Structures on the conformal manifold in six dimensional theories

Abstract

The tensors which may be defined on the conformal manifold for six dimensional CFTs with exactly marginal operators are analysed by considering the response to a Weyl rescaling of the metric in the presence of local couplings. It is shown that there are three symmetric two index tensors only one of which satisfies any positivity conditions. The general results are specialised to the six dimensional conformal theory defined by free two-forms and also to the interacting scalar φ 3 theory at two loops which preserves conformal invariance to this order. All three two index tensor contributions are present.

C T for higher derivative conformal fields and anomalies of (1, 0) superconformal 6d theories

“…The value of C T,6 is the same (1.14) as quoted in the Introduction, found earlier by other methods in [31,56]. To also reproduce the correct value a = 275 8×7!…”

“…The general expression for the Weyl-covariant 6-derivative scalar operator in curved background can be found, e.g., in [56]. Ignoring terms with derivatives of the curvature and specifying to d = 6 it can be written as 23) where the Schouten tensor P µν and its trace P are in general defined as…”

“…Here we shall follow [56,64] and consider a non-unitary theory described by 2-derivative vector field with conformal but not gauge-invariant action for d = 4. 31 The corresponding Weyl-invariant curved space action is…”

On the conformal properties of topological terms in even dimensions

“…A practical realization of this program is a necessary step in integrating conformal anomaly in D = 6 and higher even dimensions, and also may help to approaching the solution of one of the mathematical puzzles related to conformal anomaly. It is important that integrating the trace anomaly requires not only conformal operator [8,11] (see also [7,34,35,36] for other publications on the subject), but also the relation between conformal operators and topological structures, e.g., expressed in the form (40). This kind of formula is critically important for integrating anomaly in D = 2 and D = 4 and hence the proof of the Conjecture 2 would be a decisive step forward in completing the same program in higher even dimensions.…”

“…In the mathematical literature one can find a general theory for constructing conformal operators [5,6,7,8], which can be used to obtain explicit examples. For instance, the analog of Paneitz operator with six derivatives, ∆ 6 , in D = 6 can be found in [9], consequent paper [10] and in [11], where the generalization to D = 6 was also obtained.…”