ELLIPTIC HYPERGEOMETRIC INTEGRALS, SUPERCONFORMAL INDICES AND - PDF document

ELLIPTIC HYPERGEOMETRIC INTEGRALS, SUPERCONFORMAL INDICES AND DUALITY Vyacheslav P. Spiridonov Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna and Max-Planck-Institute for Mathematics, Bonn String Theory Seminar DAMTP, Cambridge, 2 December 2010 Unity of Physics and Mathematics: from Newton’s binomial theorem to quark confinement 1

Hypergeometric functions are the most popular special functions. Re-edited “Abramowitz-Stegun” handbook: NIST Handbook of Mathematical Functions Editors: F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, Cambridge University Press, 2010. Volume: 968 pp., 422 Fig., 100 Tab., Size 279 × 215 mm, Weight 2 . 9 kg. Digital Library of Mathematical Functions, National Institute of Standards and Technology, http://dlmf.nist.gov/ PLAIN HYPERGEOMETRIC FUNCTIONS THE BEGINNING: Cambridge ! – John Wallis (1655, “Arithmetica Infinitorum” ) Introduced the term “Hypergeometric Series” – Isaak Newton (1665) The binomial theorem: ∞ � ( a ) n n ! x n = (1 − x ) − a , | x | < 1 , a ∈ C , 1 F 0 ( a ; x ) = n =0 where ( a ) n = a ( a + 1) · · · ( a + n − 1) the Pochhammer symbol

– Leonhard Euler (1729 and later on) the gamma function Γ( x ): � ∞ t x − 1 e − t dt, Γ( x ) = Re( x ) > 0 , 0 the beta function (integral) B ( x, y ): � 1 t x − 1 (1 − t ) y − 1 dt = Γ( x )Γ( y ) B ( x, y ) = Γ( x + y ) , Re( x ) , Re( y ) > 0 , 0 the 2 F 1 -series: ∞ � ( a ) n ( b ) n x n , | x | < 1 , 2 F 1 ( a, b ; c ; x ) = n !( c ) n n =0 the integral representation: � 1 Γ( c ) t b − 1 (1 − t ) c − b − 1 (1 − xt ) − a dt, 2 F 1 ( a, b ; c ; x ) = Γ( c − b )Γ( b ) 0 where Re( c ) > Re( b ) > 0 and x / ∈ [1 , ∞ [.

– Gauss (1812), Riemann (1857), Barnes (1908): a detailed investigation of the 2 F 1 -function The hypergeometric equation in the canonical form: x (1 − x ) y ′′ ( x ) + ( c − ( a + b + 1) x ) y ′ ( x ) − aby ( x ) = 0 , y ( x ) = 2 F 1 ( a, b ; c ; x ) — the solution analytical near x = 0. Riemann’s P -symbol for a general solution of ODE with 3 regular singular points x = α, β, γ :   α β γ   P a 1 b 1 c 1 x   a 2 b 2 c 2 Barnes representation: � i ∞ Γ( c ) Γ( a + u )Γ( b + u )Γ( − u ) ( − x ) u du 2 F 1 ( a, b ; c ; x ) = Γ( a )Γ( b ) Γ( c + u ) − i ∞ the poles u = − a − k, − b − k and u = k , k = 0 , 1 , ... , are separated by the integration contour

– Selberg (1944): a multidimensional generalization of the Euler beta integral � 1 � 1 N � � x α − 1 (1 − x j ) β − 1 | x i − x j | 2 γ dx 1 . . . dx N . . . j 0 0 j =1 1 ≤ i<j ≤ N N � Γ( α + ( j − 1) γ )Γ( β + ( j − 1) γ )Γ(1 + jγ ) = , Γ( α + β + ( N + j − 2) γ )Γ(1 + γ ) j =1 � 1 � n, Re( α ) n − 1 , Re( β ) Re( α ) , Re( β ) > 0 , Re( γ ) > − min . n − 1 Important applications in the theory of multivariable orthogo- nal polynomials, random matrices (matrix models) and quantum mechanics.

q -HYPERGEOMETRIC FUNCTIONS – Euler : q -exponential functions ∞ � x n 1 | q | < 1 , | x | < 1 , = , ( q ; q ) n ( x ; q ) ∞ n =0 ∞ � q n ( n − 1) / 2 ( − x ) n = ( x ; q ) ∞ , | q | < 1 , ( q ; q ) n n =0 n − 1 � (1 − xq k ) ( x ; q ) n = the q -Pochhammer symbol k =0 – Gauss : q -binomial theorem ∞ � ( t ; q ) n x n = ( tx ; q ) ∞ | x | , | q | < 1 . 1 ϕ 0 ( t ; q, x ) = , ( q ; q ) n ( x ; q ) ∞ n =0 – Heine (1847): q -analogue of the 2 F 1 -function ∞ � ( s ; q ) n ( t ; q ) n x n , 2 ϕ 1 ( s, t ; w ; q, x ) = ( q ; q ) n ( w ; q ) n n =0 2 ϕ 1 ( q a , q b ; q c ; q, x ) → 2 F 1 ( a, b ; c ; x ) for q → 1 . – Ramanujan (1920): mock theta functions

The theory was developing more than 300 years with a belief that � u 1 , . . . , u r +1 � ∞ � ( u 1 ) n · · · ( u r +1 ) n x n r +1 F r ; x = n !( v 1 ) n · · · ( v r ) n v 1 , . . . , v r n =0 and � t 1 , . . . , t r +1 � ∞ � ( t 1 ; q ) n . . . ( t r +1 ; q ) n x n . r +1 ϕ r ; q, x = w 1 , . . . , w r ( q ; q ) n ( w 1 ; q ) n . . . ( w r ; q ) n n =0 + multivariable extensions + integral representations capture all hypergeometric functions with nice properties. Turn of the millenium (2000): discovery of the ELLIPTIC HYPERGEOMETRIC FUNCTIONS

First examples of terminating elliptic hypergeometric series: – elliptic solutions of the Yang-Baxter equation (I. Frenkel, Tu- raev, 1997), – special solution of the Lax pair equations for a discrete-time integrable system (V.S., Zhedanov, 1999). Recognition of the general structure ( balancing , well-poised- ness , very-well-poisedness ) of such series (V.S., 2001): ∞ r − 4 � � θ ( t 0 q 2 n ; p ) θ ( t m ) n q n , r +1 V r ( t 0 ; t 1 , . . . , t r − 4 ; q, p ) = θ ( qt 0 t − 1 θ ( t 0 ; p ) m ) n n =0 m =0 with the termination condition t m = q − N and balancing condition r − 4 � t k = t ( r − 5) / 2 q ( r − 7) / 2 . 0 k =1 The elliptic Pochhammer symbol (Zolotarev, 1878) n − 1 � θ ( zq k ; p ) , θ ( z ; p ) = ( z ; p ) ∞ ( pz − 1 ; p ) ∞ θ ( z ) n = k =0 p → 0 r +1 V r = very-well poised r − 1 ϕ r − 2 -series . lim Terminating r +1 V r -series are elliptic functions of a special form. Elliptic functions = meromorphic double periodic functions. Infinite series do not converge in general.

The principally new class of functions: Elliptic Hypergeometric Integrals (V.S., 2000, 2003) Univariate case: contour integrals � ∆( u ) du, C where ∆( u ) satisfies a first order finite difference equation ∆( u + ω 1 ) = h ( u ; ω 2 , ω 3 )∆( u ) , with h ( u ; ω 2 , ω 3 ) – an elliptic function: h ( u + ω 2 ) = h ( u + ω 3 ) = h ( u ) . Here ω 1 , 2 , 3 ∈ C , Im( ω 2 /ω 3 ) � = 0. THE ELLIPTIC BETA INTEGRAL Theorem (V.S., 2000). Let | p | , | q | , | t j | < 1, � 6 j =1 t j = pq . Then � 6 � j =1 Γ( t j z ± 1 ; p, q ) � ( p ; p ) ∞ ( q ; q ) ∞ dz z = Γ( t j t k ; p, q ) , Γ( z ± 2 ; p, q ) 4 π i T 1 ≤ j<k ≤ 6 where T is the unit circle, ∞ � 1 − z − 1 p j +1 q k +1 | p | , | q | < 1 , Γ( z ; p, q ) = , 1 − zp j q k j,k =0 is the elliptic gamma function.

Conventions Γ( t 1 , . . . , t k ; p, q ) := Γ( t 1 ; p, q ) · · · Γ( t k ; p, q ) , Γ( tz ± 1 ; p, q ) := Γ( tz ; p, q )Γ( tz − 1 ; p, q ) . This was a principally new exactly computable integral: – obeys W ( E 6 ) group of symmetries – generalizes q -beta integrals of Askey-Wilson, Rahman, ..., Eu- ler’s beta integral – many multidimensional extensions to integrals on root systems In a wide sense this is the elliptic binomial theorem . arXiv surveys: math.CA/0511579 and 0805.3135. An outstanding discovery (Dolan, Osborn, 2008): the elliptic beta integral describes the confinement phenomenon in the simplest 4 d supersymmetric quantum chromodynamics ... A completely unexpected physical application ! How it works ? Through the equality of superconformal indices (KMMR, 2005; R¨ omelsberger, 2005) for SUSY gauge theories related by the Seiberg duality (Seiberg, 1994).

Seiberg duality: SU ( N c ) gauge group example “Electric” theory: SU ( N c ) SU ( N f ) l SU ( N f ) r U (1) B U (1) R ˜ Q f f 1 1 N c /N f � ˜ Q f 1 f -1 N c /N f V adj 1 1 0 1 “Magnetic” theory: SU ( ˜ N c ) SU ( N f ) l SU ( N f ) r U (1) B U (1) R N c / ˜ q f f 1 N c N c /N f − N c / ˜ q f 1 f N c N c /N f � 2 ˜ M 1 f f 0 N c /N f ˜ V adj 1 1 0 1 where ˜ N c = N f − N c and 3 N c / 2 < N f < 3 N c (conformal window). Seiberg conjectured that these two N = 1 SYM theories have the same physics at their IR fixed points. Consistency checks: • The global anomalies match (’t Hooft anomaly matching) • Matching of the reductions N f → N f − 1 • The moduli spaces have the same dimensions and the gauge invariant operators match

Superconformal index SU (2 , 2 | 1) space-time symmetry group: J i , J i ( SU (2) subgroups generators, or Lorentz rotations), P µ , Q α , Q ˙ α (supertranslations), K µ , S α , S ˙ α (special superconformal transformations), H (dilations) and R ( U (1) R -rotations). Internal symmetries: a local gauge group G c (generators G a ) and a global flavor group F (generators F k ). For Q = Q 1 and Q † = − S 1 , one has { Q, Q † } = 2 H , H = H − 2 J 3 − 3 R/ 2 , and the superconformal index is defined as the matrix integral � � ( − 1) F p R / 2+ J 3 q R / 2 − J 3 I ( p, q, f k ) = dµ ( g ) Tr G c k f k F k e − β H � � � a g a G a e × e R = H − R/ 2 , , where dµ ( g ) is the Haar G c -invariant measure; F – the fermion number; p, q, g a , f k , β are group parameters (chemical potentials). It counts BPS states H| ψ � = 0 or cohomology of Q , Q † operators (hence, no β -dependence).