Constraints on chiral operators in \( \mathcal{N}=2 \) SCFTs
Matthew Buican
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Takahiro Nishinaka
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Constantinos Papageorgakis
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String Theories
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Open Access
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c The Authors
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Queen Mary University of London
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E1 4NS, U.K
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Rutgers University
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Piscataway, NJ 08854
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U.S.A
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CRST and School of Physics and Astronomy
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NHETC and Department of Physics and Astronomy
We study certain higher-spin chiral operators in N = 2 superconformal field theories (SCFTs). In Lagrangian theories, or in theories related to Lagrangian theories by generalized Argyres-Seiberg-Gaiotto duality (type A theories in our classification), we give a simple superconformal representation theory proof that such operators do not exist. This argument is independent of the details of the superconformal index. We then use the index to show that if a theory is not of type A but has an N = 2-preserving deformation by a relevant operator that takes it to a theory of this type in the infrared, the ultraviolet theory cannot have these higher-spin operators either. As an application of this discussion, we give a simple prescription to extract the 2a c conformal anomaly directly from the superconformal index. We also comment on how this procedure works in
1 Introduction 2 3 4
Constraints on exotic chiral operators
Theories of type A
The N = 2 superconformal index
The chiral Ub(1)R limit
The chiral U(1)R limit
Theories of type B
Theories of type C
Examples of type C theories
Conformal anomalies and the index
Summary and conclusions
N = 2 superconformal multiplet conventions
B RG flow from (G, G0) theories
Deformation of the singularity
B.2 Reduction to Seiberg-Witten description
B.3 RG flow by relevant deformations
Introduction
In the space of quantum field theories (QFTs), conformal field theories (CFTs) form a
special subspace of enhanced symmetry. The resulting conformal symmetry gives rise
to important simplifications. For instance, conformal invariance allows us to describe a
CFT by a tightly constrained set of data: its spectrum and operator product expansion
(OPE) coefficients.
While it is usually difficult to solve a given CFT (by which we mean to derive its
spectrum and OPE coefficients), one can use general principles to restrict the space of
allowed CFTs. For example, one can study the constraints imposed by associativity of
the OPE and find bounds on operator dimensions and OPE coefficients; see, e.g., [1, 2].
The restrictions on the space of superconformal field theories (SCFTs) are potentially even
more powerful; see, e.g., [3, 4] and references therein.
In this note, we describe new constraints (not derived from associativity of the OPE)
on the operator spectra of three very broad (and overlapping) classes of four-dimensional
N = 2 SCFTs:
(A) Theories, T , that have Lagrangian descriptions or theories related to Lagrangian
SCFTs by generalized Argyres-Seiberg-Gaiotto duality [5, 6]: in the latter case, we
gauge a global symmetry G of T in an exactly marginal fashion (adding additional
matter or additional interacting SCFTs charged under the gauge group as necessary)
and find a dual description in terms of a Lagrangian.
(B) N = 2 SCFTs with a Coulomb branch.
preserving renormalization group (RG) flows to Lagrangian theories (or, more
generally, theories of type A) in the infrared (IR).
Although it is clear that many theories of type A are also theories of class S (the TN
theories are a prototypical set of examples), we will not assume that this is the case in
are in class S or not) are of type B, with the exception of theories consisting of free
hypermultiplets. Note that one class of type B theories not of type A is the set of (G, G0)
(generalized) Argyres-Douglas (AD) SCFTs [710].1 Finally, the class of theories of type
C is also very broad. For example, it includes all the (G, G0) AD SCFTs. In fact, it is not
immediately clear to us if there are any theories that are of type B but not of type A or of
The main claims of our paper are:
1. Theories of type A and C do not have a certain class of higher-spin chiral (and
anti-chiral) primaries.2
2. If these operators are present in theories of type B, then they necessarily satisfy a
non-trivial set of operator relations we will describe in detail.
3. The pole structure of a particular chiral limit of the superconformal index encodes
all N = 2 SCFTs.
Our operators of interest in items 1, 2 above are defined by the shortening conditions
o = 0 , I = 1, 2 ,
where we have a commutator or anti-commutator depending on whether the SU(2)1 Lorentz
spin, j1, is even or odd, and I is an SU(2)R index labeling the two sets of Poincare
supercharges. An interesting subset of the operators we study satisfy an additional constraint
o = 0 ,
= 0 ,
j1 = 0 ,
I = 1, 2 ,
I, J = 1, 2 .
1This statement is rigorously true for theories with at least one chiral operator with non-integer scaling
2An argument against the existence of such operators in certain theories of class S was given in [11].
satisfying (1.1) as exotic chiral prim (...truncated)