Twisted characters and holomorphic symmetries
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Twisted characters and holomorphic symmetries Ingmar Saberi1 · Brian R. Williams2 Received: 8 July 2019 / Revised: 14 July 2020 / Accepted: 21 July 2020 © The Author(s) 2020
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 βγ 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)-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. Keywords Holomorphic quantum field theory · Special functions · Supersymmetry · Infinite-dimensional Lie algebras · Character formulas Mathematics Subject Classification 81T60 · 17B65 · 17B69 · 81T70 · 33E05 · 17B81
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Brian R. Williams [email protected] Ingmar Saberi [email protected]
1
Mathematisches Institut der Universität Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Deutschland
2
School of Mathematics, University of Edinburgh, Edinburgh, UK
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I. Saberi, B. R. Williams
Contents 1 Introduction . . . . . . . . . . . . . . . . 2 N = 1 supersymmetry in four dimensions . 3 The chiral multiplet and auxiliary fields . . 4 Twisted chiral matter: the βγ system . . . 5 Holomorphic characters . . . . . . . . . . 6 Turning on a superpotential . . . . . . . . 7 Holomorphic flavor symmetry . . . . . . . 8 Dimensional reduction . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . .
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