Elliptic gamma functions, gerbes and triptic curves Giovanni - PowerPoint PPT Presentation

Elliptic gamma functions, gerbes and triptic curves Giovanni Felder, ETH Zurich Paris, 18 January 2007 1

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 Andr´ e Henriques, Carlo A. Rossi and Chenchang Zhu 2

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 / Z x 1 + Z x 2 + Z x 3 . Today we consider the simplest case of such a difference equa- tion, the functional equation of the elliptic gamma function. 3

Jacobi’s infinite product In his Fundamenta nova Jacobi introduced the function ∞ (1 − q n +1 /t )(1 − q n t ) , � Θ( t, q ) = t � = 0 , | q | < 1 . n =0 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 ∞ (1 − q − n /t ) − 1 (1 − q − n − 1 t ) − 1 , � Θ( t, q ) = | q | > 1 . n =0 Jacobi and Hermite discovered transformation properties of Θ under q → q 4 π/ ln q , t → t − 4 π/ ln q and more generally under SL 2 ( Z ) ⋉ Z 2 4

Geometric content: elliptic curves Let x 1 , x 2 ∈ C be linearly independent over R . E ( x 1 ,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 = ∈ SL 2 ( Z ) c d Moduli space of oriented elliptic curves: M = Y/ SL 2 ( Z ) Y = { ( x 1 : x 2 ) ∈ C P 1 | x 1 , x 2 R -linearly independent } = C P 1 − R P 1 = H + ∪ H − 5

Universal oriented elliptic curve The group ISL 2 ( Z ) = SL 2 ( Z ) ⋉ Z 2 acts on x 2 ) � = 0 } / C × X = { ( w, x 1 , x 2 ) | Im( x 1 ¯ x 2 ✁ ✁ via ( A, n ) · ( w, x ) = ( w + n 1 x 1 + n 2 x 2 , Ax ) ✁ ✁ ✁ ✁ ✁ ✁ ✁ ✁ E = X/ ISL 2 ( Z ) universal curve · w ✁ ✁ ✁ ✁ ↓ ✁ ✁ 0 x 1 M = Y/ SL 2 ( Z ) moduli space Remarks: 1. X is the total space of the line bundle O (1) → C P 1 − R P 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

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 coordi- nates: ∞ (1 − q n +1 /t )(1 − q n t ) , t = e 2 πiz , q = e 2 πiτ � θ ( z, τ ) = n =0 Extend to Im τ � = 0 by θ ( − z, − τ ) = θ ( z, τ ) − 1 . � w x 2 , x 1 � Then ( w, x 1 , x 2 ) → θ is a meromorphic function on X , a x 2 covering space of the universal elliptic curve X/ ISL 2 ( Z ). 7

Transformation properties under G = ISL 2 ( Z ) , x ′ w ′ � � � � w , x 1 = e 2 πiQ g ( w,x ) θ 1 θ ( ∗ ) x ′ x ′ x 2 x 2 2 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 2 πiQ g ( w,x ) ) g ∈ G defines a G -equivariant line bundle L on G ( X, O × X (a class in H 1 X )) (b) θ is a G -equivariant meromorphic section of L . Namely if M denotes the sheaf of meromorphic functions, θ ∈ C 0 G ( X, M × ) and (*) means δθ = φ . (In this case everything reduces to group cohomology) 8

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

“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( z, τ, σ ) = Γ( z + τ, τ, τ + σ )Γ( z, τ + σ, σ ) . � � � � � � w , x 1 , x 2 w , x 2 , x 3 w , x 3 , x 1 = e − πiP 3 ( w,x ) / 3 , Γ Γ Γ x 3 x 3 x 3 x 1 x 1 x 1 x 2 x 2 x 2 P 3 ( w, x ) = w 3 w 2 + e 2 − 3 e 1 1 + e 2 w − e 1 e 2 . e 3 2 e 3 2 e 3 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

Geometric content: triptic curves A triptic curve is a stack of the form E x = C / Z x 1 + Z x 2 + Z x 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 = C P 2 − R P 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 ) universal triptic curve ↓ M = Y/ SL 3 ( Z ) moduli space This time Y is topologically non-trivial: it retracts to the 2- sphere x 2 1 + x 2 2 + x 2 3 = 0. 11

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 = C P 2 − R P 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 ∈ Λ prim is a good ISL 3 ( Z ) equivariant open cover of X . C ( U , O × ), ˇ C ( U , M × ) be the ˇ Let ˇ Cech complex of U with values in the sheaf of invertible holomorphic/meromorphic functions. 12

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

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 ˇ Cech cocycle in C 2 ( U , O × ) on X = O (1) → C P 2 − R P 2 . It defines a holomorphic ˇ gerbe on the stack X/ SL 3 ( Z ). (b) Γ = (Γ a,b ) is a meromorphic section of this gerbe, namely an C 1 ( U , M × ) such that δ Γ = φ invariant cochain in ˇ 14

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 holo- morphic 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) → C P 2 − R P 2 . The gerbe is defined by an equivariant cocycle φ . 15

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

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 Γ a,b φ a,b ( g ; ) ↑ ˇ δ ∆ a ( g ; ) φ a ( g, h ; ) δ G − → 17

Characteristic class Theorem The Dixmier–Douady class [ φ ] ∈ H 2 G ( X, O × ) of the gamma gerbe maps to a non-trivial class c ∈ H 3 G ( X, Z ) . There is an exact sequence 0 → Z → H 3 G ( X, Z ) / torsion → H 3 ( 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 com- patible with the complex structure and thus admits a connective structure. 18