Quantising noncompact Spin c -manifolds

Transcription

1 Quantising noncompact Spin c -manifolds Peter Hochs University of Adelaide Workshop on Positive Curvature and Index Theory National University of Singapore, 20 November 2014 Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

2 Joint work with Mathai Varghese: Geometric quantization and families of inner products, ArXiv: (symplectic case) Quantising proper actions on Spin c -manifolds, ArXiv: (Spin c -case) Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

3 1 Indices of Spin c -Dirac operators 2 Quantisation commutes with reduction 3 Noncompact groups and manifolds 4 An analytic approach in the compact case 5 The invariant, transversally L 2 -index 6 An analytic approach in the noncompact case Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

4 1. Indices of Spin c -Dirac operators Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

5 Connections on spinor bundles Let M be a Spin c -manifold. Let S M be the spinor bundle; L M be the determinant line bundle associated to the Spin c -structure. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

6 Connections on spinor bundles Let M be a Spin c -manifold. Let S M be the spinor bundle; L M be the determinant line bundle associated to the Spin c -structure. Locally, on small enough open sets U M, one has where S U 0 S U = S U 0 (L U ) 1/2, U is the spinor bundle associated to a local Spin-structure. Consider the Clifford connection on S U defined by a chosen connection L on L and the connection on S0 U induced by the Levi Civita connection. This yields a globally well-defined connection S on S. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

7 The Spin c -Dirac operator Using the connection S on S and the Clifford action c by TM on S, one can define the Spin c -Dirac operator D : Γ (S) S Ω 1 (M; S) = Γ (TM S) c Γ (S). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

8 The Spin c -Dirac operator Using the connection S on S and the Clifford action c by TM on S, one can define the Spin c -Dirac operator Properties: D : Γ (S) S Ω 1 (M; S) = Γ (TM S) c Γ (S). 1 Its principal symbol is the Clifford action σ D (ξ) = c(ξ) for ξ T M. Since c(ξ) 2 = ξ 2, the operator D is elliptic. 2 If dim(m) is even, then D is odd, i.e. splits as D ± : Γ (S ± ) Γ (S ). 3 If M is complete, D is essentially self-adjoint. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

9 The Spin c -Dirac operator Using the connection S on S and the Clifford action c by TM on S, one can define the Spin c -Dirac operator Properties: D : Γ (S) S Ω 1 (M; S) = Γ (TM S) c Γ (S). 1 Its principal symbol is the Clifford action σ D (ξ) = c(ξ) for ξ T M. Since c(ξ) 2 = ξ 2, the operator D is elliptic. 2 If dim(m) is even, then D is odd, i.e. splits as D ± : Γ (S ± ) Γ (S ). 3 If M is complete, D is essentially self-adjoint. Hence if M is compact and even-dimensional, we can define index(d + ) = dim ker(d + ) dim ker(d ) Z. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

10 The equivariant index in the compact case Let G be a compact Lie group acting on an even-dimensional compact manifold M with a G-equivariant Spin c -structure P M. Then ker(d ± ) is a finite-dimensional representation of G. Hence G-index(D) := [ker(d + )] [ker(d )] R(G), where R(G) = {[V ] [W ]; V and W finite-dimensional rep of G}. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

11 The equivariant index in the compact case Let G be a compact Lie group acting on an even-dimensional compact manifold M with a G-equivariant Spin c -structure P M. Then ker(d ± ) is a finite-dimensional representation of G. Hence G-index(D) := [ker(d + )] [ker(d )] R(G), where R(G) = {[V ] [W ]; V and W finite-dimensional rep of G}. Indices of Dirac operators are a rich source of representations. A natural question is: what are the coefficients m V Z in G-index(D) = V Ĝ m V [V ]? This can be answered using quantisation commutes with reduction. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

12 2. Quantisation and reduction Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

13 Background in physics Classical mechanics with symmetry quantisation Q Quantum mechanics with symmetry reduction R Classical mechanics (use symmetry to simplify) Q R Quantum mechanics Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

14 Background in physics Classical mechanics with symmetry quantisation Q Quantum mechanics with symmetry reduction R Classical mechanics (use symmetry to simplify) Q R Quantum mechanics Quantisation commutes with reduction: Q R = R Q, or [Q, R] = 0. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

15 Geometric quantisation Let (M, ω) be a compact symplectic manifold, acted on by a compact Lie group G which preserves ω. Then M has a G-equivariant Spin c -structure. Suppose the determinant line bundle L satisfies c 1 (L) = 2[ω] H 2 (M). Definition The Spin c -quantisation of the action by G on (M, ω) is Q Spinc G (M) = G-index(D) R(G). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

16 Reduced spaces Consider a momentum map such that for all X g, µ : M g, 2πi µ, X = L X M L L X End(L) = C (M), with X M the vector field induced by X, L a connection on L with ( L ) 2 = 2πi 2ω, and L L the Lie derivative. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

17 Reduced spaces Consider a momentum map such that for all X g, µ : M g, 2πi µ, X = L X M L L X End(L) = C (M), with X M the vector field induced by X, L a connection on L with ( L ) 2 = 2πi 2ω, and L L the Lie derivative. Definition Let ξ g. The symplectic reduction of the action at ξ is the space Theorem (Marsden Weinstein) M ξ := µ 1 (G ξ)/g. If ξ is a regular value of µ, and G acts freely (properly) on µ 1 (ξ), then M ξ is a symplectic manifold (orbifold). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

18 Quantum reduction On the quantum side, reduction means taking the dimension of the G-invariant part of a representation: Q G (M, ω) G = dim(ker D + ) G dim(ker D ) G Z. One can also take multiplicities of other irreducible representations than the trivial one. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

19 Quantisation commutes with reduction Suppose M and G are compact. Then for a slightly different definition of quantisation, Meinrenken (1998), Tian Zhang (1998) and Paradan (2001) proved that quantisation commutes with reduction (after a conjecture by Guillemin Sternberg in 1982): G (M, ω) R Q [ker D + ] [ker D ] dim(ker D) G dim(ker D ) G R (M 0, ω 0 ) Q dim(ker D + M 0 ) dim(ker(d M0 ) ) Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

20 Quantisation commutes with reduction Suppose M and G are compact. Then for a slightly different definition of quantisation, Meinrenken (1998), Tian Zhang (1998) and Paradan (2001) proved that quantisation commutes with reduction (after a conjecture by Guillemin Sternberg in 1982): G (M, ω) R Q [ker D + ] [ker D ] dim(ker D) G dim(ker D ) G R (M 0, ω 0 ) Q dim(ker D + M 0 ) dim(ker(d M0 ) ) For Spin c -quantisation as defined earlier, Paradan (2012) proved that Q Spinc G (M) G = Q(M ρ ). Here ρ is half the sum of the positive roots. (Q Spinc (Ad (G)ρ) is the trivial representation.) Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

21 Reduction at nontrivial representations In the compact symplectic case, quantisation commutes with reduction at the trivial representation implies more generally that Q Spinc G (M) = V Ĝ Q Spinc (M λv +ρ)[v ], where λ V is the highest weight of V. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

22 Reduction at nontrivial representations In the compact symplectic case, quantisation commutes with reduction at the trivial representation implies more generally that Q Spinc G (M) = V Ĝ Q Spinc (M λv +ρ)[v ], where λ V is the highest weight of V. This is based on the shifting trick: M ξ = ( M Ad (G)( ξ) ) 0. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

23 Reduction at nontrivial representations In the compact symplectic case, quantisation commutes with reduction at the trivial representation implies more generally that Q Spinc G (M) = V Ĝ Q Spinc (M λv +ρ)[v ], where λ V is the highest weight of V. This is based on the shifting trick: M ξ = ( M Ad (G)( ξ) ) 0. Meinrenken Sjamaar defined Q Spinc (M ξ ) if ξ is a singular value of µ. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

24 Spin c -reduction Now only assume M is only Spin c rather than symplectic. As in the symplectic case, one can define a Spin c -momentum map µ : M g by for X g. 2πi µ, X = L X M L L X, Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

25 Spin c -reduction Now only assume M is only Spin c rather than symplectic. As in the symplectic case, one can define a Spin c -momentum map µ : M g by for X g. Definition The reduced space at ξ g is 2πi µ, X = L X M L L X, M ξ := µ 1 (G ξ)/g. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

26 Spin c -structures on reduced spaces In the symplectic case, Marsden and Weinstein s theorem stated that reduced spaces at regular values of µ are symplectic orbifolds. In the Spin c -setting, we have the following. Proposition (Paradan Vergne) Let ξ g be a regular value of µ. 1 If G is a torus, then M ξ inherits a Spin c -structure from M. 2 If G is not a torus (but still compact), then M ξ can be realised as a reduced space for a torus action, so the first point applies. Again, one can handle the case where ξ is not a regular value. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

27 Spin c -quantisation commutes with reduction; compact case Write Q Spinc G (M) = V Ĝ m V [V ] R(G), with m V Z. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

28 Spin c -quantisation commutes with reduction; compact case Write Q Spinc G (M) = V Ĝ m V [V ] R(G), with m V Z. Let λ V the highest weight of an irreducible representation V ˆK. Theorem (Paradan Vergne, 2014) If there are points with Abelian stabilisers, then m V = Q Spinc (M λv +ρ). In general, m V is expressed as a sum of quantisations of reduced spaces. This answers the question posed earlier for compact M and G. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

29 3. Noncompact groups and manifolds Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

30 The noncompact setting Natural question: can this be generalised to noncompact G and M? Would give insight in representation theory of noncompact groups. Many phase spaces in classical mechanics (symplectic manifolds) are noncompact, e.g. cotangent bundles. In general, (equivariant) index theory of Spin c -Dirac operators is important in geometry, e.g. positive scalar curvature. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

31 Existing results In the symplectic setting, there are [Q, R] = 0 results for noncompact groups and/or manifolds. For G compact and µ proper, there are results by Ma Zhang (2009) and Paradan (2011). For M/G compact, Landsman (2005) formulated a [Q, R] = 0 conjecture, and results in this context were obtained by Landsman, H. (2008, 2009, 2014), and Mathai Zhang (2010). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

32 Existing results In the symplectic setting, there are [Q, R] = 0 results for noncompact groups and/or manifolds. For G compact and µ proper, there are results by Ma Zhang (2009) and Paradan (2011). For M/G compact, Landsman (2005) formulated a [Q, R] = 0 conjecture, and results in this context were obtained by Landsman, H. (2008, 2009, 2014), and Mathai Zhang (2010). There were no results in cases where both M/G and G are noncompact; or M is only Spin c and M or G is noncompact. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

33 Noncompact example: free particle on a line G = R acting on M = R 2 by addition on the first coordinate Now G, M and M/G are noncompact, so outside the scope of the existing approaches. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

34 Goals and method Goals: generalise [Q, R] = 0 to cases where 1 M, G and M/G may be noncompact; 2 M is only Spin c. Method: generalise the analytic approach developed by Tian Zhang. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

35 4. An analytic approach in the compact case Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

36 Localising and decomposing Consider the symplectic setting, and suppose M and G are compact. Idea: 1 consider a deformed version D t of the Dirac operator D, with a real deformation parameter t; 2 localise the kernel of D t to a neighbourhood U of µ 1 (0) in a suitable sense, for t large enough; 3 on U, decompose D t into a part on µ 1 (0) and a part normal to µ 1 (0). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

37 Localising and decomposing Consider the symplectic setting, and suppose M and G are compact. Idea: 1 consider a deformed version D t of the Dirac operator D, with a real deformation parameter t; 2 localise the kernel of D t to a neighbourhood U of µ 1 (0) in a suitable sense, for t large enough; 3 on U, decompose D t into a part on µ 1 (0) and a part normal to µ 1 (0). In this talk we focus on the localisation of the kernel of D t, where noncompactness plays the biggest role. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

38 Deforming the Dirac operator Tian and Zhang used an Ad (G)-invariant inner product on g, which exists for compact groups G. Then they defined: the function H := µ 2 C (M) G ; the vector field v by with µ : M g dual to µ; v m := 2 ( µ (m) ) M m, the deformed Spin c -Dirac operator for t R. D t = D + t 1 2 c(v), The vector field v is the Hamiltonian vector field of H. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

39 A Bochner formula Theorem (Tian Zhang) On G-invariant sections, one has D 2 t = D 2 + ta + 4πtH + t2 4 v 2, where A is a vector bundle endomorphism. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

40 A Bochner formula Theorem (Tian Zhang) On G-invariant sections, one has D 2 t = D 2 + ta + 4πtH + t2 4 v 2, where A is a vector bundle endomorphism. By carefully analysing the operator A, this allowed Tian and Zhang to localise (ker D t ) G to µ 1 (0) for large t. Note that µ 1 (0) = H 1 (0). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

41 5. The invariant, transversally L 2 -index Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

42 Quantisation in the noncompact setting The first question is how to define the quantisation of a noncompact Spin c -manifold M. We will define R Q(M) using the invariant, transversally L 2 -index. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

43 Quantisation in the noncompact setting The first question is how to define the quantisation of a noncompact Spin c -manifold M. We will define R Q(M) using the invariant, transversally L 2 -index. Let M be a complete Riemannian manifold (for integration on M, we will use the Riemannian density); G be a unimodular Lie group, acting properly and isometrically on M; dg be a Haar measure on G; E M be a Hermitian G-vector bundle. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

44 Transversally L 2 -sections A cutoff function is a function f C (M) such that for all m M, the intersection G m supp(f ) is compact, and G f (g m) 2 dg = 1. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

45 Transversally L 2 -sections A cutoff function is a function f C (M) such that for all m M, the intersection G m supp(f ) is compact, and G f (g m) 2 dg = 1. The space of transversally L 2 -sections of E is L 2 T (E) := {s Γ(E); fs L2 (E) for all cutoff functions f }/ = a.e.. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

46 Transversally L 2 -sections A cutoff function is a function f C (M) such that for all m M, the intersection G m supp(f ) is compact, and G f (g m) 2 dg = 1. The space of transversally L 2 -sections of E is L 2 T (E) := {s Γ(E); fs L2 (E) for all cutoff functions f }/ = a.e.. For any (differential) operator D on Γ (E), we have the transversally L 2 -kernel of D: ker L 2 (D) := {s Γ (E) L 2 T T (E); Ds = 0}. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

47 Invariant sections We mainly consider G-invariant transversally L 2 -sections s. Then fs L 2 (E) is independent of the cutoff function f. Hence L 2 T (E)G is a Hilbert space. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

48 A Fredholm criterion Let D be an elliptic, essentially self-adjoint, first order differential operator on a E. Let f be a cutoff function. One has the following generalisation of a result by Anghel (1993) and Gromov Lawson (1983). Proposition (Mathai H.) Suppose there is a cocompact subset K M and a number C > 0, such that for all s Γ (E) G with fs compactly supported outside K, fds L 2 (E) C fs L 2 (E). Then (ker L 2 T D) G is finite-dimensional. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

49 The invariant, transversally L 2 -index As on the previous slide, suppose D is an elliptic, essentially self-adjoint, first order differential operator on E M, such that fds L 2 (E) C fs L 2 (E), for s supported outside a cocompact set K M. Suppose also that E is Z 2 -graded, and D is odd. Definition The G-invariant, transversally L 2 -index of D is index G L 2 T (D) := dim(ker L 2 T D + ) G dim(ker L 2 T D ) G Z. Here D ± is the restriction of D to sections of the even and odd parts of E. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

50 Special cases If G is compact, f 1 is a cutoff function, so index G L (D) = dim(ker 2 L 2 D + ) G dim(ker L 2 D ) G. T If M/G is compact, cutoff functions have compact supports, so index G L (D) = dim(ker D + ) G dim(ker D ) G. 2 T Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

51 5. An analytic approach in the noncompact case Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

52 The noncompact case Idea: use invariant, transversally L 2 -index theory and generalise Tian Zhang s localisation arguments both to define quantisation and to prove [Q, R] = 0. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

53 The noncompact case Idea: use invariant, transversally L 2 -index theory and generalise Tian Zhang s localisation arguments both to define quantisation and to prove [Q, R] = 0. Issues: If M/G is noncompact, need an argument to show that the invariant, transversally L 2 -index applies. If G is noncompact, there is no Ad (G)-invariant inner product on g in general, so the deformed Dirac operator may not be G-equivariant. If M/G is noncompact, the operator A in D 2 t = D 2 + ta + 4πtH + t2 4 v 2, may be unbounded (and H and v may go to zero at infinity). If M is only Spin c, the expression for the operator A becomes less explicit. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

54 The noncompact case Idea: use invariant, transversally L 2 -index theory and generalise Tian Zhang s localisation arguments both to define quantisation and to prove [Q, R] = 0. Issues: If M/G is noncompact, need an argument to show that the invariant, transversally L 2 -index applies. If G is noncompact, there is no Ad (G)-invariant inner product on g in general, so the deformed Dirac operator may not be G-equivariant. If M/G is noncompact, the operator A in D 2 t = D 2 + ta + 4πtH + t2 4 v 2, may be unbounded (and H and v may go to zero at infinity). If M is only Spin c, the expression for the operator A becomes less explicit. Tool: families of inner products on g, parametrised by M. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

55 Families of inner products Let {(, ) m } m M be a smooth family of inner products on g, with the invariance property that for all m M, g G and ξ, ξ g, ( Ad (g)ξ, Ad (g)ξ ) g m = (ξ, ξ ) m. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

56 Families of inner products Let {(, ) m } m M be a smooth family of inner products on g, with the invariance property that for all m M, g G and ξ, ξ g, ( Ad (g)ξ, Ad (g)ξ ) g m = (ξ, ξ ) m. Then we define the function H C (M) G by the map µ : M g by H(m) = µ(m) 2 m; ξ, µ (m) = (ξ, µ(m)) m for all ξ g and m M; the vector field v as before v m := 2 ( µ (m) ) M m ; the deformed Dirac operator (which is equivariant) 1 D t = D + t 2 c(v). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

57 Main assumption We defined: v m := 2 ( µ (m) ) M m. The main assumption is that Zeros(v)/G is compact. Since µ 1 (0) Zeros(v), this implies that M 0 is compact. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

58 Main assumption We defined: v m := 2 ( µ (m) ) M m. The main assumption is that Zeros(v)/G is compact. Since µ 1 (0) Zeros(v), this implies that M 0 is compact. Other assumptions: G is unimodular, and acts properly on M. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

59 Invariant quantisation Theorem (Mathai H. in symplectic case; generalised by Braverman) If Zeros(v)/G is compact, then there is a family of inner products on g such that for t 1, the operator D t satisfies the Fredholm criterion for the G-invariant, transverally L 2 -index, so dim ( ker L 2 T D t ) G <. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

60 Invariant quantisation Theorem (Mathai H. in symplectic case; generalised by Braverman) If Zeros(v)/G is compact, then there is a family of inner products on g such that for t 1, the operator D t satisfies the Fredholm criterion for the G-invariant, transverally L 2 -index, so dim ( ker L 2 T D t ) G <. Suppose M is even-dimensional, and Zeros(v)/G is compact. Definition The G-invariant Spin c -quantisation of the action by G on M is Q Spinc (M) G = index G L 2 T (D t ) = dim ( ker L 2 T D + t ) G dim ( kerl 2 T D t ) G, for t 1 and a family of inner products on g as in the theorem. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

61 Quantisation commutes with reduction Theorem (Mathai H.) Suppose M 0 is a smooth Spin c -manifold. Then there is a class of Spin c -structures on M, for which Q Spinc (M) G = Q Spinc (M 0 ). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

62 Quantisation commutes with reduction Theorem (Mathai H.) Suppose M 0 is a smooth Spin c -manifold. Then there is a class of Spin c -structures on M, for which Remarks: Q Spinc (M) G = Q Spinc (M 0 ). The class of Spin c -structures in the theorem corresponds to using high enough tensor powers of the determinant line bundle. In the symplectic analogue of this result, one does not need high tensor powers of the line bundle if G is compact, or the action is locally free. There are conditions that imply M 0 is a Spin c -manifold: 0 is a Spin c -regular value of µ, and G acts freely on µ 1 (0). E.g. G is semisimple and 0 is a regular value of µ. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

63 A Bochner formula for families of inner products As in the compact case, the proofs of the results start with an explicit expression for D 2 t. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

64 A Bochner formula for families of inner products As in the compact case, the proofs of the results start with an explicit expression for D 2 t. Theorem On G-invariant sections, one has D 2 t = D 2 + ta + 2πtH + t2 4 v 2, with A a vector bundle endomorphism. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

65 A Bochner formula for families of inner products As in the compact case, the proofs of the results start with an explicit expression for D 2 t. Theorem On G-invariant sections, one has D 2 t = D 2 + ta + 2πtH + t2 4 v 2, with A a vector bundle endomorphism. The expression for A is different from the compact symplectic case, because of extra terms due to the use of a family of inner products on g ; the fact that M is only assumed to be Spin c. In addition, one has no control over the behaviour of A, H and v at infinity. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

66 Choosing the family of inner products on g Solution to issues arising in the noncompact/spin c -case: a suitable choice of the family of inner products on g. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

67 Choosing the family of inner products on g Solution to issues arising in the noncompact/spin c -case: a suitable choice of the family of inner products on g. Let U be a G-invariant, relatively cocompact neighbourhood of Zeros(v) η be any G-invariant smooth function on M Proposition The family of inner products on g can be chosen in such a way that for all m M \ U, H(m) 1; v m η(m), and there is a C > 0, such that for all m M, A m C( v m 2 + 1). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

68 Choosing the family of inner products on g Solution to issues arising in the noncompact/spin c -case: a suitable choice of the family of inner products on g. Let U be a G-invariant, relatively cocompact neighbourhood of Zeros(v) η be any G-invariant smooth function on M Proposition The family of inner products on g can be chosen in such a way that for all m M \ U, H(m) 1; v m η(m), and there is a C > 0, such that for all m M, A m C( v m 2 + 1). This turns out to be enough to localise (ker L 2 T D t ) G, and get [Q, R] = 0. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

69 Conclusion Suppose M carries a G-Spin c -structure. Suppose Zeros(v)/G is compact. Quantisation after reduction: Q R(M) = Q ( µ 1 (0)/G ) = dim(ker D + M 0 ) dim(ker(d M0 ) ) Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

70 Conclusion Suppose M carries a G-Spin c -structure. Suppose Zeros(v)/G is compact. Quantisation after reduction: Q R(M) = Q ( µ 1 (0)/G ) Reduction after quantisation: = dim(ker D + M 0 ) dim(ker(d M0 ) ) R Q(M) = dim ( ker L 2 T D + t ) G dim ( kerl 2 T D t ) G. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

71 Conclusion Suppose M carries a G-Spin c -structure. Suppose Zeros(v)/G is compact. Quantisation after reduction: Q R(M) = Q ( µ 1 (0)/G ) Reduction after quantisation: = dim(ker D + M 0 ) dim(ker(d M0 ) ) R Q(M) = dim ( ker L 2 T D + t ) G dim ( kerl 2 T D t ) G. One can prove that R Q(M) = Q R(M) for certain Spin c -structures, by localising to neighbourhoods of µ 1 (0), using and A C( v 2 + 1). D 2 t = D 2 + ta + 4πtH + t2 4 v 2 Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

72 Final comments Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

73 K-theory and nontrivial representations Now suppose M/G is compact. In the symplectic setting, Landsman formulated a conjecture for reduction at the trivial representation in terms of the K-theory of the C -algebra of G, by defining Q G (M) K (C G). This was solved to a large extent by Mathai Zhang. Our current result generalises theirs to the Spin c -setting. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

74 K-theory and nontrivial representations Now suppose M/G is compact. In the symplectic setting, Landsman formulated a conjecture for reduction at the trivial representation in terms of the K-theory of the C -algebra of G, by defining Q G (M) K (C G). This was solved to a large extent by Mathai Zhang. Our current result generalises theirs to the Spin c -setting. With this definition, one can also consider reduction at K-theory generators representing nontrivial representations. Using different techniques and Paradan Vergne s result, we computed the multiplicities m λ in Q Spinc G (M) = m λ [λ] K (Cr G). λ Here λ runs over the dominant integral weights of a maximal compact subgroup of G, and [λ] are generators of K (C r G). Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

75 Positive scalar curvature...? We saw that for the Spin c -structures where our result applies, we have a topological expression for the (analytic) invariant, transversally L 2 -index of the deformed Dirac operator: Q Spinc (M) G = Q Spinc (M 0 ) = e 1 2 c 1(L 0 ) Â(M 0 ). M 0 This may yield obstructions to G-invariant metrics of positive scalar curvature on the noncompact manifold M, in terms of topological properties of M 0. Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35

76 Thank you Peter Hochs (UoA) Noncompact Spin c -quantisation November / 35