Equivariant Spectral Geometry. II Giovanni Landi Vanderbilt; May 7-16, 2007
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
A recent general strategy for isospectral Dirac operators on c.q.gs. via a Drinfel d twist Neshveyev-Tuset, OA/0703161 ; Beggs-Majid, QA/0506450 the quantum Dirac operator is of the form D q = F D F 1, Spec(D q ) = Spec(D)
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).
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
(A, H, D), γ
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
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).
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
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).
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
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 := ( ) 1 0 0 1 antipode: S ( a ) b qb a = ( a qb b a )
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
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
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)
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
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
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
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
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.
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 ;
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
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])
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.
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
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
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
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
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
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
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
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)
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.
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
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
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, + 1 8 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 1 1 2 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, + 1 4 a 0 δ(a 1 ) δ(a 2 ) δ(a 3 )F D 3.
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 2 + 2 a 0 δ 3 (a 1 )P D 3, 3
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
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 2 + 2 Ukl 3 δ3 (U lk )P D 3,
(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 0 + 2 3 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.