W -Constraints for Simple Singularities Bojko Bakalov Todor Milanov North Carolina State University Supported in part by the National Science Foundation Quantized Algebra and Physics Chern Institute of Mathematics July 23-26, 2009 Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 1 / 16
Outline 1 Introduction Gromov Witten Theory Main Questions Previous Work Main Result 2 Singularities and Root Systems Simple Singularities Milnor Fibration Monodromy Representation 3 Lattice Vertex Algebras and W -Algebras Affine Lie Algebras Lattice Vertex Algebras The Vertex Algebra W XN Main Results 4 Idea for Proof Givental s Formula Computation Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 2 / 16
Gromov Witten Theory Gromov Witten invariants of a manifold X generating series D X total descendant potential. Witten Kontsevich function D pt intersection theory on M g,n. τ-function for the KdV hierarchy. Virasoro constraints Ln D pt = 0 (n 1). String eq. L 1 D pt = 0. Genus 0 Gromov Witten invariants of X quantum cup product on H (X) Frobenius algebra + flat connection Frobenius manifold. Givental s formula expresses D X in terms of D pt and the semisimple Frobenius structure. Virasoro constraints for Dpt Virasoro constraints for D X. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 3 / 16
Main Questions Questions Does D X satisfy an integrable hierarchy? Does D X satisfy Virasoro or W -constraints? Generalized Witten Conjecture: The total descendant potential for h-spin curves is a τ-function for the h-th Gelfand Dickey hierarchy and satisfies W h -constraints. W h = W(2, 3,..., h) is the Zamolodchikov Fateev Lukyanov W -algebra. W2 = Virasoro algebra. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 4 / 16
Previous Work Adler van Moerbeke: There is a unique τ-function for the h-th Gelfand Dickey hierarchy solving the string equation. It satisfies the W h -constraints. K. Saito: Singularity theory semisimple Frobenius structure total descendant potential D XN defined by Givental s formula. XN is the type of the singularity. XN = A N, D N, E 6, E 7 or E 8 for simple singularities. Fan Jarvis Ruan: D AN is the total descendant potential for Gromov Witten invariants of h-spin curves (h = N + 1). Generalization for types D and E. E. Frenkel Givental Milanov: D XN (X = A, D, E) is a τ-function for the Kac Wakimoto hierarchy of type X N in the principal realization. Type A N h-th Gelfand Dickey hierarchy (h = N + 1). This implies the generalized Witten conjecture. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 5 / 16
Main Result Theorem The total descendant potential D XN (X = A, D, E) of a simple singularity satisfies W XN -constraints. Remarks We can prove both the integrable hierarchy and W -constraints. The proof works more generally for weighted homogeneous singularities. The proof uses singularity theory, Givental s formula, and vertex algebras. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 6 / 16
Simple Singularities f (x 1, x 2, x 3 ) homogeneous polynomial of degree h (deg x i = a i ) ( f ) with an isolated critical point at x = 0 (0) = 0. x i Type h a 1, a 2, a 3 f (x) Exponents A N N +1 1, a, N +1 a x N+1 1 +x 2 x 3 1, 2,..., N D N 2N 2 2, N 2, N 1 x N 1 1 +x 1 x2 2+x 3 2 1, 3,..., 2N 3, N 1 E 6 12 3, 4, 6 x1 4+x 2 3+x 3 2 1, 4, 5, 7, 8, 11 E 7 18 4, 6, 9 x 3 1 x 2+x 3 2 +x 2 3 1, 5, 7, 9, 11, 13, 17 E 8 30 6, 10, 15 x 5 1 +x 3 2 +x 2 3 1, 7, 11, 13, 17, 19, 23, 29 The exponents m 1,..., m N satisfy q m 1 + + q m N = (qh q a 1)(q h q a 2)(q h q a 3) q h (q a 1 1)(q a 2 1)(q a 3 1). Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 7 / 16
Milnor Fibration Miniversal deformations: f t (x) = f (x) + N i=1 t ig i (x). t = (t 1,..., t N ) T = C N. g i (x) homogeneous polynomials giving a basis of C[x]/( f (x)). Milnor fibration: T C 3 T C, (t, x) (t, f t (x)). Fibers X s (s = (t, λ)) consist of (t, x) T C 3 such that f t (x) = λ. Smooth fibration outside the discriminant Σ consisting of s such that X s is singular. All smooth fibers are diffeomorphic to X 1, 1 = (0, 1) T C. Milnor lattice: Q = H 2 (X 1 ; Z) with negative the intersection form = root lattice of type X N. Homomorphism Q H 2 (X s ; Z) for generic s Σ and every path from 1 to s avoiding Σ. α Q is a vanishing cycle if α 0 and (α α) = 2. The set R of vanishing cycles is a root system of type X N. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 8 / 16
Monodromy Representation Fundamental group of (T C) \ Σ acts on Q by orthogonal transformations. Image = Weyl group W of type X N. Picard Lefschetz: Small loop around s Σ reflection r α W (α R vanishing over s). r α (β) = β (α β)α. W is generated by r α (α R). Classical monodromy: Big loop around Σ Coxeter element σ W. σ = rα1 r αn where α 1,..., α N is a basis of simple roots. σ = h is the Coxeter number. σ is diagonalizable on h = C Z Q with eigenvalues e 2πim k /h. σ has no fixed points in h. Generalized root system (K. Saito): A lattice Q with a subset of roots R invariant under the action of the reflection group W generated by r α (α R). Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 9 / 16
Affine Lie Algebras g finite-dimensional Lie algebra with a symmetric invariant bilinear form ( ). E.g., Killing form, or (a b) = tr(ab) for g gl n. Affine Lie algebra ĝ = g[t, t 1 ] CK. [a m, b n ] = [a, b] m+n + mδ m, n (a b)k, where a m = at m. K is central. Verma module M(Λ 0 ) = Indĝ g[t] CK C of level 1 (K = 1). Basic representation V (Λ 0 ) irreducible quotient of M(Λ 0 ). V (Λ0 ) has the structure of a vertex algebra. Definition (Borcherds) A vertex algebra is a vector space V (space of states) with a vacuum vector 1 V and state-field correspondence Y (, z) : V V V ((z)) + axioms. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 10 / 16
Lattice Vertex Algebras Notation Q integral (even) lattice. h = C Z Q vector space with symmetric bilinear form ( ). ĥ = h[t, t 1 ] CK Heisenberg Lie algebra. B = M(Λ 0 ) = S(h[t 1 ]t 1 ) (bosonic) Fock space. C ε [Q] = span C {e α α Q} ε-twisted group algebra. e α e β = ε(α, β)e α+β. ε: Q Q {±1} bimultiplicative; ε(α, α) = ( 1) (α α)/2. Definition (Borcherds) Lattice vertex algebra V Q = B C ε [Q]. Y (a 1 1, z) = n Z a n z n 1 for a h, a n = at n. ( Y (e α, z) = e α z α 0 exp ) ( n<0 α n z n exp n n>0 α n z n n ). Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 11 / 16
The Vertex Algebra W XN I. Frenkel Kac: When Q is a root lattice of type X N (X = A, D, E), V Q = V (Λ0 ) basic representation of ĝ. g acts by derivations on V Q via a a 0. Screening operators (e α ) 0 = Res z=0 Y (e α, z) are derivations of V Q. E. Frenkel Kac Radul Wang: The W -algebra W XN (with central charge N) = kernel of screening operators (B V Q ) = g-invariant part of V (Λ 0 ). WXN B is fixed pointwise by the Weyl group W. WXN is freely generated by fields of conformal weights m 1 + 1,..., m N + 1 (Feigin E. Frenkel). WXN contains a Virasoro field; m 1 + 1 = 2. WAN = W N+1 (central charge N); freely generated by fields of conformal weights 2,..., N + 1. WA1 = Virasoro vertex algebra (central charge 1). Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 12 / 16
Main Results Theorem 1 The total descendant potential D XN (X = A, D, E) belongs to a σ-twisted representation of the lattice vertex algebra V Q. Y (σv, z) = Y (v, e 2πi z) for v V Q. D XN (untwisted) representation of W XN. Theorem 2 D XN is a vacuum vector for the representation of W XN, i.e., Y (v, z)d XN is regular in z for all v W XN. Virasoro constraints: Y (ω, z)d XN = n Z z n 2 L n D XN is regular L n D XN = 0 for n 1. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 13 / 16
Givental s Formula D A1 D pt = D pt (ε, Q) is a formal power series in ε ±1 and Q = (Q 0, Q 1, Q 2,... ), t 2k+1 = Q k /(2k + 1)!!. Dpt N depends on ε and Q i = (Q0 i, Qi 1, Qi 2,... ), i = 1,..., N. Givental s formula: D XN (ε, q) = C t Ŝ 1 t ˆR t D N pt. t T generic, q = (q0, q 1, q 2,... ), q k = (q 1 k,..., qn k ). D XN C((ε))[[q 0 t, q 1 + 1, q 2,... ]]. ε, Q i in D N pt need to be rescaled (by i ). Ŝt and ˆR t = ˆΨ t ˆRt eût /z are certain operators. ˆΨt is a change of variables (Q 1,..., Q N ) q. C t = const so that D XN is independent of t (omitted from now on). Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 14 / 16
Computation Ancestor potential A t = ˆR t D N pt D XN = Ŝ 1 t A t. Ŝ t Y (v, λ)d XN = ŜtY (v, λ)ŝ 1 t A t = Y t (v, λ)a t. Regular at λ = regular at the critical values λ = u i (t) (i = 1,..., N) of f t (x) for a generic t T. Givental: Y t (e α, λ) = ˆR t Y A1 (e α, λ) i ˆR 1 t α R is a cycle vanishing over (t, ui (t)) Σ. for λ near u i (t). YA1 (e α, λ) corresponds to the A 1 root lattice Zα. Y A1 (e α, λ) i denotes action on the i-th factor in D N pt. Y (e α, λ)d XN = Ŝ 1 t Y t (e α, λ) ˆR t Dpt N = Ŝ 1 t ˆR t Y A1 (e α, λ) i Dpt N. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 15 / 16
Computation Y (v, λ) = k Y (v k, λ)y (v k α, λ), v W XN. v k Fock space of α h. v k α Fock space of Cα. v k α Ker Res z=0 Y A1 (e ±α, λ) = W A1 = Virasoro vertex algebra. Y (v k, λ) regular at λ = u i(t) since r α v k = v k (no monodromy). Y (v k α, λ)d XN = Ŝ 1 t ˆR t Y A1 (v k α, λ) i D N pt Regular at λ = ui (t) because D pt satisfies Virasoro constraints. Y (v, λ)d XN is regular at λ = u i (t) for v W XN. Bakalov, Milanov (NC State Univ.) W -Constraints for Simple Singularities July 25, 2009 16 / 16