Frobenius Manifolds and Integrable Hierarchies Sofia, September 28-30, 2006 Boris DUBROVIN SISSA (Trieste) 1. 2D topological field theory, Frobenius algebras and WDVV associativity equations 2. Coupling to topological gravity: first examples of integrable systems. Frobenius manifolds and their properties 3. Integrable hierarchies of the topological type 4. Universality problems 1
2D topological field theory, WDVV associativity equations, and Frobenius manifolds QFT on Σ, dim = D: local fields φ α (x), x Σ classical action Partition function S[φ] = Z Σ = Σ L(φ, φ x) [dφ] e S[φ] and correlators φ α (x)φ β (y)... Σ = [dφ] φ α (x)φ β (y)... e S[φ] 2
Topological invariance δs δg ij (x) 0. correlators are numbers depending only on the genus g = g(σ) φ α (x)φ β (y)... Σ φ α φ β... g 3
Matter sector of a 2D TFT is specified by: 1. The space of local physical states A. dim A = n <. 2. An assignment (Σ, Σ) v (Σ, Σ) A (Σ, Σ) for any oriented 2-surface Σ with an oriented boundary Σ that depends only on the topology of the pair (Σ, Σ) A (Σ, Σ) = { C if Σ = A 1... A k if Σ = C 1... C k 4
A i = A A orientation of C i is coherent to the orientation of Σ otherwise factor- The assignment satisfies the following three axioms ization rules
1. Normalization: 5
2. Multiplicativity: if (Σ, Σ) = (Σ 1, Σ 1 ) (Σ 2, Σ 2 ) (disjoint union) then v (Σ, Σ) = v (Σ1, Σ 1 ) v (Σ 2, Σ 2 ) in A (Σ, Σ) = A (Σ1, Σ 1 ) A (Σ 2, Σ 2 ). 6
3. Factorization. Let (Σ, Σ) and (Σ, Σ ) differ by 7
We require that v (Σ, Σ) = i 0 j 0 contraction of v (Σ, Σ ). Recall: ij-contraction A 1... A k A 1... Â i... Â j... A k (the i-th and the j-th factors are omitted in the r.h.s.) when A i and A j are dual using the standard pairing A A C of the i-th and j-th factors
Redenote by v g,s the vector a symmetric polylinear function on the space of the states. 8
Choosing a basis φ 1,..., φ n A we obtain the components of the polylinear function v g,s ( φα1... φ αs ) =:< φα1... φ αs > g that by definition are called the genus g correlators of the fields φ α1,..., φ αs.
Theorem. The space of the states A carries a natural structure of a Frobenius algebra. All the correlators can be expressed in a pure algebraic way in terms of this algebra < φ α1... φ αk > g =< φ α1... φ αk, H g > (in the r.h.s. the product in the algebra A). 9
The structure constant tensor commutative associative algebra 10
The invariant bilinear form, x y, z = x, y z does not degenerate 11
The unity 12
The handle operator 13
Proof. Commutativity of the multiplication is obvious since we can interchange the legs of the pants by a homeomorphism. Similarly, we obtain the symmetry of the inner product <, >. 14
Associativity follows from 15
In particular, the k-product is determined by the k-leg pants
The unity
Nondegenerateness of η. We put and prove that η = η 1. This follows from 16
Compatibility of the multiplication with the inner product is proved on 17
The proof of the second part is given on the following picture 18
Physical setup: primary chiral fields φ 1 = 1, φ 2,..., φ n Frobenius algebra A genus zero correlators η αβ = φ α φ β 0, c αβγ = φ α φ β φ γ 0. Handle operator H = η αβ φ α φ β A Conserved U(1)-charge A is graded Frobenius algebra deg φ α = q α, φ α, φ β = 0 only if q α + q β = d. 19
Realization of topological invariance: Hilbert space of states H with a nilpotent symmetry Q : H H, Q 2 = 0 {Physical observables} = Ker ad Q A := Ker Q/Im Q Q-invariance {Q, ψ} φ... = 0 20
Topological invariance comes from dφ α (x) = {Q, φ (1) α }, dφ (1) α (x) = {Q, φ (2) α } φ α (x) functions, φ (1) α (x) 1-forms, φ (2) α (x) 2-forms Then d x φ α (x)φ β (y)... = {Q, φ (1) α (x)} φ β (y)... = 0.
Nonlocal observables {Q, {Q, C φ(1) α } = Σ φ(2) α } = 0 C dφ α = 0 Topological perturbations S S(t) := S n α=1 t α Σ φ(2) α Perturbed primary correlators φ α (x)φ β (y)... (t) := [dφ] φ α (x)φ β (y)... e S(t) 21
Theorem (Dijkgraaf, Verlinde, Verlinde) The perturbation preserves the topological invariance. The perturbed primary chiral algebra A t satisfies WDVV η αβ = φ α φ β 0 (t) = const, c αβγ (t) = φ α φ β φ γ 0 (t) = 3 F (t) t α t β t γ for some function F (t) (primary free energy). Also c 1 αβ (t) η αβ 22
Arrive at WDVV equations of associativity 3 F (t) 3 F (t) t α t β t ληλµ t µ t γ t δ = 3 F (t) 3 F (t) t δ t β t ληλµ t µ t γ t α 3 F (t) t 1 t α t β = η αβ 23
U(1)-charge quasihomogeneity E F (t) = (3 d)f (t) + quadratic terms E = t 1 1 + linear in t 2,..., t n E.g., E = n α=1 (1 q α )t α α, q 1 = 0 24
Proof based on construction by twisting of N = 2 SUSY Commutator algebra [L m, L n ] = (m n)l m+n, [J m, J n ] = d m δ n+m,0 [L m, G n ] = (m n)g m+n, [J m, G n ] = G m+n [L m, Q n ] = n Q m+n, [J m, Q n ] = Q m+n {G m, Q n } = L m+n + n J m+n + d 2 m(m + 1) δ n+m,0 [L m, J n ] = n J m+n d 2 m(m + 1) δ n+m,0 25
Obtained from N = 2 SUSY algebra by modifying the N = 2 stress tensor T (z) T (z) + 1 2 J(z) Q = Q(z) T (z) = {Q, G(z)} Q(z) = [Q, J(z)] 26
Hodge condition G 0 φ α = 0, L 0 φ α = 0 U(1) charge J 0 φ α = q α φ α Q-symmetry δ ε z = θ, δ ε z = θ Superfields Φ α = φ α + θ φ (1) α + θ φ (1) α + θ θ φ (2) α 27
Möbius group action generated by L 0, G 0 z z = az + b cz + d θ θ = θ + αz2 + βz + γ (c z + d) 2 Global Ward identities on the sphere Möbius invariance Φ α1 ( z 1, θ 1 )... Φ αn ( z n, θ n ) 0 = Φ α1 (z 1, θ 1 )... Φ αn (z n, θ n ) 0 28
Using d 2 z d 2 θ Φ α (z, z, θ, θ) = d 2 z φ (2) α (z, z) integrate out all z, θ. Can choose z 1 = 0, z 2 = 1, z 3 = to arrive at Φ α1... Φ αn 0 = φ α1 φ α2 φ α3 φ (2) α 4... φ (2) α n 0 Symmetry in α 1,... α n completes the proof. 29
Example 1 Topological sigma-models (=Gromov - Witten invariants) X smooth projective variety, dim C X = d, H odd (X) = 0, dim H (X) =: n Primary observables cohomologies φ 1 = 1, φ 2,..., φ n H (X) 30