Abstract
We investigate how topological entanglement of Chern-Simons theory is captured in a string theoretic realization. Our explorations are motivated by a desire to understand how quantum entanglement of low energy open string degrees of freedom is encoded in string theory (beyond the oft discussed classical gravity limit). Concretely, we realize the Chern-Simons theory as the worldvolume dynamics of topological D-branes in the topological A-model string theory on a Calabi-Yau target. Via the open/closed topological string duality one can map this theory onto a pure closed topological A-model string on a different target space, one which is related to the original Calabi-Yau geometry by a geometric/conifold transition. We demonstrate how to uplift the replica construction of Chern-Simons theory directly onto the closed string and show that it provides a meaningful definition of reduced density matrices in topological string theory. Furthermore, we argue that the replica construction commutes with the geometric transition, thereby providing an explicit closed string dual for computing reduced states, and Rényi and von Neumann entropies thereof. While most of our analysis is carried out for Chern-Simons on S3, the emergent picture is rather general. Specifically, we argue that quantum entanglement on the open string side is mapped onto quantum entanglement on the closed string side and briefly comment on the implications of our result for physical holographic theories where entanglement has been argued to be crucial ingredient for the emergence of classical geometry.
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References
R. Gopakumar and C. Vafa, On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys. 3 (1999) 1415 [hep-th/9811131] [INSPIRE].
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [INSPIRE].
M. Aganagic, M. Mariño and C. Vafa, All loop topological string amplitudes from Chern-Simons theory, Commun. Math. Phys. 247 (2004) 467 [hep-th/0206164] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [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].
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].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
M. Van Raamsdonk, Building up spacetime with quantum entanglement, Gen. Rel. Grav. 42 (2010) 2323 [arXiv:1005.3035] [INSPIRE].
J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781 [arXiv:1306.0533] [INSPIRE].
M. Rangamani and T. Takayanagi, Holographic entanglement entropy, Lect. Notes Phys. 931 (2017) pp.1 [arXiv:1609.01287] [INSPIRE].
X. Dong, Holographic entanglement entropy for general higher derivative gravity, JHEP 01 (2014) 044 [arXiv:1310.5713] [INSPIRE].
J. Camps, Generalized entropy and higher derivative Gravity, JHEP 03 (2014) 070 [arXiv:1310.6659] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [INSPIRE].
E. Witten, Open strings on the Rindler horizon, JHEP 01 (2019) 126 [arXiv:1810.11912] [INSPIRE].
S. He, T. Numasawa, T. Takayanagi and K. Watanabe, Notes on entanglement entropy in string theory, JHEP 05 (2015) 106 [arXiv:1412.5606] [INSPIRE].
V. Balasubramanian and O. Parrikar, Remarks on entanglement entropy in string theory, Phys. Rev. D 97 (2018) 066025 [arXiv:1801.03517] [INSPIRE].
A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090 [arXiv:1304.4926] [INSPIRE].
X. Dong, A. Lewkowycz and M. Rangamani, Deriving covariant holographic entanglement, JHEP 11 (2016) 028 [arXiv:1607.07506] [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
T. Faulkner, The entanglement Renyi entropies of disjoint intervals in AdS/CFT, arXiv:1303.7221 [INSPIRE].
D.L. Jafferis, A. Lewkowycz, J. Maldacena and S.J. Suh, Relative entropy equals bulk relative entropy, JHEP 06 (2016) 004 [arXiv:1512.06431] [INSPIRE].
M. Headrick, V.E. Hubeny, A. Lawrence and M. Rangamani, Causality & holographic entanglement entropy, JHEP 12 (2014) 162 [arXiv:1408.6300] [INSPIRE].
B. Czech, J.L. Karczmarek, F. Nogueira and M. Van Raamsdonk, The gravity dual of a density matrix, Class. Quant. Grav. 29 (2012) 155009 [arXiv:1204.1330] [INSPIRE].
A.C. Wall, Maximin surfaces and the strong subadditivity of the covariant holographic entanglement entropy, Class. Quant. Grav. 31 (2014) 225007 [arXiv:1211.3494] [INSPIRE].
X. Dong, D. Harlow and A.C. Wall, Reconstruction of bulk operators within the entanglement wedge in gauge-gravity duality, Phys. Rev. Lett. 117 (2016) 021601 [arXiv:1601.05416] [INSPIRE].
J. Cotler et al., Entanglement wedge reconstruction via universal recovery channels, Phys. Rev. X 9 (2019) 031011 [arXiv:1704.05839] [INSPIRE].
N. Engelhardt and A.C. Wall, Quantum extremal surfaces: holographic entanglement entropy beyond the classical regime, JHEP 01 (2015) 073 [arXiv:1408.3203] [INSPIRE].
X. Dong and A. Lewkowycz, Entropy, extremality, Euclidean variations and the equations of motion, JHEP 01 (2018) 081 [arXiv:1705.08453] [INSPIRE].
W. Donnelly and G. Wong, Entanglement branes in a two-dimensional string theory, JHEP 09 (2017) 097 [arXiv:1610.01719] [INSPIRE].
W. Donnelly and G. Wong, Entanglement branes, modular flow and extended topological quantum field theory, JHEP 10 (2019) 016 [arXiv:1811.10785] [INSPIRE].
E. Witten, Chern-Simons gauge theory as a string theory, Prog. Math. 133 (1995) 637 [hep-th/9207094] [INSPIRE].
J. Gomis and T. Okuda, Wilson loops, geometric transitions and bubbling Calabi-Yau’s, JHEP 02 (2007) 083 [hep-th/0612190] [INSPIRE].
J. Gomis and T. Okuda, D-branes as a bubbling Calabi-Yau, JHEP 07 (2007) 005 [arXiv:0704.3080] [INSPIRE].
H. Ooguri and C. Vafa, World sheet derivation of a large N duality, Nucl. Phys. B 641 (2002) 3 [hep-th/0205297] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting topological order in a ground state wave function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
A. Pakman and A. Parnachev, Topological entanglement entropy and holography, JHEP 07 (2008) 097 [arXiv:0805.1891] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the canonical quantization of the Chern-Simons-Witten theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological Entanglement Entropy in Chern-Simons Theories and Quantum Hall Fluids, JHEP 05 (2008) 016 [arXiv:0802.3231] [INSPIRE].
V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar, Multi-boundary entanglement in Chern-Simons theory and link invariants, JHEP 04 (2017) 061 [arXiv:1611.05460] [INSPIRE].
V. Balasubramanian et al., Entanglement entropy and the colored Jones polynomial, JHEP 05 (2018) 038 [arXiv:1801.01131] [INSPIRE].
L. McGough and H. Verlinde, Bekenstein-Hawking entropy as topological entanglement entropy, JHEP 11 (2013) 208 [arXiv:1308.2342] [INSPIRE].
E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].
M. Mariño, Chern-Simons theory and topological strings, Rev. Mod. Phys. 77 (2005) 675 [hep-th/0406005] [INSPIRE].
M. Mariño, Chern-Simons theory, matrix models and topological strings, Int. Ser. Monogr. Phys. 131 (2005) 1 [INSPIRE].
W.R. Lickorish, A representation of orientable combinatorial 3-manifolds, Ann. Math. 76 (1962) 531.
V.V. Prasolov and A.B. Sossinsky, Knots, links, braids, and 3-manifolds: an introduction to the new invariants in low-dimensional topology, American Mathematical Society, U.S.A. (1997).
X. Wen, S. Matsuura and S. Ryu, Edge theory approach to topological entanglement entropy, mutual information and entanglement negativity in Chern-Simons theories, Phys. Rev. B 93 (2016) 245140 [arXiv:1603.08534] [INSPIRE].
D. Das and S. Datta, Universal features of left-right entanglement entropy, Phys. Rev. Lett. 115 (2015) 131602 [arXiv:1504.02475] [INSPIRE].
X. Wen, P.-Y. Chang and S. Ryu, Topological entanglement negativity in Chern-Simons theories, JHEP 09 (2016) 012 [arXiv:1606.04118] [INSPIRE].
G. Wong, A note on entanglement edge modes in Chern Simons theory, JHEP 08 (2018) 020 [arXiv:1706.04666] [INSPIRE].
P.V. Buividovich and M.I. Polikarpov, Entanglement entropy in gauge theories and the holographic principle for electric strings, Phys. Lett. B 670 (2008) 141 [arXiv:0806.3376] [INSPIRE].
H. Casini, M. Huerta and J.A. Rosabal, Remarks on entanglement entropy for gauge fields, Phys. Rev. D 89 (2014) 085012 [arXiv:1312.1183] [INSPIRE].
W. Donnelly and A.C. Wall, Entanglement entropy of electromagnetic edge modes, Phys. Rev. Lett. 114 (2015) 111603 [arXiv:1412.1895] [INSPIRE].
R.M. Soni and S.P. Trivedi, Aspects of entanglement entropy for gauge theories, JHEP 01 (2016) 136 [arXiv:1510.07455] [INSPIRE].
S. Ghosh, R.M. Soni and S.P. Trivedi, On the entanglement entropy for gauge theories, JHEP 09 (2015) 069 [arXiv:1501.02593] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
P. Hayden et al., Holographic duality from random tensor networks, JHEP 11 (2016) 009 [arXiv:1601.01694] [INSPIRE].
B. Zwiebach, Closed string field theory: quantum action and the B-V master equation, Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
S.F. Moosavian and R. Pius, Hyperbolic geometry and closed bosonic string field theory. Part I. The string vertices via hyperbolic Riemann surfaces, JHEP 08 (2019) 157 [arXiv:1706.07366] [INSPIRE].
S.F. Moosavian and R. Pius, Hyperbolic geometry and closed bosonic string field theory. Part II. The rules for evaluating the quantum BV master action, JHEP 08 (2019) 177 [arXiv:1708.04977] [INSPIRE].
M. Kontsevich, Enumeration of rational curves via Torus actions, hep-th/9405035 [INSPIRE].
L. Evens and D.S. Kahn, Chern classes of certain representations of symmetric groups, Trans. Amer. Math. Soc. 245 (1978) 309.
W. Fulton and R. MacPherson, Characteristic classes of direct image bundles for covering maps, Ann. Math. 125 (1987) 1.
C.H. Taubes, Lagrangians for the Gopakumar-Vafa conjecture, Adv. Theor. Math. Phys. 5 (2001) 139 [math/0201219] [INSPIRE].
M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math. 68 (1989) 175 [INSPIRE].
D.V. Fursaev, Proof of the holographic formula for entanglement entropy, JHEP 09 (2006) 018 [hep-th/0606184] [INSPIRE].
D. Harlow, The Ryu–Takayanagi formula from quantum error correction, Commun. Math. Phys. 354 (2017) 865 [arXiv:1607.03901] [INSPIRE].
X. Dong, D. Harlow and D. Marolf, Flat entanglement spectra in fixed-area states of quantum gravity, arXiv:1811.05382 [INSPIRE].
C. Akers and P. Rath, Holographic Renyi entropy from quantum error correction, JHEP 05 (2019) 052 [arXiv:1811.05171] [INSPIRE].
R. Gopakumar and C. Vafa, Topological gravity as large N topological gauge theory, Adv. Theor. Math. Phys. 2 (1998) 413 [hep-th/9802016] [INSPIRE].
O. Aharony, G. Gur-Ari and R. Yacoby, d = 3 bosonic vector models coupled to Chern-Simons gauge theories, JHEP 03 (2012) 037 [arXiv:1110.4382] [INSPIRE].
S. Giombi et al., Chern-Simons theory with vector fermion matter, Eur. Phys. J. C 72 (2012) 2112 [arXiv:1110.4386] [INSPIRE].
M. Aganagic, K. Costello, J. McNamara and C. Vafa, Topological Chern-Simons/matter theories, arXiv:1706.09977 [INSPIRE].
O. Aharony, A. Feldman and M. Honda, A string dual for partially topological Chern-Simons-Matter theories, JHEP 06 (2019) 104 [arXiv:1903.06433] [INSPIRE].
G.W. Gibbons and S.W. Hawking, Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 [INSPIRE].
L. Susskind and J. Uglum, Black hole entropy in canonical quantum gravity and superstring theory, Phys. Rev. D 50 (1994) 2700 [hep-th/9401070] [INSPIRE].
E. Witten, Topological quantum field theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
K. Hori et al., Mirror symmetry, Clay mathematics monographs volume 1, American Mathematical Society, U.S.A. (2003).
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Hubeny, V.E., Pius, R. & Rangamani, M. Topological string entanglement. J. High Energ. Phys. 2019, 239 (2019). https://doi.org/10.1007/JHEP10(2019)239
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DOI: https://doi.org/10.1007/JHEP10(2019)239