The singularities of Selberg- and Dotsenko-Fateev-like integrals. (English) Zbl 07901618
Summary: We discuss the meromorphic continuation of certain hypergeometric integrals modeled on the Selberg integral, including the 3-point and 4-point functions of BPZ’s minimal models of 2D CFT as described by Felder & Silvotti and Dotsenko & Fateev (the “Coulomb gas formalism”). This is accomplished via a geometric analysis of the singularities of the integrands. In the case that the integrand is symmetric (as in the Selberg integral itself) or, more generally, what we call “DF-symmetric,” we show that a number of apparent singularities are removable, as required for the construction of the minimal models via these methods.
MSC:
| 32A20 | Meromorphic functions of several complex variables |
| 33C60 | Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions) |
| 33C90 | Applications of hypergeometric functions |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
References:
| [1] | Aomoto, K., Kita, M.: Theory of hypergeometric functions. Springer Monographs in Mathematics. With an appendix by Toshitake Kohno, Translated from the Japanese by Kenji Iohara. Springer-Verlag, Tokyo (2011). doi:10.1007/978-4-431-53938-4 · Zbl 1229.33001 |
| [2] | Alba, VA, On combinatorial expansion of the conformal blocks arising from AGT conjecture, Lett. Math. Phys., 98, 1, 33-64, 2011 · Zbl 1242.81119 · doi:10.1007/s11005-011-0503-z |
| [3] | Aomoto, K., On the complex Selberg integral, Quart. J. Math. Oxford Ser. (2), 38, 152, 385-399, 1987 · Zbl 0639.33002 · doi:10.1093/qmath/38.4.385 |
| [4] | Belavin, AA; Polyakov, AM; Zamolodchikov, AB, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B, 241, 2, 333-380, 1984 · Zbl 0661.17013 · doi:10.1016/0550-3213(84)90052-X |
| [5] | Dotsenko, VS; Fateev, VA, Conformal algebra and multipoint correlation functions in 2D statistical models, Nucl. Phys. B, 240, 3, 312-348, 1984 · doi:10.1016/0550-3213(84)90269-4 |
| [6] | Casali, E.; Mizera, S.; Tourkine, P., Monodromy relations from twisted homology, J. High Energy Phys., 12, 087, 2019 · Zbl 1431.83163 · doi:10.1007/jhep12(2019)087 |
| [7] | de la Cruz, L.; Kniss, A.; Weinzierl, S., Properties of scattering forms and their relation to associahedra, J. High Energy Phys., 3, 064, 2018 · Zbl 1388.81929 · doi:10.1007/jhep03(2018)064 |
| [8] | Dotsenko, VS; Fateev, VA, Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge \(c\le 1\), Nucl. Phys. B, 251, 691-734, 1985 · doi:10.1016/S0550-3213(85)80004-3 |
| [9] | Dotsenko, VS; Fateev, VA, Operator algebra of two-dimensional conformal theories with central charge \(c\le 1\), Phys. Lett. B, 154, 4, 291-295, 1985 · doi:10.1016/0370-2693(85)90366-1 |
| [10] | Felder, G., BRST approach to minimal models, Nuclear Phys. B, 317, 1, 215-236, 1989 · doi:10.1016/0550-3213(89)90568-3 |
| [11] | Flores, SM; Kleban, P., A solution space for a system of null-state partial differential equations: Part 1, Commun. Math. Phys., 333, 1, 389-434, 2015 · Zbl 1314.35188 · doi:10.1007/s00220-014-2189-4 |
| [12] | Flores, SM; Kleban, P., A solution space for a system of null-state partial differential equations: Part 2, Commun. Math. Phys., 333, 1, 435-481, 2015 · Zbl 1314.35189 · doi:10.1007/s00220-014-2185-8 |
| [13] | Flores, SM; Kleban, P., A solution space for a system of null-state partial differential equations: Part 3, Commun. Math. Phys., 333, 2, 597-667, 2015 · Zbl 1311.35314 · doi:10.1007/s00220-014-2190-y |
| [14] | Flores, SM; Kleban, P., A solution space for a system of null-state partial differential equations: Part 4, Commun. Math. Phys., 333, 2, 669-715, 2015 · Zbl 1314.35190 · doi:10.1007/s00220-014-2180-0 |
| [15] | Felder, G.; Silvotti, R., Free field representation of minimal models on a Riemann surface, Phys. Lett. B, 231, 4, 411-416, 1989 · Zbl 0693.30037 · doi:10.1016/0370-2693(89)90685-0 |
| [16] | Felder, G.; Silvotti, R., Conformal blocks of minimal models on a Riemann surface, Commun. Math. Phys., 144, 1, 17-40, 1992 · Zbl 0746.57003 · doi:10.1007/BF02099189 |
| [17] | Forrester, PJ; Warnaar, SO, The importance of the Selberg integral, Bull. Am. Math. Soc. (N.S.), 45, 4, 489-534, 2008 · Zbl 1154.33002 · doi:10.1090/S0273-0979-08-01221-4 |
| [18] | Geyer., W: On Tamari lattices. Discrete Math. 133(1-3), 99-122 (1994). doi:10.1016/0012-365X(94)90019-1 · Zbl 0811.06005 |
| [19] | Gelüffand, I.M., Shilov, G.E.: Generalized Functions. Vol. I: Properties and Operations. Academic Press, London (1964) · Zbl 0115.33101 |
| [20] | Hassell, A.; Mazzeo, R.; Melrose, RB, A signature formula for manifolds with corners of codimension two, Topology, 36, 5, 1055-1075, 1997 · Zbl 0883.58038 · doi:10.1016/S0040-9383(96)00043-2 |
| [21] | Kadell, KWJ, An integral for the product of two Selberg-Jack symmetric polynomials, Compositio Math., 87, 1, 5-43, 1993 · Zbl 0788.33007 |
| [22] | Kadell, KWJ, The Selberg-Jack symmetric functions, Adv. Math., 130, 1, 33-102, 1997 · Zbl 0885.33009 · doi:10.1006/aima.1997.1642 |
| [23] | Kanie, Y.; Tsuchiya, A., Fock space representations of the Virasoro algebra. Intertwining operators, Publ. Res. Inst. Math. Sci., 22, 2, 259-327, 1986 · Zbl 0604.17008 · doi:10.2977/prims/1195178069 |
| [24] | Kanie, Y., Tsuchiya, A.: Fock space representations of Virasoro algebra and intertwining operators. Proc. Jpn. Acad. Ser. A Math. Sci. 62(1), 12-15 (1986) · Zbl 0611.17006 |
| [25] | Lenells, J.; Viklund, F., Asymptotic analysis of Dotsenko-Fateev integrals, Ann. Henri Poincare, 20, 11, 3799-3848, 2019 · Zbl 1428.30040 · doi:10.1007/s00023-019-00849-5 |
| [26] | Melrose, R.: Differential analysis on manifold with corners |
| [27] | Mizera, S., Combinatorics and topology of Kawai-Lewellen-Tye relations, J. High Energy Phys., 8, 097-150, 2017 · Zbl 1381.83126 · doi:10.1007/jhep08(2017)097 |
| [28] | Mizera, S.: Aspects of scattering amplitudes and moduli space localization: Springer Theses. Springer (2020). doi:10.1007/978-3-030-53010-5 · Zbl 1457.81067 |
| [29] | Melrose, R., Singer, M.: Scattering configuration spaces (2008). arXiv: 0808.2022 |
| [30] | Markl, M., Shnider, S., Stasheff, J.: Operads in algebra, topology and physics. Mathematical Surveys and Monographs 96. American Mathematical Society (2002). doi:10.1090/surv/096 · Zbl 1017.18001 |
| [31] | Mimachi, K.; Yoshida, M., Intersection numbers of twisted cycles and the correlation functions of the conformal field theory, Commun. Math. Phys., 234, 2, 339-358, 2003 · Zbl 1029.81062 · doi:10.1007/s00220-002-0766-4 |
| [32] | Mimachi, K.; Yoshida, M., Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory, Commun. Math. Phys., 250, 1, 23-45, 2004 · Zbl 1069.32015 · doi:10.1007/s00220-004-1138-z |
| [33] | Philippe Francesco, P., Mathieu, D.S.: Conformal Field Theory. Graduate Texts in Contemporary Physics. Springer, New York(1997). doi:10.1007/978-1-4612-2256-9 · Zbl 0869.53052 |
| [34] | Postnikov, A., Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN, 6, 1026-1106, 2009 · Zbl 1162.52007 · doi:10.1093/imrn/rnn153 |
| [35] | Selberg, A., Remarks on a multiple integral, Norsk Mat. Tidsskr., 26, 71-78, 1944 · Zbl 0063.06870 |
| [36] | Stasheff, J.D.: Homotopy associativity of \(H\)-spaces. I, II. Trans. Amer. Math. Soc. 108, 275-292 (1963); ibid. 108 (1963), 293-312. doi:10.1090/s0002-9947-1963-0158400-5 · Zbl 0114.39402 |
| [37] | Tamari, D., The algebra of bracketings and their enumeration, Nieuw Arch. Wisk. (3), 10, 131-146, 1962 · Zbl 0109.24502 |
| [38] | Tarasov, V.; Varchenko, A., Selberg-type integrals associated with \(\mathfrak{sl}_3\), Lett. Math. Phys., 65, 3, 173-185, 2003 · Zbl 1055.33016 · doi:10.1023/B:MATH.0000010712.67685.9d |
| [39] | Varchenko, A.: Multidimensional hypergeometric functions and representation theory of Lie algebras and quantum groups. Advanced Series in Mathematical Physics 21. World Scientific Publishing Co., (1995). doi:10.1142/2467 · Zbl 0951.33001 |
| [40] | Warnaar, SO, A Selberg integral for the Lie algebra \(A_n\), Acta Math., 203, 2, 269-304, 2009 · Zbl 1243.33053 · doi:10.1007/s11511-009-0043-x |
| [41] | Yoshida, M.: A geometric interpretation of the Selberg integral. Int. J. Modern Phys. A 18(24), 4343-4359 (2003). doi:10.1142/S0217751X03015192 · Zbl 1229.33005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
