The spectral action for Dirac operators with torsion

Transcription

1 The spectral action for Dirac operators with torsion Christoph A. Stephan joint work with Florian Hanisch & Frank Pfäffle Institut für athematik Universität Potsdam Tours, ai 2011

2 1 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 2 Dirac Operators with Torsion 3 Spectral Action in presence of Torsion 4 Standard odel with Torsion 5 Spectral Triples 6 Questions

3 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 1 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 2 Dirac Operators with Torsion 3 Spectral Action in presence of Torsion 4 Standard odel with Torsion 5 Spectral Triples 6 Questions

4 Torsion Geometry, Einstein-Cartan-Theory & Holst Action The set-up Consider 4-dim. Riemannian manifolds (, g), closed and spin. General orthogonal connection for vector fields has the form X Y = g X Y + A(X, Y) with g Levi-Civita connection, A a (2, 1)-tensor field. The torsion (3, 0)-tensor Define torsion (3, 0)-tensor field A XYZ = g(a(x, Y), Z) for any X, Y, Z T p

5 Torsion Geometry, Einstein-Cartan-Theory & Holst Action The space of all torsion tensors: T (T p ) = { A 3 T p AXYZ = A XZY } X, Y, Z T p Theorem (E. Cartan) One has the following decomposition of T (T p ) into irreducible O(T p )-subrepresentations: T (T p ) = V(T p ) A(T p ) C(T p ). (orthogonal decomposition w.r.t. natural scalar product)

6 Torsion Geometry, Einstein-Cartan-Theory & Holst Action Where we define the vectorial torsion V(T p ) = { A T (T p ) V s.t. X, Y, Z : A XYZ = g(x, Y) g(v, Z) g(x, Z) g(v, Y) }, the totally anti-symmetric torsion A(T p ) = { A T (T p ) } X, Y, Z : AXYZ = A YXZ, and the torsion of Cartan-type C(T p ) = { A T (T p ) X, Y, Z : AXYZ + A YZX + A ZXY = 0 4 and A(e a, e a, Z) = 0 } a=1 with e 1,...e 4 orthonormal basis of T p ().

7 Torsion Geometry, Einstein-Cartan-Theory & Holst Action Corollary For any orth. connection X Y = g X Y + A(X, Y) exist a unique vector field V, a unique 3-form T and a unique (3, 0)-tensor field C with C p C(T p ) such that A(X, Y) = g(x, Y)V g(v, Y)X + T(X, Y, ) + C(X, Y, ). Remark Let be compatible with the Riemannian metric g. Then has same geodesics as g if and only if A(X, Y) = T(X, Y, ) is totally anti-symmetric.

8 Torsion Geometry, Einstein-Cartan-Theory & Holst Action Einstein-Cartan-Hilbert theory Let be an orthogonal connection. Its scalar curvature is R = R g + 6 div g (V) 6 V 2 T C 2 where R g is the scalar curvature g. The Einstein-Cartan-Hilbert functional is ( R dvol = R g 6 V 2 T C 2) dvol. Critical Points the Einstein-Cartan-Hilbert functional Variations over all metrics and all orthogonal connections: (, g, ) is a critical point of E-C-H functional iff (, g) is an Einstein manifold, and = g (i.e. V 0, T 0, C 0).

9 Torsion Geometry, Einstein-Cartan-Theory & Holst Action Decomposition of C Cartan-type torsion C can be decomposed into a selfdual part C + and an anti-selfdual part C. Note that C Λ 1 Λ 2 = Λ 1 (Λ 2 + Λ2 ), with Λ 2 ± the ±1-eigenspaces of the Hodge star : Λ2 Λ 2. Defining C ± = C Λ 1 Λ 2 ± we have T = V A C + C Lemma The scalar curvature decomposes as R = R g + 6 div g (V) 6 V 2 T C C 2

10 Torsion Geometry, Einstein-Cartan-Theory & Holst Action The Holst density Let θ 1,...,θ 4 be the dual basis of e 1,..., e 4 and Riem(X, Y)Z = X Y Z Y X Z [X,Y] Z Define the Holst density ρ H = 4 a,b=1 g(riem(, )e a, e b ) θ a θ b For any orthogonal connection one finds ( ρ H = 6 dt + 12 T, V 3 dvol 1 2 C + 2 C 2) dvol where, 3 is the scalar product on 3-forms induced by g.

11 Torsion Geometry, Einstein-Cartan-Theory & Holst Action The Holst action Let be a closed 4-dim. manifold with Riemannian metric g and orthogonal connection. The Holst action is ( 1 I H = R dvol + 1 ) 16πG γ ρ H = 1 ( R g 6 V 2 T πG γ T, V (1 + 1 γ ) C (1 1 γ ) C 2) dvol. Loop Quantum Gravity The Holst action I H is one of the starting points for the loop quantisation of gravity. The parameter γ is called Barbero-Immirzi parameter.

12 Torsion Geometry, Einstein-Cartan-Theory & Holst Action Critical Points of the Holst action for γ ±1 Assume that γ ±1. Variations over all metrics and all orthogonal connections: (, g, ) is a critical point of the Holst action iff = g (i.e. V = 0, T = 0, C = 0). (, g) is an Einstein manifold Critical Points of the Holst action for γ = ±1 Assume that γ = ±1. Variations over all metrics and all orthogonal connections: (, g, ) is a critical point of the Holst action iff C arbitrary, C ± = 0 and V arbitrary, T = V, (, g) is an Einstein manifold

13 Dirac Operators with Torsion 1 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 2 Dirac Operators with Torsion 3 Spectral Action in presence of Torsion 4 Standard odel with Torsion 5 Spectral Triples 6 Questions

14 Dirac Operators with Torsion The spinor connection Let e 1,.., e 4 be an orthonormal frame. The spinor connection induced by is locally given by X ψ = g X ψ A(X, e i, e j ) e i e j ψ. The Dirac operator The Dirac operator for is Dψ = D g ψ A(e i, e j, e k ) e i e j e k ψ, i,j,k=1 i<j where D g is the Dirac operator for g.

15 Dirac Operators with Torsion Lemma For the Dirac operator the Cartan type component C(T p ) of the torsion is invisible. Selfadjointness The Dirac operator D associated to is formally selfadjoint iff the (3, 0)-torsion tensor A has no vectorial component, i.e. A p A(T p ) C(T p ).

16 Dirac Operators with Torsion The Dirac operator & its square For the calculation of the spectral action we need the square of the Dirac operator associated to an orthonormal connection. Bochner formula for D D Dψ = ψ Rg ψ dt ψ 3 4 T 2 ψ divg (V)ψ 9 2 V 2 ψ ( ) +9 T V ψ + (V T) ψ where = is the Laplacian associated to spin connection X ψ = g X ψ (X T) ψ 3 2 V X ψ 3 2 g(v, X)ψ,

17 Spectral Action in presence of Torsion 1 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 2 Dirac Operators with Torsion 3 Spectral Action in presence of Torsion 4 Standard odel with Torsion 5 Spectral Triples 6 Questions

18 Spectral Action in presence of Torsion Chamseddine-Connes Spectral Action The bosonic spectral action for Dirac operator D is the number of eigenvalues of D D in the interval [0,Λ 2 ] (with Λ R): ( D ) D I = Tr f, where Tr is the L 2 -trace over the space of spinor fields, and f is cut-off function with support in [0,+1] which is constant near 0. Spectral action on left- (or right-)handed Spinors: ( D ) D I L/R = Tr f P L/R Λ 2, Λ 2

19 Spectral Action in presence of Torsion Asymptotic expansion of Spectral Action From the heat trace asymptotics for t 0 ( ) Tr e t D D t n 2 a 2n (D D) n 0 (with Seeley-deWitt coefficients a 2n (D D)) one gets (by t = Λ 2 ) an asymptotics for the spectral action I Λ 4 f 4 a 0 (D D) + Λ 2 f 2 a 2 (D D) + Λ 0 f 0 a 4 (D D) as Λ with f 4, f 2, f 0 moments of cut-off function f. Similarly for I L/R using P L/R = 1 2 (id γ 5).

20 Spectral Action in presence of Torsion Spectral Action I up to Λ 2 D with vectorial and totally anti-symmetric torsion a 2 (D 1 D) = 1 16π 2 3 Rg + 3 T V 2 dvol I = Λ4 f 4 2 dvol + Λ2 f 2 32π Rg + 3 T V 2 dvol Critical Points for I (, g, ) is a critical point of the I iff X Y = g X Y + C(X, Y, ), C arbitrary, V 0, T 0 (, g) is an Einstein manifold

21 Spectral Action in presence of Torsion Spectral Action I L up to Λ 2 D with vectorial torsion and P L = 1 2 (id γ 5) a 2 (D 1 D) = 1 16π 2 3 Rg + 3 T V 2 dvol a 2 (γ 5 D 1 D) = 36 T, V 16π 2 3 dvol I L = Λ4 f Λ2 f 2 32π 2 dvol + Λ2 f 2 32π 2 36 T, V 3 dvol Holst-action without Cartan-type torsion Barbero-Immirzi parameter γ = 1 critical points as for I H 1 3 Rg + 3 T V 2 dvol

22 Spectral Action in presence of Torsion Assumption: D selfadjoint ( V 0) Theorem The a 4 (D 2 )-term in the heat trace for the Dirac operator associated with the connection whose torsion T is totally anti-symmetric is a 4 (D 2 ) = χ() 1 320π 2 Spectral Action W 2 dvol 3 32π 2 δt 2 dvol I = Λ 4 f 4 dvol + Λ2 f 2 16π f χ() f 0 320π ( R g 9 T 2) dvol W 2 dvol 3f 0 32π 2 δt 2 dvol

23 Spectral Action in presence of Torsion Equations of motion (Proca equation) Variation of I (with fixed volume) yields c 1 G c 2 B = S(T) c 3 T = c 4 dδt, (= dt = 0), with Einstein tensor G and Bach tensor B of g, and stress-energy tensor S given by ( S(T) = c 3 6g T 1 2 T 2 g ) ( c 4 2g d T 1 2 δt 2 g ) and g η (X, Y) = X η, Y η g for any pair of tangent vectors X, Y. Note: S(0) = 0.

24 Spectral Action in presence of Torsion Observations Spectral action does not give the E-H-C action. Spectral action does not constrain Cartan-type torsion. (, g) Einstein manifold, T = 0 and C arbitrary is a critical point. Expect critical points with T 0, yet, still no examples. Exclusion result Let = N S 1 with N a 3-dim. compact, constant curvature and metric g = f(t)g N + dt 2 (so B = 0). Ansatz: T = τ(x, y, z, t)dvol N. Then (, g, T) critical iff T = 0.

25 Standard odel with Torsion 1 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 2 Dirac Operators with Torsion 3 Spectral Action in presence of Torsion 4 Standard odel with Torsion 5 Spectral Triples 6 Questions

26 Standard odel with Torsion Specific particle model The ingredients: internal Hilbert space: H f with connection H f full Hilbert space: H S = L 2 (, S) H f ψ χ twisted connection (with totally anti-symmetric torsion): S = id Hf + id S H f associated Dirac operator: D b S field Φ of endomorphisms of H f (encodes Higgs boson, Yukawa couplings, etc.) The Chamseddine-Connes Dirac operator D Φ (ψ χ) = D b S (ψ χ) + γ 5 ψ Φ χ

27 Standard odel with Torsion Bochner formula for D Φ (D Φ ) 2 (ψ χ) = (ψ χ) E Φ (ψ χ) with E Φ (ψ χ) = ( ) ( 3 2 dt 1 4 Rg T 2 )ψ χ n i=1 i j ( ei e j ψ ) ( Ω H ij ) χ γ 5 e i ψ [ H f e i,φ]χ ψ (Φ 2 )χ

28 Standard odel with Torsion The Bosonic Lagrangian I = 24Λ4 f 4 π 2 + Λ2 f 2 π 2 dvol { 12 (Rg 9 T 2 ) a ϕ 2 12 c } dvol f 0 2 χ() f π 2 + f 0 2π 2 W 2 dvol f 0 9 π 2 { g 2 3 G 2 + g 2 2 F g2 1 B 2 + a D ν ϕ 2 + b ϕ 4 + 2e ϕ )} d (Rg 9 T 2 ) (a ϕ c dvol. δt 2 dvol

29 Standard odel with Torsion The full Standard odel action ( ) D 2 I S = Tr f Φ + Ψ, D Λ 2 Φ Ψ J with Ψ H S Observations Torsion couples to the Higgs field ϕ. Torsion couples to the fermions. Existence of the derivative terms of T. Torsion becomes dynamical Expect critical points of spectral action with non-zero torsion.

30 Spectral Triples 1 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 2 Dirac Operators with Torsion 3 Spectral Action in presence of Torsion 4 Standard odel with Torsion 5 Spectral Triples 6 Questions

31 Spectral Triples Even, real spectral triple (A, H, D) A real, associative, unital pre-c -algebra A A Hilbert space H on which the algebra A is faithfully represented via a representation ρ A self-adjoint operator D with compact resolvent, the Dirac operator An anti-unitary operator J on H, the real structure A unitary operator γ on H, the chirality

32 Spectral Triples The Axioms of NCG Axiom 1: Classical Dimension n Axiom 2: Regularity Axiom 3: Finiteness Axiom 4: Reality Axiom 5: First Order of the Dirac Operator Axiom 6: Orientability

33 Spectral Triples Remark Let (, g, ) be a closed Riemannian spin manifold with orthogonal connection. If the vectorial component of the torsion of is zero, then (A, H, D ) satisfies the axioms for spectral triples. Reconstruction of Let dim() be even. Then the totally anti-symmetric torsion can be reconstructed from (A, H, D ). Not true for dim() odd: Let dim() = 3 and T = dvol. Then dvol ψ = id ψ = ψ for all ψ H.

34 Questions 1 Torsion Geometry, Einstein-Cartan-Theory & Holst Action 2 Dirac Operators with Torsion 3 Spectral Action in presence of Torsion 4 Standard odel with Torsion 5 Spectral Triples 6 Questions

35 Questions Some Questions Critical points with non-zero anti-symmetric torsion? Spectral triples for non-symmetric Dirac operators? Connection NCG <=> LQG via Holst-action? Holst Action + Standard odel? The Spectral Action for Dirac Operators with skew-symmetric Torsion F. Hanisch, F. Pfäffle, C.S., CP 300:877 (2010) Gravity, Torsion and the Spectral Action Principle F. Pfäffle, C.S. arxiv: v2 [math-ph] The Holst Action by the Spectral Action Principle F. Pfäffle, C.S. arxiv: v2 [math-ph] to appear in CP