Dirac operators on the solid torus with global boundary conditio

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1 Dirac operators on the solid torus with global boundary conditions University of Oklahoma October 27, 2013

2 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 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 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, , Mishchenko, A. V. and Sitenko, Yu, Spectral Boundary Conditions and Index Theorem for Two-Dimensional Compact Manifold with Boundary, Annals of Physics, 218, , 1992.

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6 Setup, Hilbert Space and a Short Exact Sequence Let D = {z C : z 1}, S 1 = {e iθ T 2 the 2-dimensional torus : 0 θ 2π}, and

7 Setup, Hilbert Space and a Short Exact Sequence Let D = {z C : z 1}, S 1 = {e iθ T 2 the 2-dimensional torus : 0 θ 2π}, and ST 2 the solid torus, ST 2 = D S 1 C S 1, and T 2 boundary of ST 2.

8 Setup, Hilbert Space and a Short Exact Sequence Let D = {z C : z 1}, S 1 = {e iθ T 2 the 2-dimensional torus : 0 θ 2π}, and ST 2 the solid torus, ST 2 = D S 1 C S 1, and T 2 boundary of ST 2. H = L 2 (ST 2, C 2 ) = L 2 (ST 2 ) C 2

9 Setup, Hilbert Space and a Short Exact Sequence Let D = {z C : z 1}, S 1 = {e iθ T 2 the 2-dimensional torus : 0 θ 2π}, and ST 2 the solid torus, ST 2 = D S 1 C S 1, and T 2 boundary of ST 2. H = L 2 (ST 2, C 2 ) = L 2 (ST 2 ) C 2 Inner product of H denoted F, G for F, G H.

10 Setup, Hilbert Space and a Short Exact Sequence Let D = {z C : z 1}, S 1 = {e iθ T 2 the 2-dimensional torus : 0 θ 2π}, and ST 2 the solid torus, ST 2 = D S 1 C S 1, and T 2 boundary of ST 2. H = L 2 (ST 2, C 2 ) = L 2 (ST 2 ) C 2 Inner product of H denoted F, G for F, G H. 0 C 0 (D) C(S 1 ) C(ST 2 ) C(S 1 ) C(S 1 ) 0

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

12 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.

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. Let D have a special decomposition structure so that it extends naturally to the infinite cylinder.

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

15 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)

16 Dirac Operator and Boundary Conditions D = ( 1 i θ 2 z 2 z 1 i θ )

17 Dirac Operator and Boundary Conditions D = ( 1 i θ 2 z 2 z 1 i θ ) dom(d) = {F H 1 (ST 2 ) C 2 : F ext H 1 loc ((C S 1 )\ST 2 ) C 2 }

18 Dirac Operator and Boundary Conditions D = ( 1 i θ 2 z 2 z 1 i θ ) dom(d) = {F H 1 (ST 2 ) C 2 : F ext H 1 loc ((C S 1 )\ST 2 ) C 2 } (1) F ext T 2 = F T 2, (2) DF ext = 0, (3) F ext L 2 ((C S 1 )\ST 2 ) C 2

19 Fourier Decomposition z = re iϕ for F L 2 (ST 2 ) C 2

20 Fourier Decomposition z = re iϕ for F L 2 (ST 2 ) C 2 F = m,n Z ( fm,n (r) g m,n (r) ) e inϕ+imθ

21 Fourier Decomposition z = re iϕ for F L 2 (ST 2 ) C 2 F = m,n Z ( fm,n (r) g m,n (r) ) e inϕ+imθ F 2 = F, F = 1 m,n Z 0 ( fm,n (r) 2 + g m,n (r) 2) rdr

22 Kernel of D (without boundary conditions) Ker(D) consists of those functions F L 2 (ST 2 \ ({0} S 1 )) C 2 such that

23 Kernel of D (without boundary conditions) Ker(D) consists of those functions F L 2 (ST 2 \ ({0} S 1 )) C 2 such that m 0, n Z : f m,n+1 (r) = m m ( A m,ni n+1 ( m r) + B m,n K n+1 ( m r)) g m,n (r) = A m,n I n ( m r) + B m,n K n ( m r)

24 Kernel of D (without boundary conditions) Ker(D) consists of those functions F L 2 (ST 2 \ ({0} S 1 )) C 2 such that m 0, n Z : f m,n+1 (r) = m m ( A m,ni n+1 ( m r) + B m,n K n+1 ( m r)) g m,n (r) = A m,n I n ( m r) + B m,n K n ( m r) m = 0, n Z: f 0,n (r) = A 0,n r n and g 0,n (r) = B 0,n r n

25 Boundary Condition Equivalence Let F dom(d), then m K n+1 ( m )g m,n (1) mk n ( m )f m,n+1 (1) = 0 m 0, n Z f 0,n (1) = 0 m = 0, n 0 g 0,n (1) = 0 m = 0, n 0

26 Boundary Condition Equivalence Let F dom(d), then m K n+1 ( m )g m,n (1) mk n ( m )f m,n+1 (1) = 0 m 0, n Z f 0,n (1) = 0 m = 0, n 0 g 0,n (1) = 0 m = 0, n 0 Subject to the boundary conditions, Ker(D) = {0} and D = D. Also there is an operator Q such that QD = DQ = I, that is QDF = F for F dom(d) and DQF = F for F L 2 (ST 2 ) C 2, i.e. D is invertible.

27 The Inverse I G = m,n Z ( pm,n (r) q m,n (r) ) e inϕ+imθ

28 The Inverse I QG := m Z\{0},n Z G = m,n Z ( fm,n (r) g m,n (r) ( pm,n (r) q m,n (r) ) e inϕ+imθ ) e inϕ+imθ + ( f0,n (r) g 0,n (r) n Z ) e inϕ

29 The Inverse II for m 0, let K i,j (x, y) = mi i (x)k j (y) f m,n+1 (r) = r 1 r r 0 r 0 K n+1,n ( m r, m ρ) q m,n (ρ)ρdρ K n+1,n+1 ( m r, m ρ)p m,n+1 (ρ)ρdρ K n,n+1 ( m ρ, m r) q m,n (ρ)ρdρ K n+1,n+1 ( m ρ, m r)p m,n+1 (ρ)ρdρ

30 The Inverse III g m,n (r) = + 1 r 1 r r 0 r 0 K n,n ( m r, m ρ)q m,n (ρ)ρdρ K n,n+1 ( m r, m ρ) p m,n+1 (ρ)ρdρ K n,n ( m ρ, m r)q m,n (ρ)ρdρ K n+1,n ( m ρ, m r) p m,n+1 (ρ)ρdρ

31 The Inverse IV for m = 0 f 0,n (r) = g 0,n (r) = r 0 1 ρ n r ρ n r n+1 q 0,n(ρ)ρdρ n 0 r n+1 q 0,n(ρ)ρdρ n < 0 1 r r 0 r n ρ n+1 p 0,n+1(ρ)ρdρ n 0 r n ρ n+1 p 0,n+1(ρ)ρdρ n < 0

32 Results Theorem The inverse Q of the Dirac operator defined by D, is bounded. Moreover Q is a compact operator, this means this is an elliptic boundary value problem.

33 Results Theorem The inverse Q of the Dirac operator defined by D, is bounded. Moreover Q is a compact operator, this means this is an elliptic boundary value problem. Theorem The inverse Q is a p-th Schatten-class operator for p > 3.

34 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, , Klimek, S. and McBride, M., Global boundary conditions for a Dirac operator on the solid torus. Jour. Math. Phys., 52, 1-14, 2011.

35 The End Thank You