Einstein-Hilbert action on Connes-Landi noncommutative manifolds

Transcription

1 Einstein-Hilbert action on Connes-Landi noncommutative manifolds Yang Liu MPIM, Bonn Analysis, Noncommutative Geometry, Operator Algebras Workshop June 2017

2 Motivations and History Motivation: Explore notions of metric and curvature for certain class of noncommutative manifolds. Quantum gravity On noncommutative two torus: Connes s pseudo differential calculus 80 s. The computation was initiated at the same time; Gauss-Bonnet theorem (2011): Connes-Tretkoff, Khalkhali-Fathizadeh; Conformal geometry (2014): Connes-Moscovici, Khalkhali-Fathizadeh, Lesch, Moscovici-Lesch and others. This talk is based on my work arxiv (to appear in JNCG), arxiv

3 Connes-Landi noncommutative manifolds C (M θ ) Let M be a Riemannian manifold such that T n Diff(M). C (M) becomes a T n -module through the dual action: for t T n, U t : C (M) C (M): U t (f )(x) = f (t 1 x), f C (M). Isotypic decomposition: f C (M), f = ˆ f r, with f r = e 2πir t U t (f )dt. r Z n T n In particular, fh = r,l f r h l.

4 θ-deformation (M. Rieffel) For any n n skew symmetric matrix θ, denote by bicharacter on Z Z: χ θ (r, l) = e 2πi θr,l. One can construct a new family of multiplication for C (M) by twisting the convolution: let f = r f r and g = l g l, f θ h = r,l χ θ (r, l)f r h l. The new algebra C (M θ ) (C (M), θ ) represents the smooth structure of M θ. Natural vector bundles over M θ (C (M θ )-bimodules): f smooth function, X vector field, ω one form: f θ X, ω θ f, X θ ω etc. Deforming linear operators P : C (M) C (M): π θ (P)(f ) P θ f.

5 Quantum mixed with Riemannian Quantum side In general, the θ -multiplication is highly nonlocal. For example, the evaluation of (f U ) θ (h U ) at some point x U depends on U in such a obscure way that the value has no compatibility when U is shrinking around x. Riemannian side If one of h and f is T n -invariant, then f θ h = fh. In particular, For a T n -invariant metric g, the associated connection and spinor Dirac are invariant under the deformation: π θ ( ) =, π θ ( /D) = /D. Leibniz property holds when computing (X θ ω). The spectral triple (C (M θ ), L 2 (/S), /D) satisfies Connes s axioms of spin geometry.

6 Scalar Curvature functional (Spectral Geometry) For a given metric g with the associated spinor Dirac /D, the local geometry is encoded in the heat kernel asymptotic: Tr(fe t /D2 ) t 0 j=0 V j (f, /D 2 )t j m 2 m = dim M. Coefficients are functionals: f C (M) V j (f, /D 2 a) C. To localize the functionals, introduce ϕ 0 ( ) = M Tr x( )dg Functional density V j (x) C (M) is defined by the property V j (f, /D 2 ) = ϕ 0 (f (x)v j (x)) In general, V j (x) is a polynomial whose arguments are components of the curvature tensor and the derivatives.

7 What we need V 2 (x) = 1 12 S g. where S g is the scalar curvature. V 2 (, /D 2 ): scalar curvature functional. The definition is intrinsic with respect to the spectral data: Define ˆ P = Res z=0 Tr Ä P /D zä. The functional ϕ 0 is proportional to ˆ ( ) /D m, m = dim M. The scalar curvature functional is proportional to ˆ ( ) /D m+2, m = dim M.

8 Symmetry Breaking T n -invariant metrics behave like the Riemannian version The spectral triple (C (M θ ), L 2 (/S), /D) satisfies Connes s axioms of spin geometry. Leibniz property holds when computing (X θ ω). The rest of the metrics seems intangible, because there is no straightforward calculus. Widom s work tells us that any connection will lead to a pseudo differential calculus. My work started with the observation: if = g for some T n -invariant metric g, such calculus can be deformed to study the spectral geometry of M θ.

9 Quantizing metrics Let g = e 2h g for a real-valued h C (M). We would like to compare the two spinor Dirac /D g and /D g. There is a canonical and unitary identification between the L 2 -spinor sections β g g : L 2 (/S g ) L 2 (/S g ). Proposition The g -spinor Dirac /D g is unitary equivalent to β g g /D g β g g = e h/2 /D g e h/2.

10 Conformal Geometry Let us fix a T n -invariant metric g as a background metric. The conformal class [g] can be parametrized by (g, h) with h = h C (M). (M, g) e 2h with h C (M θ ) real-valued (C (M θ ), L 2 (/S), /D) e 2h with h = h C (M θ ) self-adjoint g = e 2h g D h = e h/2 /De h/2 Scalar curvature for g : S g R Dh C (M θ ): functional density of V 2 (, Dh 2) Table: Conformal change of metric and the associated scalar curvature in the noncommutative setting.

11 Modular scalar curvature Weyl factor (conformal factor), k = e h with h = h C (M θ ). Modular derivation (noncommutative differential): x h = ad h = [, h]. Modular operator: y h = exp x h = k 1 ( )k. Theorem (Liu, 16) Let dim M = 4. For g = e 2h g or /D D h = e h/2 /De h/2, upto a constant factor, the modular scalar curvature defined in Table (1) is given by R g = e ( m+2)h Ä K(x h )( 2 h) + H(x (1) h + cs g, x(2) h )( h h)ä g 1 Notations: write H as a Fourier transform: H = Fβ, H(x (1) h, x(2) h ˆR )(ψ ϕ) = β(u, v)(y h ) iu (ψ)(y h ) iv (ϕ)dudv. 2

12 The one-variable function K Assume dim M = 4. K(u) = 1 2 sinh (u/2) eu, (u/2) which shows striking similarity to the AS local index formula. The Â-genus part comes from the fact that the deformation start on the tangent space of the metrics: e h = f 1 (ad h )( h) = e h ead h 1 ( h). ad h In local index formula, the Â-genus naturally through the Jacobian of the exponential map of a Lie group G: in exponential coordinates d exp h = j g (h)dh, where g is the Lie algebra, dh is a Haar measure and Ç å j g = det g ( f 1 (ad h )) = det g ead h 1. ad h

13 Einstein Hilbert action Recall R g = e ( m+2)h Ä K(x h )( 2 h) + H(x (1) h, x(2) h )( h h)ä g 1 + cs g For each g = e 2h g. We have the scalar curvature functional (on C (M)): f V 2 (f, D 2 h) = ϕ 0 (fr g ). The noncommutative analogue of the Einstein Hilbert action is given by F EH (h) V 2 (1, D 2 h) which has a local expression in dim four: F EH (h) = ϕ 0 (R g ) = ϕ 0 Å e 2h [T (x h )( h) ( h)]g e 2h S g ã.

14 Local curvature functions in dim 4 Modular scalar curvature: K(2u) = 2e u 2 sinh ( ) u 4, u H(2u, 2v) = 4e ( 1 4 ( 3)(u+v) u ( e v/2) + ( e u/2 1 ) e v/2 v + u ) c = K(0)/6. Einstein Hilbert action F EH : uv(u + v) T (2u) = 2K(0)f 1 ( 2u) + H(2u, 2u) = e u 1 u Riemannian case is the commutative limit 2u 4e u u 2., K(0) = 1 2, H(0, 0) = e mh S g = R g xh =0

15 Variational Calculus For any a = a C (M θ ), for ε > 0, denote h ε = h + εa. Set δ a = d/dε ε=0. The gradient at h: grad h F EH is define by the equation: for any a = a C (M θ ), δ a F EH (h) = ϕ 0 (a grad h F EH ). By differentiating δ a Tr(e td2 h ), we obtain: Proposition Let m = dim M, δ a V 2 (1, Dε) 2 = 2 m ˆ 1 V 2 ( e uh/2 ae uh/2 du, D 2 h). 2 1

16 Proposition Let m = dim M and j 0 = (m 2)/2. The gradient grad h F EH is given by where grad h F EH = e 2j0h Ä K EH (x h )( 2 h) + H EH (x (1) h + ( 2j 0 )c(m)e 2j0h S, sinh(s/4) K EH (2s) = 8j 0 K(2s), s sinh((s + t)/4) H EH (2s, 2t) = 8j 0 H(2s, 2t) s + t, x(2) h )( h h)ä g 1

17 On the other hand, we can compute the gradient grad h F EH directly from the local expression ï Å δ a [ϕ 0 (R g )] = δ a ϕ 0 e 2h [T (x h )( h) ( h)]g 1 1 ãò 12 e 2h S g. Theorem (Liu, 16) The internal relations are given by K EH (s) = (T + T )(s), H EH (s, t) = (L + M P Q)(s, t), where T (s) = T ( s)e s and functions L, M, P and Q are given in the next page.

18 Some two-variable functions Recall: f 1 (u) = (e u 1)/u and T = T ( s)e s. Divided difference Then [u, v]t L(s, t) = 2T (s)f 1 ( (s + t)), T (u) T (v). u v M(s, t) = 2e (s+t) ( e t [t, s](t ) [s, t](t ) ), P(s, t) = 2f 1 ( s)t (t) 2[s + t, t](t ) + 2[s + t, s](t ), Q(s, t) = 2f 1 ( s) T (t) 2[s + t, t]( T ) + 2[s + t, s]( T ) (Lesch): the appearance of 2[s + t, t](t ) + 2[s + t, s](t ) is due to the commutator of commutative and noncommutative derivations: [, T (x h )](ψ).

19 Independence of the background metric Let g be the noncommutative metric in question, g 0 and g 1 are two background metrics (T n -invariant). g = e 2h g 1 = e 2(h+a) g 0 with a = a C (M θ ) commutative and g 1 = e 2a g 0. We can quantize g in g 1 and g 0 : g e h/2 /D g1 e h/2 or g e (h+a)/2 /D g0 e (h+a)/2. Since /D g1 is unitary equivalent to e a/2 /D g0 e a/2 and e (h+a)/2 = e h/2 e a/2 provided [h, a] = 0. Sum up, e (h+a)/2 /D g0 e (h+a)/2 e h/2 /D g1 e h/2.

20 From local expression side In g 1 -calculus, In g 0 -calculus, K(x h )( g1 h) + H(x (1) h K(x h )( g0 (h + a)) + H(x (1) h, x(2) h, x(2) h )( h h) g 1 )( (h + a) (h + a)) g 1 For them to be the same, crossed terms with h a should cancel out with each other. They come from replacing g1 g0 and expanding (h + a) (h + a). Vanishing of the coefficients leads to the following equation (in dim M = 4) H(u, 0) + H(0, u) = 2K(u) Unfortunately, this functional relation follows from those shown in a few slides before. 1 0