Elliptic gamma functions, gerbes and triptic curves

Transcription

1 Elliptic gamma functions, gerbes and triptic curves Giovanni Felder, ETH Zurich Paris, 18 January

2 Table of contents 0. Introduction 1. Two periods: Jacobi s infinite products, elliptic curves, SL 2 (Z) 2. Three periods: Ruijsenaars s elliptic gamma functions 3. The moduli stack of triptic curves and SL 3 (Z) 4. The gamma gerbe and its Dixmier Douady class based on joint work with Alexander Varchenko and with André Henriques, Carlo A. Rossi and Chenchang Zhu 2

3 Introduction In conformal field theory based on quantum groups and statistical mechanics there appear linear difference equations with elliptic coefficients. Idea: the step plays the role of a third period. Geometrically, one is lead to consider triptic curves C/Zx 1 + Zx 2 + Zx 3. Today we consider the simplest case of such a difference equation, the functional equation of the elliptic gamma function. 3

4 Jacobi s infinite product In his Fundamenta nova Jacobi introduced the function Θ(t, q) = n=0 (1 q n+1 /t)(1 q n t), t 0, q < 1. The Jacobi product obeys the functional equation Θ(qt, q) = t 1 Θ(t, q). This equation holds also for q > 1 if we set Θ(t, q) = n=0 (1 q n /t) 1 (1 q n 1 t) 1, q > 1. Jacobi and Hermite discovered transformation properties of Θ under q q 4π/ ln q, t t 4π/ ln q and more generally under SL 2 (Z) Z 2 4

5 Geometric content: elliptic curves Let x 1, x 2 C be linearly independent over R. E (x1,x 2 ) = C/Z x 1+ Z x 2 is an oriented elliptic curve. ( a b ) E x E x iff x = λax, λ C, A = c d SL 2 (Z) Moduli space of oriented elliptic curves: M = Y/SL 2 (Z) Y = {(x 1 : x 2 ) CP 1 x 1, x 2 R-linearly independent} = CP 1 RP 1 = H + H 5

6 Universal oriented elliptic curve The group ISL 2 (Z) = SL 2 (Z) Z 2 acts on X = {(w, x 1, x 2 ) Im(x 1 x 2 ) 0}/C via (A, n) (w, x) = (w + n 1 x 1 + n 2 x 2, Ax) E = X/ISL 2 (Z) M = Y/SL 2 (Z) universal curve moduli space x 2 w 0 x 1 Remarks: 1. X is the total space of the line bundle O(1) CP 1 RP 1. (It is actually a trivial bundle over the union of contractible spaces H + H ) 2. These spaces are mildly singular. They should be treated as stacks. 6

7 The Jacobi product as a section of a line bundle over the universal elliptic curve For Im τ > 0, let us write the theta product in additive coordinates: θ(z, τ) = n=0 (1 q n+1 /t)(1 q n t), t = e 2πiz, q = e 2πiτ Extend to Im τ 0 by θ( z, τ) = θ(z, τ) 1. Then (w, x 1, x 2 ) θ ( w x2, x 1 x 2 ) is a meromorphic function on X, a covering space of the universal elliptic curve X/ISL 2 (Z). 7

8 Transformation properties under G = ISL 2 (Z) θ ( w x, x 1 2 x 2 ) = e 2πiQ g(w,x) θ ( w, x ) 1 x 2 x 2 ( ) w = w + n 1 x 1 + n 2 x 2, x = Ax, g = (A, n) G = ISL 2 (Z) Q g (w, x) Q(x 1, x 2 )[w] of degree 2 in w. Meaning: (a) φ = (e 2πiQ g(w,x) ) g G defines a G-equivariant line bundle L on X (a class in H 1 G (X, O X )) (b) θ is a G-equivariant meromorphic section of L. Namely if M denotes the sheaf of meromorphic functions, θ CG 0(X, M ) and (*) means δθ = φ. (In this case everything reduces to group cohomology) 8

9 Rational, trigonometric and elliptic gamma function Euler 1729: Γ(z + 1) = z Γ(z) z! = Γ(z + 1) = j=1 j 1 z (j + 1) z j + z Jackson 1912: Γ(z + σ, σ) = (1 e 2πiz )Γ(z, σ) Γ(z, σ) = j=0 1 1 r j t, r = e2πiσ, t = e 2πiz Ruijsenaars 1997: Γ(z + σ, τ, σ) = θ(z, τ)γ(z, τ, σ) Γ(z, τ, σ) = j,k=0 1 q j+1 r k+1 t 1 1 q j r k, q = e 2πiτ, r = e 2πiσ, t = e 2πiz t 9

10 Modular properties Extend the definition of Γ(z, τ, σ) to a meromorphic function on C (C R) (C R): Γ(z, τ, σ) = Γ(z + τ, τ, σ) 1, Γ(z, τ, σ) = Γ(z + σ, τ, σ) 1. Then (G. F., A. Varchenko 2000) Γ Γ(z, τ, σ) = Γ(z + τ, τ, τ + σ)γ(z, τ + σ, σ). ( w, x 1, x ) ( 2 w Γ, x 2, x ) ( 3 w Γ, x 3, x ) 1 x 3 x 3 x 3 x 1 x 1 x 1 x 2 x 2 x 2 = e πip 3(w,x)/3, P 3 (w, x) = w3 e 3 3 e 1 2 e 3 w 2 + e2 1 + e 2 2 e 3 w e 1 e 2 4 e 3. e 1 = x 1 + x 2 + x 3, e 2 = x 1 x 2 + x 1 x 3 + x 2 x 3, e 3 = x 1 x 2 x 3. 10

11 Geometric content: triptic curves A triptic curve is a stack of the form E x = C/Zx 1 + Zx 2 + Zx 3, where x 1, x 2, x 3 C span C over R. E x E x iff x = λax λ C, A SL 3 (Z). The moduli space of oriented triptic curves is Y/SL 3 (Z), Y = CP 2 RP 2. ISL 3 (Z) = SL 3 (Z) Z 3 acts on X = {(w, x) C C 3 C R 3 }/C = total space of O(1) Y. E = X/ISL 3 (Z) M = Y/SL 3 (Z) universal triptic curve moduli space This time Y is topologically non-trivial: it retracts to the 2- sphere x x2 2 + x2 3 = 0. 11

12 An ISL 3 (Z)-equivariant cover of X There is a good open cover of X labeled by Λ prim, the set of primitive vectors in Λ = Z 3 C 3. If a Λ prim let H(a) be the oriented hyperplane in the dual lattice Λ with equation δ, a = 0. U a = {x Y = CP 2 RP 2 Im( α, x β, x ) > 0} for any oriented basis α, β of H(a). Let V a = p 1 (U a ) X. Lemma U = (V a ) a Λprim is a good ISL 3 (Z) equivariant open cover of X. Let Č(U, O ), Č(U, M ) be the Čech complex of U with values in the sheaf of invertible holomorphic/meromorphic functions. 12

13 Gamma functions associated to pairs of primitive vectors For a, b Λ prim linearly independent set Γ a,b (w, x) = H(a) H(b) = Z γ. Set Γ a,±a = 1. δ C + (a,b)/z γ (1 e 2πi( δ,x w)/ γ,x ) δ C + (a,b)/z γ (1 e+2πi( δ,x w) / γ,x ). H(b) Γ a,b is a meromorphic function on V a V b. It reduces C + _ b to Γ ( w x3, x 1 x, x ) 2 3 x if (a, b) = 3 γ (e 1, e 2 ). a C_ + H(a) 13

14 Theorem Γ a,b = Γ 1 b,a and on V a V b V c, Γ a,b (w, x)γ b,c (w, x)γ c,a (w, x) = e πip a,b,c(w,x)/3 for some polynomial P a,b,c (w, x) Q(x 1, x 2, x 3 )[w] of degree 3 in w with rational coefficients, holomorphic on V a V b V c. Moreover Γ ga,gb (w, gx) = Γ a,b (w, x), g SL 3 (Z). Consequences (a) The invertible holomorphic functions φ a,b,c = e πip a,b,c/3, a, b, c Λ prim on V a V b V c form an SL 3 (Z)-invariant Čech cocycle in Č 2 (U, O ) on X = O(1) CP 2 RP 2. It defines a holomorphic gerbe on the stack X/SL 3 (Z). (b) Γ = (Γ a,b ) is a meromorphic section of this gerbe, namely an invariant cochain in Č 1 (U, M ) such that δγ = φ 14

15 Including the translation subgroup Let µ Λ = Z 3. Then Γ a,b (w, x) Γ a,b (w + µ, x, x) = φ a,b(µ; w, x) b(µ; w, x) a (µ; w, x), (w, x) V a V b, for some meromorphic functions a (µ; ) M (V a ) and holomorphic functions φ a,b (µ; ) O (V a V b ). These identitities are part of a system of identities stating that (Γ, ) define a G-equivariant meromorphic section of the gamma gerbe G on the total space X of the line bundle O(1) CP 2 RP 2. The gerbe is defined by an equivariant cocycle φ. 15

16 The gamma gerbe Let G = ISL 3 (Z) = SL 3 (Z) Z 3. The complex C n G (U, F) = p+q=nc p (G, Č q (U, F)), n = 0, 1, 2,... with total differential D = δ G + ( 1) pˇδ computes the equivariant cohomology of X with values in F = O or M. Theorem φ CG 2(U, O ) = C 0,2 C 1,1 C 2,0 is a 2-cocycle and thus defines a gerbe G on the stack X/G. The meromorphic cochain (Γ, ) CG 1(U, M ) = C 0,1 C 1,0 obeys D(Γ, ) = φ and thus defines a meromorphic section of G. 16

17 Explicit formulae In explicit terms, we have identities φ a,b,c (y)γ a,c (y) = Γ a,b (y)γ b,c (y), y V a V b V c, φ a,b (g; y)γ g 1 a,g 1 b (g 1 y) b (g; y) = a (g; y)γ a,b (y), y V a V b, φ a (g, h; y) a (gh; y) = a (g; y) g 1 a (h; g 1 y), y V a, for all a, b, c I, g, h G. φ a,b,c ˇδ Γ a,b φ a,b (g; ) a (g; ) φ a (g, h; ) δ G 17

18 Characteristic class Theorem The Dixmier Douady class [φ] HG 2(X, O ) of the gamma gerbe maps to a non-trivial class c HG 3 (X, Z). There is an exact sequence 0 Z HG 3 (X, Z)/torsion H3 (Z 3, Z) 0, and c maps to a generator of H 3 (Z 3, Z) Z. It is well-known that the theta function bundle is hermitian. The same holds for the gamma gerbe: Theorem The gamma gerbe G has a hermitian structure compatible with the complex structure and thus admits a connective structure. 18