NONINTEGER FLUXES, DOLBEAULT COMPLEXES, AND SUPERSYMMETRIC QUANTUM MECHANICS based on [1104.3986] and [1105.3935] Hannover, August 1, 2011
WHAT IS WRONG WITH NONINTEGER FLUX? Quantization of Dirac monopole charge (eq = n/2) follows from 1. quantization of the angular momentum of the qe - system 2. Uniqueness of the electron wave function near the Dirac string: Ψ(x) Ψ(x)e 4iπeq after going around. Consider the 2-dim motion over S 2. Dirac string a point x 0. If Ψ(x 0 ) = 0, no contradiction!
For a mathematician A fiber bundle on S 2 implies Two overlapping maps {x µ } and {x µ} The relation A µ = Ω 1 (x)[a µ + i µ ]Ω(x) with Ω(x) well defined in the overlap. Topological charge (= π 1 (gauge group)) coincides with the flux.
Let us be STUPID! and CONSIDER /D and H = /D 2 on S 2 \{ } with a noninteger flux. One FINDS H is Hermitian on the Hilbert space involving nonsingular on S 2 functions. For certain ψ H, /Dψ does not belong to H anymore breaking of supersymmetry. Witten index = magnetic flux. Gauge field cannot be deformed to zero.
A related problem: motion over R 2 in the field of a solenoid with fractional flux [Aharonov + Casher, 1979] continuous spectrum regularizing with nonlocal APS boundary conditions. not my subject today.
TECHNICALITIES Consider the chiral (0+1) superfields W = w+ 2θψ iθ θẇ, W = w 2 θ ψ +iθ θ w ( DW = D W = 0). and the action S = dtd 2 θ [ ] DW D W 4(1 + W W ) + G( W, W ) 2, G = q 2 ln(1 + W W ). q is the magnetic flux.
Describes the motion over S 2 ds 2 = with the gauge field 2dwd w (1 + ww) 2. A w, w = ( i G, i G), F w w = F ww = 2i G.
SUPERCHARGES [ Q = i(1 + ww)ψ [ Q = i(1 + ww) ψ ] w(1 q) 2(1 + ww) ] w(1 + q) 2(1 + ww) /D(1 σ 3 )/2 /D(1 + σ 3 )/2. and the HAMILTONIAN H = { Q, Q} act on Ψ cov = Ψ F =0 ( w, w) + ψψ F =1 ( w, w) normalized with the measure µ d wdw = g d wdw = d wdw (1 + ww) 2,
In the sector F = 0, H F =0 = (1+ ww) 2 +κ(1+ ww)( w w )+κ 2 ww+κ, with κ = 1 q 2. Spectrum: MONOPOLE HARMONICS [Wu + Yang, 1976] (q) F =0 Ψ mn = e imφ (1 z) m /2 (1 + z) m+2κ /2 P n m, m+2κ (z) (m = 0, ±1,..., n = 0, 1,... ). where Pn α,β (z) are the Jacobi polynomials, P α,β n (z) = 1 2 n n k=0 ( n + α k ) ( n + β n k (orthogonal polynomials on the interval z ( 1, 1) with the weight µ = (1 z) α (1 + z) β ). ) (1 + z) k (z 1) n k. (1)
For integer q: Square integrable eigenfunctions are regular on S 2. The result of the action of Q and Q on an eigenfunction is also regular on S 2 supersymmetry of the spectrum The Witten index n (0) F =0 n(0) F =1 of this system is equal to q. For fractional q: Square integrable Hilbert space non-hermiticity of the Hamiltonian Hilbert space of regular functions H is Hermitian. For some such functions QΨ or QΨ is singular breaking of supersymmetry.
γ m=0 m= 2 m= 1 m=0 m=1 m= 1 1 1 1 2 q 1 m=0 m=1 Figure 1: The eigenstates of H F =0 for different q. Solid lines regular functions. Dashed lines singular but square integrable functions. γ is the asymptotic behavior, Ψ w γ..
An ancient elementary example [Shifman + Vainshtein + A.S., 1988] H = p2 2 + [W (x)] 2 2 + W (x) 2 [ ψ, ψ] with W = ωx + 1/x. is In the bosonic sector F = 0, the Hamiltonian H B = p2 2 + ω2 x 2 2 3ω 2 The ground state Ψ 0 (x) exp{ ω 2 x 2 /2} has the negative energy E = ω. It does not belong to the domain of Q p + iw (x).
DOLBEAULT WONDERS AND MYSTERIES de Rahm complex Hilbert space of p-forms. d - exterior derivative d = d - its conjugate d 2 = (d ) 2 = 0, {d, d } = cov Large Dolbeault complex Hilbert space of (p, q)-forms. Holomorphic exterior derivatives, and their conjugates. for Kähler manifolds, 2 = 2 = ( ) 2 = ( ) 2 = {, } = {, } = 0, {, } = {, } = cov
δ (0,0) (1,0) (0,1) δ (2,0) (1,1) (0,2) (2,1) (1,2) δ (2,2) δ Figure 2: DOLBEAULT KAHLER DIAMOND describes d 4 θk( W j, W j ) [Zumino, 1984]. A restricted Dolbeault complex acting on the space of (p, 0) - forms and involving only and can be defined for any complex manifold. {, } Dolbeault {, } = anti Dolbeault cov.
MOTIVATION An SQM model describing the restricted Dolbeault complex [E.Ivanov + A.S., 2010] L = 1 [g µν ẋ µ ẋ ν + ig µν ψ µ ψ ν 16 ] 2 αc µνβ ψ α ψ µ ψ ν ψ β where C µνβ is some special torsion and ψ ν = ψ ν + Γ ν αβẋα ψ β (with Γ ν αβ involving this torsion). 4-fermion term makes it difficult a direct calculation of the index. +A µ ẋ µ i 2 F µνψ µ ψ ν
in some mathematical sources (including EGH) one can find d λ α /2π I = e λ α/2π 1, α=1 where λ α are eigenvalues of the antisymmetric matrix R µν = 1 2 R µνρσ dx ρ dx σ. in the Kähler case, R µν R k j = R k j p l dz p d z l. Symmetric polynomials of λ α are expressed into the products of Tr{R m }. Makes no sense in the non-kähler case!
One should instead untwist the torsion (a smooth deformation!) and reduce the problem to evaluating the index of the standard torsionless Dirac operator... CORRECT FORMULA cf. [Bismut, 1989 ] I Dolbeault = ( d α=1 ) λ α /4π sinh λ α /4π exp { } 2i π ( kw j ) dz j d z k with W = (1/4) ln det h. ( ) is even in λ α and is expressed into Tr{R 2m }.
An AMUSING toy: S 4 \{ } In contrast to S 2, S 4 is not a complex manifold. S 2 can be covered by 2 maps with ds 2 = 2dwd w/(1 + ww) 2, ds 2 = 2dw d w /(1 + w w ) 2 and w = 1/w. Holomorphic gluing. For S 4, one can write ds 2 = 2dwj d w j (1 + w k w k ), 2 ds2 = 2dw j d w j (1 + w k w k ), 2 but the gluing function w j = w j /( w k w k ) is not holomorphic. A non-integrable almost complex structure for S 6.
SQM description of Dolbeault on S 4 \{ }. Supercharges Q = i(1 + ww)ψ j j + iψ j ψ k ψj w k, Q = i ψ j [ (1 + ww) j 2w j ] + i ψj ψk ψ j w k. Hamiltonian in the F = 0 sector H F =0 = (1 + ww) 2 j j + 2(1 + ww)w j j. To compare with S 4 = (1+ ww) 2 j j +(1+ ww)(w j j + w j j ).
The RESULTS: One cannot define the Dolbeault complex on S 4 \{ } on the space of nonsingular on S 4 functions: the Hamiltonian is not Hermitian. One can define the Dolbeault complex on S 4 \{ } on the space of square integrable functions. No problems with supersymmetry! The (square integrable) zero modes of the Dolbeault laplacian are Ψ = 1 and Ψ = w 1,2. This gives IS Dolbeault 4 \{ } = 1 + 2 = 3. No idea of an integral from which it can be obtained. I wrong S = 2. 4 The same for S 6 \{ }. 1+3 + 6 = 10 square integrable zero modes: Ψ = 1, w 1,2,3, w j w k. I wrong S = 9/2 6 for higher-dimensional spheres S2d \{ } I = C d 1 2d 1
TECHICALITIES-2 Ansatz for the eigenfunctions of H Ψ ms = S ms F ( ww), with S ms (m = 0, ±1,... ; s = 0, 1,...) being mutually orthogonal tensor structures, annihilated by j j S 00 = 1, S 01 = w j w k ww 2 δ jk, S 10 = w j, S 11 = w i w j w k ww 3 (w iδ jk + w j δ ik ), S 1,0 = w j,... (m - angular momentum); S ms has 2s+ m +1 independent components (and the corresponding 2s + m + 1 degeneracy in the spectrum).
Introduce z = cos θ = 1 ww 1 + ww The coefficients of S ms satisfy the spectral equations (z 2 1)F (z) + 2(2z + m + 2s)F 4(m + s) (z) + F (z) 1 + z = λf (z), m 0, (z 2 1)F (z) + 2(2z + m + 2s)F (z) + 4s F (z) 1 + z = λf (z), m 0, Their solutions are expressed into Jacobi polynomials.