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:

1F0(a; x) = ∞

• n=0

(a)n n! xn = (1 − x)−a, |x| < 1, a ∈ C, where (a)n = a(a + 1) · · · (a + n − 1) the Pochhammer symbol

– Leonhard Euler (1729 and later on) the gamma function Γ(x): Γ(x) = ∞ tx−1e−tdt, Re(x) > 0, the beta function (integral) B(x, y): B(x, y) = 1 tx−1(1−t)y−1dt = Γ(x)Γ(y) Γ(x + y), Re(x), Re(y) > 0, the 2F1-series:

2F1(a, b; c; x) = ∞

• n=0

(a)n(b)n n!(c)n xn, |x| < 1, the integral representation:

2F1(a, b; c; x) =

Γ(c) Γ(c − b)Γ(b) 1 tb−1(1 − t)c−b−1(1 − xt)−adt, where Re(c) > Re(b) > 0 and x / ∈ [1, ∞[.

– Gauss (1812), Riemann (1857), Barnes (1908): a detailed investigation of the 2F1-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) = 2F1(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: P    α β γ a1 b1 c1 x a2 b2 c2    Barnes representation:

2F1(a, b; c; x) =

Γ(c) Γ(a)Γ(b) i∞

−i∞

Γ(a + u)Γ(b + u)Γ(−u) Γ(c + u) (−x)udu 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

• j=1

xα−1

j

(1 − xj)β−1

• 1≤i<j≤N

|xi − xj|2γdx1 . . . dxN =

N

• j=1

Γ(α + (j − 1)γ)Γ(β + (j − 1)γ)Γ(1 + jγ) Γ(α + β + (N + j − 2)γ)Γ(1 + γ) , Re(α), Re(β) > 0, Re(γ) > − min 1 n, Re(α) n − 1, Re(β) 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

∞

• n=0

xn (q; q)n = 1 (x; q)∞ , |q| < 1, |x| < 1,

∞

• n=0

qn(n−1)/2 (q; q)n (−x)n = (x; q)∞, |q| < 1, (x; q)n =

n−1

• k=0

(1 − xqk) the q-Pochhammer symbol – Gauss: q-binomial theorem

1ϕ0(t; q, x) = ∞

• n=0

(t; q)n (q; q)n xn = (tx; q)∞ (x; q)∞ , |x|, |q| < 1. – Heine (1847): q-analogue of the 2F1-function

2ϕ1(s, t; w; q, x) = ∞

• n=0

(s; q)n(t; q)n (q; q)n(w; q)n xn,

2ϕ1(qa, qb; qc; q, x) → 2F1(a, b; c; x)

for q → 1. – Ramanujan (1920): mock theta functions

The theory was developing more than 300 years with a belief that

r+1Fr

u1, . . . , ur+1 v1, . . . , vr ; x

• =

∞

• n=0

(u1)n · · · (ur+1)n n!(v1)n · · · (vr)n xn and

r+1ϕr

t1, . . . , tr+1 w1, . . . , wr ; q, x

• =

∞

• n=0

(t1; q)n . . . (tr+1; q)n (q; q)n(w1; q)n . . . (wr; q)n xn. + 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+1Vr(t0; t1, . . . , tr−4; q, p) = ∞

• n=0

θ(t0q2n; p) θ(t0; p)

r−4

• m=0

θ(tm)n θ(qt0t−1

m )n

qn, with the termination condition tm = q−N and balancing condition

r−4

• k=1

tk = t(r−5)/2 q(r−7)/2. The elliptic Pochhammer symbol (Zolotarev, 1878) θ(z)n =

n−1

• k=0

θ(zqk; p), θ(z; p) = (z; p)∞(pz−1; p)∞ lim

p→0 r+1Vr = very-well poised r−1ϕr−2-series.

Terminating r+1Vr-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

• C

∆(u)du, where ∆(u) satisfies a first order finite difference equation ∆(u + ω1) = h(u; ω2, ω3)∆(u), with h(u; ω2, ω3) – an elliptic function: h(u + ω2) = h(u + ω3) = h(u). Here ω1,2,3 ∈ C, Im(ω2/ω3) = 0. THE ELLIPTIC BETA INTEGRAL Theorem (V.S., 2000). Let |p|, |q|, |tj| < 1, 6

j=1 tj = pq. Then

(p; p)∞(q; q)∞ 4πi

• T

6

j=1 Γ(tjz±1; p, q)

Γ(z±2; p, q) dz z =

• 1≤j<k≤6

Γ(tjtk; p, q), where T is the unit circle, Γ(z; p, q) =

∞

• j,k=0

1 − z−1pj+1qk+1 1 − zpjqk , |p|, |q| < 1, is the elliptic gamma function.

Conventions Γ(t1, . . . , tk; p, q) := Γ(t1; p, q) · · · Γ(tk; 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(E6) 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 4d supersymmetric quantum chromodynamics ... A completely unexpected physical application ! How it works ? Through the equality of superconformal indices (KMMR, 2005; R¨

• melsberger, 2005) for SUSY gauge theories related by the

Seiberg duality (Seiberg, 1994).

Seiberg duality: SU(Nc) gauge group example “Electric” theory: SU(Nc) SU(Nf)l SU(Nf)r U(1)B U(1)R Q f f 1 1 ˜ Nc/Nf

• Q

f 1 f

• 1

˜ Nc/Nf V adj 1 1 1 “Magnetic” theory: SU( ˜ Nc) SU(Nf)l SU(Nf)r U(1)B U(1)R q f f 1 Nc/ ˜ Nc Nc/Nf

• q

f 1 f −Nc/ ˜ Nc Nc/Nf M 1 f f 2 ˜ Nc/Nf ˜ V adj 1 1 1 where ˜ Nc = Nf−Nc and 3Nc/2 < Nf < 3Nc (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 Nf → Nf − 1
• The moduli spaces have the same dimensions and the gauge

invariant operators match

Superconformal index SU(2, 2|1) space-time symmetry group: Ji, Ji (SU(2) subgroups generators, or Lorentz rotations), Pµ, Qα, Q ˙

α (supertranslations),

Kµ, Sα, S ˙

α (special superconformal transformations),

H (dilations) and R (U(1)R-rotations). Internal symmetries: a local gauge group Gc (generators Ga) and a global flavor group F (generators F k). For Q = Q1 and Q† = −S1, one has {Q, Q†} = 2H, H = H − 2J3 − 3R/2, and the superconformal index is defined as the matrix integral I(p, q, fk) =

• Gc

dµ(g) Tr

• (−1)FpR/2+J3qR/2−J3

× e

• a gaGae
• k fkF ke−βH

, R = H − R/2, where dµ(g) is the Haar Gc-invariant measure; F – the fermion number; p, q, ga, fk, β are group parameters (chemical potentials). It counts BPS states H|ψ = 0 or cohomology of Q, Q† operators (hence, no β-dependence).

“Computation” (simple examples, guesswork, plethystic machin- ery; R¨

• melsberger, 2007):

I(y; p, q) =

• Gc

dµ(z) exp ∞

• n=1

1 nind

• pn, qn, zn, yn

with the “single particle states index” ind(p, q, z, y) = 2pq − p − q (1 − p)(1 − q)χadjG(z) +

• j

(pq)rjχRF ,j(y)χRG,j(z) − (pq)1−rjχ ¯

RF ,j(y)χ ¯ RG,j(z)

(1 − p)(1 − q) . χRF ,j(y) and χRG,j(z) are characters of the respective represen- tations (yj ∝ efj, za ∝ ega), and rj are halves of the R-charges. For the unitary group SU(N), z = (z1, . . . , zN), N

j=1 za = 1,

• SU(N)

dµ(z) = 1 N!

• TN−1 ∆(z)∆(z−1)

N−1

• a=1

dza 2πiza , ∆(z) =

• 1≤a<b≤N

(za − zb), the Vandermonde determinant.

Explicit identification with the elliptic beta integral (Dolan, Os- born, 2008) with tk = (pq)1/6yk, k = 1, . . . , 6. The left-hand side: G = SU(2), F = SU(6), the representation content 1) (adj, 1), χSU(2),adj(z) = z2 + z−2 + 1, 2) (f, f), χSU(2),f(z) = z + z−1, rf = 1/6, χSU(6),f(y) =

6

• k=1

yk, χSU(6), ¯

f(y) = 6

• k=1

y−1

k , 6

• k=1

yk = 1. The right-hand side: G = 1, F = SU(6) with the single repre- sentation TA : Φij = −Φji, χSU(6),TA(y) =

• 1≤i<j≤6

yiyj, rTA = 1/3. A Wess-Zumino type theory for the confined colored particles! CONCLUSION: Explicit computability of the elliptic hypergeomet- ric integrals = confinement in 4d N = 1 SUSY gauge theories. The process of integrals’ computation = transition from UV (weak coupling) to IR (strong coupling) physics. This is a new conceptual paradigm for exact mathe- matical formulas.

The general Seiberg duality. The electric theory index: IE = κNc

• TNc−1

Nf

i=1

Nc

j=1 Γ(sizj, t−1 i z−1 j )

• 1≤i<j≤Nc Γ(ziz−1

j , z−1 i zj) Nc−1

• j=1

dzj 2πizj ,

Nc

• j=1

zj = 1, κN = (p; p)N−1

∞

(q; q)N−1

∞

N! . The magnetic theory: IM = κ ˜

Nc

Nf

i,j=1 Γ(sit−1 j )×

×

• T

Nc−1

Nf

i=1 Nc j=1 Γ(S

1

• Ncs−1

i xj, T − 1

• Nctix−1

j )

• 1≤i<j≤

Nc Γ(xix−1 j , x−1 i xj)

• Nc−1
• j=1

dxj 2πixj , where ˜

Nc j=1 xj = 1,

˜ Nc = Nf − Nc, S = Nf

i=1 si,

T = Nf

i=1 ti,

ST −1 = (pq)Nf−Nc. Theorem: IE = IM For Nc = 2, Nf = 4 (V.S., 2003), for general Nc, Nf (Rains, 2003), equality of superconformal indices (Dolan, Osborn, 2008) Joint work with G.S. Vartanov (+ in preparation): – Nucl. Phys. B824 (2010), 192 – Phys. Rev. Lett. 105 (2010) 061603 – Elliptic hypergeometry of supersymmetric dualities, Commun.

• Math. Phys. (2011), to appear, arXiv:0910.5944

– Superconformal indices of N = 4 SYM field theories, arXiv:1005.4196.

Summary of the main results:

• ’t Hooft anomaly matching ←

→ the total ellipticity condition for elliptic hypergeometric terms: ratios of integral kernels satisfy a set of linear first order q- difference equations with coefficients which are elliptic func- tions (with modulus p) of all variables zk, xl, tj, sj, and q

• All known identities lead to totally elliptic hypergeometric

terms → conjecture: this property is necessary for com- putability/nice symmetry of integrals

• SC indices are invariants of the conformal manifold against

the exactly marginal deformations. The effects of certain non-marginal deformations are traced through giving spe- cial values to chemical potentials. For instance, reduction in the number of flavors by adding mass terms M k

k Qk

Qk with M k

k → ∞ in the electric theory leads to Higgsing of the

gauge group on the magnetic side, so that SU(Nf − Nc) → SU(Nf−Nc−1). It is equivalent to the restriction skt−1

k

= pq for indices (a simple substitution for IE and a residue calculus for IM)

• “Vanishing” (delta function behavior) of superconformal in-

dices ← → chiral symmetry breaking

• About 15 new computable elliptic beta integrals on root sys-

tems and a similar number of new symmetry transforma- tions for higher order elliptic hypergeometric integrals with SU(N), SP(2N), G2, E6, F4 gauge groups (conjectures)

• About 15 new pairs of N = 1 dual field theories (e.g., mul-

tiple dualities for Gc = SP(2N) with 8 flavors and Gc = SU(N) with 4 + 4 flavors, new confining theories)

• Analysis/conjectures of about 20 dualities/integral relations

for SO(N) gauge group (in preparation).

• There are non-trivial dualities lying outside the conformal

window (for different dual gauge groups)

• For N = 4 SYM theories – derivation of explicit forms of

indices for all simple gauge groups and exact computation

• f them in a particular (“p = s2 = 0”) limit. Possible con-

sequences for the AdS/CFT correspondence for finite rank gauge groups.

• Partial confirmation of the equality of indices for N = 4 SYM

SP(2N) ← → SO(2N + 1) duality (GPRR, 2010)

• Discovery of many new relations between dualities (some of

them are deducible from the others)

• It is conjectured that there are infinitely (countably) many

supersymmetric dualities and corresponding elliptic hyperge-

• metric integral identities.

Elliptic Selberg integral dualities Cho, Kraus (1996); Cs´ aki, Skiba, Schmaltz (1997): gauge group Gc = SP(2N), flavor group F = SU(6) × U(1) SP(2N) SU(6) U(1) U(1)R Q f f N − 1 2r = 1

3

A TA 1

• 3

Ak 1 −3k QAmQ TA 2(N − 1) − 3m

2 3

where k = 2, . . . , N and m = 0, . . . , N − 1. Equality of the superconformal indices is equivalent to the BCN- elliptic beta integral of type II (the elliptic Selberg integral) (van Diejen, V.S., 2000): for |p|, |q|, |t|, |tm| < 1 and t2N−2 6

m=1 tm = pq,

• TN
• 1≤j<k≤N

Γ(tz±1

j z±1 k ; p, q)

Γ(z±1

j z±1 k ; p, q) N

• j=1

6

m=1 Γ(tmz±1 j ; p, q)

Γ(z±2

j ; p, q)

dzj 2πizj = 2NN! (p; p)N

∞(q; q)N ∞ N

• j=1
• Γ(tj; p, q)

Γ(t; p, q)

• 1≤m<s≤6

Γ(tj−1tmts; p, q)

• .

+ two more new dual theories associated with W(E6) symmetry ! (the multiple duality case, V.S., Vartanov, 2008)

Elliptic beta integrals and solvable models of statistical mechanics V.S., arXiv:1011.3798 [hep-th]

• Abstract. The univariate elliptic beta integral was discovered

by the author in 2000. Recently Bazhanov and Sergeev have inter- preted it as a star-triangle relation (STR). This important observa- tion is discussed in more detail in connection to author’s previous work on the elliptic modular double and supersymmetric dualities. We describe also a new Faddeev-Volkov type solution of STR, con- nections with the star-star relation, and higher-dimensional ana- logues of such relations. In this picture Seiberg dualities are de- scribed by symmetries of the elliptic hypergeometric integrals (in- terpreted as superconformal indices) which, in turn, represent STR and Kramers-Wannier type duality transformations for elementary partition functions in solvable models of statistical mechanics.

CONCLUSION

• 1. Elliptic hypergeometric functions are universal objects with

applications in various fields of mathematics (combinatorics, har- monic analysis on root systems, representation theory, theory of au- tomorphic forms, approximation theory, continued fractions, topol-

• gy, etc) and theoretical physics (supersymmetric dualities, Calo-

gero-Sutherland type quantum mechanical models, solvable models

• f statistical mechanics, random matrices and stochastic determi-

nantal processes, etc).

• 2. They unify special functions (of elliptic and hypergeometric

types) under one roof and make them firm, unique, undeformable

• bjects living in the Platonic world of ideal bodies.
• 3. Work in progress / open questions

– higher genus Riemann surface generalization – proofs of the existing conjectures – the universal elliptic beta integral evaluation / symmetry trans- formation – automorphic properties – uncovering of full physical information hidden in integrals (in- terpreted as superconformal indices or partition functions of sta- tistical mechanics spin systems).