Central charges of 2d superconformal defects

Abstract

In conformal field theories (CFTs) of dimension d > 3, two-dimensional (2d) conformal defects are characterised in part by central charges defined via the defect's contribution to the trace anomaly. However, in general for interacting CFTs these central charges are difficult to calculate. For superconformal 2d defects in supersymmetric (SUSY) CFTs (SCFTs), we show how to compute these defect central charges from the SUSY partition function either on S d with defect along S 2 , or on S 1 × S d−1 with defect along S 1 × S 1. In the latter case we propose that defect central charges appear in an overall normalisation factor, as part of the SUSY Casimir energy. For 2d half-BPS defects in 4d N = 2 SCFTs and in the 6d N = (2, 0) SCFT we obtain novel, exact results for defect central charges using existing results for partition functions computed using SUSY localisation, SUSY indices, and correspondences to 2d Liouville, Toda, and q-deformed Yang-Mills theories. Some of our results for defect central charges agree with those obtained previously via holography, showing that the latter are not just large-N and/or strongcoupling limits, but are exact. Our methods can be straightforwardly extended to other superconformal defects, of various codimension, as we demonstrate for a 4d defect in the 6d N = (2, 0) SCFT.

Defect CFT techniques in the 6d $$ \mathcal{N} $$ = (2, 0) theory

“…In particular, the second identity agrees with the explicit holographic calculations of [8,30,31] and was conjectured to come from supersymmetry in [27]. The two remaining anomaly coefficients a and b 1 were calculated at N = 1 in [8] and for N > 1 using holographic entanglement entropy in the presence of surface operators [32][33][34][35], and the superconformal index [36]. Finally, in section 5 we expand our scope and consider the analog of the operator product expansion but for bulk operators in the presence of a defect-the defect operator expansion (dOE) [37,38].…”

“…At large N one can use holography to calculate the expectation values, in the JHEP03(2021)261 presence of the defect, of operators in the traceless symmetric representation of so(5) R [69], which contains in particular O IJ in the stress tensor multiplet. Since the AGT correpondence can be used to calculate the expectation value of the stress tensor [36], it might also calculate expectation values for this larger class of operators at finite N .…”

Surface operators in superspace

“…Surface operators exhibit a conformal anomaly [27], as expected for all even dimensional defects. The anomaly in six dimensions has been determined perturbatively [17,19,28,29] and holographically [24,30], and the results are consistent with what has been obtained from the entanglement entropy for the bubbling M5/M2 geometry [31,32], and exactly from the determination of the corresponding superconformal index [33].…”

Defect CFT in the 6d (2,0) theory from M2 brane dynamics in AdS7 × S4

“…In this case the classical M2-brane minimal surface is the same AdS 3 but one is to impose the Neumann boundary condition on S 4 fluctuations (and average over an expansion point in the sphere) to preserve the SO(5) symmetry. 18 An exact expression for another anomaly coefficient is derived in [55] from the computation of the associated superconformal index. 19 If one assumes that the series in (3.7) terminates at 1/N order then the coefficient of this term can be of course fixed by requiring that the full expression should vanish for N = 1.…”