Modular curvatures for toric noncommutative manifolds Yang Liu - PowerPoint PPT Presentation

Modular curvatures for toric noncommutative manifolds Yang Liu Ohio State University July 27, 2016 Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 1 / 25

Noncommutative spaces In noncommutative geometry, a geometric space is implemented by a spectral triple ( A , H , D ): The algebra A represents the “coordinate functions” on the underlying space, elements in A are bounded operators on H that do not necessary commute with each other as in quantum physics. D is an self-adjoint unbounded operator on H with the first order condition: all the commutators [ a , D ] are bounded where a ∈ A . A typical example is the spectral triple for Dirac model: + ) , / − ) ⊕ L 2 ( / ( C ∞ ( M ) , L 2 ( / D ) , S S where M is a closed Spin manifold with spinor bundle S = S + ⊕ S − , and / D is the associated Dirac operator. Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 2 / 25

Global geometry General algebraic-topological and analytical tools for global treatment of the usual spaces have been successfully adapted and upgraded to the noncommutative context, such as: K -theory; cyclic cohomology; Morita equivalence; operator-theoretic index theorems; Hopf algebra symmetry, etc. Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 3 / 25

Local Geometry By contrast, the fundamental local geometric concepts, in particular, the notion of intrinsic curvature, which lies at the very core of geometry, has only recently begun to be comprehended via the study of modular geometry on noncommutative two tori. Proposed in A. Connes and H. Moscovici’s recent work(2014) “It is the high frequency behavior of the spectrum of D coupled with the action of the algebra A in H which detects the local curvature of the geometry.” Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 4 / 25

Spectral Geometry On a closed Riemannian manifold M , let ∆ be a Laplacian type operator, the Schwartz kernel of the heat operator operator has the following asymptotic expansion on the diagonal: e − t ∆ ( x , x , t ) ∽ t ց 0 � V j ( x ) t ( j − d ) / 2 , d = dim M . j ≥ 0 The coefficients V j are polynomial functions on in the curvature tensor and its covariant derivatives. For example, let ∆ be the scalar Laplacian, then upto a factor (4 π ) − d / 2 : V 2 ( x , ∆) = 1 6 S ∆ 1 ∆ − 2 | Ric | 2 + 2 | R | 2 ä Ä − 12∆ S ∆ + 5 S 2 V 4 ( x , ∆) = 360 here S ∆ is the scalar curvature function, Ric and R are the Ricci curvature tensor and the full curvature tensor respectively. Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 5 / 25

Scalar curvature functional The diagonal of the heat kernel e − t ∆ ( x , x , t ) does not make sense for our noncomutative spaces. The operator-theoretic counterpart is the trace functional f �→ Tr( fe − t ∆ ) , ∀ f ∈ C ∞ ( M ) . As before, it has an asymptotic expansion as t → 0 f �→ Tr � fe − t ∆ � ∽ t ց 0 � V j ( f , ∆) t ( j − d ) / 2 , d = dim M , f ∈ C ∞ ( M ) . j ≥ 0 Definition If we take the Laplacian operator ∆ as the definition of a “Riemannian metric”. The we will call the functional density R ∈ C ∞ ( M ) of the second heat coefficient functional ˆ f R , f ∈ C ∞ ( M ) V j ( f , ∆) = M as the associated scalar curvature. Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 6 / 25

Guassian Equations Conformal change of metric g ′ = e − h g , the Laplacian operators are linked by ∆ g ′ = e − h ∆ g . the Gaussian curvatures are related by the Guassian equation (2∆ g ( h ) + K g ) e h = K g ′ . Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 7 / 25

Yamabe Equations 4 n − 2 g for some positve Let n = dim M ≥ 3. Conformal change of metric: g u = u function u . The scalar curvatures R g u and R g are related by the Yamabe equation n − 2 n +2 n − 2 , L g u = 4( n − 1) R g u u where n − 2 L g = − ∆ g + 4( n − 1) R g is the conformal Laplacian operator defined on ( M , g ) with n ≥ 3. Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 8 / 25

In dimension four: under the conformal change of metric: g u = u 2 g , the scalar curvatures are related as follows: R g u = − 6 u 1 / 3 (∆ g u ) + u 1 / 3 R g . Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 9 / 25

Conformal change of metric in noncommutative setting In Riemannian geometry, the Hilbert spaces of L 2 ( M , g ) of L 2 -functions depends on the metric g . When a family of metrics is considered, for instance, when studying variation problems, we often choose to fix the Hilbert space. The price to pay is a purturbation of the Laplacian operator. Conformal perturbation of the Laplacian operator Now on our noncommutative spaces, the conformal factor k = e h is implemented by exponentiate a self-adjoint operator h . The resulting operator k is invertible and positive. The new Laplacian, upto a conjugation by k is of the form: ∆ k = k ∆ + lower order terms . Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 10 / 25

Toric manifolds Let M be a smooth manifold and T n ⊂ Diff ( M ). Then C ∞ ( M ) be come a smooth T n -module via the pull-back action: ( U t ( f ))( x ) � f ( t − 1 · x ) , x ∈ M , f ( x ) ∈ C ∞ ( M ) , t ∈ T n . (1) The notation U t stands for “unitary” because later we will assume that the torus acts on M as isometries, then U t admits a unitary extension to L 2 ( M ). The smoothness means that for any fixed f ∈ C ∞ ( M ), the function t �→ U t ( f ) belongs to C ∞ ( T n , C ∞ ( M )). By Fourier theory on T n , any elements in C ∞ ( M ) has a isotypical decomposition: let T n ∼ = R n / 2 π Z n , ˆ � T n e − 2 π ir · t U t ( f ) dt f = f r , f r = (2) r ∈ Z n Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 11 / 25

Deformation of C ∞ ( M ) Given a n × n skew symmetric matrix Θ, we denote a bicharacter: χ Θ : Z n × Z n → S 1 : χ Θ ( r , l ) = � r , Θ l � . We deformed C ∞ ( M ) with respect to Θ, the resulting new algebra is denoted by C ∞ ( M Θ ) = ( C ∞ ( M ) , × Θ ) which is identical to C ∞ ( M ) as a topological vector space with the pointwise multiplication replaced via a twisted convolution: � f × Θ g � f , g ∈ C ∞ ( M ) , χ Θ ( r , l ) f r g l , (3) r , l ∈ Z n where f r , g l are isotypical components of f and g . Since the torus action can be quickly extends to the cotangent bundle T ∗ M , the deformed algebra is defined in a similar way: C ∞ ( T ∗ M Θ ) = ( C ∞ ( T ∗ M ) , × Θ ) Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 12 / 25

Noncommutative two tori Å 0 − θ/ 2 ã Let Θ = , θ ∈ R \ Q . θ/ 2 0 Consider T 2 acts on itself via translations: t · ( e 2 π is 1 , e 2 π is 2 ) = ( e 2 π i ( s 1 − t 1 ) , e 2 π i ( s 2 − t 2 ) ) , t = ( t 1 , t 2 ) ∈ R 2 . Take u = e 2 π is 1 and v = e 2 π is 1 in C ∞ ( T 2 ), thus U t ( u ) = e 2 π it 1 u , U t ( v ) = e 2 π it 2 v . We recover the noncommutative relation which defines the commutative two torus: u × Θ v = e 2 π i θ v × Θ u . The deformed algebra C ∞ ( T 2 Θ ) is called a smooth noncommutative two torus. Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 13 / 25

Noncommtutative four spheres Å 0 − θ/ 2 ã Let Θ = , θ ∈ R \ Q . θ/ 2 0 Let T 2 act on R 5 via rotations on the first four components, namely, Ñ e 2 π it 1 é e 2 π it 2 t = ( t 1 , t 2 ) �→ ∈ SO(5) , 1 the induced action on S 4 gives rise to the noncommutative four sphere C ∞ ( S 4 θ ). Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 14 / 25

Deforming operators Now we assume that the torus acts on M as isometries: T n ⊂ Iso ( M ) and M is a closed Riemannian manifold. Let H = L 2 ( M ). ∀ t ∈ T n , U t extends to a unitary operator on H Observation: the representation C ∞ ( M ) ⊂ B ( H ) of left-multiplication is equivariant: L U t ( f ) = U t L f U − 1 , f ∈ C ∞ ( M ) . t We impose a T n -module on B ( H ) via the adjoint action: Ad t : B ( H ) → B ( H ) : P �→ U t PU − 1 , t ∈ T n . t For any g ∈ C ∞ ( M ), we define the deformed operator π Θ ( L f ) π Θ ( L f )( g ) � � χ Θ ( r , l )( L f ) r g l , r , l ∈ Z n which recovers the left × Θ -multiplication. Yang Liu (Ohio State University) Modular curvatures for toric noncommutative manifolds July 27, 2016 15 / 25