Non-compact duality, super-Weyl invariance and effective actions
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Springer
Received: June 19, 2020 Accepted: June 27, 2020 Published: July 30, 2020
Sergei M. Kuzenko Department of Physics M013, The University of Western Australia, 35 Stirling Highway, Perth, W.A. 6009, Australia
E-mail: [email protected] Abstract: In both N = 1 and N = 2 supersymmetry, it is known that Sp(2n, R) is the maximal duality group of n vector multiplets coupled to chiral scalar multiplets τ (x, θ) that parametrise the Hermitian symmetric space Sp(2n, R)/U(n). If the coupling to τ is introduced for n superconformal gauge multiplets in a supergravity background, the action is also invariant under super-Weyl transformations. Computing the path integral over the gauge prepotentials in curved superspace leads to an effective action Γ[τ, τ¯] with the following properties: (i) its logarithmically divergent part is invariant under super-Weyl and rigid Sp(2n, R) transformations; (ii) the super-Weyl transformations are anomalous upon renormalisation. In this paper we describe the N = 1 and N = 2 locally supersymmetric “induced actions” which determine the logarithmically divergent parts of the corresponding effective actions. In the N = 1 case, superfield heat kernel techniques are used to compute the induced action of a single vector multiplet (n = 1) coupled to a chiral dilaton-axion multiplet. We also describe the general structure of N = 1 super-Weyl anomalies that contain weight-zero chiral scalar multiplets ΦI taking values in a K¨ahler manifold. Explicit anomaly calculations are carried out in the n = 1 case. Keywords: Supergravity Models, Superspaces, Supersymmetric Effective Theories, Supersymmetry and Duality ArXiv ePrint: 2006.00966
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP07(2020)222
JHEP07(2020)222
Non-compact duality, super-Weyl invariance and effective actions
Contents 1 Introduction
1
2 Superconformal higher-derivative actions 2.1 Local N = 1 supersymmetry 2.2 Local N = 2 supersymmetry 2.3 Relating the N = 2 and N = 1 actions
3 4 5 6 7 7 10
4 Quantisation
10
5 Heat kernel calculations (I) 5.1 Generalised Schwinger-DeWitt representation 5.2 Evaluation of the heat kernel in flat superspace 5.3 Super-Weyl anomaly
13 13 14 16
6 Heat kernel calculations (II) 6.1 Generalised Schwinger-DeWitt representation 6.2 Chiral and super-Weyl anomalies
16 17 20
7 Concluding comments
21
A Super-Weyl transformations
22
1
Introduction
It is well known that the group of electromagnetic duality rotations of free Maxwell’s equations is the compact group U(1), assuming the duality invariance of the energymomentum tensor. Almost forty years ago, it was shown by Gaillard and Zumino [1, 2] that the non-compact group Sp(2n, R) is the maximal duality group of n vector field I = −F I in the presence of scalar τ i parametrising the homogeneous space strengths Fab ba Sp(2n, R)/U(n).1 In the absence of scalars, the largest duality group proves to be U(n), the maximal compact subgroup of Sp(2n, R). These results admit a natural extension to the case when th
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