Modular constraints on superconformal field theories

Abstract

We constrain the spectrum of N = (1, 1) and N = (2, 2) superconformal field theories in two-dimensions by requiring the NS-NS sector partition function to be invariant under the Γ θ congruence subgroup of the full modular group SL(2, Z). We employ semi-definite programming to find constraints on the allowed spectrum of operators with or without U (1) charges. Especially, the upper bounds on the twist gap for the non-current primaries exhibit interesting peaks, kinks, and plateau. We identify a number of candidate rational (S)CFTs realized at the numerical boundaries and find that they are realized as the solutions to modular differential equations associated to Γ θ . Some of the candidate theories have been discussed by Höhn in the context of self-dual extremal vertex operator (super)algebra. We also obtain bounds for the charged operators and study their implications to the weak gravity conjecture in AdS 3 . #1 Throughout this paper, we only consider parity invariant CFTs so that left-moving and rightmoving fermions have the same boundary condition.

Anomalies and bounds on charged operators

“…We argue that the same relation between the existence of a universal bound and the 't Hooft anomaly also holds true if the symmetry group is U (1). Indeed, previous universal bounds for the lightest U (1) charged operator in the literature [7][8][9] are restricted to U (1) global symmetries generated by holomorphic currents, which are always anomalous (i.e. cannot be gauged).…”

“…Another pragmatic way to detect the anomaly (α = −1) is by the ambiguity/inconsistency in constructing the torus partition function of the Z 2 orbifold theory. 9 Let us attempt to compute the torus partition function of the would-be orbifold theory, which can be written as a sum of four terms (times a factor of 1 2 ). The first two terms account for the contributions from the Z 2 even states in the H, while the last two terms are from the Z 2 even states in the defect Hilbert space H L .…”

“…We also ignore the dependence on the spin h −h, and only keep track of the scaling dimension ∆ = h +h of the operators. We set τ = −τ = it, and expand the torus partition function in each sector as 9) where (N ± ) ∆ ∈ Z ≥0 and (N L ) ∆ ∈ Z ≥0 are the degeneracies of all states (not necessarily primaries) with scaling dimension ∆, in H ± and in H L , respectively. Here, g ∆ (t) is the scaling character that counts the contribution from a single operator of dimension ∆:…”

“…In [7][8][9], a bound on the lightest charged operator is derived for a holomorphic U (1), which always has an 't Hooft anomaly. Hence, it is consistent with the above statement.…”

Notes on superconformal representations in two dimensions

“…A multiplet is long, or massive, if its primary [j] h lies above the unitarity bound, and it is short, or massless, if the primary lies on the unitarity bound. The unitarity bound is described by a set of line segments (see [38])…”