Abstract
Zero modes of modular Hamiltonian of one interval are found in momentum space for two dimensional massless free scalar theory. Finite correlators are extracted from separate region connected correlation functions with the insertion of zero modes. Correlators of (n, 1)-type are claimed to be conformal block up to a set of theory dependent constants. We fix the correlators of (2, 1)-type with the coefficients of three point function in 2d CFTs.
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L. Bombelli, R.K. Koul, J. Lee and R.D. Sorkin, A Quantum Source of Entropy for Black Holes, Phys. Rev.D 34 (1986) 373 [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett.71 (1993) 666 [hep-th/9303048] [INSPIRE].
C.G. Callan Jr. and F. Wilczek, On geometric entropy, Phys. Lett.B 333 (1994) 55 [hep-th/9401072] [INSPIRE].
H. Araki, Relative Entropy of States of Von Neumann Algebras, Publ. Res. Inst. Math. Sci. Kyoto1976 (1976) 809 [INSPIRE].
R. Haag, Local quantum physics: fields, particles, algebras, Springer, Berlin, Germany (1992).
J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys.17 (1976) 303 [INSPIRE].
P.D. Hislop and R. Longo, Modular Structure of the Local Algebras Associated With the Free Massless Scalar Field Theory, Commun. Math. Phys.84 (1982) 71 [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
H. Casini and M. Huerta, Entanglement entropy in free quantum field theory, J. Phys.A 42 (2009) 504007 [arXiv:0905.2562] [INSPIRE].
D.L. Jafferis and S.J. Suh, The Gravity Duals of Modular Hamiltonians, JHEP09 (2016) 068 [arXiv:1412.8465] [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett.96 (2006) 181602 [hep-th/0603001] [INSPIRE].
D. Kabat and G. Lifschytz, Local bulk physics from intersecting modular Hamiltonians, JHEP06 (2017) 120 [arXiv:1703.06523] [INSPIRE].
T. Faulkner and A. Lewkowycz, Bulk locality from modular flow, JHEP07 (2017) 151 [arXiv:1704.05464] [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys.38 (1999) 1113 [hep-th/9711200] [INSPIRE].
T. Banks, M.R. Douglas, G.T. Horowitz and E.J. Martinec, AdS dynamics from conformal field theory, hep-th/9808016 [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett.B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys.2 (1998) 253 [hep-th/9802150] [INSPIRE].
J. Long, Correlation function of modular Hamiltonians, JHEP11 (2019) 163 [arXiv:1907.00646] [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys.B 241 (1984) 333 [INSPIRE].
E. Hijano, P. Kraus and R. Snively, Worldline approach to semi-classical conformal blocks, JHEP07 (2015) 131 [arXiv:1501.02260] [INSPIRE].
E. Hijano, P. Kraus, E. Perlmutter and R. Snively, Witten Diagrams Revisited: The AdS Geometry of Conformal Blocks, JHEP01 (2016) 146 [arXiv:1508.00501] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, U.S.A. (1997).
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A Stereoscopic Look into the Bulk, JHEP07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP08 (2016) 162 [arXiv:1606.03307] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys.B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys.B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
I. Bakas and E. Kiritsis, Bosonic Realization of a Universal W-Algebra and Z∞Parafermions, Nucl. Phys.B 343 (1990) 185 [Erratum ibid.B 350 (1991) 512] [INSPIRE].
J. Long, On co-dimension two defect operators, arXiv:1611.02485 [INSPIRE].
B. Chen and J. Long, Rényi mutual information for a free scalar field in even dimensions, Phys. Rev.D 96 (2017) 045006 [arXiv:1612.00114] [INSPIRE].
B. Chen, L. Chen, P.-x. Hao and J. Long, On the Mutual Information in Conformal Field Theory, JHEP06 (2017) 096 [arXiv:1704.03692] [INSPIRE].
A.B. Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal Quantum Field Theory, Theor. Math. Phys.65 (1985) 1205 [INSPIRE].
J. Long, Higher Spin Entanglement Entropy, JHEP12 (2014) 055 [arXiv:1408.1298] [INSPIRE].
I.S. Gradshteyn and I.M. Ryzhik, Table of integrals, series, and products, Seventh edition, Elsevier/Academic Press, Amsterdam, The Netherlands, (2007).
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Long, J. Correlation function with the insertion of zero modes of modular Hamiltonians. J. High Energ. Phys. 2020, 173 (2020). https://doi.org/10.1007/JHEP01(2020)173
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DOI: https://doi.org/10.1007/JHEP01(2020)173