From scalar fields on quantum spaces to blobbed topological recursion. (English) Zbl 1519.81473
Summary: We review the construction of the \(\lambda\phi^4\)-model on noncommutative geometries via exact solutions of Dyson-Schwinger equations and explain how this construction relates via (blobbed) topological recursion to problems in algebraic and enumerative geometry.
MSC:
| 81T75 | Noncommutative geometry methods in quantum field theory |
| 05C62 | Graph representations (geometric and intersection representations, etc.) |
| 14A22 | Noncommutative algebraic geometry |
Keywords:
quantum field theory; noncommutative geometry; matrix models; (blobbed) topological recursion; renormalisation; ribbon graphsReferences:
| [1] | Aizenman, M.; Duminil-Copin, H., Marginal triviality of the scaling limits of critical 4D Ising and \(####\) models, Ann. Math., 194, 163-235 (2021) · Zbl 1489.60150 · doi:10.4007/annals.2021.194.1.3 |
| [2] | Aizenman, M., Proof of the triviality of \(####\) field theory and some mean field features of Ising models for d > 4, Phys. Rev. Lett., 47, 1-4 (1981) · doi:10.1103/physrevlett.47.1 |
| [3] | Borot, G.; Charbonnier, S.; Garcia-Failde, E., Topological recursion for fully simple maps from ciliated maps (2021) |
| [4] | Bonzom, V.; Dub, N., Blobbed topological recursion for correlation functions in tensor models (2020) |
| [5] | Bychkov, B.; Dunin-Barkowski, P.; Kazarian, M.; Shadrin, S., Topological recursion for Kadomtsev-Petviashvili tau functions of hypergeometric type (2020) |
| [6] | Borot, G.; Eynard, B.; Orantin, N., Abstract loop equations, topological recursion and new applications, Commun. Numer. Theor. Phys., 9, 51-187 (2015) · Zbl 1329.14074 · doi:10.4310/cntp.2015.v9.n1.a2 |
| [7] | Bernardi, O.; Fusy, E., Bijections for planar maps with boundaries, J. Combin. Theory A, 158, 176-227 (2018) · Zbl 1391.05025 · doi:10.1016/j.jcta.2018.03.001 |
| [8] | Borot, G.; Garcia-Failde, E., Simple maps, Hurwitz numbers, and topological recursion, Commun. Math. Phys., 380, 581-654 (2020) · Zbl 1457.37055 · doi:10.1007/s00220-020-03867-1 |
| [9] | Branahl, J.; Hock, A.; Wulkenhaar, R., Perturbative and geometric analysis of the quartic Kontsevich Model, Symmetry, Integrability Geometry Methods Appl., 17, 085 (2021) · Zbl 1480.81098 · doi:10.3842/sigma.2021.085 |
| [10] | Branahl, J.; Hock, A.; Wulkenhaar, R., Blobbed topological recursion of the quartic Kontsevich model: I. Loop equations and conjectures, Commun. Math. Phys., 393, 1529-1582 (2022) · Zbl 1507.81160 · doi:10.1007/s00220-022-04392-z |
| [11] | Bouchard, V.; Mariño, M., Hurwitz numbers, matrix models and enumerative geometry, From Hodge Theory to Integrability and TQFT: tt*-Geometry, 263-283 (2008), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1151.14335 |
| [12] | Borot, G., Formal multidimensional integrals, stuffed maps, and topological recursion, Ann. Inst. Henri Poincaré, 1, 225-264 (2014) · Zbl 1297.05013 · doi:10.4171/aihpd/7 |
| [13] | Borot, G., Summary of results in topological recursion. Notes on Gaëtan Borot’s personal homepage (2018) |
| [14] | Borot, G.; Shadrin, S., Blobbed topological recursion: properties and applications, Math. Proc. Camb. Phil. Soc., 162, 39-87 (2017) · Zbl 1396.14031 · doi:10.1017/s0305004116000323 |
| [15] | Connes, A.; Douglas, M. R.; Schwarz, A. S., Noncommutative geometry and matrix theory, J. High Energy Phys. (1998) · doi:10.1088/1126-6708/1998/02/003 |
| [16] | Chekhov, L.; Eynard, B.; Orantin, N., Free energy topological expansion for the two-matrix model, J. High Energy Phys. (2006) · Zbl 1226.81250 · doi:10.1088/1126-6708/2006/12/053 |
| [17] | Connes, A.; Kreimer, D., Renormalization in quantum field theory and the Riemann-Hilbert problem: I. The Hopf algebra structure of graphs and the main theorem, Commun. Math. Phys., 210, 249-273 (2000) · Zbl 1032.81026 · doi:10.1007/s002200050779 |
| [18] | Chepelev, I.; Roiban, R., Renormalization of quantum field theories on noncommutative Bbb \(####\): I. Scalars, J. High Energy Phys. (2000) · Zbl 0990.81756 · doi:10.1088/1126-6708/2000/05/037 |
| [19] | Doplicher, S.; Fredenhagen, K.; Roberts, J. E., The quantum structure of spacetime at the Planck scale and quantum fields, Commun. Math. Phys., 172, 187-220 (1995) · Zbl 0847.53051 · doi:10.1007/bf02104515 |
| [20] | Disertori, M.; Gurau, R.; Magnen, J.; Rivasseau, V., Vanishing of beta function of non-commutative \(####\) theory to all orders, Phys. Lett. B, 649, 95-102 (2007) · Zbl 1248.81253 · doi:10.1016/j.physletb.2007.04.007 |
| [21] | de Jong, J.; Hock, A.; Wulkenhaar, R., Nested Catalan tables and a recurrence relation in noncommutative quantum field theory, Ann. Henri. Poincaré, 9, 47-72 (2022) · Zbl 1495.81075 · doi:10.4171/aihpd/113 |
| [22] | Disertori, M.; Rivasseau, V., Two- and three-loops beta function of non-commutative \(####\) theory, Eur. Phys. J. C, 50, 661-671 (2007) · Zbl 1248.81254 · doi:10.1140/epjc/s10052-007-0211-0 |
| [23] | Eynard, B.; Orantin, N., Mixed correlation functions in the two-matrix model, and the Bethe ansatz, J. High Energy Phys. (2005) · doi:10.1088/1126-6708/2005/08/028 |
| [24] | Eynard, B.; Orantin, N., Invariants of algebraic curves and topological expansion, Commun. Numer. Theor. Phys., 1, 347-452 (2007) · Zbl 1161.14026 · doi:10.4310/cntp.2007.v1.n2.a4 |
| [25] | Eynard, B., Large N expansion of the two-matrix model, J. High Energy Phys. (2003) · Zbl 1226.82029 · doi:10.1088/1126-6708/2003/01/051 |
| [26] | Eynard, B., (Counting Surfaces Progress in Mathematical Physics vol 70) (2016), Basel: Birkhäuser, Basel · Zbl 1338.81005 |
| [27] | Filk, T., Divergencies in a field theory on quantum space, Phys. Lett. B, 376, 53-58 (1996) · Zbl 1190.81096 · doi:10.1016/0370-2693(96)00024-x |
| [28] | Fröhlich, J., On the triviality of \(####\) theories and the approach to the critical point ind ⩾ 4 dimensions, Nucl. Phys. B, 200, 281-296 (1982) · doi:10.1016/0550-3213(82)90088-8 |
| [29] | Grosse, H.; Hock, A.; Wulkenhaar, R., Solution of all quartic matrix models (2019) |
| [30] | Grosse, H.; Hock, A.; Wulkenhaar, R., Solution of the self-dual Φ^4 QFT-model on four-dimensional Moyal space, J. High Energy Phys. (2020) · Zbl 1434.81046 · doi:10.1007/JHEP01(2020)081 |
| [31] | Grosse, H.; Klimčík, C.; Prešnajder, P., Finite quantum field theory in noncommutative geometry, Int. J. Theor. Phys., 35, 231-244 (1996) · Zbl 0846.58015 · doi:10.1007/bf02083810 |
| [32] | Grosse, H.; Madore, J., A noncommutative version of the Schwinger model, Phys. Lett. B, 283, 218-222 (1992) · doi:10.1016/0370-2693(92)90011-r |
| [33] | Grosse, H.; Wulkenhaar, R., The β-function in duality covariant noncommutative ϕ^4-theory, Eur. Phys. J. C, 35, 277-282 (2004) · Zbl 1191.81207 · doi:10.1140/epjc/s2004-01853-x |
| [34] | Grosse, H.; Wulkenhaar, R., Renormalisation of ϕ^4-theory on noncommutative \(####\) in the matrix base, Commun. Math. Phys., 256, 305-374 (2005) · Zbl 1075.82005 · doi:10.1007/s00220-004-1285-2 |
| [35] | Grosse, H.; Wulkenhaar, R., Power-counting theorem for non-local matrix models and renormalisation, Commun. Math. Phys., 254, 91-127 (2005) · Zbl 1079.81049 · doi:10.1007/s00220-004-1238-9 |
| [36] | Grosse, H.; Wulkenhaar, R., Renormalisation of ϕ^4-theory on noncommutative \(####\) to all orders, Lett. Math. Phys., 71, 13-26 (2005) · Zbl 1115.81055 · doi:10.1007/s11005-004-5116-3 |
| [37] | Grosse, H.; Wulkenhaar, R., Progress in solving a noncommutative quantum field theory in four dimensions (2009) |
| [38] | Grosse, H.; Wulkenhaar, R., Self-dual noncommutative ϕ^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory, Commun. Math. Phys., 329, 1069-1130 (2014) · Zbl 1305.81129 · doi:10.1007/s00220-014-1906-3 |
| [39] | Hock, A., Matrix field theory, PhD Thesis (2020) · Zbl 1439.81003 |
| [40] | Hock, A.; Wulkenhaar, R., Blobbed topological recursion of the quartic Kontsevich model: II. Genus = 0 (2021) |
| [41] | Kontsevich, M., Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys., 147, 1-23 (1992) · Zbl 0756.35081 · doi:10.1007/bf02099526 |
| [42] | Krajewski, T.; Wulkenhaar, R., Perturbative quantum gauge fields on the noncommutative torus, Int. J. Mod. Phys. A, 15, 1011-1029 (2000) · Zbl 0963.81055 · doi:10.1142/s0217751x00000495 |
| [43] | Landau, L. D.; Abrikosov, A. A.; Khalatnikov, I. M., On the removal of infinities in quantum electrodynamics (in Russian), Dokl. Akad. Nauk SSSR, 95, 497-500 (1954) · Zbl 0059.21904 · doi:10.1016/B978-0-08-010586-4.50083-3 |
| [44] | Lindner, J., Two approaches for a perturbative expansion in blobbed topological recursion: I (2022) |
| [45] | Langmann, E.; Szabo, R. J., Duality in scalar field theory on noncommutative phase spaces, Phys. Lett. B, 533, 168-177 (2002) · Zbl 0994.81116 · doi:10.1016/s0370-2693(02)01650-7 |
| [46] | Mirzakhani, M., Simple geodesics and Weil-Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math., 167, 179-222 (2006) · Zbl 1125.30039 · doi:10.1007/s00222-006-0013-2 |
| [47] | Martín, C. P.; Sánchez-Ruiz, D., The one loop UV divergent structure of U(1) Yang-Mills theory on noncommutative \(####\), Phys. Rev. Lett., 83, 476-479 (1999) · Zbl 0958.81199 · doi:10.1103/physrevlett.83.476 |
| [48] | Minwalla, S.; Raamsdonk, M.; Seiberg, N., Noncommutative perturbative dynamics, J. High Energy Phys. (2000) · Zbl 0959.81108 · doi:10.1088/1126-6708/2000/02/020 |
| [49] | Osterwalder, K.; Schrader, R., Axioms for Euclidean Green’s functions II, Commun. Math. Phys., 42, 281-305 (1975) · Zbl 0303.46034 · doi:10.1007/bf01608978 |
| [50] | Panzer, E.; Wulkenhaar, R., Lambert-W solves the noncommutative Φ^4-model, Commun. Math. Phys., 374, 1935-1961 (2020) · Zbl 1436.81091 · doi:10.1007/s00220-019-03592-4 |
| [51] | Rivasseau, V., Non-commutative renormalization, Quantum Spaces, 19-107 (2007), Basel: Birkhäuser, Basel · Zbl 1139.81047 |
| [52] | Schürmann, J.; Wulkenhaar, R., An algebraic approach to a quartic analogue of the Kontsevich model, Mathematical Proceedings of the Cambridge Philosophical Society (2022) · Zbl 1515.81197 · doi:10.1017/S0305004122000366 |
| [53] | Szabo, R. J., Quantum field theory on noncommutative spaces, Phys. Rep., 378, 207-299 (2003) · Zbl 1042.81581 · doi:10.1016/s0370-1573(03)00059-0 |
| [54] | Thürigen, J., Renormalization in combinatorially non-local field theories: the BPHZ momentum scheme, Symmetry, Integrability Geometry Methods Appl., 17, 094 (2021) · Zbl 1480.81097 · doi:10.3842/sigma.2021.094 |
| [55] | Thürigen, J., Renormalization in combinatorially non-local field theories: the Hopf algebra of two-graphs, Math. Phys. Anal. Geom., 24, 19 (2021) · Zbl 1470.81050 · doi:10.1007/s11040-021-09390-6 |
| [56] | Tricomi, F., Integral Equations (1985), New York: Dover, New York |
| [57] | Witten, E., Two-dimensional gravity and intersection theory on moduli space, Surveys in Differential Geometry (Cambridge, MA, 1990), 243-310 (1991), Bethlehem, PA: Lehigh University, Bethlehem, PA · Zbl 0757.53049 |
| [58] | Wulkenhaar, R., Field theories on deformed spaces, J. Geom. Phys., 56, 108-141 (2006) · Zbl 1083.81014 · doi:10.1016/j.geomphys.2005.04.019 |
| [59] | Wulkenhaar, R.; Chamseddine, A.; Consani, C.; Higson, N.; Khalkhali, M.; Moscovici, H.; Yu, G., Quantum field theory on noncommutative spaces, Advances in Noncommutative Geometry, 607-690 (2019), Berlin: Springer, Berlin · Zbl 1482.81038 |
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