Table Of Content

arXiv:15mm.nnnnn PreprinttypesetinJHEPstyle-HYPERVERSION CERN-PH-TH/2015-002 Wall-crossing made smooth Boris Pioline 5 CERN PH-TH, Case C01600, CERN, CH-1211 Geneva 23, Switzerland 1 0 2 Sorbonne Universités, UPMC Université Paris 6, UMR 7589, F-75005 Paris, France n a J Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, 7 Université Pierre et Marie Curie, 4 place Jussieu, F-75252 Paris cedex 05, France ] e-mail: [email protected] h t - p e Abstract: In D = 4,N = 2 theories on R3,1, the index receives contributions not only from h single-particleBPSstates, countedbytheBPSindices, butalsofrommulti-particlestatesmade [ ofBPSconstituents. Inarecentwork[1], ageneralformulaexpressingtheindexintermsofthe 1 v BPS indices was proposed, which is smooth across walls of marginal stability and reproduces 3 theexpectedsingle-particlecontributions. Inthisnote, Ianalyzethetwo-particlecontributions 4 6 predicted by this formula, and show agreement with the spectral asymmetry of the continuum 1 0 ofscattering statesin thesupersymmetric quantum mechanics oftwonon-relativistic, mutually . 1 non-local dyons. This provides a physical justification for the error function profile used in the 0 mathematics literature on indefinite theta series, and in the physics literature on black hole 5 1 partition functions. : v i X r a Contents 1. Introduction 1 2. Two-particle contribution to the index 3 3. Supersymmetric electron-monopole quantum mechanics 5 4. Discussion 10 A. Robustness of the spectral asymmetry 11 1. Introduction The recent work [1] proposed a general formula for the index I(R,u,C) in four-dimensional field theories on R3,1 with N = 2 supersymmetry. This index can be understood as the partition function on R3 times an Euclidean circle of radius R, with periodic boundary conditions for fermions, chemical potentials C conjugate to the electromagnetic charge γ, and with an insertion of a suitable four-fermion vertex so as to saturate fermionic zero-modes. Equivalently, it can be defined as a trace I(R,u,C) = −1 Tr (−1)2J3(2J )2σ e−2πRH−2πi(cid:104)γ,C(cid:105), (1.1) 2 H(u) 3 γ overthefullHilbertspaceH(u)ofthetheoryonR3 (hereJ istheangularmomentumoperator 3 around a fixed axis, and σ is a charge-dependent sign, satisfying the quadratic refinement γ property σ σ = (−1)(cid:104)γ,γ(cid:48)(cid:105)σ ). γ γ(cid:48) γ+γ(cid:48) Unlike the BPS indices Ω(γ,u), which count single-particle BPS states and exhibit dis- continuities across walls of marginal stability, I(R,u,C) is a smooth function of the Coulomb branch moduli u, away from the loci where additional states become massless. This is possi- ble because I(R,u,C) receives contributions not only from single-particle BPS states, but also from the continuum of multi-particle states. Indeed, while multi-particle states do not saturate the BPS bound M = |Z |, the density of bosonic and fermionic states are not necessarily equal, γ as noted early on in [2, 3] (see [4, 5, 6] for recent discussions in the context of two-dimensional superconformal field theories). Still, only multi-particle states made of BPS constituents can contribute, so one expects that the index can be expressed in terms of the BPS indices Ω(γ,u). In [1], using insight from the study of the hyperkähler metric on the Coulomb branch in the theory compactified down to three dimensions [7], and by analogy with a similar construction – 1 – in the context of the hypermultiplet moduli space in string vacua [8, 9], we proposed a general formula for the index1 I(R,u,C) = −R2 (cid:61)(X¯ΛF )+ R (cid:88)Ω(γ,u)(cid:90) dt (cid:0)t−1Z −tZ¯ (cid:1)log(1−X (t)). (1.2) 2 Λ 16iπ2 t γ γ γ γ (cid:96)γ where (cid:96) are the BPS rays {t ∈ C× : Z /t ∈ iR−} , and X are the solutions to the system of γ γ γ integral equations [7] (cid:34) (cid:35) (cid:88) Ω(γ(cid:48),u) (cid:90) dt(cid:48)t+t(cid:48) X = Xsf exp (cid:104)γ,γ(cid:48)(cid:105) log(1−X (t(cid:48))) , (1.3) γ γ 4πi t(cid:48) t−t(cid:48) γ(cid:48) γ(cid:48) (cid:96)γ(cid:48) with Xsf providing the ‘semi-flat’, large R approximation to X , γ γ Xsf = σ e−πiR(t−1Zγ−tZ¯γ)−2πi(cid:104)γ,C(cid:105). (1.4) γ γ The X ’s are holomorphic functions on the twistor space Z of the Coulomb branch M (R), γ 3 which provide canonical Darboux coordinates for the holomorphic symplectic structure on Z. Theycanalsobeunderstoodasvevsofcertaininfraredlineoperators[11]. InthelimitR → ∞, a formal solution to the system (1.3) is obtained by substituting X → Xsf on the r.h.s. and γ γ iterating. This leads an expansion of the form   (cid:88) (cid:89) (cid:89) Xγ = Xγsf exp (cid:104)αi,αj(cid:105) Ω(αi,u)gT (1.5) T (i,j)∈T1 i∈T0 (cid:80) where T runs over trees decorated by charges α such that γ = α , g are certain iterated i i T contour integrals [7, 12], and Ω(γ,u) = (cid:80) 1 Ω(γ/d,u) are the ‘rational BPS indices’ [13, 14, d|γ d2 15], which arise from expanding the log in (1.3). Substituting in (1.2), one obtains a formal expansion (cid:88) (cid:88) I = I(0) + I(1) + I(2) +... (1.6) γ γ,γ(cid:48) γ γ,γ(cid:48) where I(0) stands for the first term in (1.2), while I(n) , proportional to Ω(γ ,u)...Ω(γ ,u) γ1,...,γn 1 n is interpreted as the contribution of a multi-particle state of charge {γ ,...γ } to the index. 1 n In particular, the one-particle contribution is obtained by replacing X → Xsf in (1.2), leading γ γ to R I(1) = σ Ω(γ,u)|Z |K (2πR|Z |)e−2πi(cid:104)γ,C(cid:105) (1.7) γ 4π2 γ γ 1 γ In [1], we matched this result with the index of a relativistic particle of charge γ and mass |Z |. To define the index, we regulated the infrared divergences by switching on a chemical γ potential θ for the rotations J in the xy plane, restricting the z direction to a finite interval 3 of length L, and removing the regulators as follows, (cid:20)sin2(θ/2) (cid:21) (cid:0) (cid:1) I(1) = 2R lim ∂2 Tr σ e−2πRH+iθJ3−2πi(cid:104)γ,C(cid:105) . (1.8) γ θ→2π θ πL H1(u) γ L→∞ whereH (u)istheone-particleHilbertspace. Thesameregulatorshouldthenbeusedtodefine 1 the full index (1.1). Our aim in this note is to perform a similar check for the two-particle contribution. 1Various refinements of this index have been introduced in [10], but lie beyond the scope of this note. – 2 – 2. Two-particle contribution to the index According to the conjecture (1.2), the contribution of a two-particle state with charges {γ,γ(cid:48)} to the index is obtained by inserting the one-particle approximation to (1.3) in (1.2), I(2) = − R (cid:88)(cid:104)γ,γ(cid:48)(cid:105)Ω¯(γ)Ω¯(γ(cid:48)) (cid:90) dt (cid:90) dt(cid:48)t+t(cid:48) (cid:0)t−1Z −tZ¯ (cid:1)Xsf(t)Xsf(t(cid:48))+(γ ↔ γ(cid:48)) γ,γ(cid:48) 64π3 t t(cid:48) t−t(cid:48) γ γ γ γ(cid:48) γ,γ(cid:48) (cid:96)γ (cid:96)(cid:48)γ (2.1) Definingψγ = argZγ, ψγγ(cid:48) = ψγ−ψγ(cid:48) andchangingtheintegrationvariablestot = ieiψγex++x−, t(cid:48) = ieiψγ(cid:48)ex+−x−, this can be rewritten as ∞ ∞ (cid:90) (cid:90) (cid:18) (cid:19) iR i I(2) = (−1)(cid:104)γ,γ(cid:48)(cid:105)(cid:104)γ,γ(cid:48)(cid:105)Ω¯(γ)Ω¯(γ(cid:48))σ dx dx coth x + ψ γ,γ(cid:48) 16π3 γ+γ(cid:48) + − − 2 γγ(cid:48) −∞ −∞ ×(cid:2)|Zγ|cosh(x+ +x−)+|Zγ(cid:48)|cosh(x+ −x−)(cid:3)e−2πR|Zγ|cosh(x++x−)−2πR|Zγ(cid:48)|cosh(x+−x−)−2πi(cid:104)γ+γ(cid:48),C(cid:105). (2.2) We shall be interested in the behavior of I(2) in the vicinity of the wall of marginal stability γ,γ(cid:48) whereψ → 0, inthelimitR → ∞. Weassumethatγ andγ(cid:48) areprimitivevectorsgenerating γ,γ(cid:48) ¯ the positive cone of BPS states whose central charges align at the wall, so that Ω(γ,u) and Ω¯(γ(cid:48),u) are constant across the wall, and equal to Ω(γ,u) and Ω(γ(cid:48),u). Away from the wall, the integrals over x and x are dominated by saddle points at x = x = 0, producing2 + − + − (cid:18) (cid:19) 1 |Z |+|Z | 1 Iγ(2,γ)(cid:48) ≈ 32π3(−1)(cid:104)γ,γ(cid:48)(cid:105)(cid:104)γ,γ(cid:48)(cid:105)Ω¯(γ)Ω¯(γ(cid:48))σγ+γ(cid:48) |Zγ ||Z γ|(cid:48) cot 2ψγγ(cid:48) e−2πR(|Zγ|+|Zγ(cid:48)|)−2πi(cid:104)γ+γ(cid:48),C(cid:105). γ γ(cid:48) (2.3) However, the saddle point approximation breaks down near the wall where ψ → 0, as the γγ(cid:48) saddle point collides with the pole at x = −iψ . To deal with this, we first perform the − 2 γγ(cid:48) integral over x , which is dominated by a saddle point at + |Z |−|Z | x ∼ γ(cid:48) γ x +O(x2) . (2.4) + |Z |+|Z | − − γ γ(cid:48) In the limit R → ∞, I(2) is well approximated by γ,γ(cid:48) i (cid:113) I(2) ≈ (−1)(cid:104)γ,γ(cid:48)(cid:105)(cid:104)γ,γ(cid:48)(cid:105)Ω¯(γ)Ω¯(γ(cid:48))σ R(|Z |+|Z |) γ,γ(cid:48) 16π3 γ+γ(cid:48) γ γ(cid:48) × (cid:90)∞dx− coth(cid:18)x− + i ψγγ(cid:48)(cid:19)e−2πR(|Zγ|+|Zγ(cid:48)|)−4π|ZRγ|Z|+γ||Z|Zγγ(cid:48)|(cid:48)|x2−−2πi(cid:104)γ+γ(cid:48),C(cid:105). (2.5) 2 −∞ In the limit ψ → 0, we can further approximate coth(x) ∼ 1/x and evaluate the integral γγ(cid:48) using the formula [17, 4.18], valid for α and β real and non-zero, (cid:90) ∞ dz e−β2z2 = iπsgn(α)eα2β2Erfc(|αβ|) (2.6) z −iα −∞ 2The analysis in this section bears some similarities with the one in [16]. – 3 – where Erfc is the complementary error function. Noting that 1 |Z |+|Z |−|Z | ∼ m ψ2 , (2.7) γ γ(cid:48) γ+γ(cid:48) 2 γγ(cid:48) γγ(cid:48) where m = |Zγ||Zγ(cid:48)| is the reduced mass of the two-particle system, we find γγ(cid:48) |Zγ|+|Zγ(cid:48)| 1 (cid:113) I(2) ≈ (−1)(cid:104)γ,γ(cid:48)(cid:105)Ω¯(γ)Ω¯(γ(cid:48))(cid:104)γ,γ(cid:48)(cid:105)σ R(|Z |+|Z |) γ,γ(cid:48) 16π2 γ+γ(cid:48) γ γ(cid:48) (2.8) (cid:16) (cid:17) ×sgn(ψγγ(cid:48))Erfc |ψγγ(cid:48)|(cid:112)πRmγγ(cid:48) e−2πR|Zγ+γ(cid:48)|−2πi(cid:104)γ+γ(cid:48),C(cid:105) The two-particle contribution is discontinuous across the wall: as ψ goes from negative to γγ(cid:48) positive, I(2) jumps by γ,γ(cid:48) 1 (cid:113) ∆Iγ(2,γ)(cid:48) ≈ 8π2(−1)(cid:104)γ,γ(cid:48)(cid:105)Ω¯(γ)Ω¯(γ(cid:48))(cid:104)γ,γ(cid:48)(cid:105)σγ+γ(cid:48) R(|Zγ|+|Zγ(cid:48)|)e−2πR|Zγ+γ(cid:48)|−2πi(cid:104)γ+γ(cid:48),C(cid:105) . (2.9) On the other hand, the one-particle contribution I(1) is also discontinuous across the wall, γ,γ(cid:48) due to the fact that the one-particle index Ω¯(γ +γ(cid:48)) jumps [18]:3 Ω¯(γ +γ(cid:48),u) = Ω¯+(γ +γ(cid:48))−(−1)(cid:104)γ,γ(cid:48)(cid:105)|(cid:104)γ,γ(cid:48)(cid:105)|Ω¯(γ)Ω¯(γ(cid:48))Θ((cid:104)γ,γ(cid:48)(cid:105)ψ ) . (2.10) γγ(cid:48) The first term in (2.10) corresponds to the one-particle index on the side where (cid:104)γ,γ(cid:48)(cid:105)ψ < 0, γγ(cid:48) so that the two states of charge γ and γ(cid:48) cannot form a BPS bound state, while the second term is the contribution of the BPS bound state which exists on the side where (cid:104)γ,γ(cid:48)(cid:105)ψ > 0. γγ(cid:48) Inserting (2.10) in I(1) , and taking the limit R → ∞, we find γ+γ(cid:48) (cid:104) (cid:105) I(1) ≈ Ω¯+(γ +γ(cid:48))−(−1)(cid:104)γ,γ(cid:48)(cid:105)|(cid:104)γ,γ(cid:48)(cid:105)|Ω¯(γ)Ω¯(γ(cid:48))Θ((cid:104)γ,γ(cid:48)(cid:105)ψ ) γ+γ(cid:48) γγ(cid:48) (cid:112) (2.11) R|Z | ×σγ+γ(cid:48) 8π2γ+γ(cid:48) e−2πR|Zγ+γ(cid:48)|−2πi(cid:104)γ+γ(cid:48),C(cid:105) , whose jump exactly compensates (2.9). In fact, neglecting the difference between |Z | and γ+γ(cid:48) |Z |+|Z | under the square root (which cannot be told apart in our approximation), the sum γ γ(cid:48) of (2.8) and (2.11) can be written as (cid:110) I(1) +I(2) ≈ Ω¯+(γ +γ(cid:48)) γ+γ(cid:48) γ,γ(cid:48) − 1(−1)(cid:104)γ,γ(cid:48)(cid:105)|(cid:104)γ,γ(cid:48)(cid:105)|Ω¯(γ)Ω¯(γ(cid:48))(cid:16)1+Erf(cid:16)sgn((cid:104)γ,γ(cid:48)(cid:105))ψ (cid:112)πRm (cid:17)(cid:17)(cid:111) γγ(cid:48) γγ(cid:48) 2 (cid:112) R|Z | ×σγ+γ(cid:48) 8π2γ+γ(cid:48) e−2πR|Zγ+γ(cid:48)|−2πi(cid:104)γ+γ(cid:48),C(cid:105) , (2.12) where we have used the identity Erf(x) = sgn(x)(1−Erfc(|x|). In plain words, the addition of the two-particle contribution to I(1) has converted the step function Θ(x) in (2.10) into γ+γ(cid:48) the smooth function 1[1 + Erf(x)]. This shows that the sum of the one and two-particle 2 contributions is not only continuous, but also differentiable across the wall (see Figure 1 for (cid:112) illustration), which acquires a finite width of order 1/ Rm as a function of the relative γ,γ(cid:48) phase ψ between the central charges Z and Z . It would be interesting to generalize this γγ(cid:48) γ γ(cid:48) computation to the case of non-primitive wall-crossing, and to relax the non-relativistic limit R → ∞. 3Here Θ(x) denotes the Heaviside step function, equal to 1 when x>0 and 0 otherwise. – 4 – �� -� -� � � -� -� � � -� -� � � -� -� -� -� -� -� Figure 1: Behavior of the one-particle contribution (−2Θ(x)), two-particle contribution (sign(x)Erfc(|x|)) and their sum (−1−Erf(x)) to the index with total charge γ +γ(cid:48) across a wall where the phases of Z and Z align (x → 0). γ γ(cid:48) 3. Supersymmetric electron-monopole quantum mechanics Our goal in the remainder of this note is to derive the two-particle contribution (2.8) from the supersymmetric quantum mechanics of a system of two non-relativistic particles with mutually non-local primitive charges γ,γ(cid:48). After factoring out the center of mass degrees of freedom, which can be treated as in (1.8), and the internal degrees of freedom, counted by Ω¯(γ)Ω¯(γ(cid:48)), the system is described by N = 4 quantum mechanics with Hamiltonian [19, 20]4 1 q 1 (cid:16) q(cid:17)2 H = (p(cid:126)−qA(cid:126))2 − B(cid:126) ·(cid:126)σ ⊗(1 −σ )+ ϑ− . (3.1) 2 3 2m 2m 2m r where q = 1(cid:104)γ,γ(cid:48)(cid:105) is half the Dirac-Schwinger-Zwanziger product of the electromagnetic 2 charges, B(cid:126) = (cid:126)r is the magnetic field of a unit charge magnetic monopole sitting at the r3 (cid:126) origin, A is the corresponding gauge potential, (cid:126)σ are the Pauli matrices, and m = m is the γγ(cid:48) reduced mass of the two-particle system. Classically, the system has bound states for qϑ > 0, no bound states for qϑ < 0, and a continuum of scattering states with energy E ≥ E = ϑ2. c 2m The parameter ϑ is fixed by equating E with the binding energy, c ϑ2 = |Z |+|Z |−|Z |, (3.2) γ γ(cid:48) γ+γ(cid:48) 2m so ϑ ∼ mψ near the wall, cf. (2.7). Quantum mechanically, H describes two bosonic γγ(cid:48) degrees of freedom with helicity h = 0, and one fermionic doublet with helicity h = ±1/2 and gyromagnetic ratio g = 4. This unusual value is fixed by the requirement of supersymmetry, and can be understood as the combined effect of electromagnetic and scalar interactions [22]. (cid:126) (cid:126) Indeed, the Hamiltonian (3.1) commutes with the four supercharges (here Π = p(cid:126) − qA) [19, 20, 21] (cid:32) (cid:33) 1 0 −i(cid:0)ϑ− q(cid:1)+(cid:126)σ ·Π(cid:126) Q = √ r (3.3) 4 2m i(cid:0)ϑ− q(cid:1)+(cid:126)σ ·Π(cid:126) 0 r 1 (cid:18) 0 −(cid:0)ϑ− q(cid:1)σ −iΠ +(cid:15) Π σ (cid:19) Q = √ r a a abc b c . (3.4) a 2m −(cid:0)ϑ− q(cid:1)σ +iΠ +(cid:15) Π σ 0 r a a abc b c 4N = 4 supersymmetry allows a position-dependent rescaling of the kinetic term [21], but the spectral asymmetry is independent of this deformation, as long as it goes to one at spatial infinity. – 5 – which satisfy the algebra (here m = 1,2,3,4) {Q ,Q } = 2Hδ . (3.5) m n mn The complete spectrum of this Hamiltonian was analyzed in [23], but unfortunately these authors stopped short of computing the density of states in the continuum. We shall revisit this computation, adapting the classic treatment of the electron-monopole system without potential in [24]. The Hamiltonian (3.1) commutes with the total angular momentum operator [24] (cid:126)r 1 (cid:126) (cid:126) J =(cid:126)r∧(p(cid:126)−qA)−q + (cid:126)σ ⊗(1 −σ ), [J ,J ] = i(cid:15) J . (3.6) 2 3 a b abc c r 4 The Schro¨dinger equation HΨ = EΨ can be solved by separating the angular and radial dependence. For this we diagonalize J and J(cid:126)2 denoting by m and j(j +1) their eigenvalues. 3 For the spin 0 part (corresponding to the first two entries of the eigenvector Ψ), we write Ψ = f(r)Y , j ≥ |q|, j −q ∈ Z, (3.7) q,j,m where Y are the monopole harmonics (also known as spin-weighted spherical harmonics), q,l,m given in the patch around θ = 0 by [25] (cid:115) (2l+1)(l−m)!(l+m)! q+m q−m Y = 2m (1−cosθ)− 2 (1+cosθ) 2 P−q−m,q−m(cosθ)ei(m+q)φ. q,l,m 4π(l−q)!(l+q)! l+m (3.8) Here Pα,β(x) are the Legendre polynomials n (−1)n dn (cid:2) (cid:3) Pα,β(x) = (1−x)−α(1+x)−β (1−x)α+n(1+x)β+n . (3.9) n 2nn! dxn Using 1 1 (cid:16) (cid:17)2 (p(cid:126)−qA(cid:126))2 = − ∂2r+ (cid:126)r∧(p(cid:126)−qA(cid:126)) (3.10) r r r2 (cid:16) (cid:17)2 1 1 (cid:126)r∧(p(cid:126)−qA(cid:126)) = L(cid:126)2 −q2 = − ∂ sinθ∂ − (∂ +iq(cosθ−1))2 , (3.11) sinθ θ θ sin2θ φ one can show that the radial wave function satisfies [25] (cid:20) 1 1 ν2 −q2 − 1 1 (cid:16) q(cid:17)2 (cid:21) − ∂2r+ 4 + ϑ− −E f(r) = 0, (3.12) 2mr r 2mr2 2m r where ν = j + 1. 2 For the spin 1/2 part (corresponding to the last two entries of the eigenvector Ψ), the angular dependence is a linear combination of modes with orbital momentum j − 1 and j + 1 2 2 [24], (cid:113)   (cid:113)  j+mY − j−m+1Y φ(1) = (cid:113) 2j q,j−21,m−12 , φ(2) =  (cid:113) 2j+2 q,j+12,m−12 , j −q ∈ Z+ 1 (3.13) j,m  j−mY  j,m  j+m+1Y  2 1 1 1 1 2j q,j− ,m+ 2j+2 q,j+ ,m+ 2 2 2 2 – 6 – The first set of modes occurs for j ≥ |q|+ 1 while the second occurs for j ≥ |q|− 1. In order 2 2 (cid:126) (cid:126) to diagonalize the action of (cid:126)σ ·(cid:126)r and (cid:126)σ ·(p(cid:126)−qA), which commute with J, it is convenient to introduce the linear combinations (for j ≥ |q|+ 1) [24] 2 ξ(+) = c φ(1) −c φ(2), ξ(−) = c φ(1) +c φ(2) (3.14) j,m + j,m − j,m j,m − j,m + j,m where the coefficients √ √ (cid:0) (cid:1) q 2j +1+2q ± 2j +1−2q c = (3.15) ± (cid:112) |q| 2(4j +2) √ (cid:18) (cid:19) Y 2Y satisfy c2 +c2 = 1. Using (cid:126)r·(cid:126)σ = 2r(cid:112)π √0,1,0 0,1,−1 , and the multiplication rule + − 3 − 2Y −Y 0,1,1 0,1,0 (cid:115) j(cid:88)1+j2 (2j +1)(2j +1) 1 2 Y Y = (cid:104)j ,−q ,j ,−q |j ,−q −q (cid:105) q1,j1,m1 q2,j2,m2 4π(2j +1) 1 1 2 2 3 1 2 3 j3=max(|j1−j2|,|m1+m2|) ×(cid:104)j ,m ,j ,m |j ,m +m (cid:105)Y 1 1 2 2 3 1 2 q1+q2,j3,m1+m2 (3.16) for monopole harmonics, one can show that these combinations satisfy, for any f(r), ((cid:126)σ ·(cid:126)r)f(r)ξ(±) =−rf(r)ξ(∓), j,m j,m (3.17) (cid:126)σ ·(p(cid:126)−qA(cid:126))f(r)ξ(±) =i(cid:0)∂ +r−1(1∓µ)(cid:1) f(r)ξ(∓), j,m r j,m where we defined (cid:113) µ = (j + 1)2 −q2. (3.18) 2 Using the fact that the Hamiltonian in the spin 1/2 sector can be written as 1 (cid:16) (cid:17)2 q 1 (cid:16) q(cid:17)2 (cid:126) (cid:126) H = (cid:126)σ ·(p(cid:126)−qA) − B ·(cid:126)σ + ϑ− (3.19) 1/2 2m 2m 2m r and the identity (cid:18) (cid:19)(cid:18) (cid:19) 1±µ 1∓µ 1 µ(µ∓1) ∂ + ∂ + = ∂2r− , (3.20) r r r r r r r2 we find that its action on f(r)ξ(±) is given by j,m (cid:32) (cid:33) (cid:32) (cid:33) ξ(+) (cid:20) 1 1 µ2 1 (cid:16) q(cid:17)2(cid:21) ξ(+) H ·f(r) j,m = − ∂2r+ + ϑ− ·f(r) j,m 1/2 ξ(−) 2mr r 2mr2 2m r ξ(−) j,m j,m (3.21) (cid:32) (cid:33) 1 (cid:18)−µ q(cid:19) ξ(+) + · f(r) j,m . 2mr2 q µ ξ(−) j,m (cid:112) The 2×2 matrix on the second line has eigenvalues ± µ2 +q2 = ±(j + 1), and eigenvectors 2 ξ˜(±) = (µ∓(cid:112)µ2 +q2)ξ(+) −qξ(−) (3.22) j,m j,m j,m – 7 – Noting that the coefficient of the centrifugal 1/r2 term in the potential is proportional to (cid:112) (cid:18) 1 1(cid:19)2 1 µ2 ± µ2 +q2 = j + ± −q2 − , (3.23) 2 2 4 we find that the radial wavefunctions f (r) associated to the eigenmodes ξ˜(±) satisfy the same ± j,m equation as (3.12) with ν = j+1 (for the + sign, which we refer to as the helicity h = 1 mode) 2 or ν = j (for the − sign, which we refer to as the helicity −1 mode). 2 Finally, for j = |q|−1/2, the space of eigenmodes of J2,J is one-dimensional, spanned by 3 (cid:18) (cid:19) j−m j+m−1 sinθ η ≡ φ(2) ∝ (1+cosθ) 2 (1−cosθ) 2 ei(m+j)φ . (3.24) m j,m eiφ(1−cosθ) One has, in place of (3.17), q q ((cid:126)σ ·(cid:126)r)η = r η , (cid:126)σ ·(p(cid:126)−qA(cid:126))f(r)η = −i (∂ +r−1)f(r)η , (3.25) m m m r m |q| |q| leading to the same equation (3.12) with ν = j. In summary, the radial equation is given by (3.12) with ν = j +h+ 1 , j = |q|+h+(cid:96) (3.26) 2 with h = 0 for the two bosonic degrees of freedom and h = ±1 for the spin 1/2 degree of 2 freedom, and (cid:96) ∈ N in all cases. Solutions to (3.12) with energy E ≡ k2 > ϑ2 are linear 2m 2m combinations5 (cid:16) √ (cid:17) (cid:16) √ (cid:17) rf(r) = βM 2ir k2 −ϑ2 +γW 2ir k2 −ϑ2 , (3.27) −√iqϑ ,ν −√iqϑ ,ν k2−ϑ2 k2−ϑ2 where M (z) and W (z) are Whittaker functions, which are solutions of the second order λ,ν λ,ν differential equation (cid:20) 1 λ 1 −ν2(cid:21) D ·w(z) ≡ ∂2 − + + 4 w(z) = 0 (3.28) λ,ν z 4 z z2 satisfying 1 M (z) ∼ zν+2 , W (z) ∼ zλe−z/2. (3.29) λ,ν λ,ν z→0 |z|→∞ The solution proportional to M is regular at r = 0, while the solution proportional to W describes an outgoing spherical wave. Since Γ(2ν +1) Γ(2ν +1) M (z) = eiπλW (eiπz)+ eiπ(λ−ν−1)W (z), (3.30) λ,ν Γ(ν −λ+ 1) −λ,ν Γ(ν +λ+ 1) 2 λ,ν 2 2 5It helps to note that (3.12) is isomorphic to the Schrödinger equation of the hydrogen atom, whose radial wave-functions are linear combinations of M (2ikr) and W (2ikr) where η = q q m/k, E = k2/2m. 1 1 1 2 iη,(cid:96)+ iη,(cid:96)+ 2 2 (see e.g. [26, Chap. 14.6]). – 8 – we find that the S-matrix in angular momentum channel (cid:96) and helicity h is (cid:16) (cid:17) Γ(ν +λ+ 1) Γ |q|+(cid:96)+2h+1−i√ qϑ S (k) = 2 = k2−ϑ2 . (3.31) h,(cid:96) Γ(ν −λ+ 12) Γ(cid:16)|q|+(cid:96)+2h+1+i√ qϑ (cid:17) k2−ϑ2 In particular, bound states correspond to poles of the S-matrix, and occur only when qϑ > 0, with energy ϑ2 (cid:18) q2 (cid:19) E = 1− , (3.32) h,(cid:96),n 2m (|q|+(cid:96)+2h+n+1)2 with n ≥ 0. The energy depends only on the sum N = (cid:96) + n + 2h, so the spectrum has additional degeneracies beyond those predicted by rotational symmetry and supersymmetry [19, 23]. The supersymmetric ground state occurs in the h = −1 sector with n = (cid:96) = 0 and 2 (cid:16) (cid:17) has degeneracy 2|q| = |(cid:104)γ ,γ (cid:105)|. Its wave function is 0 , in agreement with [20, 4.16]. 1 2 rq−1e−ϑrηm The density of states (minus the density of states for a free particle in R3) is the derivative of the scattering phase, ρ(k)dk = 1 dlogS(k). The canonical partition function for states of 2πi helicity h is therefore ∞ ∞ (cid:88) (cid:88) Tr e−2πRH =Θ(qϑ) (2(cid:96)+2|q|+2h+1)e−2πREh,(cid:96),n h (cid:96)=0 n=0 (cid:16) (cid:17) +(cid:88)∞ (2(cid:96)+2|q|+2h+1) (cid:90) ∞ dk∂k log Γ |q|+(cid:96)+2h+1−i√kq2ϑ−ϑ2 e−πRk2, (cid:16) (cid:17) m k=|ϑ| 2πi Γ |q|+(cid:96)+2h+1+i√ qϑ (cid:96)=0 k2−ϑ2 (3.33) where the first term, corresponding to discrete bound states, contributes only when qϑ > 0. Summing over all types h weighted by fermionic parity (−1)2h, only the BPS bound state with E = 0 contributes from the first term, while the contribution of the continuum of scattering states simplifies to ∞ ∞ (cid:90) (cid:20) (cid:21) (cid:88) dk∂k (2(cid:96)+2|q|)log z(cid:96) −(2(cid:96)+2|q|+2)log z(cid:96)+1 e−πRmk2, (3.34) 2πi z¯ z¯ (cid:96) (cid:96)+1 (cid:96)=0 k=|ϑ| where qϑ z = |q|+(cid:96)−i√ . (3.35) (cid:96) k2 −ϑ2 Cancelling the terms in the sum, only the contribution (cid:96) = 0 remains, leading to 2|q| (cid:90)∞ dk∂k log(cid:34)|q|−i√kq2ϑ−ϑ2(cid:35)e−πRmk2 =2qϑ (cid:90)∞ √ dk e−πRmk2 . (3.36) 2πi |q|+i√ qϑ π k k2 −ϑ2 k2−ϑ2 k=|ϑ| k=|ϑ| This is in fact the standard spectral asymmetry predicted by Callias’ theorem [27]6. In- deed, the result (3.36) does not depend on the details of the S-matrix, but only on the ra- 6While the Callias theorem is valid a smooth monopole background, the extension to singular monopoles wasworkedoutin[28],andleadstothesamespectralasymmetry. IthankA.RoystonandD.vandenBleeken for discussions on this matter. – 9 –