M ETRIC , TORSION AND MINIMAL OPERATORS ON NONCOMMUTATIVE TORI . Andrzej Sitarz Institute of Physics Institute of Mathematics, Jagiellonian University Polish Academy of Sciences Kraków Warsaw 19.08.2014, Warsaw
1 M ETRIC IN N ONCOMMUTATIVE G EOMETRY A proposition: spectral geometry a la Connes Measuring geometry through spectral data The ubiquitous heat kernel What is the metric ? 2 T HE N ONCOMMUTATIVE T ORI The usual Dirac and its conformal rescaling The asymmetric noncommutative torus Higher dimensional cases and minimality Dirac operators on principal U ( 1 ) bundles 3 C ONCLUSIONS
S PECTRAL TRIPLES D EFINITION : THE SPECTRAL TRIPLE Algebra A , its faithful representation π on a Hilbert space H , a selfadjoint operator D , satisfying several conditions: ∀ a ∈ A [ D , π ( a )] ∈ B ( H ) , D − 1 is compact 1 even ST: ∃ γ ∈ A ′ : γ 2 = 1 , γ = γ † , γ D + D γ = 0, 2 ∃ J , antilinear J 2 = ± 1 , JJ † = 1 3 J γ = ± γ J , JD = ± DJ , [ J π ( a ) J , π ( b )] = 0 , [[ D , a ] , J π ( b ) J ] = 0 ( D : first order differential operator) 4 ...+ conditions of „analysis” type 5 T HEOREM [C ONNES ] If A = C ∞ ( M ) , M a spin Riemannian compact manifold, H = L 2 ( S ) (sections of spinor bundle) and D the Dirac operator on M then to ( A , H , D ) is a spectral triple (with a real structure).
E XAMPLES OF SPECTRAL GEOMETRIES The Noncommutative Torus: UV = e 2 π i θ VU Dirac operator the same as on the torus [Connes] Finite matrix algebras ( M n ( C ) ⊕ M k ( C ) ⊕ · · · Dirac operator is a finite hermitian matrix [AS & Paschke, Krajewski] Quantum spaces ( q -deformations of spheres) Interesting Dirac operators [Dabrowski, AS, Landi, Varilly, et al] Moyal deformation [ x µ , x ν ] = θ µν The usual Dirac [Gracia-Bondia et al] κ -deformation [ x 0 , x i ] = 1 κ x i Doubly Special Relativity [Matassa] H OW TO CONSTRUCT THEM ? There is so far no general method. There are very few examples.
G EOMETRY FROM SPECTRAL DATA (1) T HE SPECTRAL PROPERTIES OF THE D IRAC OPERATOR Classically it is known that the spectrum of the Dirac is discrete (separate eigenvalues), with finite multiplicities and no point of convergence apart form ∞ . Roughly speaking - spectrum of Dirac squared is like the spectrum of Laplace operator. The properties do not change if we modify the operator (change the metric, add connections, torsion etc etc.) S O ONE CAN COMPUTE SOME SPECTRAL FUNCTIONALS � � f ( D 2 ) S ( D , f ) = tr , where f is a suitable function such that the expression makes sense and tr is the usual trace on the Hilbert space.
G EOMETRY FROM SPECTRAL DATA (2) T HE EXOTIC TRACE Although the usual trace does not extend to the operators like Dirac (or its powers) there are some exotic traces which might be interpreted as regularized traces (something of the form of ζ -function regularization). T HE SPECTRAL FUNCTIONAL (2) Using this exotic traces we can postulate the spectral functional to be (for example) � � Λ n − D − n , S ( D , Λ) = n � where is that exotic trace. For most n that would be 0 but some terms would be nonzero.
H EAT - KERNEL ASYMPTOTICS T HEOREM (G ILKEY ) For a Riemannian n-dimensional manifold M, with a metric compatible connection, ∇ , on a vector bundle E over M and a Laplace-type operator is of the form: L = − g ab ∇ a ∇ b − Z , we have: � 1 � k − n n 2 � tr e − tL = � a [ k ] ( x ) + o ( t ) . 4 π t M k = 0 where a [ k ] ( x ) are functions on M (De Witt–Seeley–Gilkey).
H EAT - KERNEL ASYMPTOTICS a [ 2 ] = tr ( − 1 a [ 0 ] = rank ( E ) , a [ 1 ] = 0 , 6 R + Z ) . where R is the scalar curvature. In case the genuine or minimal scalar Laplace operator we have: a [ 2 ] = − 1 6 R , so from the first two terms we recover volume and the (integrated) curvature. R EMARK First term gives nothing else but the so-called Weyl’s theorem about the growth of eigenvalues of the Laplace operator.
S O , WHAT IS THE METRIC ? W HERE IS THE METRIC HIDDEN ? The Dirac operator encodes the metric: d ( p , q ) = sup | f ( p ) − f ( q ) | , || [ D , f ] ||≤ 1 WARNING: easy but not computable ! W HAT WE CAN COMPUTE ? We can compute the volume, the volume functional, the norms of various objects: √ gf ) � � − f | D | − d , f �→ ( M We can compute the curvature and the curvature funcional: √ gR ( g ) f ) � � − f | D | − d + 2 , f �→ ( M
T HE QUESTION ! A LL D IRACS ? What is the space of all possible Dirac operators ? so far, there is no definitive answer generalize what is first order differential operator [ D , a ] bounded is not enough ? D IFFERENTIAL AND PSEUDODIFFERENTIAL OPERATORS The metric is equivalently given by the principal symbol of the Laplace operator but it is a second order operator. Its naive square root is a first order pseudodifferential operator only and √ [ ∆ , a ] is still bounded.
T HE PROBLEM : I F WE HAVE A FAMILY OF D IRAC OPERATORS ... Then: how can we identify the metric ? how can we compute some geometric quantities (like curvature) ? how do we identify minimal operators (like the true Laplace operator ? F OR EXAMPLE : Classically, all of the operators written below, for a function, h > 0, give the same metric: 1 1 2 ( h 4 D 2 + D 2 h 4 ) . 2 ( h 2 D + Dh 2 ) 2 , ( hDh ) 2 , h 2 D 2 h 2 , but which one is the genuine (spinor) Laplace operator ??
T HE NONCOMMUTATIVE PROBLEM : H OW TO CREATE A FAMILY OF L APLACE - TYPE OPERATORS ? Take a real spectral triple with a Dirac operator D take h > 0 in the commutant of the algebra: h ∈ J A J , consider operators: 1 1 2 ( h 4 D 2 + D 2 h 4 ) . 2 ( h 2 D + Dh 2 ) 2 , ( hDh ) 2 , h 2 D 2 h 2 , compute the spectral functionals using the generalization of Wodzicki residue: � − P = Res s = 0 ( tr P | D | s ) . try to identify the geometric meaning of them T HE PROBLEM There is no way to identify torsion or torsion-like objects through some other means.
A PERFECT EXAMPLE : NC T ORUS Noncommutative Torus (Tori): C ∞ ( T n Θ ) , is an algebra, which is generated by two ( n ) unitary operators, which commute up to a scalar phase: U i U l = e 2 π i θ kl U l U k , where θ kl is a real antisymmetric matrix (for n = 2 i , l = 1 , 2). Let t be the trace on C ∞ ( T n K ∈ Z n a K U k � � � Θ ) , t := a 0 and H t be the GNS Hilbert space obtained by completion of C ∞ ( T n Θ ) with respect of the norm induced by the scalar product � a , b � := t ( a ∗ b ) . On H t we consider the left regular representation of C ∞ ( T n Θ ) by bounded operators. Let δ µ , µ ∈ { 1 , . . . , n } , be the n (pairwise commuting) canonical derivations, defined by δ µ ( U K ) := K µ U K .
S PECTRAL TRIPLE ON NC T ORI ( N - DIMENSIONAL ) Θ ) acting on H := H t ⊗ C 2 m with n = 2 m Let A Θ := C ∞ ( T n or n = 2 m + 1, Each element of A Θ is represented on H as L ( a ) ⊗ 1 2 m where L (resp. R ) is the left (resp. right) multiplication. The Tomita conjugation J 0 ( a ) := a ∗ satisfies [ J 0 , δ µ ] = 0 and we define J := J 0 ⊗ C 0 where C 0 is an operator on C 2 m . The Dirac operator is given by D := − i δ µ ⊗ γ µ , And it has been shown that this is (basically) the unique equivariant Dirac operator on the noncommutative torus. What about nonequivariant Dirac operators ?
T HEOREM There exists a family of conformally rescaled Dirac operators on the noncommutative 2 -torus for which the Gauss-Bonnet formula holds, that is ζ D ( 0 ) = 0 , where ζ D ( z ) = Tr ( | D | z ) . √ gR ( g ) = 0 . � Classically this means that: T 2 First family of operators of the type (and conformally rescaled Laplace operators) h 2 D 2 h 2 , D h = hDh , where h ∈ JC ∞ ( T 2 Θ ) J , so it is in the commutant, h > 0, was introduced by Connes and Tretkoff. 4-dimensional version was studied by Fatzizadeh + Khalkhali and AS. n -dimensional version is possible
A SYMMETRIC TORUS Take a torus with the metric dx 2 + k − 2 ( x , y ) dy 2 (that is, for instance the usual „round" torus embedded in R 3 which has k − 1 = c + cos y ). Torus embedded in R 3 Asymmetric torus in R 3 The scalar curvature of the torus with such metric reads R = 2 k − 1 ∂ 2 x ( k ) − 4 k − 2 ( ∂ x ( k )) 2 .
A SYMMETRIC TORUS The Dirac operator is: D = − i σ 1 δ 1 − i σ 2 � k δ 2 + 1 � 2 δ 2 ( k ) , T HEOREM (L.D ABROWSKI +AS) The scalar curvature functional for the asymmetric torus is: √ gR = F 11 ( δ 1 ( k ) , δ 1 ( k )) + F ′ 11 ( δ 1 ( k ) 2 ) + F 22 ( δ 2 ( k ) , δ 2 ( k )) + F ′ 22 ( δ 2 ( k ) 2 ) + F 1 ( δ 11 ( k )) + F 2 ( δ 22 ( k )) ,
where ( 2 s 2 + 4 st + 4 s + 3 + 8 t + 3 t 2 ) F 11 ( s , t ) = − 2 π , 3 k 3 ( t + 1 ) 3 ( s + 1 )( s + t ) 11 ( s ) = 4 π 1 F ′ ( s + 1 ) 3 , 3 k 3 ( t 2 − 6 t + 1 ) F 22 ( s , t ) = π , ( t + 1 ) 3 2 k ( s 2 − 6 s + 1 ) 2 ( s ) = − π F ′ , ( s + 1 ) 3 2 k and F 1 ( s ) = 2 π 1 ( s + 1 ) 2 . 3 k 2 F 2 ( s ) = 0 . and its trace vanishes.
First, the square of D reads D 2 = � ( δ 1 ) 2 + k 2 ( δ 1 ) 2 � � � 3 2 k δ 2 ( k ) + 1 2 δ 2 ( k ) k + i σ 3 δ 1 ( k ) + δ 2 � 4 ( δ 2 ( k )) 2 + 1 � 1 2 i σ 3 δ 12 ( k ) + 1 + 2 k δ 22 ( k ) . and its symbol is σ ( D 2 ) = a 0 + a 1 + a 2 , where � � ξ 2 1 + k 2 ξ 2 a 0 = 2 � � 3 2 k δ 2 ( k ) + 1 2 δ 2 ( k ) k + i σ 3 δ 1 ( k ) a 1 = ξ 2 � 4 ( δ 2 ( k )) 2 + 1 � 1 2 i σ 3 δ 12 ( k ) + 1 a 2 = 2 k δ 22 ( k ) .
Recommend
More recommend
