D-bar Operators in Commutative and Noncommutative Domain

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1 D-bar Operators in Commutative and Noncommutative Domains University of Oklahoma October 19, 2013

2 Introduction and Relavent Papers Atiyah, M. F., Patodi, V. K. and Singer I. M., Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Phil. Soc., 77, 43-69, 1975.

3 Introduction and Relavent Papers Atiyah, M. F., Patodi, V. K. and Singer I. M., Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Phil. Soc., 77, 43-69, Carey, A. L., Klimek, S. and Wojciechowski, K. P., Dirac operators on noncommutative manifolds with boundary, Lett. Math. Phys. 93, , 2010.

4 Commutative Disk and Annulus

5 Spaces and Operator Let D w + = {z C : z w + } and A w,w + = {z C : 0 < w z w + }.

6 Spaces and Operator Let D w + = {z C : z w + } and A w,w + = {z C : 0 < w z w + }. Notice D w + = S 1 and A w,w + = S 1 S 1.

7 Spaces and Operator Let D w + = {z C : z w + } and A w,w + = {z C : 0 < w z w + }. Notice D w + = S 1 and A w,w + = S 1 S 1. Let D = z be the operator acting on the space of H 1 functions.

8 Two Short Exact Sequences Let σf (ϕ) = f (w + e iϕ ) for z = re iϕ.

9 Two Short Exact Sequences Let σf (ϕ) = f (w + e iϕ ) for z = re iϕ. 0 C 0 (D w +) C(D w +) σ C(S 1 ) 0

10 Two Short Exact Sequences Let σf (ϕ) = f (w + e iϕ ) for z = re iϕ. 0 C 0 (D w +) C(D w +) σ C(S 1 ) 0 Let σ ± f (ϕ) = f (w ± e iϕ ).

11 Two Short Exact Sequences Let σf (ϕ) = f (w + e iϕ ) for z = re iϕ. 0 C 0 (D w +) C(D w +) σ C(S 1 ) 0 Let σ ± f (ϕ) = f (w ± e iϕ ). 0 C 0 (A w,w +) C(A w,w +) σ+ σ C(S 1 ) C(S 1 ) 0

12 APS Setup Let M be a closed manifold with boundary Y and let D be a Dirac operator defined on M.

13 APS Setup Let M be a closed manifold with boundary Y and let D be a Dirac operator defined on M. Let M have a product structure near the boundary so that an infinite cylinder can be attached.

14 APS Setup Let M be a closed manifold with boundary Y and let D be a Dirac operator defined on M. Let M have a product structure near the boundary so that an infinite cylinder can be attached. Let D have a special decomposition structure so that it extends naturally to the infinite cylinder.

15 APS Boundary Condition Study D with domain: F H 1 (M)

16 APS Boundary Condition Study D with domain: F H 1 (M) There is a F ext H 1 loc (cylinder) such that DF ext = 0, F ext Y = F Y and F ext L 2 (cylinder)

17 Boundary Conditions Let D D = D and D A = D where

18 Boundary Conditions Let D D = D and D A = D where dom(d D ) consists of a H 1 (D w +) such that there is a ext H 1 loc (C \ D w +) such that aext S 1 = a S 1, Da ext = 0, a ext L 2 (C \ D w +)

19 Boundary Conditions Let D D = D and D A = D where dom(d D ) consists of a H 1 (D w +) such that there is a ext H 1 loc (C \ D w +) such that aext S 1 = a S 1, Da ext = 0, a ext L 2 (C \ D w +) and dom(d A ) consists of a H 1 (A w,w +) such that there is a ext Hloc 1 (C \ A w,w +) such that aext S = a 1 S 1 S 1 S 1, Da ext = 0, a ext L 2 (C \ A w,w +).

20 Boundary Conditions Let D D = D and D A = D where dom(d D ) consists of a H 1 (D w +) such that there is a ext H 1 loc (C \ D w +) such that aext S 1 = a S 1, Da ext = 0, a ext L 2 (C \ D w +) and dom(d A ) consists of a H 1 (A w,w +) such that there is a ext Hloc 1 (C \ A w,w +) such that aext S = a 1 S 1 S 1 S 1, Da ext = 0, a ext L 2 (C \ A w,w +). D D and D A have parametrices, i.e. they are almost invertible.

21 A Fourier Series and Boundary Conditions Equivalence a = e inϕ f n (r) + g n (r)e inϕ n=0 n=1

22 A Fourier Series and Boundary Conditions Equivalence a = e inϕ f n (r) + g n (r)e inϕ n=0 n=1 Let a dom(d D ), then f n (w + ) = 0 for n 0.

23 A Fourier Series and Boundary Conditions Equivalence a = e inϕ f n (r) + g n (r)e inϕ n=0 n=1 Let a dom(d D ), then f n (w + ) = 0 for n 0. Let a dom(d A ), then f n (w + ) = 0 for n 0 and g n (w ) = 0 for n 1.

24 Dirac Operator in Polar Form and Parametrix Decomposition D = eiϕ 2 ( r + i ) r ϕ

25 Dirac Operator in Polar Form and Parametrix Decomposition Qa = D = eiϕ 2 w + e inϕ n=0 + ( r + i ) r ϕ r f n+1 (ρ) r n 1 ρ n dρ r e inϕ g n 1 (ρ) ρn 1 w r n dρ n=1

26 Results Theorem The operators D D and D A are unbounded Fredholm operators. Moreover their respective parametrices Q D and Q A are compact operators. This also means these are elliptic boundary value problems.

27 Quantum Disk

28 Gelfand-Naimark Theorem Let X compact topological space and C(X ) the continuous functions on X. Can associate C(X ) with X

29 Gelfand-Naimark Theorem Let X compact topological space and C(X ) the continuous functions on X. Can associate C(X ) with X C(X ) commutative C algebra with unit

30 Gelfand-Naimark Theorem Let X compact topological space and C(X ) the continuous functions on X. Can associate C(X ) with X C(X ) commutative C algebra with unit GN says if A is a commutative C algebra with unit, then there is a X, compact topological space such that A = C(X )

31 Gelfand-Naimark Theorem Let X compact topological space and C(X ) the continuous functions on X. Can associate C(X ) with X C(X ) commutative C algebra with unit GN says if A is a commutative C algebra with unit, then there is a X, compact topological space such that A = C(X ) We think of a noncommutative (quantum) space as a noncommutative C algebra.

32 Some Weights Let {e k } be the canonical basis for l 2 (N).

33 Some Weights Let {e k } be the canonical basis for l 2 (N). Define U W e k = w(k)e k+1 where {w(k)} k N is an increasing sequence of positive real numbers such that exists. w + := lim k w(k)

34 The Quantum Disk and a Short Exact Sequence Let C (U W ) be the C algebra generated by U W.

35 The Quantum Disk and a Short Exact Sequence Let C (U W ) be the C algebra generated by U W. 0 K C (U W ) σ C(S 1 ) 0

36 The Quantum Disk and a Short Exact Sequence Let C (U W ) be the C algebra generated by U W. 0 K C (U W ) σ C(S 1 ) 0 σ(u) = e iϕ, σ(u ) = e iϕ, σ(compact) = 0

37 The Quantum Disk and a Short Exact Sequence Let C (U W ) be the C algebra generated by U W. 0 K C (U W ) σ C(S 1 ) 0 σ(u) = e iϕ, σ(u ) = e iϕ, σ(compact) = 0 This C algebra is the quantum disk.

38 Informal Idea About Noncommutative Spaces Classical D C(D) C algebra generated by z and z Quantum D q C (U W ), generated by unilateral shift

39 Informal Idea About Noncommutative Spaces Classical D C(D) C algebra generated by z and z A C(A) C algebra generated by z and z Quantum D q C (U W ), generated by unilateral shift A q C (U W ), generated by bilateral shift

40 A Formal Series I Ke k = ke k, and let a (n) (k) be an inversely summable sequence whose sum goes to zero at n.

41 A Formal Series I Ke k = ke k, and let a (n) (k) be an inversely summable sequence whose sum goes to zero at n. Let a C (U W ) define a series := U n f n (K) + g n (K)(U ) n n=0 n=1

42 A Formal Series I Ke k = ke k, and let a (n) (k) be an inversely summable sequence whose sum goes to zero at n. Let a C (U W ) define a series := U n f n (K) + g n (K)(U ) n n=0 n=1 f n (k) = e k, (U ) n ae k, g n (k) = e k, au n e k

43 A Formal Series II a series 2 = k=0 n=0 1 a (n) (k) f n(k) 2 + k=0 n=1 1 a (n) (k) g n(k) 2

44 A Formal Series II a series 2 = k=0 n=0 1 a (n) (k) f n(k) 2 + k=0 n=1 1 a (n) (k) g n(k) 2 This series looks very similar to the Fourier series for the classical case. Think of k as the discretization of the radial variable r where we divide up the unit interval into infinitely many subintervals so the 1/a (n) (k) appear as the differential term in the integral for the norm.

45 Hilbert Space Let H be the Hilbert space consisting of the formal series a series such that a series is finite.

46 Hilbert Space Let H be the Hilbert space consisting of the formal series a series such that a series is finite. Proposition If a C (U W ), then a series converges to a in H and moreover C (U W ) is dense in H.

47 A Commutator and Trace Define S := [U W, U W ].

48 A Commutator and Trace Define S := [U W, U W ]. S is hyponormal, injective, and trace class with tr S = (w + ) 2.

49 A Commutator and Trace Define S := [U W, U W ]. S is hyponormal, injective, and trace class with tr S = (w + ) 2. S is also invertible with unbounded inverse.

50 A Commutator and Trace Define S := [UW, U W ]. S is hyponormal, injective, and trace class with tr S = (w + ) 2. S is also invertible with unbounded inverse. Set a (n) (k) = S 1/2 (k)s 1/2 (k + n)

51 The Operator Da = S 1/2 [a, U W ]S 1/2

52 The Operator Da = S 1/2 [a, U W ]S 1/2 dom(d) = {a H : Da H}

53 Boundary Conditions Let a = U n f n (K) + g n (K)(U ) n n=0 n=1

54 Boundary Conditions Let a = U n f n (K) + g n (K)(U ) n n=0 n=1 If a dom(d), then f n ( ) = 0 for n 0.

55 Boundary Conditions Let a = U n f n (K) + g n (K)(U ) n n=0 n=1 If a dom(d), then f n ( ) = 0 for n 0. D also has a parametrix Q.

56 Unbounded Jacobi Operators ( ) A (n) h(k) = a (n) (k) h(k) c (n) (k 1)h(k 1) ( ) A (n) h(k) = a (n+1) (k) h(k) c (n) (k)h(k + 1)

57 Unbounded Jacobi Operators ( ) A (n) h(k) = a (n) (k) h(k) c (n) (k 1)h(k 1) ( ) A (n) h(k) = a (n+1) (k) h(k) c (n) (k)h(k + 1) where c (n) (k) = w(k)/w(k + n + 1)

58 Operator Decomposition Da = U n+1 A (n) W (n) f n (K) + n=0 W (n 1) A (n 1) g n (K)(U ) n 1 n=1

59 The Parametrix Qa = m=0 n=1 U n k + i=k j=1 n 1 i=0 j=0 n w(k + j) w(i + j) S 1/2 (i)s 1/2 (i + n + 1) f n+1 (i) w(k + n) w(i + j) w(k + j) S 1/2 (i)s 1/2 (i + n 1) g n 1 (i) (U ) n w(i + n 1)

60 Results Theorem The operator D is an unbounded Fredholm operator. Moreover it s parametrix Q is a compact operator and hence this is an elliptic boundary value problem.

61 Bibliography Atiyah, M. F., Patodi, V. K. and Singer I. M., Spectral asymmetry and Riemannian geometry I. Math. Proc. Camb. Phil. Soc., 77, 43-69, Klimek, S. and McBride, M., D-bar Operators on Quantum Domains. Math. Phys. Anal. Geom., 13, , 2010.

62 The End Thank You

Dirac operators on the solid torus with global boundary conditio

Dirac operators on the solid torus with global boundary conditions University of Oklahoma October 27, 2013 and Relavent Papers Atiyah, M. F., Patodi, V. K. and Singer I. M., Spectral asymmetry and Riemannian