Calabi-Yau CFTs and random matrices

Afkhami-Jeddi, Nima; Ashmore, Anthony; Córdova, Clay

doi:10.1007/jhep02(2022)021

Title: Calabi-Yau CFTs and random matrices

Journal Article · 03 February 2022 · Journal of High Energy Physics (Online)

DOI: https://doi.org/10.1007/jhep02(2022)021 · OSTI ID:1976443

Afkhami-Jeddi, Nima ^[1];

^[2]; Córdova, Clay ^[3]

University of Chicago, IL (United States). Enrico Fermi Institute and Kadanoff Center for Theoretical Physics; Univ. of Chicago, IL (United States)
University of Chicago, IL (United States). Enrico Fermi Institute and Kadanoff Center for Theoretical Physics; Sorbonne University, Paris (France)
University of Chicago, IL (United States). Enrico Fermi Institute and Kadanoff Center for Theoretical Physics

Using numerical methods for finding Ricci-flat metrics, we explore the spectrum of local operators in two-dimensional conformal field theories defined by sigma models on Calabi-Yau targets at large volume. Focusing on the examples of K3 and the quintic, we show that the spectrum, averaged over a region in complex structure moduli space, possesses the same statistical properties as the Gaussian orthogonal ensemble of random matrix theory.

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Research Organization:: University of Chicago, IL (United States)

Sponsoring Organization:: European Union’s Horizon 2020; Marie Skłodowska-Curie Grant; USDOE Office of Science (SC)

Grant/Contract Number:: SC0021432

OSTI ID:: 1976443

Journal Information:: Journal of High Energy Physics (Online), Journal Name: Journal of High Energy Physics (Online) Journal Issue: 2 Vol. 2022; ISSN 1029-8479

Publisher:: Springer NatureCopyright Statement

Country of Publication:: United States

Language:: English

References (91)

Machine Learning Calabi–Yau Metrics Ashmore, Anthony; He, Yang‐Hui; Ovrut, Burt A. Fortschritte der Physik, Vol. 68, Issue 9 https://doi.org/10.1002/prop.202000068	journal	August 2020
Quantum Signatures of Chaos Haake, Fritz Springer Series in Synergetics https://doi.org/10.1007/978-3-642-05428-0	book	January 2010
On the ergodic properties of nowhere dispersing billiards Bunimovich, L. A. Communications in Mathematical Physics, Vol. 65, Issue 3 https://doi.org/10.1007/BF01197884	journal	October 1979
Geometrical structure and ultraviolet finiteness in the supersymmetric ?-model Alvarez-Gaume, Luis; Freedman, Daniel Z. Communications in Mathematical Physics, Vol. 80, Issue 3 https://doi.org/10.1007/BF01208280	journal	September 1981
Finiteness of Ricci flatN=2 supersymmetric σ-models Alvarez-Gaumé, L.; Coleman, S.; Ginsparg, P. Communications in Mathematical Physics, Vol. 103, Issue 3 https://doi.org/10.1007/BF01211757	journal	September 1986
Finiteness of Ricci flat supersymmetric non-linear σ-models Alvarez-Gaumé, L.; Ginsparg, P. Communications in Mathematical Physics, Vol. 102, Issue 2 https://doi.org/10.1007/BF01229382	journal	June 1985
Free partition functions and an averaged holographic duality Afkhami-Jeddi, Nima; Cohn, Henry; Hartman, Thomas Journal of High Energy Physics, Vol. 2021, Issue 1 https://doi.org/10.1007/JHEP01(2021)130	journal	January 2021
The wasteland of random supergravities Marsh, David; McAllister, Liam; Wrase, Timm Journal of High Energy Physics, Vol. 2012, Issue 3 https://doi.org/10.1007/JHEP03(2012)102	journal	March 2012
Special geometry on the 101 dimesional moduli space of the quintic threefold Aleshkin, Konstantin; Belavin, Alexander Journal of High Energy Physics, Vol. 2018, Issue 3 https://doi.org/10.1007/JHEP03(2018)018	journal	March 2018
Exact Kähler potential from gauge theory and mirror symmetry Gomis, Jaume; Lee, Sungjay Journal of High Energy Physics, Vol. 2013, Issue 4 https://doi.org/10.1007/JHEP04(2013)019	journal	April 2013
AdS3 gravity and random CFT Cotler, Jordan; Jensen, Kristan Journal of High Energy Physics, Vol. 2021, Issue 4 https://doi.org/10.1007/JHEP04(2021)033	journal	April 2021
Exact results in D = 2 supersymmetric gauge theories Doroud, Nima; Gomis, Jaume; Le Floch, Bruno Journal of High Energy Physics, Vol. 2013, Issue 5 https://doi.org/10.1007/JHEP05(2013)093	journal	May 2013
(2, 2) superconformal bootstrap in two dimensions Lin, Ying-Hsuan; Shao, Shu-Heng; Wang, Yifan Journal of High Energy Physics, Vol. 2017, Issue 5 https://doi.org/10.1007/JHEP05(2017)112	journal	May 2017
N $$ \mathcal{N} $$ = 4 superconformal bootstrap of the K3 CFT Lin, Ying-Hsuan; Shao, Shu-Heng; Simmons-Duffin, David Journal of High Energy Physics, Vol. 2017, Issue 5 https://doi.org/10.1007/JHEP05(2017)126	journal	May 2017
Numerical metrics, curvature expansions and Calabi-Yau manifolds Cui, Wei; Gray, James Journal of High Energy Physics, Vol. 2020, Issue 5 https://doi.org/10.1007/JHEP05(2020)044	journal	May 2020
Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning Anderson, Lara B.; Gerdes, Mathis; Gray, James Journal of High Energy Physics, Vol. 2021, Issue 5 https://doi.org/10.1007/JHEP05(2021)013	journal	May 2021
The refined Swampland Distance Conjecture in Calabi-Yau moduli spaces Blumenhagen, Ralph; Klaewer, Daniel; Schlechter, Lorenz Journal of High Energy Physics, Vol. 2018, Issue 6 https://doi.org/10.1007/JHEP06(2018)052	journal	June 2018
Onset of random matrix behavior in scrambling systems Gharibyan, Hrant; Hanada, Masanori; Shenker, Stephen H. Journal of High Energy Physics, Vol. 2018, Issue 7 https://doi.org/10.1007/JHEP07(2018)124	journal	July 2018
Refined swampland distance conjecture and exotic hybrid Calabi-Yaus Erkinger, David; Knapp, Johanna Journal of High Energy Physics, Vol. 2019, Issue 7 https://doi.org/10.1007/JHEP07(2019)029	journal	July 2019
Universal dynamics of heavy operators in CFT2 Collier, Scott; Maloney, Alexander; Maxfield, Henry Journal of High Energy Physics, Vol. 2020, Issue 7 https://doi.org/10.1007/JHEP07(2020)074	journal	July 2020
A universal inequality for CFT and quantum gravity Hellerman, Simeon Journal of High Energy Physics, Vol. 2011, Issue 8 https://doi.org/10.1007/JHEP08(2011)130	journal	August 2011
2D CFT partition functions at late times Dyer, Ethan; Gur-Ari, Guy Journal of High Energy Physics, Vol. 2017, Issue 8 https://doi.org/10.1007/JHEP08(2017)075	journal	August 2017
Modular bootstrap revisited Collier, Scott; Lin, Ying-Hsuan; Yin, Xi Journal of High Energy Physics, Vol. 2018, Issue 9 https://doi.org/10.1007/JHEP09(2018)061	journal	September 2018
Constraints on 2d CFT partition functions Friedan, Daniel; Keller, Christoph A. Journal of High Energy Physics, Vol. 2013, Issue 10 https://doi.org/10.1007/JHEP10(2013)180	journal	October 2013
Bootstrap bounds on closed Einstein manifolds Bonifacio, James; Hinterbichler, Kurt Journal of High Energy Physics, Vol. 2020, Issue 10 https://doi.org/10.1007/JHEP10(2020)069	journal	October 2020
Averaging over Narain moduli space Maloney, Alexander; Witten, Edward Journal of High Energy Physics, Vol. 2020, Issue 10 https://doi.org/10.1007/JHEP10(2020)187	journal	October 2020
Chaos, complexity, and random matrices Cotler, Jordan; Hunter-Jones, Nicholas; Liu, Junyu Journal of High Energy Physics, Vol. 2017, Issue 11 https://doi.org/10.1007/JHEP11(2017)048	journal	November 2017
Unitarization from geometry Bonifacio, James; Hinterbichler, Kurt Journal of High Energy Physics, Vol. 2019, Issue 12 https://doi.org/10.1007/JHEP12(2019)165	journal	December 2019
Critical Points and Supersymmetric Vacua I Douglas, Michael R.; Shiffman, Bernard; Zelditch, Steve Communications in Mathematical Physics, Vol. 252, Issue 1-3 https://doi.org/10.1007/s00220-004-1228-y	journal	October 2004
An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces Strohmaier, Alexander; Uski, Ville Communications in Mathematical Physics, Vol. 317, Issue 3 https://doi.org/10.1007/s00220-012-1557-1	journal	August 2012
Modular Constraints on Calabi-Yau Compactifications Keller, Christoph A.; Ooguri, Hirosi Communications in Mathematical Physics, Vol. 324, Issue 1 https://doi.org/10.1007/s00220-013-1797-8	journal	September 2013
Two-Sphere Partition Functions and Gromov–Witten Invariants Jockers, Hans; Kumar, Vijay; Lapan, Joshua M. Communications in Mathematical Physics, Vol. 325, Issue 3 https://doi.org/10.1007/s00220-013-1874-z	journal	January 2014
Elliptic Genera of Two-Dimensional $${\mathcal{N} = 2}$$ N = 2 Gauge Theories with Rank-One Gauge Groups Benini, Francesco; Eager, Richard; Hori, Kentaro Letters in Mathematical Physics, Vol. 104, Issue 4 https://doi.org/10.1007/s11005-013-0673-y	journal	November 2013
The background field method and the ultraviolet structure of the supersymmetric nonlinear σ-model Alvarez-Gaumé, Luis; Freedman, Daniel Z.; Mukhi, Sunil Annals of Physics, Vol. 134, Issue 1 https://doi.org/10.1016/0003-4916(81)90006-3	journal	June 1981
Chaos on the pseudosphere Balazs, N. L.; Voros, A. Physics Reports, Vol. 143, Issue 3 https://doi.org/10.1016/0370-1573(86)90159-6	journal	November 1986
Conformal invariance of supersymmetric σ-models on Calabi-Yau manifolds Nemeschansky, D.; Sen, A. Physics Letters B, Vol. 178, Issue 4 https://doi.org/10.1016/0370-2693(86)91394-8	journal	October 1986
Character formulas for the N = 4 superconformal algebra Eguchi, Tohru; Taormina, Anne Physics Letters B, Vol. 200, Issue 3 https://doi.org/10.1016/0370-2693(88)90778-2	journal	January 1988
Constraints on supersymmetry breaking Witten, Edward Nuclear Physics B, Vol. 202, Issue 2 https://doi.org/10.1016/0550-3213(82)90071-2	journal	July 1982
Ultra-violet finiteness of supersymmetric non-linear sigma models Hull, C. M. Nuclear Physics B, Vol. 260, Issue 1 https://doi.org/10.1016/0550-3213(85)90317-7	journal	October 1985
Superstring modifications of Einstein's equations Gross, David J.; Witten, Edward Nuclear Physics B, Vol. 277 https://doi.org/10.1016/0550-3213(86)90429-3	journal	January 1986
Operator content of two-dimensional conformally invariant theories Cardy, John L. Nuclear Physics B, Vol. 270 https://doi.org/10.1016/0550-3213(86)90552-3	journal	January 1986
Superconformal algebras and string compactification on manifolds with SU(n) holonomy Eguchi, Tohru; Ooguri, Hirosi; Taormina, Anne Nuclear Physics B, Vol. 315, Issue 1 https://doi.org/10.1016/0550-3213(89)90454-9	journal	March 1989
A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory Candelas, Philip; De La Ossa, Xenia C.; Green, Paul S. Nuclear Physics B, Vol. 359, Issue 1 https://doi.org/10.1016/0550-3213(91)90292-6	journal	July 1991
A new supersymmetric index Cecotti, Sergio; Fendley, Paul; Intriligator, Ken Nuclear Physics B, Vol. 386, Issue 2 https://doi.org/10.1016/0550-3213(92)90572-S	journal	November 1992
Elliptic genera and N = 2 superconformal field theory Kawai, Toshiya; Yamada, Yasuhiko; Yang, Sung-Kil Nuclear Physics B, Vol. 414, Issue 1-2 https://doi.org/10.1016/0550-3213(94)90428-6	journal	February 1994
Random-matrix theories in quantum physics: common concepts Guhr, Thomas; Müller–Groeling, Axel; Weidenmüller, Hans A. Physics Reports, Vol. 299, Issue 4-6 https://doi.org/10.1016/S0370-1573(97)00088-4	journal	June 1998
Counting dyons in N = 4 string theory Dijkgraaf, Robbert; Verlinde, Erik; Verlinde, Herman Nuclear Physics B, Vol. 484, Issue 3 https://doi.org/10.1016/S0550-3213(96)00640-2	journal	January 1997
Supersymmetry and the cohomology of (hyper) Kähler manifolds Figueroa-O'Farrill, J. M.; Köhl, C.; Spence, B. Nuclear Physics B, Vol. 503, Issue 3 https://doi.org/10.1016/S0550-3213(97)00548-8	journal	October 1997
Numerical Weil–Petersson metrics on moduli spaces of Calabi–Yau manifolds Keller, Julien; Lukic, Sergio Journal of Geometry and Physics, Vol. 92 https://doi.org/10.1016/j.geomphys.2015.02.018	journal	June 2015
Spectral properties of the Laplacian and random matrix theories Bohigas, O.; Giannoni, M. J.; Schmit, C. Journal de Physique Lettres, Vol. 45, Issue 21 https://doi.org/10.1051/jphyslet:0198400450210101500	journal	January 1984
Periodic Orbits and Classical Quantization Conditions Gutzwiller, Martin C. Journal of Mathematical Physics, Vol. 12, Issue 3 https://doi.org/10.1063/1.1665596	journal	March 1971
Numerical Calabi–Yau metrics Douglas, Michael R.; Karp, Robert L.; Lukic, Sergio Journal of Mathematical Physics, Vol. 49, Issue 3 https://doi.org/10.1063/1.2888403	journal	March 2008
Numerical Ricci-flat metrics on K3 Headrick, Matthew; Wiseman, Toby Classical and Quantum Gravity, Vol. 22, Issue 23 https://doi.org/10.1088/0264-9381/22/23/002	journal	November 2005
Is There Really a de Sitter/CFT Duality Dyson, Lisa; Lindesay, James; Susskind, Leonard Journal of High Energy Physics, Vol. 2002, Issue 08 https://doi.org/10.1088/1126-6708/2002/08/045	journal	August 2002
Eternal black holes in anti-de Sitter Maldacena, Juan Journal of High Energy Physics, Vol. 2003, Issue 04 https://doi.org/10.1088/1126-6708/2003/04/021	journal	April 2003
The statistics of string/M theory vacua Douglas, Michael R. Journal of High Energy Physics, Vol. 2003, Issue 05 https://doi.org/10.1088/1126-6708/2003/05/046	journal	May 2003
Very long time scales and black hole thermal equilibrium Barbón, José L. F.; Rabinovici, Eliezer Journal of High Energy Physics, Vol. 2003, Issue 11 https://doi.org/10.1088/1126-6708/2003/11/047	journal	November 2003
Counting Flux Vacua Ashok, Sujay K.; Douglas, Michael R. Journal of High Energy Physics, Vol. 2004, Issue 01 https://doi.org/10.1088/1126-6708/2004/01/060	journal	January 2004
Distributions of flux vacua Denef, Frederik; Douglas, Michael R. Journal of High Energy Physics, Vol. 2004, Issue 05 https://doi.org/10.1088/1126-6708/2004/05/072	journal	May 2004
Distributions of nonsupersymmetric flux vacua Denef, Frederik; Douglas, Michael R. Journal of High Energy Physics, Vol. 2005, Issue 03 https://doi.org/10.1088/1126-6708/2005/03/061	journal	March 2005
Calabi-Yau metrics for quotients and complete intersections Braun, Volker; Brelidze, Tamaz; Douglas, Michael R. Journal of High Energy Physics, Vol. 2008, Issue 05 https://doi.org/10.1088/1126-6708/2008/05/080	journal	May 2008
Eigenvalues and eigenfunctions of the scalar Laplace operator on Calabi-Yau manifolds Braun, Volker; Brelidze, Tamaz; Douglas, Michael R. Journal of High Energy Physics, Vol. 2008, Issue 07 https://doi.org/10.1088/1126-6708/2008/07/120	journal	July 2008
Generalized Berezin quantization, Bergman metrics and fuzzy laplacians Lazaroiu, Calin Iuliu; McNamee, Daniel; Sämann, Christian Journal of High Energy Physics, Vol. 2008, Issue 09 https://doi.org/10.1088/1126-6708/2008/09/059	journal	September 2008
Disorder on the landscape Podolsky, Dmitry; Majumder, Jaydeep; Jokela, Niko Journal of Cosmology and Astroparticle Physics, Vol. 2008, Issue 05 https://doi.org/10.1088/1475-7516/2008/05/024	journal	May 2008
A new approach for computing the geometry of the moduli spaces for a Calabi–Yau manifold Aleshkin, Konstantin; Belavin, Alexander Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 5 https://doi.org/10.1088/1751-8121/aa9e7a	journal	January 2018
Ergodic theory and the geodesic flow on surfaces of constant negative curvature Hopf, Eberhard Bulletin of the American Mathematical Society, Vol. 77, Issue 6 https://doi.org/10.1090/S0002-9904-1971-12799-4	journal	November 1971
Mathematics of Quantum Chaos in 2019 Zelditch, Steve Notices of the American Mathematical Society, Vol. 66, Issue 09 https://doi.org/10.1090/noti1958	journal	October 2019
Level Clustering in the Regular Spectrum Berry, M. V.; Tabor, M. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 356, Issue 1686 https://doi.org/10.1098/rspa.1977.0140	journal	September 1977
"Repulsion of Energy Levels" in Complex Atomic Spectra Rosenzweig, Norbert; Porter, Charles E. Physical Review, Vol. 120, Issue 5 https://doi.org/10.1103/PhysRev.120.1698	journal	December 1960
"Repulsion of Energy Levels" in Complex Atomic Spectra Trees, R. E. Physical Review, Vol. 123, Issue 4 https://doi.org/10.1103/PhysRev.123.1293	journal	August 1961
Random matrices, fractional statistics, and the quantum Hall effect Callaway, David J. E. Physical Review B, Vol. 43, Issue 10 https://doi.org/10.1103/PhysRevB.43.8641	journal	April 1991
Sachdev-Ye-Kitaev model and thermalization on the boundary of many-body localized fermionic symmetry-protected topological states You, Yi-Zhuang; Ludwig, Andreas W. W.; Xu, Cenke Physical Review B, Vol. 95, Issue 11 https://doi.org/10.1103/PhysRevB.95.115150	journal	March 2017
Moduli-dependent KK towers and the swampland distance conjecture on the quintic Calabi-Yau manifold Ashmore, Anthony; Ruehle, Fabian Physical Review D, Vol. 103, Issue 10 https://doi.org/10.1103/PhysRevD.103.106028	journal	May 2021
Kahler geometry and the renormalization of supersymmetricσmodels Alvarez-Gaumé, Luis; Freedman, Daniel Z. Physical Review D, Vol. 22, Issue 4 https://doi.org/10.1103/PhysRevD.22.846	journal	August 1980
Kaluza-Klein towers on general manifolds Hinterbichler, Kurt; Levin, Janna; Zukowski, Claire Physical Review D, Vol. 89, Issue 8 https://doi.org/10.1103/PhysRevD.89.086007	journal	April 2014
Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model García-García, Antonio M.; Verbaarschot, Jacobus J. M. Physical Review D, Vol. 94, Issue 12 https://doi.org/10.1103/PhysRevD.94.126010	journal	December 2016
Spectral form factor in a random matrix theory Brézin, E.; Hikami, S. Physical Review E, Vol. 55, Issue 4 https://doi.org/10.1103/PhysRevE.55.4067	journal	April 1997
Local Operators in the Eternal Black Hole Papadodimas, Kyriakos; Raju, Suvrat Physical Review Letters, Vol. 115, Issue 21 https://doi.org/10.1103/PhysRevLett.115.211601	journal	November 2015
Nonlinear Models in 2 + ε Dimensions Friedan, D. Physical Review Letters, Vol. 45, Issue 13 https://doi.org/10.1103/PhysRevLett.45.1057	journal	September 1980
Fluctuation Properties of Nuclear Energy Levels: Do Theory and Experiment Agree? Haq, R. U.; Pandey, A.; Bohigas, O. Physical Review Letters, Vol. 48, Issue 16 https://doi.org/10.1103/PhysRevLett.48.1086	journal	April 1982
Characterization of Chaotic Quantum Spectra and Universality of Level Fluctuation Laws Bohigas, O.; Giannoni, M. J.; Schmit, C. Physical Review Letters, Vol. 52, Issue 1 https://doi.org/10.1103/PhysRevLett.52.1	journal	January 1984
Arithmetical chaos and violation of universality in energy level statistics Bolte, J.; Steil, G.; Steiner, F. Physical Review Letters, Vol. 69, Issue 15 https://doi.org/10.1103/PhysRevLett.69.2188	journal	October 1992
Gapless spin-fluid ground state in a random quantum Heisenberg magnet Sachdev, Subir; Ye, Jinwu Physical Review Letters, Vol. 70, Issue 21 https://doi.org/10.1103/PhysRevLett.70.3339	journal	May 1993
The Spectral Form Factor Is Not Self-Averaging Prange, R. E. Physical Review Letters, Vol. 78, Issue 12 https://doi.org/10.1103/PhysRevLett.78.2280	journal	March 1997
Random-matrix physics: spectrum and strength fluctuations Brody, T. A.; Flores, J.; French, J. B. Reviews of Modern Physics, Vol. 53, Issue 3 https://doi.org/10.1103/RevModPhys.53.385	journal	July 1981
Exact Computation of the Special Geometry for Calabi–Yau Hypersurfaces of Fermat Type Aleshkin, K.; Belavin, A. JETP Letters, Vol. 108, Issue 10 https://doi.org/10.1134/S0021364018220010	journal	November 2018
On the Metrical Transitivity of the Geodesics on Closed Surfaces of Constant Negative Curvature Hedlund, Gustav A. The Annals of Mathematics, Vol. 35, Issue 4 https://doi.org/10.2307/1968495	journal	October 1934
Characteristic Vectors of Bordered Matrices With Infinite Dimensions Wigner, Eugene P. The Annals of Mathematics, Vol. 62, Issue 3 https://doi.org/10.2307/1970079	journal	November 1955
Geodesics on Calabi-Yau manifolds and winding states in non-linear sigma models Gao, Peng; Douglas, Michael R. Frontiers in Physics, Vol. 1 https://doi.org/10.3389/fphy.2013.00026	journal	January 2013
Energy functionals for Calabi-Yau metrics Headrick, Matthew; Nassar, Ali Advances in Theoretical and Mathematical Physics, Vol. 17, Issue 5 https://doi.org/10.4310/ATMP.2013.v17.n5.a1	journal	January 2013
Some Numerical Results in Complex Differential Geometry Donaldson, S. K. Pure and Applied Mathematics Quarterly, Vol. 5, Issue 2 https://doi.org/10.4310/PAMQ.2009.v5.n2.a2	journal	January 2009

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