Singularities, Root Systems, and W-Algebras

Transcription

1 Singularities, Root Systems, and W-Algebras Bojko Bakalov North Carolina State University Joint work with Todor Milanov Supported in part by the National Science Foundation Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

2 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 Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

3 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 L n D pt = 0 (n 1). String eq. L 1 D pt = 0. The KdV hierarchy + string eq. determine uniquely Dpt. 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 D pt Virasoro constraints for D X. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

4 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. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

5 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. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

6 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. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

7 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 , 4, 6 x1 4+x 2 3+x 3 2 1, 4, 5, 7, 8, 11 E , 6, 9 x 3 1 x 2+x 3 2 +x 2 3 1, 5, 7, 9, 11, 13, 17 E , 10, 15 x 5 1 +x 3 2 +x 2 3 1, 7, 11, 13, 17, 19, 23, 29 Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

8 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. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

9 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 rα (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. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

10 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 gln. Loop algebra g[t, t 1 ] = polynomial maps from S 1 to g. [a(t), b(t)] = [a, b](t), [am, b n ] = [a, b] m+n, where a m = at m. Affine Lie algebra ĝ = g[t, t 1 ] CK is a central extension of the loop algebra: K is central, [am, b n ] = [a, b] m+n + mδ m, n (a b)k, g ĝ via a a0. Basic representation V (Λ 0 ) irreducible representation of ĝ: V (Λ0 ) is generated by a vector 1 such that g[t]1 = {0}, K acts as the identity on V (Λ0 ), V (Λ 0 ) has the structure of a vertex algebra. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

11 Lattice Vertex Algebras 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. 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 = V (Λ 0 ) = S(h[t 1 ]t 1 ) bosonic Fock space. B α = B as an ĥ module except that the h-action has weight α Q. Definition (Borcherds) Lattice vertex algebra V Q = α Q B α. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

12 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 VQ via a a 0. E. Frenkel Kac Radul Wang: The W-algebra W XN (with central charge N) = 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). W XN contains a Virasoro field; m = 2. W AN = W N+1 (central charge N); freely generated by fields of conformal weights 2,..., N + 1. W A1 = Virasoro vertex algebra (central charge 1). Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13

13 Main Results Theorem 1 The total descendant potential D XN (X = A, D, E) lies in 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. Bojko Bakalov (NCSU) Singularities, Root Systems, and W-Algebras February 18, / 13