Tetrahedron equation: algebra, topology, and integrability

D. V. Talalaev

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 76 (2021) no. 4, pp. 685-721

Résumé

The Zamolodchikov tetrahedron equation inherits almost all the richness of structures and topics in which the Yang–Baxter equation is involved. At the same time, this transition symbolizes the growth of the order of the problem, the step from the Yang–Baxter equation to the local Yang–Baxter equation, from the Lie algebra to the 2-Lie algebra, from ordinary knots in $\mathbb{R}^3$ to 2-knots in $\mathbb{R}^4$. These transitions are followed in several examples, and there are also discussions of the manifestation of the tetrahedron equation in the long-standing question of integrability of the three-dimensional Ising model and a related model of neural network theory: the Hopfield model on a two-dimensional lattice. Bibliography: 82 titles.

Détail

Keywords: 2-knots, integrable models of statistical physics, Hopfield model.
Mots-clés : tetrahedron equation

Bibliographie

[1] A. B. Zamolodchikov, “Tetrahedra equations and integrable systems in three-dimensional space”, Soviet Phys. JETP, 52:2 (1980), 325–336 | MR

[2] V. F. R. Jones, “A polynomial invariant for knots via von Neumann algebras”, Bull. Amer. Math. Soc. (N.S.), 12:1 (1985), 103–111 | DOI | MR | Zbl

[3] K. Kassel, Kvantovye gruppy, Grad. Texts in Math., 155, Fazis, M., 1999, xxi+663 per. s angl.: pp. | DOI | MR | Zbl

[4] V. G. Drinfel'd, “On some unsolved problems in quantum group theory”, Quantum groups (Leningrad, 1990), Lecture Notes in Math., 1510, Springer, Berlin, 1992, 1–8 | DOI | MR | Zbl

[5] L. A. Takhtadzhyan, L. D. Faddeev, “The quantum method of the inverse problem and the Heisenberg $XYZ$ model”, Russian Math. Surveys, 34:5 (1979), 11–68 | DOI | MR

[6] A. P. Veselov, “Integrable maps”, Russian Math. Surveys, 46:5 (1991), 1–51 | DOI | MR | Zbl

[7] A. P. Veselov, “Yang–Baxter maps and integrable dynamics”, Phys. Lett. A, 314:3 (2003), 214–221 | DOI | MR | Zbl

[8] V. V. Bazhanov, S. M. Sergeev, “Yang–Baxter maps, discrete integrable equations and quantum groups”, Nuclear Phys. B, 926 (2018), 509–543 | DOI | MR | Zbl

[9] V. M. Bukhshtaber, “The Yang–Baxter transformation”, Russian Math. Surveys, 53:6 (1998), 1343–1345 | DOI | DOI | MR | Zbl

[10] I. M. Krichever, “Baxter's equations and algebraic geometry”, Funct. Anal. Appl., 15:2 (1981), 92–103 | DOI | MR | Zbl

[11] V. G. Gorbunov, C. Korff, C. Stroppel, “Yang–Baxter algebras, convolution algebras, and Grassmannians”, Russian Math. Surveys, 75:5 (2020), 791–842 | DOI | DOI | MR | Zbl

[12] B. Sutherlend, “Two-dimensional hydrogen bonded crystals without the ice rule”, J. Math. Phys., 11:11 (1970), 3183–3186 | DOI

[13] R. J. Baxter, “One-dimensional anisotropic Heisenberg chain”, Phys. Rev. Lett., 26:14 (1971), 834 | DOI

[14] R. J. Baxter, “One-dimensional anisotropic Heisenberg chain”, Ann. Physics, 70:2 (1972), 323–337 | DOI | MR

[15] D. V. Talalaev, “Zamolodchikov tetrahedral equation and higher Hamiltonians of $2d$ quantum integrable systems”, SIGMA, 13 (2017), 031, 14 pp. | DOI | MR | Zbl

[16] J. M. Maillet, F. Nijhoff, “Integrability for multidimensional lattice models”, Phys. Lett. B, 224:4 (1989), 389–396 | DOI | MR

[17] N. Reshetikhin, V. G. Turaev, “Invariants of 3-manifolds via link polynomials and quantum groups”, Invent. Math., 103 (1991), 547–597 | DOI | MR | Zbl

[18] V. M. Buchstaber, “Semigroups of maps into groups, operator doubles, and complex cobordisms”, Topics in topology and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 170, Adv. Math. Sci., 27, Amer. Math. Soc., Providence, RI, 1995, 9–31 | DOI | MR | Zbl

[19] V. G. Drinfeld, “Almost cocommutative Hopf algebras”, Leningrad Math. J., 1:2 (1990), 321–342 | MR | Zbl

[20] I. G. Korepanov, G. I. Sharygin, D. V. Talalaev, “Cohomologies of $n$-simplex relations”, Math. Proc. Cambridge Philos. Soc., 161:2 (2016), 203–222 | DOI | MR | Zbl

[21] Yu. I. Manin, V. V. Schechtman, “Arrangements of hyperplanes, higher braid groups and higher Bruhat orders”, Algebraic number theory – in honour of K. Iwasawa, Adv. Stud. Pure Math., 17, Academic Press, Boston, MA, 1989, 289–308 | DOI | MR | Zbl

[22] Yu. I. Manin, V. V. Schechtman, “Arrangements of real hyperplanes and Zamolodchikov equations”, Group theoretical methods in physics (Yurmala, 1985), v. 1, VNU Sci. Press, Utrecht, 1986, 151–165 | MR | Zbl

[23] M. M. Kapranov, V. A. Voevodsky, “2-categories and Zamolodchikov tetrahedra equations”, Algebraic groups and their generalizations: quantum and infinite-dimensional methods (University Park, PA, 1991), Proc. Sympos. Pure Math., 56, Part 2, Amer. Math. Soc., Providence, RI, 1994, 177–259 | MR | Zbl

[24] V. I. Danilov, A. V. Karzanov, G. A. Koshevoy, “Cubillages of cyclic zonotopes”, Russian Math. Surveys, 74:6 (2019), 1013–1074 | DOI | DOI | MR | Zbl

[25] F. Gray, Pulse code communication, U.S. Patent 2632058, 1953, 18 pp.

[26] L. D. Faddeev, N. Yu. Reshetikhin, L. A. Takhtajan, “Quantization of Lie groups and Lie algebras”, Algebraic analysis, v. I, Academic Press, Boston, MA, 1988, 129–139 | DOI | MR | Zbl

[27] S. MacLane, “Categorical algebra'”, Bull. Amer. Math. Soc., 71 (1965), 40–106 | DOI | MR | Zbl

[28] R. Fenn, M. Jordan-Santana, L. Kauffman, “Biquandles and virtual links”, Topology Appl., 145:1-3 (2004), 157–175 | DOI | MR | Zbl

[29] W. Rump, “A decomposition theorem for square-free unitary solutions of the quantum Yang–Baxter equation”, Adv. Math., 193:1 (2005), 40–55 | DOI | MR | Zbl

[30] W. Rump, “Braces, radical rings, and the quantum Yang–Baxter equation”, J. Algebra, 307:1 (2007), 153–170 | DOI | MR | Zbl

[31] V. Lebed, L. Vendramin, “On structure groups of set-theoretic solutions to the Yang–Baxter equation”, Proc. Edinb. Math. Soc. (2), 62:3 (2019), 683–717 | DOI | MR | Zbl

[32] V. V. Sokolov, “Classification of constant solutions of the associative Yang–Baxter equation on $\operatorname{Mat}_3$”, Theoret. and Math. Phys., 176:3 (2013), 1156–1162 | DOI | DOI | MR | Zbl

[33] M. Van den Bergh, “Double Poisson algebras”, Trans. Amer. Math. Soc., 360:11 (2008), 5711–5769 | DOI | MR | Zbl

[34] M. M. Preobrazhenskaya, D. V. Talalaev, “Group extensions, fiber bundles, and a parametric Yang–Baxter equation”, Theoret. and Math. Phys., 207:2 (2021), 670–677 | DOI | DOI | MR | Zbl

[35] V. M. Buchstaber, S. Igonin, S. Konstantinou-Rizos, M. Preobrazhenskaia, “Yang–Baxter maps, Darboux transformations, and linear approximations of refactorisation problems”, J. Phys. A, 53:50 (2020), 504002, 23 pp. | DOI | MR

[36] A. S. Crans, Lie 2-algebras, Ph.D. Thesis, Univ. of California, Riverside, 2004, v+114 pp., arXiv: math/0409602

[37] E. Neher, Jordan triple systems by the grid approach, Lecture Notes in Math., 1280, Springer-Verlag, Berlin, 1987, xii+193 pp. | DOI | MR | Zbl

[38] J. M. Maillet, “On pentagon and tetrahedron equations”, Algebra i analiz, 6:2 (1994), 206–214 ; St. Petersburg Math. J., 6:2 (1995), 375–383 | MR | Zbl

[39] R. M. Kashaev, S. M. Sergeev, “On pentagon, ten-term, and tetrahedron relations”, Comm. Math. Phys., 195:2 (1998), 309–319 | DOI | MR | Zbl

[40] D. Bar-Natan, “On Khovanov's categorification of the Jones polynomial”, Algebr. Geom. Topol., 2 (2002), 337–370 | DOI | MR | Zbl

[41] D. Roseman, “Reidemeister-type moves for surfaces in four-dimensional space”, Knot theory (Warsaw, 1995), Banach Center Publ., 42, Polish Acad. Sci. Inst. Math., Warsaw, 1998, 347–380 | DOI | MR | Zbl

[42] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, M. Saito, “Quandle cohomology and state-sum invariants of knotted curves and surfaces”, Trans. Amer. Math. Soc., 355:10 (2003), 3947–3989 | DOI | MR | Zbl

[43] P. Cotta-Ramusino, M. Martellini, “BF theories and 2-knots”, Knots and quantum gravity (Riverside, CA, 1993), Oxford Lecture Ser. Math. Appl., 1, Oxford Sci. Publ., Oxford Univ. Press, New York, 1994, 169–189 | MR | Zbl

[44] L. Crane, D. Yetter, “A categorical construction of 4d topological quantum field theories”, Quantum topology, Ser. Knots Everything, 3, World Sci. Publ., River Edge, NJ, 1993, 120–130 | DOI | MR | Zbl

[45] R. J. Baxter, Exactly solved models in statistical mechanics, Academic Press, Inc., London, 1982, xii+486 pp. | MR | MR | Zbl

[46] S. Istrail, “Statistical mechanics, three-dimensionality and NP-completeness. I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces”, Proceedings of the thirty-second annual ACM symposium on theory of computing (STOC '00) (Portland, OR, 2000), ACM, New York, 2000, 87–96 | DOI | MR | Zbl

[47] S. El-Showk, M. F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi, “Solving the 3D Ising model with the conformal bootstrap”, Phys. Rev. D, 86:2 (2012), 025022, 32 pp. | DOI

[48] D. V. Talalaev, “Towards integrable structure in 3d Ising model”, J. Geom. Phys., 148 (2020), 103545, 10 pp. | DOI | MR | Zbl

[49] M. Minsky, S. A. Papert, Perceptrons. An introduction to computational geometry, The MIT Press, Cambridge, Mass.–London, 1969, 258 pp. | Zbl | Zbl

[50] W. A. Little, “The existence of persistent states in the brain”, Math. Biosci., 19:1-2 (1974), 101–120 | DOI | Zbl

[51] J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities”, Proc. Nat. Acad. Sci. U.S.A., 79:8 (1982), 2554–2558 | DOI | MR | Zbl

[52] S. Haykin, Neural networks and learning machines, 3rd ed., Prentice Hall, Upper Saddle River, NJ, 2008, 906 pp.

[53] S. Recanatesi, M. Katkov, S. Romani, M. Tsodyks, “Neural network model of memory retrieval”, Front. Comput. Neurosci., 9 (2015), 149 | DOI

[54] S. Romani, M. Tsodyks, “Short-term plasticity based network model of place cells dynamics”, Hippocampus, 25 (2015), 94–105 | DOI

[55] H. Sompolinsky, I. Kanter, “Temporal association in asymmetric neural networks”, Phys. Rev. Lett., 57:22 (1986), 2861–2864 | DOI

[56] A. A. Frolov, D. Husek, I. P. Muraviev, “Informational capacity and recall quality in sparsely encoded Hopfield-like neural network: analytical approaches and computer simulation”, Neural Netw., 10:5 (1997), 845–855 | DOI

[57] R. Roscher, B. Bohn, M. F. Duarte, J. Garcke, “Explainable machine learning for scientific insights and discoveries”, IEEE Access, 8 (2020), 42200–42216 | DOI

[58] N. Pospelov, S. Nechaev, K. Anokhin, O. Valba, V. Avetisov, A. Gorsky, “Spectral peculiarity and criticality of a human connectome”, Phys. Life Rev., 31 (2019), 240–256 | DOI

[59] D. J. Amit, H. Gutfreund, H. Sompolinsky, “Statistical mechanics of neural networks near saturation”, Ann. Physics, 173:1 (1987), 30–67 | DOI

[60] J. Hietarinta, “Permutation-type solutions to the Yang–Baxter and other $n$-simplex equations”, J. Phys. A, 30:13 (1997), 4757–4771 | DOI | MR | Zbl

[61] V. V. Bazhanov, S. M. Sergeev, “Zamolodchikov's tetrahedron equation and hidden structure of quantum groups”, J. Phys. A, 39:13 (2006), 3295–3310 ; (2005), 19 pp., arXiv: hep-th/0509181 | DOI | MR | Zbl

[62] R. M. Kashaev, I. G. Korepanov, S. M. Sergeev, “Functional tetrahedron equation”, Theoret. and Math. Phys., 117:3 (1998), 1402–1413 | DOI | DOI | MR | Zbl

[63] S. M. Sergeev, “Supertetrahedra and superalgebras”, J. Math. Phys., 50:8 (2009), 083519, 21 pp. | DOI | MR | Zbl

[64] V. V. Bazhanov, V. V. Mangazeev, S. M. Sergeev, “Quantum geometry of three-dimensional lattices”, J. Stat. Mech. Theory Exp., 2008:7 (2008), P07004, 27 pp. | DOI | MR | Zbl

[65] I. G. Korepanov, “Tetrahedral Zamolodchikov algebras corresponding to Baxter's $L$-operators”, Comm. Math. Phys., 154:1 (1993), 85–97 | DOI | MR | Zbl

[66] R. M. Kashaev, “On discrete three-dimensional equations associated with the local Yang–Baxter relation”, Lett. Math. Phys., 38:4 (1996), 389–397 ; (1995), 10 pp., arXiv: solv-int/9512005 | DOI | MR | Zbl

[67] S. M. Sergeev, “Solutions of the functional tetrahedron equation connected with the local Yang–Baxter equation for the ferro-electric condition”, Lett. Math. Phys., 45:2 (1998), 113–119 ; Solutions of the functional tetrahedron equation connected with the local Yang–Baxter equation for the ferro-electric, 1997, 7 pp., arXiv: solv-int/9709006 | DOI | MR | Zbl

[68] J. Hietarinta, “Labelling schemes for tetrahedron equations and dualities between them”, J. Phys. A, 27:17 (1994), 5727–5748 | DOI | MR | Zbl

[69] A. Berenstein, S. Fomin, A. Zelevinsky, “Parametrizations of canonical bases and totally positive matrices”, Adv. Math., 122:1 (1996), 49–149 | DOI | MR | Zbl

[70] R. M. Kashaev, “On discrete three-dimensional equations associated with the local Yang–Baxter relation”, Lett. Math. Phys., 38:4 (1996), 389–397 | DOI | MR | Zbl

[71] V. Gorbounov, D. Talalaev, “Electrical varieties as vertex integrable statistical models”, J. Phys. A, 53 (2020), 454001, 28 pp. | DOI | MR

[72] B. Bychkov, A. Kazakov, D. Talalaev, “Functional relations on anisotropic Potts models: from Biggs formula to the tetrahedron equation”, SIGMA, 17 (2021), 035, 30 pp. | DOI | MR | Zbl

[73] E. C. Zeeman, “Twisting spun knots”, Trans. Amer. Math. Soc., 115 (1965), 471–495 | DOI | MR | Zbl

[74] J. S. Carter, M. Elhamdadi, M. Saito, “Homology theory for the set-theoretic Yang–Baxter equation and knot invariants from generalizations of quandles”, Fund. Math., 184 (2004), 31–54 ; (2004 (v1 – 2002)), 22 pp., arXiv: math/0206255 | DOI | MR | Zbl

[75] S. V. Matveev, “Distributive groupoids in knot theory”, Math. USSR-Sb., 47:1 (1984), 73–83 | DOI | MR | Zbl

[76] I. G. Korepanov, D. V. Talalaev, G. I. Sharygin, “Integrable 3D statistical models on six-valent graphs”, Proc. Steklov Inst. Math., 302 (2018), 198–216 | DOI | DOI | MR | Zbl

[77] J. M. Maillet, “Lax equations and quantum groups”, Phys. Lett. B, 245:3-4 (1990), 480–486 | DOI | MR

[78] D. V. Osipov, “The Krichever correspondence for algebraic varieties”, Izv. Math., 65:5 (2001), 941–975 | DOI | DOI | MR | Zbl

[79] D. V. Osipov, “The unramified two-dimensional Langlands correspondence”, Izv. Math., 77:4 (2013), 714–741 | DOI | DOI | MR | Zbl

[80] M. V. Fedoryuk, Obyknovennye differentsialnye uravneniya, 2-e izd., pererab. i dop., Nauka, M., 1985, 448 pp. | MR | Zbl

[81] D. V. Talalaev, “Hopfield neural network and anisotropic Ising model”, NEUROINFORMATICS 2020: Advances in neural computation, machine learning, and cognitive research IV, Stud. Comput. Intell., 925, Springer, Cham, 2021, 381–386 | DOI

[82] Yi Da, Ge Xiurun, “An improved PSO-based ANN with simulated annealing technique”, Neurocomputing, 63 (2005), 527–533 | DOI