Warped Weyl fermion partition functions

Published for SISSA by Springer Received: September 17, 2015 Accepted: October 23, 2015 Published: November 19, 2015 Alejandra Castro,a Diego M. Hofmana and Gábor Sárosib,c a Institute for Theoretical Physics, University of Amsterdam, Science Park 904, Postbus 94485, 1090 GL Amsterdam, The Netherlands b Department of Theoretical Physics, Institute of Physics, Budapest University of Technology, H-1521 Budapest, Hungary c Kavli Institute for Theoretical Physics, Kohn Hall, University of California, Santa Barbara CA 93106-4030, U.S.A. E-mail: , , Abstract: Warped conformal field theories (WCFTs) are a novel class of non-relativistic theories. A simple, yet non-trivial, example of such theory is a massive Weyl fermion in (1 + 1)-dimensions, which we study in detail. We derive general properties of the spectrum and modular properties of partition functions of WCFTs. The periodic (Ramond) sector of this fermionic system is non-trivial, and we build two novel partition functions for this sector which have no counterpart in a CFT2 . The thermodynamical properties of WCFTs are revisited in the canonical and micro-canonical ensemble. Keywords: Field Theories in Lower Dimensions, Conformal and W Symmetry, Holography and condensed matter physics (AdS/CMT) ArXiv ePrint: 1508.06302 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP11(2015)129 JHEP11(2015)129 Warped Weyl fermion partition functions Contents 1 2 Warped Weyl fermion: from the plane to the cylinder 2.1 Warped Weyl fermion 2.2 General properties of radial quantization 4 5 10 3 On to the torus 3.1 Warped Weyl fermion 3.2 Primary content of the Warped Weyl fermion theory 3.3 On modular invariance for trace partition functions 12 14 17 19 4 Thermodynamics 4.1 Canonical ensemble 4.2 Micro-canonical ensemble 21 22 25 5 Discussion 26 A Brief summary on WCFTs 28 B Standard modular forms and functions 30 C Characters from the fermionic path integral 31 1 Introduction Conformal field theories (CFTs) represent the building blocks of our knowledge of quantum field theory. This is particularly true when it comes to strongly coupled theories. They constitute the low energy limit of all relativistic quantum field theories that are not gapped and, as such, they universally describe physical systems of interest. As relevant as their physical applications are the mathematical properties of CFTs; these properties allow for remarkable progress in their precise understanding. One line of research where the power of CFTs has become manifest is the bootstrap program [1–4]. It has been understood that the analytic properties of four point functions in CFTs contain basic information that constrain the theory severely. It is even possible that minimal data might be used to fully determine the complete landscape of non-trivial CFTs. This would be a huge triumph of theoretical physics and might solve some important puzzles like the nature of the elusive (2,0) maximally supersymmetric CFT in six dimensions [5]. An even more powerful case is that of two dimensional CFTs. It is well known that here the conformal group is upgraded to two sets of infinite dimensional Virasoro algebras [6]. –1– JHEP11(2015)129 1 Introduction –2– JHEP11(2015)129 This allows for great mathematical control of these theories. It is possible, for example, to understand the general properties of thermal states and finite volume vacua from a simple minded analysis of the quantum theory on the two dimensional euclidean plane. Furthermore, the bootstrap program mentioned above can be fully carried out for two dimensional CFTs with central charge c < 1 and, in this case, the landscape of unitary CFTs is completely understood [7]. There is one extra feature of two dimensional CFTs which gives them great power and mathematical structure. This is the concept of modular invariance. CFTs defined on the torus have very interesting mathematical properties which strongly constrain the form of the partition function. Modular invariance is the statement that a CFT can be quantized on any of the two non trivial cycles of the torus and the resulting partition function should not be affected. While one can’t prove this feature from first principles — it is necessary to add it as an extra postulate — this is a very natural property that is suggested from the path integral formulation of the quantum theory. A striking consequence of modular invariance is that the asymptotic density of states at high energies is fully determined by the central charges. This is known as the Cardy formula [8]. It is natural to wonder: are there other quantum field theories with a similar behavior? Interestingly, it has proven very hard to extend the concept of modular invariance to more general types of quantum field theories. See, however, [9–11] for a discussion in CFTs in more than two dimensions and [12, 13] for some results in non relativistic field theories. Because of the great power of modular invariance, it would be of importance to understand larger families of quantum field theories where such a concept is available. There is of course one more crucial reason why CFTs have played such an important role in theoretical physics; namely, the AdS/CFT correspondence [14–16]. It has been long understood that quantum theories of gravity on Anti de Sitter (AdS) space-times are dual to CFTs in one less dimension. One of the main reasons to suspect the existence of such holographic principle is the fact that the entropy of black holes scales with their area and not their volume as one might have expected. This fact is, however, very universal and not connected directly to the peculiarities of AdS space-times. The obvious question then becomes: how do we understand holography in non-AdS space-times? Actually the questions raised above are heavily interconnected. One important clue in this direction was the observation that, in many holographic setups, the entropy of black holes seemed to present a Cardy-like behavior if understood as accounting for the asymptotic density of states of dual quantum field theories. The most interesting cases are given by (near) extremal rotating Kerr black holes [17, 18] and warped AdS3 space-times [19, 20]. Interestingly, these spaces do not posses the local SL(2, R)×SL(2, R) isometry group typical of AdS3 holography which implies the Cardy scaling. It was understood in [21] that a class of two dimensional non-relativistic theories indeed has enough structure: Warped Conformal Field Theories (WCFTs), first described in [22], can account for this Cardy scaling and, moreover, enjoy the global SL(2, R) × U(1) symmetry group manifest in the holographic setups mentioned above. WCFTs exhibit an infinite dimensional Virasoro-Kac-Moody u(1) algebra. As such they are in principle as powerful as traditional two dimensional CFTs. It was later understood in [23] what is a natural holographic description of WCFTs. –3– JHEP11(2015)129 (...truncated)