Twisted characters and holomorphic symmetries

Abstract

We consider holomorphic twists of arbitrary supersymmetric theories in four dimensions. Working in the BV formalism, we rederive classical results characterizing the holomorphic twist of chiral and vector supermultiplets, computing the twist explicitly as a family over the space of nilpotent supercharges in minimal supersymmetry. The BV formalism allows one to work with or without auxiliary fields, according to preference; for chiral superfields, we show that the result of the twist is an identical BV theory, the holomorphic $$\beta \gamma $$ β γ system with superpotential, independent of whether or not auxiliary fields are included. We compute the character of local operators in this holomorphic theory, demonstrating agreement of the free local operators with the usual index of free fields. The local operators with superpotential are computed via a spectral sequence and are shown to agree with functions on a formal mapping space into the derived critical locus of the superpotential. We consider the holomorphic theory on various geometries, including Hopf manifolds and products of arbitrary pairs of Riemann surfaces, and offer some general remarks on dimensional reductions of holomorphic theories along the $$(n-1)$$ ( n - 1 ) -sphere to topological quantum mechanics. We also study an infinite-dimensional enhancement of the flavor symmetry in this example, to a recently studied central extension of the derived holomorphic functions with values in the original Lie algebra, that generalizes the familiar Kac–Moody enhancement in two-dimensional chiral theories.

Feynman diagrams and $Ω$-deformed M-theory

“…One can think the infinitesimal super-translation parameter ε that appears in the global supersymmetry transformation as a rigid limit of the bosonic ghost q. For instance, in 4d N = 1 holomorphically twisted field theory [41][42][43][44], with Q paired with ε + , the supersymmetry transformation of the bottom component φ of anti-chiral superfield Ψ = (φ,ψ,F ) transforms as δφ =εψ, δψ = iε +∂φ + iε − ∂φ +εF .…”

A domain wall in twisted M-theory

“…We can recast the 4d holomorphic field theory in a first-order formalism, called a twisted formalism in [44], and find that the 4d Lagrangian resembles that of the 2d βγ system. The authors of [14] derived the 4d βγ system. In §3.1, §3.2, we will translate and explain this mathematical notion in the more familiar language to physicists using holomorphic descent, which is an analog of the topological descent defined in Witten's classical paper [1].…”

“…We can make the analogy more concrete by rewriting the Lagrangian in the descent field utilizing the BV Lagrangian [14]. The physical Lagrangian (30) admits a compact representation in terms of the following twisted superfields γ = ( φ, φ(1) , φ(2) ),…”

“…As our main intention to present this expression is to show the resemblance of the higher VOA to the usual 2d VOA, we will not try to give details on the BV formalism. The reader who is interested in this may refer to [14,44].…”

“…The protected subsector of the 2d example naturally forms a familiar vertex operator algebra. On the other hand, the holomorphic sector of 4d N = 1 theory reorganizes itself into a higher vertex operator algebra (higher VOA) [11][12][13][14], which will be the main player of this paper. Higher VOA naturally appears in twisted Mtheory [15,16], where there are 4 holomorphic directions and 7 topological directions.…”