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Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients
Ben Geloun, Joseph1; Ramgoolam, Sanjaye2
1 Laboratoire d’Informatique de Paris Nord UMR CNRS 7030 Université Sorbonne Paris Nord, 99, avenue J.-B. Clement, 93430 Villetaneuse, France. International Chair in Mathematical Physics and Applications, ICMPA–UNESCO Chair, 072 B.P. 50 Cotonou, Benin.
2 School of Physics and Astronomy Centre for Research in String Theory Queen Mary University of London London E1 4NS United Kingdom. School of Physics and Mandelstam Institute for Theoretical Physics, University of Witwatersrand, Wits, 2050, South Africa.
Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 547-594.

We define solvable quantum mechanical systems on a Hilbert space spanned by bipartite ribbon graphs with a fixed number of edges. The Hilbert space is also an associative algebra, where the product is derived from permutation group products. The existence and structure of this Hilbert space algebra has a number of consequences. The algebra product, which can be expressed in terms of integer ribbon graph reconnection coefficients, is used to define solvable Hamiltonians with eigenvalues expressed in terms of normalized characters of symmetric group elements and degeneracies given in terms of Kronecker coefficients, which are tensor product multiplicities of symmetric group representations. The square of the Kronecker coefficient for a triple of Young diagrams is shown to be equal to the dimension of a sub-lattice in the lattice of ribbon graphs. This leads to an answer to the long-standing question of a combinatorial interpretation of the Kronecker coefficients. As avenues for future research, we discuss applications of the ribbon graph quantum mechanics in algorithms for quantum computation. We also describe a quantum membrane interpretation of these quantum mechanical systems.

Received:
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Accepted:
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DOI: 10.5802/alco.254
Classification: 05E10, 05C85, 05A19, 06B99, 22D20
Keywords: Belyi maps, Kronecker coefficients, quantum physics, Ribbon graphs

Ben Geloun, Joseph 1; Ramgoolam, Sanjaye 2

1 Laboratoire d’Informatique de Paris Nord UMR CNRS 7030 Université Sorbonne Paris Nord, 99, avenue J.-B. Clement, 93430 Villetaneuse, France. International Chair in Mathematical Physics and Applications, ICMPA–UNESCO Chair, 072 B.P. 50 Cotonou, Benin.
2 School of Physics and Astronomy Centre for Research in String Theory Queen Mary University of London London E1 4NS United Kingdom. School of Physics and Mandelstam Institute for Theoretical Physics, University of Witwatersrand, Wits, 2050, South Africa.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Ben Geloun, Joseph; Ramgoolam, Sanjaye. Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients. Algebraic Combinatorics, Volume 6 (2023) no. 2, pp. 547-594. doi : 10.5802/alco.254. https://alco.centre-mersenne.org/articles/10.5802/alco.254/

[1] Ambjorn, Jan; Durhuus, Bergfinnur; Jonsson, Thordur Three-dimensional simplicial quantum gravity and generalized matrix models, Mod. Phys. Lett. A, Volume 6 (1991), pp. 1133-1146 | DOI | MR | Zbl

[2] Amburg, N.; Itoyama, H.; Mironov, Andrei; Morozov, Alexei; Vasiliev, D.; Yoshioka, R. Correspondence between Feynman diagrams and operators in quantum field theory that emerges from tensor model, Eur. Phys. J. C, Volume 80 (2020) no. 5, p. 471 | DOI

[3] Arute, Frank et al. Quantum supremacy using a programmable superconducting processor, Nature, Volume 574 (2019) no. 7779, pp. 505-510 | DOI

[4] Avohou, Remi C.; Ben Geloun, Joseph; Dub, Nicolas On the counting of O(N) tensor invariants, Adv. Theor. Math. Phys., Volume 24 (2020) no. 4, pp. 821-878 | DOI | MR | Zbl

[5] Balasubramanian, Vijay; Czech, Bartlomiej; Larjo, Klaus; Simon, Joan Integrability versus information loss: A Simple example, J. High Energy Phys., Volume 11 (2006), p. 001 | DOI

[6] Belyi, Gennadiĭ Vladimirovich On Galois Extensions of a Maximal Cyclotomic Field, Mathematics of The USSR-Izvestiya, Volume 14 (1980), pp. 247-256

[7] Ben Geloun, Joseph On the counting tensor model observables as U(N) and O(N) classical invariants, PoS, Volume CORFU2019 (2020), p. 175 | DOI

[8] Ben Geloun, Joseph; Ramgoolam, Sanjaye Counting tensor model observables and branched covers of the 2-sphere, Ann. Inst. H. Poincare D Comb. Phys. Interact., Volume 1 (2014) no. 1, pp. 77-138 | DOI | MR | Zbl

[9] Ben Geloun, Joseph; Ramgoolam, Sanjaye Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras, J. High Energy Phys., Volume 11 (2017), p. 092 | DOI | MR | Zbl

[10] Ben Geloun, Joseph; Ramgoolam, Sanjaye Quantum mechanics of bipartite ribbon graphs: Integrality, Lattices and Kronecker coefficients (2020) | arXiv | DOI

[11] Bhattacharyya, Rajsekhar; Collins, Storm; de Mello Koch, Robert Exact Multi-Matrix Correlators, J. High Energy Phys., Volume 03 (2008), p. 044 | DOI | MR

[12] Bhattacharyya, Rajsekhar; de Mello Koch, Robert; Stephanou, Michael Exact Multi-Restricted Schur Polynomial Correlators, J. High Energy Phys., Volume 06 (2008), p. 101 | DOI | MR

[13] Bonelli, Giulio; Tanzini, Alessandro; Zabzine, Maxim On topological M-theory, Adv. Theor. Math. Phys., Volume 10 (2006) no. 2, pp. 239-260 | DOI | Zbl

[14] Brown, Thomas William; Heslop, P. J.; Ramgoolam, S. Diagonal free field matrix correlators, global symmetries and giant gravitons, J. High Energy Phys., Volume 04 (2009), p. 089 | DOI | MR

[15] Burgisser, Peter; Christandl, Matthias; Ikenmeyer, Christian Nonvanishing of Kronecker coefficients for rectangular shapes, Adv. Math., Volume 227 (2011), pp. 2082-2091 | DOI | MR | Zbl

[16] Cameron, Peter J. Combinatorics: Topics, Techniques, Algorithms, Cambridge University Press, 1994 (reprinted 1996)

[17] Carrozza, Sylvain Large N limit of irreducible tensor models: O(N) rank-3 tensors with mixed permutation symmetry, J. High Energy Phys., Volume 06 (2018), p. 039 | DOI | MR | Zbl

[18] Carrozza, Sylvain; Tanasa, Adrian O(N) Random Tensor Models, Lett. Math. Phys., Volume 106 (2016) no. 11, pp. 1531-1559 | DOI | MR | Zbl

[19] Castro, Erick; Roditi, Itzhak A combinatorial matrix approach for the generation of vacuum Feynman graphs multiplicities in ϕ 4 theory, J. Phys. A, Volume 51 (2018) no. 39, p. 395202 | DOI | MR | Zbl

[20] Cohen, Henri A course in computational algebraic number theory, Graduate Texts in Mathematics, Springer Science & Business Media, 2000 | DOI

[21] Cordes, Stefan; Moore, Gregory W.; Ramgoolam, Sanjaye Lectures on 2-d Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. B Proc. Suppl., Volume 41 (1995), pp. 184-244 | DOI | MR | Zbl

[22] Corley, Steve; Jevicki, Antal; Ramgoolam, Sanjaye Exact correlators of giant gravitons from dual N=4 SYM theory, Adv. Theor. Math. Phys., Volume 5 (2002), pp. 809-839 | DOI | MR | Zbl

[23] Corteel, Sylvie; Goupil, Alain; Schaeffer, Gilles Content evaluation and class symmetric functions, Adv. Math., Volume 188 (2004), pp. 315-336 | DOI | MR | Zbl

[24] De Mello Koch, Robert; Gossman, David; Hasina Tahiridimbisoa, Nirina; Mahu, Augustine Larweh Holography for Tensor models, Phys. Rev. D, Volume 101 (2020) no. 4, p. 046004 | DOI | MR

[25] de Mello Koch, Robert; Gossman, David; Tribelhorn, Laila Gauge Invariants, Correlators and Holography in Bosonic and Fermionic Tensor Models, J. High Energy Phys., Volume 09 (2017), p. 011 | DOI | MR | Zbl

[26] de Mello Koch, Robert; Ramgoolam, Sanjaye From Matrix Models and Quantum Fields to Hurwitz Space and the absolute Galois Group (2010) | arXiv

[27] de Mello Koch, Robert; Ramgoolam, Sanjaye A double coset ansatz for integrability in AdS/CFT, J. High Energy Phys., Volume 06 (2012), p. 083 | DOI | MR | Zbl

[28] de Mello Koch, Robert; Ramgoolam, Sanjaye Strings from Feynman Graph counting : without large N, Phys. Rev. D, Volume 85 (2012), p. 026007 | DOI

[29] de Wit, B.; Hoppe, J.; Nicolai, H. On the Quantum Mechanics of Supermembranes, Nucl. Phys. B, Volume 305 (1988), p. 545 | DOI | MR

[30] Delporte, Nicolas; Rivasseau, Vincent The Tensor Track V: Holographic Tensors, 17th Hellenic School and Workshops on Elementary Particle Physics and Gravity (2018)

[31] Diaz, Pablo Tensor and Matrix models: a one-night stand or a lifetime romance?, J. High Energy Phys., Volume 06 (2018), p. 140 | DOI | MR | Zbl

[32] Diaz, Pablo Backgrounds from tensor models: A proposal, Phys. Rev. D, Volume 103 (2021) no. 6, p. 066010 | DOI | MR

[33] Diaz, Pablo; Rey, Soo-Jong Invariant Operators, Orthogonal Bases and Correlators in General Tensor Models, Nucl. Phys. B, Volume 932 (2018), pp. 254-277 | DOI | MR | Zbl

[34] Diaz, Pablo; Rey, Soo-Jong Orthogonal Bases of Invariants in Tensor Models, J. High Energy Phys., Volume 02 (2018), p. 089 | DOI | MR | Zbl

[35] Duff, Michael J. Classical and Quantum Supermembranes, Class. Quant. Grav., Volume 6 (1989), pp. 1577-1598 | DOI | MR

[36] Estes, Dennis R. Eigenvalues of symmetric integer matrices, J. Number Theory, Volume 42 (1992) no. 3, pp. 292-296 | DOI | MR | Zbl

[37] Féray, Valentin Proof of Stanley’s conjecture about irreducible character values of the symmetric group, Ann. Comb., Volume 13 (2010), p. 453-–461 | DOI | Zbl

[38] Fulton, William Young Tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, 1997

[39] GAP4, The GAP Group GAP – Groups, Algorithms, and Programming, Version 4.11.0, Lattices Algorithms and Applications (2022) https://www.gap-system.org/

[40] Gomis, Joaquim; Kamimura, Kiyoshi; Townsend, Paul K. Non-relativistic superbranes, J. High Energy Phys., Volume 11 (2004), p. 051 | DOI | MR

[41] Goodman, R.; Wallach, N.R. Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, Springer New York, 2009 | DOI

[42] Gopakumar, Rajesh What is the Simplest Gauge-String Duality? (2011) | arXiv | DOI

[43] Gopala, K. Krishna; Labelle, Patrick; Shramchenko, Vasilisa Enumeration of N-rooted maps using quantum field theory, Nucl. Phys. B, Volume 936 (2018), pp. 668-689 | DOI | MR | Zbl

[44] Gross, David J.; Taylor, Washington Two-dimensional QCD is a string theory, Nucl. Phys. B, Volume 400 (1993), pp. 181-208 | DOI | MR | Zbl

[45] Grothendieck, Alexandre Esquisse d’un Programme (1984)

[46] Gubser, S. S.; Klebanov, Igor R.; Polyakov, Alexander M. Gauge theory correlators from noncritical string theory, Phys. Lett. B, Volume 428 (1998), pp. 105-114 | DOI | Zbl

[47] Gurau, Razvan The complete 1/N expansion of colored tensor models in arbitrary dimension, Ann. Henri Poincaré, Volume 13 (2012), pp. 399-423 | DOI | MR | Zbl

[48] Gurau, Razvan Random Tensors, Oxford University Press, 2017

[49] Hamermesh, Morton Group theory and its application to Physical Problems, Dover Books on Physics, Dover, 2003

[50] Horava, Petr Topological rigid string theory and two-dimensional QCD, Nucl. Phys. B, Volume 463 (1996), pp. 238-286 | DOI | MR | Zbl

[51] Horava, Petr Membranes at Quantum Criticality, J. High Energy Phys., Volume 03 (2009), p. 020 | DOI | MR

[52] Ikenmeyer, Christian; Mulmuley, Ketan; Walter, Michael On vanishing of Kronecker coefficients, Comput. Complexity, Volume 26 (2017), pp. 949-992 | DOI | MR | Zbl

[53] Itoyama, H.; Mironov, A.; Morozov, A. Cut and join operator ring in tensor models, Nucl. Phys. B, Volume 932 (2018), pp. 52-118 | DOI | MR | Zbl

[54] Itoyama, H.; Mironov, A.; Morozov, A. Tensorial generalization of characters, J. High Energy Phys., Volume 12 (2019), p. 127 | DOI | MR | Zbl

[55] Itoyama, Hiroshi; Yoshioka, Reiji Generalized cut operation associated with higher order variation in tensor models, Nucl. Phys. B, Volume 945 (2019), p. 114681 | DOI | MR | Zbl

[56] Jones, Gareth Aneurin Regular embeddings of complete bipartite graphs: classification and enumeration, Proc. Lond. Math. Soc, Volume 101 (2010), pp. 427-453 | DOI | MR | Zbl

[57] Kemp, Garreth; Ramgoolam, Sanjaye BPS states, conserved charges and centres of symmetric group algebras, J. High Energy Phys., Volume 01 (2020), p. 146 | DOI | MR | Zbl

[58] Kimura, Yusuke Multi-matrix models and Noncommutative Frobenius algebras obtained from symmetric groups and Brauer algebras, Commun. Math. Phys., Volume 337 (2015) no. 1, pp. 1-40 | DOI | MR | Zbl

[59] Kimura, Yusuke Noncommutative Frobenius algebras and open-closed duality (2017) | arXiv

[60] Kimura, Yusuke; Ramgoolam, Sanjaye Branes, anti-branes and Brauer algebras in gauge-gravity duality, J. High Energy Phys., Volume 11 (2007), p. 078 | DOI | MR | Zbl

[61] Kimura, Yusuke; Ramgoolam, Sanjaye Enhanced symmetries of gauge theory and resolving the spectrum of local operators, Phys. Rev. D, Volume 78 (2008), p. 126003 | DOI | MR

[62] Klebanov, Igor R.; Milekhin, Alexey; Popov, Fedor; Tarnopolsky, Grigory Spectra of eigenstates in fermionic tensor quantum mechanics, Phys. Rev. D, Volume 97 (2018) no. 10, p. 106023 | DOI | MR

[63] Klebanov, Igor R.; Popov, Fedor; Tarnopolsky, Grigory TASI Lectures on Large N Tensor Models, PoS, Volume TASI2017 (2018), p. 004 | DOI

[64] Kotosevec, Vaclav OEIS Foundation Inc. Entry A000108 in The On-Line Encyclopedia of Integer Sequences https://oeis.org/A279819

[65] Krishnan, Chethan; Pavan Kumar, K. V. Exact Solution of a Strongly Coupled Gauge Theory in 0+1 Dimensions, Phys. Rev. Lett., Volume 120 (2018) no. 20, p. 201603 | DOI

[66] Lando, S.K.; Gamkrelidze, R.V.; Vassiliev, V.A.; Zvonkin, A.K. Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences, Low dimensional topology II, Springer Berlin Heidelberg, 2013

[67] Lassalle, Michel An explicit formula for the characters of the symmetric group, Math. Ann., Volume 340 (2007), pp. 383-405 | DOI | MR | Zbl

[68] Lenstra, Arjen K.; Lenstra, Hendrik W.; Lovász, László Miklós Factoring polynomials with rational coefficients, Math. Ann., Volume 261 (1982), pp. 515-534 | DOI | MR | Zbl

[69] Lin, Hai; Lunin, Oleg; Maldacena, Juan Martin Bubbling AdS space and 1/2 BPS geometries, J. High Energy Phys., Volume 10 (2004), p. 025 | DOI | MR

[70] Lorenzini, Dino Smith normal form and Laplacians, J. Combin. Theory Ser. B, Volume 98 (2008), pp. 1271-1300 | DOI | MR | Zbl

[71] Maldacena, Juan Martin The Large N limit of superconformal field theories and supergravity | Zbl

[72] Manivel, Laurent On the asymptotics of Kronecker coefficients, J. Algebraic Combin., Volume 42 (2014), pp. 999-1025 | DOI | MR

[73] Mattioli, Paolo; Ramgoolam, Sanjaye Permutation Centralizer Algebras and Multi-Matrix Invariants, Phys. Rev. D, Volume 93 (2016) no. 6, p. 065040 | DOI

[74] Micciancio, Daniele Basic algorithms, Lattices Algorithms and Applications

[75] Mulmuley, Ketan; Sohoni, Milind A. Geometric Complexity Theory I: An Approach to the P vs. NP and Related Problems, SIAM J. Comput., Volume 31 (2001), pp. 496-526 | DOI | MR | Zbl

[76] Murnaghan, Francis D. On the Representations of the Symmetric Group, Amer. J. Math., Volume 59 (1937), pp. 437-488 | DOI | Zbl

[77] Murnaghan, Francis D. The Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group, Amer. J. Math., Volume 60 (1938), pp. 761-784 | DOI | MR | Zbl

[78] Nakayama, Tadasi On some modular properties of irreducible representations of a symmetric group, I, Jpn. J. Math., Volume 18 (1941), pp. 89-108 | MR | Zbl

[79] Pak, Igor; Panova, Greta On the complexity of computing Kronecker coefficients, Comput. Complexity, Volume 26 (2017), pp. 1-36 | MR | Zbl

[80] Pak, Igor; Panova, Greta; Vallejo, Ernesto Combinatorics and complexity of Kronecker coefficients, Workshop Summary (2015) https://simons.berkeley.edu/sites/default/files/docs/1899/slidespanova.pdf

[81] Pak, Igor; Panova, Greta; Yeliussizov, Damir On the largest Kronecker and Littlewood-Richardson coefficients, J. Combin. Theory Ser. A, Volume 165 (2019), pp. 44-77 | MR | Zbl

[82] Paraoanu, G. S. Recent progress in quantum simulation using superconducting circuits, J. Low Temp. Phys., Volume 175 (2014), p. 633 | DOI

[83] Pasukonis, Jurgis; Ramgoolam, Sanjaye Quivers as Calculators: Counting, Correlators and Riemann Surfaces, J. High Energy Phys., Volume 04 (2013), p. 094 | DOI | MR | Zbl

[84] Ram, Arun Dissertation, Chapter 1: Representation theory, 2004 http://math.soimeme.org/~arunram/Preprints/dissertationChapt1.pdf

[85] Ramgoolam, Sanjaye Permutations and the combinatorics of gauge invariants for general N, PoS, Volume CORFU2015 (2016), p. 107 | DOI

[86] Rivasseau, Vincent The Tensor Track, IV, PoS, Volume CORFU2015 (2016), p. 106 | DOI

[87] Schneps, Leila The Grothendieck Theory of Dessins D’Enfants, Lecture note series, Cambridge University Press, 1994 | DOI

[88] Schrijver, Alexander Theory of Linear and Integer Programming, Wiley Series in Discrete Mathematics & Optimization, Wiley, 1998

[89] Simon, Barry Representations of finite and compact groups, American Mathematical Society, 1991

[90] Stanley, Richard P. Positivity problems and conjectures, Mathematics: frontiers and perspectives, American Mathematical Society, Providence, RI, (2000), pp. 295-319 | Zbl

[91] Stanley, Richard P. A conjectured combinatorial interpretation of the normalized irreducible character values of the symmetric group, arXiv: Combinatorics (2006) | arXiv

[92] Stanley, Richard P.; Fomin, Sergey Enumerative Combinatorics, Cambridge Studies in Advanced Mathematics, 2, Cambridge University Press, 1999 | DOI

[93] Wikipedia On Murnaghan-Nakayama rule https://en.wikipedia.org/wiki/Murnaghan%E2%80%93Nakayama_rule

[94] Witten, Edward Anti-de Sitter space and holography, Adv. Theor. Math. Phys., Volume 2 (1998), pp. 253-291 | DOI | MR | Zbl

[95] Witten, Edward An SYK-Like Model Without Disorder, J. Phys. A, Volume 52 (2019) no. 47, p. 474002 | DOI | MR | Zbl

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