Chiral algebra, localization, modularity, surface defects, and all that

Journal Article · Fri Sep 11 00:00:00 EDT 2020 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/5.0002661· OSTI ID:1802219

Dedushenko, Mykola  ^[1]; Fluder, Martin ^[2]

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1. California Institute of Technology (CalTech), Pasadena, CA (United States)
2. California Institute of Technology (CalTech), Pasadena, CA (United States); Univ. of Tokyo (Japan). Kavli Institute for the Physics and Mathematics of the Universe (WPI)

We study the 2D vertex operator algebra (VOA) construction in 4D N = 2 superconformal field theories on S³ × S¹, focusing on both old puzzles and new observations. The VOA lives on a two-torus ${\mathbb{T}}^2\subset S^3\times S^1$, it is $\frac{1}{2}\mathbb{Z}$-graded, and this torus is equipped with the natural choice of spin structure (1,0) for the $\mathbb{Z}+\frac{1}{2}$-graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserves the VOA, we find that it admits discrete deformations, which allows access to the remaining spin structures (1,1), (0,1), and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle and the two-point functions and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the T-transformation acts on our four partition functions and lifts to a large diffeomorphism on S³ × S¹. More interestingly, we generalize the four partition functions on the torus to an infinite family labeled by both the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the S-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the Ω-background.

• Research Organization: California Institute of Technology (CalTech), Pasadena, CA (United States)
• Sponsoring Organization: USDOE Office of Science (SC)
• Grant/Contract Number: SC0011632; de-sc0011632
• OSTI ID: 1802219
• Alternate ID(s): OSTI ID: 1659533
• Journal Information: Journal of Mathematical Physics, Vol. 61, Issue 9; ISSN 0022-2488
• Publisher: American Institute of Physics (AIP)
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Physics
Gauge group
Quantum field theory
Differential geometry
Algebraic structures
Conformal algebra
Superconformal algebras
Lagrangian field theories
Particle symmetry
Supersymmetric gauge theory
Operator product expansion