Moduli spaces of sheaves and the boson-fermion correspondence

Transcription

1 Moduli spaces of sheaves and the boson-fermion correspondence Alistair Savage Department of Mathematics and Statistics University of Ottawa Joint work with Anthony Licata (Stanford/MPI) Slides: asavag2 Full details: arxiv: (to appear in Selecta Math.)

2 Algebraic boson-fermion correspondence

3 Geometric representation theory Basic idea vector space (co)homology of some space(s) algebra action geometrically defined operators (e.g. via correspondences) Examples Space Lusztig quiver varieties Nakajima quiver varieties Hilb n (C 2 ) Affine Grassmannian of G Algebraic object U q (g) Reps of g Bosonic Fock space Rep theory of G L

4 Geometric representation theory Benefits and uses use rep theory to study geometry of spaces involved use geometry to study rep theory geometric realizations often produce nice bases with integrality and positivity properties geometrization categorification

5 Hilbert schemes and Heisenberg algebras (Nakajima, Grojnowksi) Hilb n (C 2 ) = Hilbert scheme of n points in C 2 Consider n H (Hilb n (C 2 )) Correspondences in Hilb n (C 2 ) Hilb n+k (C 2 ) yield action of Heisenberg algebra n H (Hilb n (C 2 )) = bosonic Fock space (1 color) Important idea: Consider all the Hilbert schemes together

6 Geometric operators via correspondences Spaces X, Y Correspondence Z X Y and natural projections X p X X Y p Y Y Define an operator H (X ) H (Y ) by H (X ) α (p Y )! (p X (α) [Z]) H (Y )

7 Overview Our goal Find space(s) such that (co)homology has natural actions of Heisenberg algebra and Clifford algebra Operators Carlsson-Okounkov: correspondences (virtual) vector bundles

8 Oscillator/Heisenberg algebra r-colored oscillator algebra s r = Cp l (m) CK m Z, l {1,...,r} commutation relations [s r, K] = 0, [p k (m), p l (n)] = 1 m δ m, nδ k,l K. Span C {p l (m), K l {1,..., r}, m Z \ {0}} is an r-colored infinite-dimensional Heisenberg algebra

9 Bosonic Fock space r-colored bosonic Fock space B = B r, B = C[p 1, p 2,... ; q, q 1 ] = Λ C[q, q 1 ] Λ = ring of symmetric functions, p n = i x n i Charge-energy decomposition: B = c Z r, j Z 0 B c j B c j = {(q c1 f 1,..., q cr f r ) f i Λ, deg f i = j}

10 Bosonic Fock space Representation of s r on B p l (m) id (l 1) p m id (r l), m > 0, p l ( m) id (l 1) 1 m p m id (r l), m > 0, p l (0) id (l 1) q q id (r l), K id. Physical interpretation p l ( m) creates a particle of color l in state m p l (m) annihilates a particle of color l in state m

11 Clifford algebra r-colored Clifford algebra Cl r generators ψ l (j), ψ l (j), j Z, 1 l r relations ψ l (i)ψ l (j) + ψ l (j) ψ l (i) = δ ij, ψ l (i)ψ l (j) + ψ l (j)ψ l (i) = 0 = ψ l (i) ψ l (j) + ψ l (j) ψ l (i), [ψ k (i), ψ l (j)] = [ψ k (i), ψ l (j) ] = [ψ k (i), ψ l (j) ] = 0

12 Fermionic Fock space Semi-infinite monomials I = i 1 i 2..., i j Z such that i 1 > i 2 > i 3 >... i k = i k 1 1 for k 0 Fermionic Fock space F = F r F = Span C {semi-infinite monomials}

13 r-colored fermionic Fock space Charge The charge of I is the integer c(i ) such that i k = c(i ) + 1 k for k 0 For I = (I 1,..., I r ), c(i) = (c(i 1 ),..., c(i r )) Z r Partitions and semi-infinite monomials For a partition I of charge c, define a partition λ(i ) by I = i 1 i 2..., i k = (c(i ) + 1 k) + λ(i ) k For I = (I 1,..., I r ), λ(i) = (λ(i 1 ),..., λ(i r ))

14 Bijection {semi-infinite monomials} {partitions} Z I (λ(i ), c(i )) Energy The energy of I is r I = λ(i k ) Z 0 k=1 Charge-energy decomposition F = c Z r, j Z 0 F c j F c j = Span C {I c(i ) = c, I = j}

15 Fermionic Fock space Wedging and contracting operators ψ(j)(i 1 i 2... ) { 0 if j = i s for some s, = ( 1) s i 1 i s j i s+1... if i s > j > i s+1. ψ(j) (i 1 i 2... ) { 0 if j i s for all s, = ( 1) s 1 i 1 i 2 i s 1 i s+1... if j = i s.

16 Fermionic Fock space Representation of Cl r on F ψ l (j) id l 1 ψ(j) id r l ψ l (j) id l 1 ψ(j) id r l We have ψ l (j)(f c ) F c+1 l ψ l (j) (F c ) F c 1 l Physical interpretation ψ l (j) creates a particle of color l in state j ψ l (j) annihilates a particle of color l in state j

17 Boson-fermion correspondence Bosonization Define an s r -structure on F by p l (n) 1 ψ l (j)ψ l (j + n), n Z \ {0}, n j Z p l (0) j>0 ψ(j)ψ(j) j 0 ψ(j) ψ(j) We have isomorphism F = s r mod B, F c j B c j Fermionization Can define Cl r -structure on B (vertex operators) Have isomorphism F = Cl r mod B

18 Algebraic boson-fermion correspondence

19 The moduli space M(r, n) Definition M(r, n) = moduli space of framed rank r torsion-free sheaves on CP 2 with c 2 = n Alternative description B V A W = C r, V = C n i W j M(r, n) = {(A, B, i, j) [A, B]+ij = 0, (A, B, i, j) is stable}/gl(v ) (A, B, i, j) is stable if proper A, B-invariant subspace of V containing im i r = 1: M(1, n) = Hilbert scheme of n points in C 2

20 Previous work Consider r = 1 case (Hilbert schemes) Nakajima/Grojnowski defined action of Heisenberg algebra on cohomology via correspondences Natural torus action fixed points are Nakajima A quiver varieties (Nakajima defines action of gl via correspondences) Localization Theorem: Equivariant cohomology of a space isomorphic to equivariant cohomology of fixed point set Yields geometric boson-fermion correspondence ([S.]) Licata defined Clifford algebra action on the cohomology of the fixed point set Drawback: Heisenberg operators defined globally but Clifford operators defined only on fixed point sets

21 Torus actions Torus Fix torus T = (C ) r C. T-action on M(r, n) For c Z r, let M c (r, n) be the moduli space with T -action (e, t) c (A, B, i, j) = (ta, t 1 B, ie 1 t c, et c j) where t c = (t c1, t c2,..., t cr ) (C ) r and (C ) r acts diagonally on W (we ve fixed a basis for W ).

22 Equivariant cohomology HT (M c(r, n)) = T -equivariant cohomology of M c (r, n) HT (M c(r, n)) is a module over HT (pt) = C[ɛ, b 1,..., b r ] where ɛ = c 2 (t), b i = c 2 (e i ) have degree 2. Localized equivariant cohomology We let HT (M c(r, n)) = HT (M c(r, n)) C[b1,...,b r,ɛ] C(b 1,..., b r, ɛ) denoted the localized equivariant cohomology of M c (r, n).

23 Fixed points and tangent spaces T -fixed points of M c (r, n) M c (r, n) T {r-colored semi-infinite monomials I of charge c} Tangent spaces T I = tangent space at I T acts on T I fix a splitting T I = T I T + I

24 Inner product Inclusion and projection maps inclusion i : M c (r, n) T M c (r, n) projection p : M c (r, n) T {pt} Inner product Define inner product on HT 2rn(M c(r, n), C) by a, b n,c = ( 1) rn p (i ) 1 (a b) and extend to an inner product on n,c H2rn T (M c(r, n), C), =, n,c n,c

25 Our geometric vector space Orthonormal classes Define [I] = i (1 I ) ctop T (T ) H2rn T (M c(r, n), C) The classes {[I]} are orthonormal Definition I A c (r, n) = Span C {[I]} H 2rn T (M c(r, n), C) A = n,c A c (r, n) Restriction of, to A is non-degenerate and C-valued Note: A is a C-lattice of localized equivariant cohomology

26 Operators on equivariant cohomology Inner product Define inner product on H 2r(n 1+n 2 ) T (M c (r, n 1 ) M d (r, n 2 )) analogously a, b c,d = ( 1) rn 2 p ((i 1 i 2 ) ) 1 (a b) Operators (Carlsson-Okounkov) A class β H 2r(n 1+n 2 ) T (M c (r, n 1 ) M d (r, n 2 )) gives a linear operator β : A c (r, n 1 ) A d (r, n 2 ) by βa, b d def = a b, β c,d

27 Tautological bundles V GL(V ) {(A, B, i, j) [A, B] + ij = 0, (A, B, i, j) stable} M c (r, n) is a T -equivariant vector bundle denote it V T -equivariant vector bundle W M c (r, n) M c (r, n) denote it W Have T -equivariant vector bundles Hom(V, V ) Hom(V, W ) Hom(W, V )

28 T -equivariant complex T -equivariant complex of vector bundles on M c (r, n 1 ) M d (r, n 2 ) t Hom(V 1, V 2 ) t 1 Hom(V 1, V 2 ) Hom(V 1, V 2 ) σ τ Hom(V 1, V 2 ). Hom(W 1, V 2 ) Hom(V 1, W 2 ) where ξa 1 A 2 ξ C σ(ξ) = ξb 1 B 2 ξ ξi 1, τ D I = ( ) [A, D] + [C, B] + i 2 J + Ij 1. j 2 ξ J Notes: 1 When c = d, n 1 = n 2, cohomology of this complex is the tangent bundle. 2 Zero set of section of a complex similar to above yields Nakajima s correspondences for Heisenberg action on Hilbert schemes (rank 1 case).

29 Geometric Heisenberg operators Definition K c,d (n 1, n 2 ) = cohomology of the above complex (vector bundle) Define operators P l (n) : A A by P l (n) Ac(r,k) = ±γ l c T top(k c,c (k, k n)), n 0 P l (0) Ac(r,k) = c l id Notes: γ l are orthogonal equivariant cohomology classes ctop T denotes top non-vanishing Chern class

30 Geometric bosonic Fock space Theorem [Licata-S.] 1 maps p l (n) P l (n), n Z, l {1,..., r}, K id define a rep of s r on A 2 linear map A = B, [I] (q c(i 1) s λ(i 1 ),..., q c(i r ) s λ(i r )) is an isometric isomorphism of s r -modules 3 under this isomorphism HT 2rn (M c(r, n)) A c (r, n) B c n

31 Geometric Clifford operators Definition Define operators by Ψ l (n) : A A Ψ l (n) Ac(r,k) = c T top(k c,c+1l (k, k + n c l 1)) Define Ψ l (n) to be adjoint to Ψ l (n)

32 Geometric fermionic Fock space Theorem [Licata-S.] 1 maps ψ l (n) Ψ l (n), ψ l (n) Ψ l (n), n Z, l {1,..., r}, define a rep of Cl r on A 2 linear map A = F, [I] I is an isometric isomorphism of Cl r -modules 3 under this isomorphism HT 2rn (M c(r, n)) A c (r, n) F c n

33 Summary r-colored Heisenberg alg r-colored Clifford alg c,n H2rn T (M c(r, n)) A bosonic Fock space fermionic Fock space Important idea: consider different T -actions together

34 Further directions homogeneous realization of basic rep of ĝl r affine Lie algebra ĝl r embeds into Cl r slight modification of complexes yields action of ĝl r on A n,c : c α =0 A c(r, n) = basic representation principal realization of basic rep? relation between these and other geometric constructions of basic rep? explicit algebraic descriptions of nice geometric bases level-rank duality other vertex operator constructions? categorification?