Abstract
We study the free fermion theory in 1+1 dimensions deformed by chemical potentials for holomorphic, conserved currents at finite temperature and on a spatial circle. For a spin-three chemical potential μ, the deformation is related at high temperatures to a higher spin black hole in hs[0] theory on AdS3 spacetime. We calculate the order μ2 corrections to the single interval Rényi and entanglement entropies on the torus using the bosonized formulation. A consistent result, satisfying all checks, emerges upon carefully accounting for both perturbative and winding mode contributions in the bosonized language. The order μ2 corrections involve integrals that are finite but potentially sensitive to contact term singularities. We propose and apply a prescription for defining such integrals which matches the Hamiltonian picture and passes several non-trivial checks for both thermal corrections and the Rényi entropies at this order. The thermal corrections are given by a weight six quasi-modular form, whilst the Rényi entropies are controlled by quasi-elliptic functions of the interval length with modular weight six. We also point out the well known connection between the perturbative expansion of the partition function in powers of the spin-three chemical potential and the Gross-Taylor genus expansion of large-N Yang-Mills theory on the torus. We note the absence of winding mode contributions in this connection, which suggests qualitatively different entanglement entropies for the two systems.
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J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys. A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. (2004) P06002 [hep-th/0405152] [INSPIRE].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. (2005) P04010 [cond-mat/0503393] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and conformal field theory, J. Phys. A 42 (2009) 504005 [arXiv:0905.4013] [INSPIRE].
J.L. Cardy and P. Calabrese, Unusual Corrections to Scaling in Entanglement Entropy, J. Stat. Mech. (2010) P04023 [arXiv:1002.4353] [INSPIRE].
M.R. Douglas, Conformal field theory techniques in large-N Yang-Mills theory, hep-th/9311130 [INSPIRE].
R. Dijkgraaf, Chiral deformations of conformal field theories, Nucl. Phys. B 493 (1997) 588 [hep-th/9609022] [INSPIRE].
M. Gutperle and P. Kraus, Higher Spin Black Holes, JHEP 05 (2011) 022 [arXiv:1103.4304] [INSPIRE].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Spacetime Geometry in Higher Spin Gravity, JHEP 10 (2011) 053 [arXiv:1106.4788] [INSPIRE].
P. Kraus and E. Perlmutter, Partition functions of higher spin black holes and their CFT duals, JHEP 11 (2011) 061 [arXiv:1108.2567] [INSPIRE].
M.R. Gaberdiel, T. Hartman and K. Jin, Higher Spin Black Holes from CFT, JHEP 04 (2012) 103 [arXiv:1203.0015] [INSPIRE].
M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, Black holes in three dimensional higher spin gravity: A review, J. Phys. A 46 (2013) 214001 [arXiv:1208.5182] [INSPIRE].
M.R. Gaberdiel, K. Jin and E. Perlmutter, Probing higher spin black holes from CFT, JHEP 10 (2013) 045 [arXiv:1307.2221] [INSPIRE].
S.F. Prokushkin and M.A. Vasiliev, Higher spin gauge interactions for massive matter fields in 3-D AdS space-time, Nucl. Phys. B 545 (1999) 385 [hep-th/9806236] [INSPIRE].
I.R. Klebanov and A.M. Polyakov, AdS dual of the critical O(N) vector model, Phys. Lett. B 550 (2002) 213 [hep-th/0210114] [INSPIRE].
S. Giombi and X. Yin, Higher Spin Gauge Theory and Holography: The Three-Point Functions, JHEP 09 (2010) 115 [arXiv:0912.3462] [INSPIRE].
S. Giombi and X. Yin, Higher Spins in AdS and Twistorial Holography, JHEP 04 (2011) 086 [arXiv:1004.3736] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, An AdS 3 Dual for Minimal Model CFTs, Phys. Rev. D 83 (2011) 066007 [arXiv:1011.2986] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Minimal Model Holography, J. Phys. A 46 (2013) 214002 [arXiv:1207.6697] [INSPIRE].
S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Higher spin entanglement entropy from CFT, JHEP 06 (2014) 096 [arXiv:1402.0007] [INSPIRE].
S. Datta, J.R. David, M. Ferlaino and S.P. Kumar, Universal correction to higher spin entanglement entropy, Phys. Rev. D 90 (2014) 041903 [arXiv:1405.0015] [INSPIRE].
J. de Boer and J.I. Jottar, Entanglement Entropy and Higher Spin Holography in AdS 3, JHEP 04 (2014) 089 [arXiv:1306.4347] [INSPIRE].
M. Ammon, A. Castro and N. Iqbal, Wilson Lines and Entanglement Entropy in Higher Spin Gravity, JHEP 10 (2013) 110 [arXiv:1306.4338] [INSPIRE].
S. Datta, Relative entropy in higher spin holography, Phys. Rev. D 90 (2014) 126010 [arXiv:1406.0520] [INSPIRE].
A. Castro and E. Llabrés, Unravelling Holographic Entanglement Entropy in Higher Spin Theories, JHEP 03 (2015) 124 [arXiv:1410.2870] [INSPIRE].
J. Long, Higher Spin Entanglement Entropy, JHEP 12 (2014) 055 [arXiv:1408.1298] [INSPIRE].
B. Chen, J. Long and J.-j. Zhang, Holographic Rényi entropy for CFT with W symmetry, JHEP 04 (2014) 041 [arXiv:1312.5510] [INSPIRE].
C.N. Pope, L.J. Romans and X. Shen, The Complete Structure of W(Infinity), Phys. Lett. B 236 (1990) 173 [INSPIRE].
E. Bergshoeff, C.N. Pope, L.J. Romans, E. Sezgin and X. Shen, The Super W ∞ Algebra, Phys. Lett. B 245 (1990) 447 [INSPIRE].
C.N. Pope, Lectures on W algebras and W gravity, hep-th/9112076 [INSPIRE].
I. Bakas and E. Kiritsis, Bosonic Realization of a Universal W Algebra and Z(infinity) Parafermions, Nucl. Phys. B 343 (1990) 185 [Erratum ibid. B 350 (1991) 512] [INSPIRE].
J. de Boer and J.I. Jottar, Boundary Conditions and Partition Functions in Higher Spin AdS 3 /CFT 2, arXiv:1407.3844 [INSPIRE].
M. Henneaux and S.-J. Rey, Nonlinear W ∞ as Asymptotic Symmetry of Three-Dimensional Higher Spin Anti-de Sitter Gravity, JHEP 12 (2010) 007 [arXiv:1008.4579] [INSPIRE].
A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields, JHEP 11 (2010) 007 [arXiv:1008.4744] [INSPIRE].
A. Campoleoni, S. Fredenhagen and S. Pfenninger, Asymptotic W-symmetries in three-dimensional higher-spin gauge theories, JHEP 09 (2011) 113 [arXiv:1107.0290] [INSPIRE].
H. Afshar, M. Gary, D. Grumiller, R. Rashkov and M. Riegler, Semi-classical unitarity in 3-dimensional higher-spin gravity for non-principal embeddings, Class. Quant. Grav. 30 (2013) 104004 [arXiv:1211.4454] [INSPIRE].
J.R. David, M. Ferlaino and S.P. Kumar, Thermodynamics of higher spin black holes in 3D, JHEP 11 (2012) 135 [arXiv:1210.0284] [INSPIRE].
M. Ferlaino, T. Hollowood and S.P. Kumar, Asymptotic symmetries and thermodynamics of higher spin black holes in AdS3, Phys. Rev. D 88 (2013) 066010 [arXiv:1305.2011] [INSPIRE].
G. Compére, J.I. Jottar and W. Song, Observables and Microscopic Entropy of Higher Spin Black Holes, JHEP 11 (2013) 054 [arXiv:1308.2175] [INSPIRE].
G. Compére and W. Song, \( \mathcal{W} \) symmetry and integrability of higher spin black holes, JHEP 09 (2013) 144 [arXiv:1306.0014] [INSPIRE].
A. Perez, D. Tempo and R. Troncoso, Higher spin gravity in 3D: Black holes, global charges and thermodynamics, Phys. Lett. B 726 (2013) 444 [arXiv:1207.2844] [INSPIRE].
A. Perez, D. Tempo and R. Troncoso, Higher spin black hole entropy in three dimensions, JHEP 04 (2013) 143 [arXiv:1301.0847] [INSPIRE].
M. Henneaux, A. Perez, D. Tempo and R. Troncoso, Chemical potentials in three-dimensional higher spin anti-de Sitter gravity, JHEP 12 (2013) 048 [arXiv:1309.4362] [INSPIRE].
J. de Boer and J.I. Jottar, Thermodynamics of higher spin black holes in AdS 3, JHEP 01 (2014) 023 [arXiv:1302.0816] [INSPIRE].
T. Faulkner, The Entanglement Renyi Entropies of Disjoint Intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
T. Barrella, X. Dong, S.A. Hartnoll and V.L. Martin, Holographic entanglement beyond classical gravity, JHEP 09 (2013) 109 [arXiv:1306.4682] [INSPIRE].
S. Datta and J.R. David, Rényi entropies of free bosons on the torus and holography, JHEP 04 (2014) 081 [arXiv:1311.1218] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal field theory, Springer, New York, U.S.A. (1997), pg. 890.
N.J. Iles and G.M.T. Watts, Modular properties of characters of the W3 algebra, arXiv:1411.4039 [INSPIRE].
C. Itzykson and J.-M. Drouffe, Statistical Field Theory, Cambridge University Press, Cambridge, U.K. (1989).
M.R. Gaberdiel, K. Jin and W. Li, Perturbations of W ∞ CFTs, JHEP 10 (2013) 162 [arXiv:1307.4087] [INSPIRE].
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, U.K. (1927).
L. Griguolo, D. Seminara and R.J. Szabo, Two-dimensional Yang-Mills theory and moduli spaces of holomorphic differentials, Phys. Lett. B 600 (2004) 275 [hep-th/0408055] [INSPIRE].
D.J. Gross, Two-dimensional QCD as a string theory, Nucl. Phys. B 400 (1993) 161 [hep-th/9212149] [INSPIRE].
D.J. Gross and W. Taylor, Two-dimensional QCD is a string theory, Nucl. Phys. B 400 (1993) 181 [hep-th/9301068] [INSPIRE].
D.J. Gross and W. Taylor, Twists and Wilson loops in the string theory of two-dimensional QCD, Nucl. Phys. B 403 (1993) 395 [hep-th/9303046] [INSPIRE].
R.E. Rudd, The string partition function for QCD on the torus, hep-th/9407176 [INSPIRE].
R. Dijkgraaf, Mirror symmetry and elliptic curves, in The Moduli Space of Curves, Birkhäuser, Prog. Math. 129 (1995) 149.
M. Kaneko and D. Zagier, A generalized Jacobi theta function and quasimodular forms, in The Moduli Space of Curves, Birkhäuser, Prog. Math. 129 (1995) 165.
A. Velytsky, Entanglement entropy in d+1 SU(N) gauge theory, Phys. Rev. D 77 (2008) 085021 [arXiv:0801.4111] [INSPIRE].
A. Gromov and R.A. Santos, Entanglement Entropy in 2D Non-abelian Pure Gauge Theory, Phys. Lett. B 737 (2014) 60 [arXiv:1403.5035] [INSPIRE].
T. Azeyanagi, T. Nishioka and T. Takayanagi, Near Extremal Black Hole Entropy as Entanglement Entropy via AdS 2 /CFT 1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].
C.P. Herzog and T. Nishioka, Entanglement Entropy of a Massive Fermion on a Torus, JHEP 03 (2013) 077 [arXiv:1301.0336] [INSPIRE].
J.J. Atick, L.J. Dixon, P.A. Griffin and D. Nemeschansky, Multiloop Twist Field Correlation Functions for Z(N) Orbifolds, Nucl. Phys. B 298 (1988) 1 [INSPIRE].
N. Koblitz, Introduction to Elliptic Curves and Modular Forms, Springer, New York, U.S.A. (1984).
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Datta, S., David, J.R. & Kumar, S.P. Conformal perturbation theory and higher spin entanglement entropy on the torus. J. High Energ. Phys. 2015, 41 (2015). https://doi.org/10.1007/JHEP04(2015)041
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DOI: https://doi.org/10.1007/JHEP04(2015)041