Triality in SU(2) Seiberg-Witten theory and Gauss hypergeometric function

Abstract

Through AGT conjecture, we show how triality observed in \N=2 SU(2) N_f=4 QCD can be interpreted geometrically as the interplay among six of Kummer's twenty-four solutions belonging to one fixed Riemann scheme in the context of hypergeometric differential equations. We also stress that our presentation is different from the usual crossing symmetry of Liouville conformal blocks, which is described by the connection coefficient in the case of hypergeometric functions. Besides, upon solving hypergeometric differential equations at the zeroth order by means of the WKB method, a curve (thrice-punctured Riemann sphere) emerges. The permutation between these six Kummer's solutions then boils down to the outer automorphism of the associated curve.

16 pages; v2: references added, minor revision; Section 3.1.1 discussing N=2* SU(2) theory associated with a pinched once-punctured torus, Jack (or Gegenbauer) polynomials and Jacobi polynomials newly added; v3: Section 4 discussing "crossing symmetry and triality" & "geometric realization of triality" newly added thanks to PRD referee advice, to be published in PRD; v4: TexStyle changed only

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Cited by in corpus (15)

• The Omega deformed B-model for rigid N=2 theories
• Proving AGT conjecture as HS duality: extension to five dimensions
• On AGT Relations with Surface Operator Insertion and Stationary Limit of Beta-Ensembles
• A direct proof of AGT conjecture at beta = 1
• Towards a proof of AGT conjecture by methods of matrix models
• Brezin-Gross-Witten model as "pure gauge" limit of Selberg integrals
• Massive Scaling Limit of beta-Deformed Matrix Model of Selberg Type
• Classical conformal blocks from TBA for the elliptic Calogero-Moser system
• N=2 Gauge Theories: Congruence Subgroups, Coset Graphs and Modular Surfaces
• Asymptotic expansions of the hypergeometric function with two large parameters $-$ application to the partition function of a lattice gas in a field of traps
• Resurgence in the Bi-Yang-Baxter Model
• Developments of theory of effective prepotential from extended Seiberg-Witten system and matrix models
• Four flavours, triality and bimodular forms
• Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings
• On non-stationary Lamé equation from WZW model and spin-1/2 XYZ chain