Applies $\mathcal{Q}$-exact deformations—specifically $L^v$, $L_H$, and $L_W$—to localize the path integral on BPS configurations, with $L^v$ for Coulomb branch and $L^v + L_H$ for Higgs branch.
• Derives localization equations for vector and chiral multiplets, including $D=0$, $D_\mu \sigma = 0$, $F_{\varphi\alpha} = 0$, and $F_{\theta\alpha} = \varepsilon F_{\theta\varphi}$, with modified constraints under Higgs branch deformation.
• Introduces a modified bosonic action $L_{\text{Bos}}^{\text{v}} + L_H + L_W$ that is positive definite and depends on the superpotential $W$, enabling localization on critical points of $W$.
• Uses the real moment map $H(\phi) = e^2(\bar{\phi}\phi - \zeta)$ to implement Higgs branch localization, with $\zeta \to \infty$ in the infrared.
• Applies the $\mathcal{Q}$-exact interpolation to surface operators and Wilson loops, connecting them to quantum K-theory and $K$-theoretic vertex functions.

Experimental results

Research questions

• RQ1 How does the 3D $\mathcal{N}=2$ cigar partition function encode the BPS spectrum and non-perturbative corrections in gauge theories?
• RQ2 What is the physical realization of stable envelopes and their role in wall-crossing via Janus interfaces?
• RQ3 How do $\mathcal{Q}$-exact deformations in the supergravity background lead to localization on Coulomb and Higgs branches?
• RQ4 In what way does the half-index relate to vertex functions and K-theoretic quasimap invariants in Nakajima varieties?
• RQ5 How do Wilson loops and surface operators in the cigar background realize quantum K-theory and connect to elliptic cohomology?

Key findings

• The cigar partition function computes the half-index, which generates the spectrum of BPS states and encodes the effective twisted superpotential in 3D $\mathcal{N}=2$ theories.
• Localization on the Coulomb branch is achieved via $L^v$, leading to a positive definite action with constraints $D=0$, $D_\mu \sigma = 0$, and $F_{\varphi\alpha} = 0$.
• Higgs branch localization is realized by adding $L_H$, modifying the vector multiplet equations to include $D_{\varphi}\sigma = D_{\alpha}\sigma = 0$ and $H(\phi) = \frac{D_\theta(\sigma \sin\theta)}{f(\theta)} - \frac{F_{\theta\varphi} \cos\theta}{\ell f(\theta) \sin\theta}$.
• The superpotential deformation $L_W$ leads to a localized action depending on $W$, with equations $F = \overline{F} = 0$, $D_\alpha \phi = \varepsilon D_\varphi \phi$, and $\frac{i}{\ell}D_\varphi \phi - \frac{\Delta}{f(\theta)}\phi - \widehat{\sigma}\phi \cos\theta = 0$.
• The $\mathcal{Q}$-exact interpolation of surface operators and Wilson loops realizes quantum K-theory, with the half-index serving as a generating function for $K$-theoretic vertex invariants.
• Wall-crossing via Janus interfaces is described by $\mathcal{Q}$-exact deformations, and the resulting transition functions are shown to match the action of stable envelopes on the Hilbert space of BPS states.