Anomalies, conformal manifolds, and spheres

Abstract

Abstract:The two-point function of exactly marginal operators leads to a universal contribution to the trace anomaly in even dimensions. We study aspects of this trace anomaly, emphasizing its interpretation as a sigma model, whose target space M is the space of conformal field theories (a.k.a. the conformal manifold). When the underlying quantum field theory is supersymmetric, this sigma model has to be appropriately supersymmetrized. As examples, we consider in some detail N = (2, 2) and N = (0, 2) supersymmetric theories in d = 2 and N = 2 supersymmetric theories in d = 4. This reasoning leads to new information about the conformal manifolds of these theories, for example, we show that the manifold is Kähler-Hodge and we further argue that it has vanishing Kähler class. For N = (2, 2) theories in d = 2 and N = 2 theories in d = 4 we also show that the relation between the sphere partition function and the Kähler potential of M follows immediately from the appropriate sigma models that we construct. Along the way we find several examples of potential trace anomalies that obey the Wess-Zumino consistency conditions, but can be ruled out by a more detailed analysis.

The Holographic F Theorem

“…Finally, there has been considerable recent interest in how much supersymmetry is required to determine uniquely partition functions on even-dimensional spheres [27]. In three dimensions, an analogous question arises at quadratic order in the source for conformal field theories deformed by operators of dimensions 1/2 and 5/2.…”

“…The regularity and asymptotic expansions are most easily extracted from (27) and (29) while (28) is most convenient for the subsequent numerical calculations.…”

Bootstrapping N = 2 $$ \mathcal{N}=2 $$ chiral correlators

“…It would be interesting to use the prescription of [50] in order to check how our bound fares with known theories, although our results are far from their optimal values (see also [75] for a recent discussion on the geometry and topology of M).…”

Conformal manifolds: ODEs from OPEs

“…Interestingly, some of these manifolds turn out to be strongly coupled at all points. There is also a growing body of work developing the non-perturbative understanding of these theories through the superconformal algebra [9][10][11][12]. The short multiplets to…”