Geometric Realizations of the Basic Representation of ĝl r

Transcription

1 Geometric Realizations of the Basic Representation of ĝl r Joel Lemay Department of Mathematics and Statistics University of Ottawa September 23rd, 2013 Joel Lemay Geometric Realizations of ĝl r Representations 1/34

2 Introduction Goal To give a geometric description of the various realizations of the basic representation of ĝl r. Outline 1 Motivation 2 Focus on the principal realization 1 Algebraic description 2 Cohomology and vector bundles 3 Geometric description 3 Generalize to other realizations Joel Lemay Geometric Realizations of ĝl r Representations 2/34

3 Motivation Representations of affine Lie algebras are related (in physics) to quantum states, bosons/fermions (particles that make up the universe), etc... Mathematics (algebra) Physics Vector space V Universe (quantum states) Operators on V Realize affine Lie algebras using operators on V Different Realizations Geometrizing: Creation/annihilation of bosons and fermions Certain creation/ annihilation processes Different "vacuum spaces" (space where nothing exists) "Algebra Geometry" = often leads to new insights. Joel Lemay Geometric Realizations of ĝl r Representations 3/34

4 Algebraic Description Definition (Heisenberg algebra) Complex Lie algebra s = k Z {0} Cα(k) Cc, [s, c] = 0, [α(k), α(j)] = kδ k+j,0 c. Action on C[x 1, x 2,... ] (bosonic Fock space) via α(k) x k, α( k) kx k, k > 0, c id. Joel Lemay Geometric Realizations of ĝl r Representations 4/34

5 Algebraic Description Basic representation of ĝ = ĝl r or ŝl r Recall ĝ = ( g C[t, t 1 ] ) Cc. Basic representation, V basic (ĝ), irreducible representation characterized by the existence of a v V basic (ĝ) such that: (g C[t]) v = 0, and c v = v. Note: s ĝ in various ways, call this a Heisenberg subalgebra (HSA). Realizations of V basic (ĝ) Given a HSA, as ĝ-modules, V basic (ĝ) = Ω C[x 1, x 2,... ], where Ω = {v V basic (ĝ) α(k) v = 0 for all k > 0} is the vacuum space. Joel Lemay Geometric Realizations of ĝl r Representations 5/34

6 Algebraic Description The problem is choice Different choices of HSA s yield different Ω. For ŝl 2: Principal HSA: α(k) e t k 1 +f t k, α( k) k 2k 1 (e t k +f t 1 k ), = dim Ω = 1 (rest. of V basic to s remains irred.). Note: in terms of Chevalley generators, α(1) = E 0 + E 1 and α( 1) = F 0 + F 1 (up to scalar mult.). Homogeneous HSA: α(k) h t k, α( k) 1 8 (h t k ), = dim Ω =. Joel Lemay Geometric Realizations of ĝl r Representations 6/34

7 Algebraic Description In general HSA s of ŝl r are parametrized by partitions of r, i.e. r = (r 1,..., r s ), s.t. r r s = r and r 1 r s. Two extreme cases: Principal HSA r = (r). Homogeneous HSA r = (1, 1,..., 1). Joel Lemay Geometric Realizations of ĝl r Representations 7/34

8 Algebraic Description Definition (Fermionic Fock space) Infinite wedge space, i.e. F := span C {i 1 i 2 i k Z, i k > i k+1, Define zero-charge subspace i k+1 = i k 1 for k 0}. F 0 := span C {i 1 i 2 F i k = 1 k for k 0}. Definition (Fermions) Operators ψ(j), ψ (j) on F: for all j Z, ψ(j) = wedge j, ψ (j) = contract j. This defines an irreducible representation of the Clifford algebra, Cl. Joel Lemay Geometric Realizations of ĝl r Representations 8/34

9 Algebraic Description How to describe different realizations? Ex: Principal realization. Define: ψ(z) := ψ(k)z k, ψ (z) := ψ (k)z k. k Z k Z The homogeneous components of : ψ(ω p z)ψ (ω q z) : ωp q, 1 p, q r, p q, 1 ωp q and : ψ(z)ψ (z) :, where ω = e 2πi/r, span a Lie algebra of operators on F 0 isomorphic to ĝl r, and F 0 = Vbasic. Principal HSA given by α(k) i Z : ψ(i)ψ (i + k) :. Joel Lemay Geometric Realizations of ĝl r Representations 9/34

10 Algebraic Description How to describe different realizations? (continued) Via the boson-fermion correspondence, F 0 = C[x1, x 2,... ] = C C[x 1, x 2,... ], (1-dim vacuum space), as ĝl r-modules. Need a slight "tweak" to get a realization for V basic (ŝl r). For other realizations... we ll see later. Joel Lemay Geometric Realizations of ĝl r Representations 10/34

11 Geometrizing in a nutshell Goal: Find geometric analogs of algebraic objects. Algebra Vector space V Geometry (co)homology of algebraic varieties Linear maps on V Geometric operators on (co)homology Realize algebras using linear maps on V Realize Geometric versions of algebras Joel Lemay Geometric Realizations of ĝl r Representations 11/34

12 Equivariant Cohomology Equivariant cohomology Let X be a (nice) 4n-dim. variety, T = (C ) d torus acting on X. HT (pt) = C[t 1,..., t d ], Localized equivariant cohomology HT(X) = HT(X) C[t1,...,t d ] C(t 1,..., t d ). H T (X) = H T (XT ). Joel Lemay Geometric Realizations of ĝl r Representations 12/34

13 Equivariant Cohomology Bilinear form on H 2n T (X) X i X T p pt. For a, b H 2n T (X), a, b X = p (i ) 1 (a b). Bilinear form on H 2(n 1+n 2 ) T (X 1 X 2 ) Given T X 1, X 2 and a, b H 2(n 1+n 2 ) T (X 1 X 2 ), a, b X1 X 2 = p ((i 1 i 2 ) ) 1 (a b). Joel Lemay Geometric Realizations of ĝl r Representations 13/34

14 Vector Bundle Operator Operator α H 2(n 1+n 2 ) T (X 1 X 2 ) gives a map α : H 2n 1 T (X 1) H 2n 2 T (X 2) with structure constants Operators from vector bundles α(a), b X2 = a b, α X1 X 2. Let E X 1 X 2 be a T-equivariant vector bundle. Then the k-th Chern class c k (E) H 2k T (X 1 X 2 ). For β H 2(n 1+n 2 k) T (X 1 X 2 ), β c k (E) : H 2n 1 T (X 1) H 2n 2 T (X 2). Joel Lemay Geometric Realizations of ĝl r Representations 14/34

15 Shopping List Need: A variety whose cohomology corresponds to F 0. A vector bundle whose Chern classes give the appropriate operators: Heisenberg algebra ŝl r and ĝl r Clifford algebra Joel Lemay Geometric Realizations of ĝl r Representations 15/34

16 Picking the Right Variety (Principal Realization) Definition (Hilbert scheme) The Hilbert scheme of n points in the plane can be defined as HS(n) := {I C[x, y] dim(c[x, y]/i) = n}. Note: dim HS(n) = 4n. Torus action Fix T = C. Action on HS(n) induced by t x = tx, and t y = t 1 y. Fact (by a result of Nakajima, Yoshioka 2005): HS(n) T basis of F0 F. n Joel Lemay Geometric Realizations of ĝl r Representations 16/34

17 Quiver Varieties Definition (Quiver) A quiver is a directed graph, i.e. Q = (Q 0, Q 1 ), where Q 0 = {vertices} and Q 1 = {arrows}. Fix Q: Q 0 = Z r, Q 1 = {k k + 1} k Zr {k k 1} k Zr r 1 Joel Lemay Geometric Realizations of ĝl r Representations 17/34

18 Quiver Varieties Definition (Nakajima quiver variety) Z r -graded vector space V, v = (dim V k ) k Zr, v = k v k. Let G v := k Z r GL(V k ). Define M := variety whose points consist of i V 0, linear maps C ± k : V k V k±1 for all k Z r, such that C + k 1 C + k 1 V k 1 V k V k+1 commutes, C k C k+1 2 i generates V under application of C ± k. The Nakajima quiver variety is QV(v) = M /G v. Joel Lemay Geometric Realizations of ĝl r Representations 18/34

19 Quiver Varieties Theorem (Nakajima/Barth) The Hilbert scheme is isomorphic to the quiver variety with 1 vertex. That is, for Q : we have HS(n) = QV(n). Observation Z r T. Thus, Z r HS(n). Fact: HS(n) Zr = v =n QV(v), (r vertices). QV(v) inherits a T-action. Joel Lemay Geometric Realizations of ĝl r Representations 19/34

20 Idea H T ( ) Define operators Heisenberg algebra HS(n) HS(n) Zr HS(n) T Clifford algebra = v QV(v) ŝl r Joel Lemay Geometric Realizations of ĝl r Representations 20/34

21 Picking the Right Vector Bundle (Principal Realization) Vector bundles on QV(v) Tautological vector bundles V k Gv M QV(v) and C QV(v) QV(v) Denote by V k (k Q 0 ) and W, respectively. Note: T-equivariant w.r.t. trivial action on V k and C. On the product QV(v 1 ) QV(v 2 ) Have Hom-bundles: Hom(Vk 1, Vj 2 ), Hom(W 1, V0 2 ), Hom(V0 1, W 2 ). Joel Lemay Geometric Realizations of ĝl r Representations 21/34

22 Picking the Right Vector Bundle (Principal Realization) On the product QV(v 1 ) QV(v 2 ) E := k Q 0 Hom(V 1 k, V 2 k ), E ± := k Q 0 Hom(V 1 k, V 2 k±1). Complex of vector bundles E σ te + Hom(W, V 2 0 ) t 1 E Hom(V 1 0, W) τ E (Similar to construction of Nakajima s Hecke correspondence) Theorem ( Licata, Savage 2009) ker τ/ im σ QV(v 1 ) QV(v 2 ) is a vector bundle. Denote this vector bundle by K(QV(v 1 ), QV(v 2 )). Joel Lemay Geometric Realizations of ĝl r Representations 22/34

23 Picking the Right Vector Bundle (Principal Realization) Observations K(HS(n 1 ), HS(n 2 )) is a vector bundle on HS(n 1 ) HS(n 2 ). K(HS(n 1 ), HS(n 2 )) Zr is a v.b. on HS(n 1 ) Zr HS(n 2 ) Zr. HS(n 1 ) Zr HS(n 2 ) Zr = v 1,v 2 QV(v1 ) QV(v 2 ). Theorem (L.) K(HS(n 1 ), HS(n 2 )) Zr = v 1,v 2 K(QV(v 1 ), QV(v 2 )). Observation Gives a geometric interpretation of α(1) = k E k. Joel Lemay Geometric Realizations of ĝl r Representations 23/34

24 Putting it all together (Principal Realization) Algebraically Geometrically α(k) changes energy n n k = α(k) in terms of K(HS(n), HS(n k)) E k, F k change weight v v 1 k = E k, F k in terms of K(QV(v), QV(v 1 k )) Principal HSA in ŝl r = "Geometric" HSA in ŝl r. Joel Lemay Geometric Realizations of ĝl r Representations 24/34

25 Putting it all together (Principal Realization) Definition (Geometric operators) α(k), E k, F k : n H 2n T (HS(n)) n H 2n T (HS(n)), restricted to H 2n T (HS(n)), α(k) := β c tnv (K(HS(n), HS(n k))), E k, F k := γ c tnv (K(QV(v), QV(v 1 k ))), ( v = n). Theorem (Licata, Savage, L.) The E k and F k satisfy the Kac-Moody relations for ŝl r. The α(k) satisfy the Heisenberg relations. Yields a geometric version of the principal realization. Joel Lemay Geometric Realizations of ĝl r Representations 25/34

26 Other Realizations (Algebraically) For a partition r = (r 1,..., r s ), Divide an r r matrix into s 2 blocks of size r i r j : r 1 r 1 r 1 r 2... r 1 r s r 2 r 1 r 2 r 2... r 2 r s... r s r 1 r s r 2... r s r s Joel Lemay Geometric Realizations of ĝl r Representations 26/34

27 Other Realizations (Algebraically) Diagonal blocks: Correspond to ĝl r i. Take s copies of the previous construction = "s-coloured" versions of our previous algebras and Fock spaces: s s i, C[x 1, x 2,... ] s, i=1 s Cl i, F s. i=1 Off-diagonal blocks: "Mixed" vertex operators. Operators on (i, j)-th block given in terms of ψ i (z) and ψ j (z). Joel Lemay Geometric Realizations of ĝl r Representations 27/34

28 Other Realizations (Geometrically) Need an "s-coloured version" of the Hilbert scheme. Definition (Moduli space M(s, n)) Let M(s, n) be the moduli space of framed torsion-free sheaves on P 2 with rank s and second Chern class n. Note: M(1, n) = HS(n). Thus, M(s, n) is a "higher rank" generalization of the Hilbert scheme. Torus action M(s, n) comes equipped with a natural T = (C ) s+1 action. Joel Lemay Geometric Realizations of ĝl r Representations 28/34

29 Other Realizations (Geometrically) Need a geometric interpretation of "dividing into blocks". C -action Define an embedding C T by z (1, z, z 2,..., z s 1, 1). = C acts on M(s, n). Theorem (Nakajima, 2001) M(s, n) C = ni =n HS(n 1 ) HS(n s ). Notation For n N s, let HS(n) = HS(n 1 ) HS(n s ). Joel Lemay Geometric Realizations of ĝl r Representations 29/34

30 Other Realizations (Geometrically) Need s-coloured versions of operators. s-coloured vector bundles Similar to the principal case, we have a vector bundle K(M(s, n), M(s, m)) on M(s, n) M(s, m). K(M(s, n), M(s, m)) C is a v.b. on C -fixed points. Can show K(n, m) := K(M(s, n ), M(s, m )) C HS(n) HS(m), is a v.b. on HS(n) HS(m). Joel Lemay Geometric Realizations of ĝl r Representations 30/34

31 s-coloured Geometric Operators Algebraically α l (k) changes energy n n k on l-th colour Geometrically = α l(k) in terms of K(n, n k1 l ) Definition (s-coloured geometric Heisenberg operators) α l (k) : n H 2 n T (HS(n)) n H 2 n T (HS(n)), restricted to H 2 n T (HS(n)), Theorem (L.) α l (k) := β c tnv (K(n, n k1 l )). The α l (k) satisfy the s-coloured Heisenberg relations. Joel Lemay Geometric Realizations of ĝl r Representations 31/34

32 s-coloured Geometric Operators s-coloured Chevalley operators R := lcm(r 1,..., r s ), (sometimes 2). Can embed Z R T such that HS(n) Z R = HS(n 1 ) Zr 1 HS(ns ) Zrs = QV(v 1 ) QV(v s ). }{{} v l =n l =: QV(v 1,...,v s ) Can show K(v 1, u 1,..., v s, u s ) := K(n, m) Z R QV(v 1,...,v s ) QV(u 1,...,u s ), is a v.b. on QV(v 1,..., v s ) QV(u 1,..., u s ). Joel Lemay Geometric Realizations of ĝl r Representations 32/34

33 s-coloured Geometric Operators Algebraically Ek l, Fl k change weight v v 1 k on l-th colour = Geometrically E l k, Fl k in terms of K(v 1, v 1,..., v l, v l 1 k,..., v s, v s ) Definition (s-coloured geometric Chevalley operators) E l k, F l k : n H 2 n T (HS(n)) n H 2 n T (HS(n)), restricted to H 2 n T (HS(n)), E l k, F l k := γ c tnv (K(v 1, v 1,..., v l, v l 1 k,..., v s, v s )). Theorem (L.) The E l k and Fl k satisfy the Kac-Moody relations for l ŝl r l. Joel Lemay Geometric Realizations of ĝl r Representations 33/34

34 Future Goals We have the diagonal block operators. Still need the off-diagonal block operators. The end. Joel Lemay Geometric Realizations of ĝl r Representations 34/34