Exact generalized partition function of 2D CFTs at large central charge
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Springer
Received: December 29, 2018 Accepted: May 7, 2019 Published: May 15, 2019
Anatoly Dymarskya,b and Kirill Pavlenkob,c a
Department of Physics and Astronomy, University of Kentucky, Lexington, KY, 40506, U.S.A. b Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Bolshoy Boulevard 30, bld. 1, Moscow 121205, Russia c Moscow Institute of Physics and Technology, Institutsky per. 9, Dolgoprudny 141700, Russia
E-mail: [email protected], [email protected] Abstract: We discuss generalized partition function of 2d CFTs on thermal cylinder decorated by higher qKdV charges. We propose that in the large central charge limit qKdV charges factorize such that generalized partition function can be rewritten in terms of auxiliary non-interacting bosons. The explicit expression for the generalized free energy is readily available in terms of the boson spectrum, which can be deduced from the conventional thermal expectation values of qKdV charges. In other words, the picture of the auxiliary non-interacting bosons allows extending thermal one-point functions to the full non-perturbative generalized partition function. We verify this conjecture for the first seven qKdV charges using recently obtained pertrubative results and find corresponding contributions to the auxiliary boson masses. We further extend these results by conjecturing the full spectrum of bosons and find an exact expression for the generalized partition function as a function of infinite tower of chemical potentials in the limit of large central charge. Keywords: Conformal and W Symmetry, Conformal Field Theory, Integrable Field Theories ArXiv ePrint: 1812.05108
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)077
JHEP05(2019)077
Exact generalized partition function of 2D CFTs at large central charge
Contents 1
2 Thermal average of Q2k−1 2.1 Thermodynamic limit 2.2 Q1 2.3 Q3 2.4 Q5 2.5 Q7 2.6 Q9 2.7 Q11 , Q13 , and beyond
3 3 4 4 5 5 6 6
3 Generalized partition function
6
4 Discussion
8
A Alternative representation of the partition function
9
B 1/c versus 1/c0 expansion
9
1
Introduction
Generalized partition function of 2d CFTs decorated by higher qKdV charges [1–3], the so-called Generalized Gibbs Ensemble, ( ∞ ) X Z = Tr exp − µ2k−1 Q2k−1 , µ1 ≡ β, Q1 ≡ H, (1.1) k=1
has been in the focus of attention recently in the context of thermalization of large c 2d conformal theories [4–15]. In this work we assume thermodynamic limit, when the size of the spatial circle goes to infinity ` → ∞ and (1.1) describes theory on a thermal cylinder. In a recent work [15] we observed that in the large central charge limit first two nontrivial qKdV charges Q3 , Q5 admit simple structure. Namely, ˜ 2k−1 , `2k−1 Q2k−1 = Pk (L0 ) + `2k−1 Q
(1.2)
˜ 2k−1 accounts where Pk is a polynomial of degree k, Pk (L0 ) = Lk0 + . . . , and the operator Q for the rest. Written in the conventional basis of conformal theory (sets {mi }, m1 ≥ m2 , . . . , ≥ mk , are arranged in dominance order), |
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