Equivariant Spectral Geometry. II

Transcription

1 Equivariant Spectral Geometry. II Giovanni Landi Vanderbilt; May 7-16, 2007

2 Some experimental findings; equivariant spectral triples on: toric noncommutative geometry (including and generalizing nc tori) the manifold of quantum SU(2) families of quantum two spheres higher dimensional quantum orthogonal spheres quantum projective spaces The guiding principle: equivariance with respect to a quantum symmetry equivariance will build all the geometry from scratch

3 A recent general strategy for isospectral Dirac operators on c.q.gs. via a Drinfel d twist Neshveyev-Tuset, OA/ ; Beggs-Majid, QA/ the quantum Dirac operator is of the form D q = F D F 1, Spec(D q ) = Spec(D)

4 There exists formulas for q-analogues of the Dirac operator on quantum groups... ; let us call Q these naive Dirac operators. Now the fundamental equation to define the thought for true Dirac operator D which we used above implicitly on the deformed 3-sphere (after suspension to the 4-sphere and for deformation parameters which are complex of modulus one) is, [D] q 2 = Q. where the symbol [x] q has the usual meaning in q-analogues,... The main point is that it is only by virtue of this equation that the commutators [D, a] will be bounded... A. Connes, G. L., Noncommutative Manifolds, the Instanton Algebra and Isospectral Deformations, CMP (2001).

5 After some initial skepticism, this programme was completed in L. Dabrowski, G. Landi, A. Sitarz, W. van Suijlekom, J.C. Varilly The Dirac operator on SU q (2), CMP (2005). An isospectral nc geometry on the manifold underlying SU q (2) The guiding principle: equivariance with respect to a quantum symmetry equivariance will build all the geometry from scratch

6 (A, H, D), γ

7 A real structure on (A, H, D) A real structure J for the spectral triple (A, H, D) is given by an antilinear isometry J on H such that [π(a), Jπ(b)J 1 ] I, [ [D, π(a)], Jπ(b)J 1 ] I, a, b A, with I an operator ideal of infinitesimals On A(SU q (2)) and A(S 2 qt ) one can takes I = OP

8 Symmetries implemented by the action of a Hopf -algebra U q (g), a quantum universal enveloping algebra; Let U = (U,, S, ε) be a Hopf -algebra and A be a left U-module -algebra, i.e., there is a left action of U on A, h xy = (h (1) x)(h (2) y), h 1 = ε(h)1, (h x) = S(h) x, notation (h) = h (1) h (2).

9 A -representation π of A on V (a dense subspace of H) is called U-equivariant if there is a -representation λ of U on V s. t. for all h U, x A and ξ V. λ(h) π(x) ξ = π(h (1) x) λ(h (2) ) ξ, This is the same as a -representation of the left crossed product -algebra A U defined as the -algebra generated by the two -subalgebras A and U with crossed commutation relations hx = (h (1) x)h (2), h U, x A A linear operator D on V is equivariant if it commutes with λ(h), for all h U and ξ V Dλ(h) ξ = λ(h)d ξ

10 An antiunitary operator J is equivariant if it leaves V invariant and if it is the antiunitary part in the polar decomposition of an antilinear (closed) operator T that satisfies the condition T λ(h) ξ = λ(s(h) )T ξ, for all h U and ξ V, where S is the antipode of U A (real graded) spectral triple (A, H, D, γ, J) is U-equivariant if the representation of A and the operators D and J are equivariant (and γ commutes with the equivariant representation).

11 The nc geometry of A(SU q (2)) The algebra: With 0 < q < 1, let A = A(SU q (2)) be the -algebra generated by a and b, with relations: ba = qab, b a = qab, bb = b b, a a + q 2 b b = 1, aa + bb = 1 these state that the defining matrix is unitary ( ) a b U = qb a

12 A(SU q (2)) is a Hopf -algebra (a quantum group) with coproduct: ( a ) b qb a := ( ) a b. qb a ( a ) b qb a counit: ε ( a ) b qb a := ( ) antipode: S ( a ) b qb a = ( a qb b a )

13 The symmetry: The quantum universal envelopping algebra U = U q (su(2)) is the -algebra generated by e, f, k, with k invertible, and relations ek = qke, kf = qfk, k 2 k 2 = (q q 1 )(fe ef), the -structure is simply k = k, f = e, e = f

14 The Hopf -algebra structure coproduct: k = k k, f = f k + k 1 f, e = e k + k 1 e counit: ɛ(k) = 1, ɛ(f) = 0, ɛ(e) = 0 antipode: Sk = k 1, Sf = qf, Se = q 1 e

15 The action of U on A A natural bilinear pairing between U and A, k, a = q 1 2, k, a = q 1 2, e, qb = f, b = 1 gives canonical left and right U-module algebra structures on A (they mutually commute): h x := x (1) h, x (2), x h := h, x (1) x (2) we use the notation (x) = x (1) x (2)

16 The invertible antipode transforms the right action into a second left action of U on A, commuting with the first h x := x S 1 (ϑ(h)), with automorphism ϑ of U q (su(2)) given by ϑ(k) := k 1, ϑ(f) := e, ϑ(e) := f

17 The representation theory of U q (su(2)) The irreducible finite dim representations σ l of U q (su(2)) are labelled by nonnegative half-integers (the spin) l 1 2 N 0: σ l (k) lm = q m lm, σ l (f) lm = [l m][l + m + 1] l, m + 1, σ l (e) lm = [l m + 1][l + m] l, m 1, on the irreducible U-module V l = span{ lm, m = l,..., l} σ l is a -representation, with respect to the hermitian scalar product on V l for which the vectors lm are orthonormal. the brackets denote q-integers [n] = [n] q := qn q n q q 1 provided q 1

18 Left regular representation of A(SU q (2)) The left regular representation of A as an equivariant representation with respect to the two left actions and of U With λ and ρ mutually commuting representations of U on V, a representation π of the -algebra A on V is (λ, ρ)-equivariant if λ(h) π(x)ξ = π(h (1) x) λ(h (2) )ξ, ρ(h) π(x)ξ = π(h (1) x) ρ(h (2) )ξ, h U, x A, ξ V As a representation space the prehilbert space V := 2l=0 V l V l, lmn := lm ln, m, n = l,..., l

19 Two copies of U q (su(2)) act via the irreps σ l : λ(h) = σ l (h) id, ρ(h) = id σ l (h) A (λ, ρ)-equivariant -repr π of A(SU q (2)) on the Hilbert space V is the left regular representation. It has the form: π(a) lmn = A + lmn l+ m + n + + A lmn l m + n +, π(b) lmn = B + lmn l+ m + n + B lmn l m + n, with suitable explicit constants A ± lmn, B± lmn. Here l ± := l ± 1 2, m± := m ± 1 2, n± := n ± 1 2

20 Spin representation We amplify the left regular representation π of A to the spinor representation π = π id on We also set ρ = ρ id on W. W := V C 2 = V V 12 But, we replace λ on V by its tensor product with σ 12 on C 2 : λ (h) := (λ σ 12 )( h) = λ(h (1) ) σ 12 (h (2) ) The spinor representation π of A on W is (λ, ρ )-equivariant. A basis { jµn, jµn } for W from its q-clebsch-gordan decomp.

21 Equivariant Dirac operator Any self-adjoint operator on H = (V C 2 ) cl, that commutes with both actions ρ and λ of U q (su(2)) is of the form D jµn = d j jµn, D jµn = d j jµn, with d j and d j real eigenvalues of D; depend only on j; Restrictions on eigenvalues comes by requiring boundedness of the commutators [D, π (x)] for x A Unbounded commutators are obtained with the naive q-dirac d j = 2 [2j + 1] q + q 1, d j = d j [BibikovKulish] q-analogues of the classical eigenvalues of D/ 1 2 ;

22 D/ is the classical ( round ) Dirac operator on the sphere S 3 ; An operator D with spectrum the one of the classical D/ [CL] With eigenvalues linear in j : d j = c 1 j + c 2, d j = c 1 j + c 2, with c 1 c 1 < 0 (for a nontrivial sign), all commutators [D, π (x)] for x A are bounded operators. up to irrelevant scaling factors the choice of c j, c j with d j = 2j + 3 2, d j = 2j 1 2, is immaterial; the spectrum of D (with multiplicity) is the classical one Essentially the only possibility for a Dirac operator satisfying a (modified) first-order condition

23 The real structure For the l.r.r. H ψ = L 2 (SU q (2), ψ), ψ the Haar state; the Tomita operator T ψ =: J ψ 1/2 ψ defines the positive modular operator ψ and the antiunitary modular conjugation J ψ. On the basis: T ψ lmn = ( 1) 2l+m+n q m+n l, m, n. With respect to the representations λ and ρ of U q (su(2)): T ψ λ(h) Tψ 1 = λ(sh), T ψ ρ(h) Tψ 1 = ρ(sh) i.e. the Tomita operator for the Haar state of the Hopf -algebra A implements the antilinear involutory automorphism h (Sh) of the (dual) Hopf -algebra U (cf. [Masuda-Nakagami-Woronowicz])

24 The commuting -antirepresentation of A on H ψ π (x) := J ψ π(x ) J 1 ψ is a -representation of the opposite algebra A(SU 1/q (2)). It is equivariant: λ(h) π (x)ξ = π ( h (2) x) λ(h (1) )ξ, ρ(h) π (x)ξ = π ( h (2) x) ρ(h (1) )ξ, for all h U, x A and ξ V. h h is the automorphism of U: k := k, f := q 1 f, ẽ := qe.

25 The real structure for spinors is not the modular conjugation J ψ J ψ for the spinor representation of A(SU q (2)) conjugation of the spinor rep. π (A(SU q (2))) by the modular operator yields a representation of the opposite algebra A(SU 1/q (2)) this force the Dirac operator D to be equivariant under the corresponding symmetry of U 1/q (su(2)) as well this extra equivariance condition force D to be a scalar operator: no equivariant 3 + -summable real spectral triple on A(SU q (2)) with the modular conjugation operator

26 The remedy is to modify J to a non-tomita conjugation operator while keeping with the symmetry of the spinor representation. However, the expected properties of real spectral triples do hold only up to compact perturbations The strategy: Construct the right representation of the algebra A on spinors by its symmetry alone, requiring (λ, ρ )-equivariance Deduce the conjugation operator J on spinors as the intertwiner of the left and right spinor representations NB. These representations do not commute any longer

27 The conjugation operator J is the antilinear operator on H defined explicitly on the orthonormal spinor basis by J jµn := i 2(2j+µ+n) j, µ, n,, J jµn := i 2(2j µ n) j, µ, n,. J is antiunitary and J 2 = 1. J is equivariant. Also, [J, D] = 0 with the isospectral D The commutant and the first-order conditions are satisfied only up to infinitesimals of arbitrary high order: [π (x), Jπ (y)j 1 ] I, [π (x), [D, Jπ (y)j 1 ]] I, I OP

28 Proposition 1. The spectral triple (A(SU q (2)), H, D) is regular and 3 + -summable. It has simple dimension spectrum given by Σ = {1, 2, 3}. Its KO-dimension is 3. The dimension spectrum is worked out by constructing a symbol map from order zero pseudodifferential operators on A(SU q (2)) to a noncommutative version of the cosphere bundle

29 For all cases: an interesting pseudo-differential calculus Formulæ for Connes-Moscovici local index thm Started by A. Connes, Cyclic cohomology, quantum group symmetries and the local index formula for SU q (2), (2004) for the singular (no limit when q 1) spectral triple in P. S. Chakraborty, A. Pal, Equivariant spectral triples on the quantum SU(2) group, (2003) A lot of experimental evidence for a general theory

30 The cosphere bundle and the dimension spectrum for SU q (2) On l 2 (N) with o.n. basis { ε x : x N }, two representation of A(SU q (2)): π ± (a) ε x := 1 q 2x+2 ε x+1, π ± (b) ε x := ±q x ε x. not faithful: b b ker π ±. The quotients A(D 2 q± ) 0 ker π ± A(SU q (2)) r ± A(D 2 q±) 0, are the algebras of two quantum disks D 2 q± ; the two hemispheres of the equatorial Podleś sphere S 2 q = S 2 qt=0

31 The subalgebra B of Ψ 0 (A) generated by all δ k (A) for k 0 is indeed generated by a set of diagonal operators ã ± := ±δ(a ± ) = P a ± P + (1 P )a ± (1 P ), b ± := ±δ(b ± ) = P b ± P + (1 P )b ± (1 P ), together with the off-diagonal operators a / := (1 P )a + P +P a (1 P ), b / := (1 P )b + P +P b (1 P ), with a natural decomposition of the spinor rep. π (a) := a + + a, π (b) := b + + b ± map the label j to j ± 1 2 of the spinor basis and P := 2 1 (1 + F ), F = Sign D

32 There is a -homomorphism ρ : B A(Dq+ 2 ) A(D2 q ) A(S 1 ) ρ(ã + ) := r + (a) r (a) u, ρ(ã ) := q r + (b) r (b ) u, ρ( b + ) := r + (a) r (b) u, ρ( b ) := r + (b) r (a ) u. ρ(a / ) := ρ(b / ) = 0 u generates A(S 1 ) Definition 2. The cosphere bundle on SU q (2), denoted by A(S q), is defined as the range of the map ρ in A(D 2 q+ ) A(D2 q ) A(S1 ). Element x B are determined by ρ(x) A(S q) up to smoothing operators (these drop out from relevant computations)

33 A symbol map σ : A(D 2 q± ) A(S1 ); on generators σ(r ± (a)) := u, σ(r ± (b)) := 0, maps D 2 q± to their common boundary S1 On the algebras A(D 2 q± ) three linear functionals τ 1 and τ 0 ; τ 1 (x) := 1 2π σ(x), S 1 τ 0 (x) := lim Tr N π (x) (N + 3 N 2 )τ 1(x), τ 0 (x) := lim Tr N π (x) (N + 1 N 2 )τ 1(x), for x A(D 2 q± ); and Tr N(T ) := N k=0 ε k T ε k.

34 With the natural restriction map, r : A(S q) A(D 2 q+ ) A(D2 q ). Proposition 3. The spectral triple (A(SU q (2)), H, D) has simple dimension spectrum by {1, 2, 3}. The corresponding residues are T D 3 = 2(τ 1 τ 1 ) ( rρ(t ) 0), T D 2 = ( τ 1 (τ 0 + τ 0 ) + (τ 0 + τ 0 ) τ 1)( rρ(t ) 0 ), T D 1 = (τ 0 τ 0 + τ 0 τ 0 )( rρ(t ) 0), with T B. ρ(t ) 0 the degree-zero part with respect to the Z-grading on A(S q) induced (in its representation) by the one-parameter group of automorphisms γ(t) generated by D

35 The local index formula for 3-dimensional geometries A Fredholm index of the operator D, ϕ : K 1 (A) Z. ϕ([u]) := Index(P up ) = dim ker P UP dim ker P U P. With F = Sign D and P = 1 2 (1 + F ). Computed by pairing K 1 (A) with a nonlocal cyclic cocycle χ 1 (a 0, a 1 ) = Tr(a 0 [F, a 1 ]) ; the Fredholm module (H, F ) over A = A(SU q (2)) is 1-summable since all commutators [F, π(x)] are trace-class

36 The C M local index theorem expresses the index map in terms of a local cocycle φ odd = (φ 1, φ 2 ) φ 1 (a 0, a 1 ) := a 0 [D, a 1 ] D 1 1 a 0 ([D, a 1 ]) D 3 4 a 0 2 ([D, a 1 ]) D 5, φ 3 (a 0, a 1, a 2, a 3 ) := 1 12 (T ) := [D 2, T ] With [F, a] traceclass for each a A, a 0 [D, a 1 ] [D, a 2 ] [D, a 3 ] D 3, φ 1 (a 0, a 1 ) = a 0 δ(a 1 )F D φ 3 (a 0, a 1, a 2, a 3 ) = 1 12 a 0 δ 2 (a 1 )F D 2 a 0 δ 3 (a 1 )F D 3, a 0 δ(a 1 ) δ(a 2 ) δ(a 3 )F D 3.

37 With the additional use of a simple dimension spectrum not containing 0 and bounded above by 3 ; the Chern character χ 1 is equal to φ odd (b + B)φ ev where the η-cochain φ ev = (φ 0, φ 2 ) is φ 0 (a) := Tr(F a D z ) φ 2 (a 0, a 1, a 2 ) := 1 24, z=0 a 0 δ(a 1 ) δ 2 (a 2 )F D 3 ; φ 1 = χ 1 + bφ 0 + Bφ 2, φ 3 = bφ 2. With the same conditions on the dimension spectrum and commutators [F, a], the local Chern character φ odd = ψ 1 (b+b)φ ev, ψ 1 (a 0, a 1 ) := 2 a 0 δ(a 1 )P D 1 a 0 δ 2 (a 1 )P D a 0 δ 3 (a 1 )P D 3, 3

38 and φ ev = (φ 0, φ 2 ), φ 0 (a) := Tr(a D z ) φ 2 (a 0, a 1, a 2 ) := 1 24, z=0 a 0 δ(a 1 ) δ 2 (a 2 )F D 3. The term in P D 3 would vanish if the latter were traceclass [Connes] (this is the statement that P has metric dimension 2) Summing up, up to coboundaries, the cyclic 1-cocycles χ 1 can be given by means of one single (b, B)-cocycle ψ 1 : χ 1 = ψ 1 bβ, where β(a) = 2 Tr(P a D z ) z=0

39 Compute Index(P U P ) when U is the defining unitary ( ) a b U = qb a, acting H C 2 via the representation π 1 2 : kernel of P U P is trivial kernel of P U P contains only elements proportional to the vector ( 0, 0, 1 2, q 1 0, 0, 1 2, leading to ϕ([u]) = Index(P UP ) = 1. ), Calculating ψ 1 (U 1, U) (with the local cyclic cocycle extended by Tr C 2), up to an overall 1 2 factor gives 2 Ukl δ(u lk)p D 1 Ukl δ2 (U lk )P D Ukl 3 δ3 (U lk )P D 3,

40 (summation over k, l = 0, 1) Since ρ(δ 2 (U kl )) = ρ(u kl ), one is left to compute the degree 0 part of ρ(u kl δ(u lk)), which using the relations of A(D 2 q± ) is Then, ρ(u kl δ(u lk)) 0 = 2(1 q 2 ) 1 r (b) 2 ψ 1 (U 1, U) = 2(1 q 2 )(2τ 0 τ τ 1 τ 1 )(1 r (b) 2 ) = 2. (τ 1 τ 0 + τ 0 τ 1)(1 1) With the proper coefficient: Index(P UP ) = 1 2 ψ 1(U 1, U) = 1.