On divergences in non-minimal ${ \newcommand{\N}{{{\mathcal N}}} \N=4}$ conformal supergravity

Abstract

We review the question of quantum consistency of N = 4 conformal supergravity in 4 dimensions. The UV divergences and anomalies of the standard ("minimal") conformal supergravity where the complex scalar ϕ is not coupled to the Weyl graviton kinetic term can be cancelled by coupling this theory to N = 4 super Yang-Mills with gauge group of dimension 4. The same turns out to be true also for the "non-minimal" N = 4 conformal supergravity with the action (recently constructed in arXiv:1609.09083) depending on an arbitrary holomorphic function f (ϕ). The special case of the "non-minimal" conformal supergravity with f = e 2ϕ appears in the twistor-string theory. We show that divergences and anomalies do not depend on the form of the function f and thus can be cancelled just as in the "minimal" f = 1 case by coupling the theory to four N = 4 vector multiplets.1 Also at Lebedev Institute, Moscow. tseytlin@imperial.ac.uk Conformal supergravities (CSGs) are N ≤ 4 supersymmetric extensions of the (C mnkl ) 2 Weyl gravity in 4 dimensions [1,2]. They are formally power-counting renormalizable with one coupling constant. The corresponding one-loop beta-functions were found to be non-zero [3] implying non-vanishing conformal anomaly. As the Weyl symmetry here is gauged, this means quantum inconsistency [4]. The same conclusion was reached also from the analysis of chiral SU (4) R-symmetry gauge anomalies [5], in agreement with the fact that all anomalies should belong to the same N = 4 superconformal multiplet.Remarkably, it was observed that the standard N = 4 CSG theory of [2] can be made UV finite [4,6,7] and thus anomaly-free [5] by coupling it [8] to exactly four N = 4 super Maxwell multiplets (e.g., to U (2) N = 4 super YM theory). This was shown directly at the one-loop order but should be true to all orders as the beta-function in N > 1 conformal supergravity and conformal anomaly of SYM may receive contributions only from the first loop (as follows from formal superspace arguments as in the SYM case, see [7]). In the present case of N = 4 there is also another reason for one-loop exactness: the conformal anomaly is tied by supersymmetry with SU (4) chiral anomaly which has one-loop origin.In the SU (1, 1) invariant N = 4 CSG of [2] the 4-derivative complex scalar ϕ = φ + iψ did not couple to Weyl graviton and SU (4) gauge field kinetic terms. It was conjectured in [4, 6, 7] that there may exist a "non-standard" (non SU (1, 1) invariant) version of N = 4 CSG. If one assumes that ϕ may "non-minimally" couple to Weyl term, f (ϕ)(C mnkl ) 2 + ... = (1 + k 1 φ + k 2 φ 2 + ...)(C mnkl ) 2 + ..., then there will be additional contributions to the beta-function that may cancel against the "minimal" N = 4 CSG beta-function. This finiteness conjecture was, however, in an apparent contradiction with the chiral anomaly count [5] as "non-minimal" couplings should not contribute to the chiral anomaly (see, e.g., [9]).This would suggest that either (i) a "non-minimal" theory does not exist as non-minimal scalar couplings are inc...

Aspects of Weyl supergravity

“…It is true that these theories are power-counting renormalizable but the one-loop beta-functions turns out to be non-vanishing [37] and therefore they suffer from a conformal anomaly. The latter leads to serious problems in Weyl gravity as the conformal symmetry is gauged and therefore leads to inconsistencies [38][39][40]. Let us also note that one may arrive at the same conclusion by the calculation of the chiral gauge anomalies of the SU(4) R-symmetry [41] and by recalling that all anomalies are arranged in the same multiplet of the N = 4 superconformal symmetry.…”

Unraveling conformal gravity amplitudes

“…[12]; however, no complete Lagrangian was given there. Recently Tseytlin considered general non-minimal N = 4 conformal supergravities, and concluded that they are free of conformal anomalies given that four vector multiplets are added to the spectrum [123]; thus agreeing with the analysis of the minimal theory [124][125][126]. A complete bosonic action for all N = 4 non-minimal conformal supergravity theories -parametrized by a free function -has been proposed by Butter, Ciceri, de Wit and Sahoo [127].…”

Bimetric, conformal supergravity and its superstring embedding

“…Althought such theories are power-counting renormalizable, their one-loop beta-functions are be non-vanishing [45] and therefore they suffer from a conformal anomaly. The latter leads to serious problems since conformal symmetry is gauged in Weyl gravity and therefore leads to inconsistencies [46][47][48]. The same conclusion can be drawn by considering the chiral gauge anomalies of the SU(4) R-symmetry [49] and recalling that all anomalies are accommodated in the same multiplet of the N = 4 superconformal symmetry.…”