A nilpotency index of conformal manifolds
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Springer
Received: March Revised: August Accepted: September Published: October
27, 18, 24, 28,
2020 2020 2020 2020
Zohar Komargodski,a Shlomo S. Razamat,b Orr Selab and Adar Sharonc a
Simons Center for Geometry and Physics, Stony Brook, New York, U.S.A. b Department of Physics, Technion, Haifa 32000, Israel c Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot 7610001, Israel
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We show that exactly marginal operators of Supersymmetric Conformal Field Theories (SCFTs) with four supercharges cannot obtain a vacuum expectation value at a generic point on the conformal manifold. Exactly marginal operators are therefore nilpotent in the chiral ring. This allows us to associate an integer to the conformal manifold, which we call the nilpotency index of the conformal manifold. We discuss several examples in diverse dimensions where we demonstrate these facts and compute the nilpotency index. Keywords: Conformal Field Theory, Supersymmetric Gauge Theory ArXiv ePrint: 2003.04579
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)183
JHEP10(2020)183
A nilpotency index of conformal manifolds
Contents 1 Introduction
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2 Vanishing expectation values 2.1 Elementary results 2.2 The main argument
3 3 5
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6 7 11 11 12 13 15
Introduction
The existence of continuous families of Conformal Field Theories (CFTs) is a surprising phenomenon that occurs mostly in supersymmetric theories. Indeed, without supersymmetry, in two dimensions it is not expected that there are continuous families of conformal field theories except in the case of c = 1 [1]. In higher dimensions, no examples are known (with any finite central charge) of non-trivial conformal manifolds. Given a marginal primary operator, for it to be exactly marginal infinitely many intricate constraints need to be satisfied (see [2–5] for the first couple of constraints) and it is quite hard to believe that such examples exist without supersymmetry. In this note we study some canonical questions about conformal manifolds that occur in supersymmetric theories with four supercharges (or more). The existence of such conformal manifolds has a long history, see e.g. [6–8], and a plethora of examples was first systematically constructed in [9] and later in [10] (see also [11]). Let P be such an SCFT. For convenience, the reader may keep in mind either N = 2 theories in 3d or N = 1 theories in 4d. An exactly marginal operator resides in a multiplet of a chiral primary operator OI of dimension d − 1. The subscript I here ranges over a basis of the space of exactly marginal operators. If we deform the action by Z dd xhI Q2 OI + c.c. , (with Q standing for the supercharge) then the theory remains conformal. We thus get a space of SCFTs with local coordinates hI . This space is called the conformal manifold
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JHEP10(2020)183
3 Simple examples 3.1 2d N = (2
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