Abstract
We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e. when the partition functions are “flavored”. We begin with a new proof of the transformation law for the modular transformation of such partition functions. Then we proceed to apply modular bootstrap techniques to constrain the spectrum of charged states in the theory. We improve previous upper bounds on the state with the greatest “mass-to-charge” ratio in such theories, as well as upper bounds on the weight of the lightest charged state and the charge of the weakest charged state in the theory. We apply the extremal functional method to theories that saturate such bounds, and in several cases we find the resulting prediction for the occupation numbers are precisely integers. Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we find that adding flavor allows the extremal functional method to solve for some partition functions that would not be accessible to it otherwise.
Highlights
Modular invariance is a powerful tool for studying two-dimensional Conformal Field Theories (CFTs)
We study the implications of modular invariance on 2d CFT partition functions with abelian or non-abelian currents when chemical potentials for the charges are turned on, i.e. when the partition functions are “flavored”
Because such theories sometimes do not saturate a bound on the full space of states but do saturate a bound in the neutral sector of states, we find that adding flavor allows the extremal functional method to solve for some partition functions that would not be accessible to it otherwise
Summary
Modular invariance is a powerful tool for studying two-dimensional Conformal Field Theories (CFTs). The resulting partition function is no longer modular invariant, but has a welldefined and theory-independent transformation law [5]: aτ + b cz This transformation law was used to constrain the spectrum of charges in general 2d CFTs in [6]. When the symmetry current Ja is non-abelian, it is more appropriate to consider bounds on the dimensions of different representations in the theory. Perhaps the most interesting aspect of this analysis is not the specific partition function for this case, but rather that fact that searching for constraints in a representation dependent manner yields structure hidden to a flavor-blind analysis. After this work was completed, the paper [11] appeared on arXiv considering modular bootstrap constraints on theories with conserved currents, though the analysis there did not use the flavored partition function
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
