The Atiyah–Patodi–Singer (APS, or spectral) boundary conditions confine massless fermions to a finite spatial region (a 'bag') while preserving chiral symmetry; they are defined nonlocally through the spectral projector of the intrinsic Dirac operator on the boundary. Abrikosov and Wipf (2007) computed this projector in closed form for the three-dimensional spherical bag and found that it equals one half of the boundary delta-function kernel minus ii times the normal gamma matrix times the massless fermion propagator restricted to the boundary. They verified the mechanism on a two-mode model and asserted that the relation is universal — independent of the bag's shape and of the space dimension — while conceding that a general proof was still to be given. The relation is verified only for the three-dimensional spherical bag and for a two-mode model; its author's 2008 follow-up explicitly defers the generalization to non-spherical domains and higher dimensions; later applications (the Nutku helicoid, a curved geometry with a genuinely non-spherical boundary; rotating cylinders; worldline computations) all fall back on the constructive mode-sum definition; and the adjacent Calderón-projection framework of elliptic boundary theory provides Green-kernel boundary projections for arbitrary smooth boundaries but neither proves nor refutes an exact identity with the APS spectral projector. The open problem is to settle the projector–propagator identity for arbitrary smooth boundaries and dimensions — specifying exactly which Green function makes it exact and characterizing the correction terms if it fails — together with its extension to boundaries whose boundary operator has zero modes and constructive, numerically implementable projector kernels for non-spherical boundaries carrying tangential gauge fields and curvature.

Background

Confining relativistic fermions to a finite spatial region — a 'bag' — is a classical device of hadron physics (the MIT bag model). Boundary conditions for the Dirac operator must keep the operator self-adjoint and, for massless fermions, should not break chiral symmetry, i.e. the invariance under ψ↦γ5ψ\psi\mapsto\gamma_5\psi that follows from the anticommutation of γ5\gamma_5 with the massless Dirac operator. Local conditions such as the MIT bag condition violate this. The spectral (APS) boundary conditions introduced by Atiyah, Patodi and Singer in their study of spectral asymmetry on manifolds with boundary do not. Near a smooth boundary ∂B\partial B, with ξ\xi the outward normal coordinate, the Euclidean Dirac operator takes the product form γ^(∂ξ+B∂)\hat{\gamma}(\partial_\xi+B_{\partial}), where γ^\hat{\gamma} is the gamma matrix along the outward normal and B∂B_{\partial} — the boundary operator — is the intrinsic self-adjoint Dirac operator on ∂B\partial B, built from the tangential gamma matrices, the tangential gauge field and the spin connection of the boundary. The constructive definition of the spectral boundary conditions expands the boundary values of the spinor in eigenfunctions of B∂B_{\partial} and annihilates the components with eigenvalues of one sign; the defining object is therefore the spectral projector P\mathcal{P} onto (say) the positive spectrum of B∂B_{\partial}, and the boundary condition reads Pψ∣∂B=0\mathcal{P}\psi|_{\partial B}=0. This definition is nonlocal — it refers to the spectrum of the boundary as a whole — and is tied to a preferred basis. For a three-dimensional spatial bag evolving in time one uses the truncated variant of these conditions introduced in Abrikosov's earlier work on the bag model, which is purely spatial and avoids causality problems of the full cylinder conditions.

Abrikosov and Wipf computed the projector explicitly for a three-dimensional spherical bag. After rotating the spinors into a 'work representation' in which the boundary operator on the sphere becomes I+2 L^⋅S^I+2\,\hat{L}\cdot\hat{S} (orbital angular momentum coupled to an effective spin), so that its eigenfunctions are ordinary spherical spinors with eigenvalues l+1l+1 and −l-l, they summed the spectral series in closed form and obtained, in the covariant form (their Eq. (45)),

P(x^,y^)=12 δ∂B(x^−y^)−i (γ^n^x) S(x^,y^),x^,y^∈∂B,\mathcal{P}(\hat{x},\hat{y})=\frac{1}{2}\,\delta_{\partial B}(\hat{x}-\hat{y})-\mathrm{i}\,(\hat{\gamma}\hat{n}_x)\,\mathcal{S}(\hat{x},\hat{y}),\qquad \hat{x},\hat{y}\in\partial B,

where δ∂B\delta_{\partial B} is the delta-function (unit) kernel on the boundary, n^x\hat{n}_x is the outward unit normal at x^\hat{x}, and S\mathcal{S} is the propagator (Green function) of the massless fermions — for the sphere, the free-space propagator restricted to the boundary. Note that the displayed identity is convention-dependent: the symbol P\mathcal{P} here is the single-sign spectral projector defined above, whereas the source's P\mathcal{P} in Eq. (45) is the chirally blocked full-spinor projector, the two readings being related by P↦I−P\mathcal{P}\mapsto\mathcal{I}-\mathcal{P} together with the sign of the i(γ^n^x)S\mathrm{i}(\hat{\gamma}\hat{n}_x)\mathcal{S} term; which object and sign convention is intended is part of question (1). They also verified the mechanism on a 'two-mode model', a half-plane carrying a single eigenvalue pair ±λ\pm\lambda of the boundary operator, where the identity reduces to the symmetric convention θ(0)=12\theta(0)=\tfrac12 for the step function at the boundary. On this basis the paper asserts that the relation is universal, 'suitable for boundaries of arbitrary shape in any dimension', and remarks that a general proof 'is not difficult' — but no such proof is given. Two gaps are visible in the construction itself: the boundary operator is assumed to have no zero eigenvalues (the authors note this holds for the sphere and, they assert without proof, for convex manifolds), and the explicit kernel is available only for the sphere.

The question sits next to a classical chapter of elliptic theory that the physics paper does not invoke. For a Dirac-type operator on a compact manifold with smooth boundary, the projection of the boundary L2L^2-space onto the Cauchy data space — the boundary values of square-integrable solutions of the homogeneous equation — is the Calderón projection. It can be computed from the Green function of a canonically doubled problem — the closed manifold obtained by gluing two copies of the bag along their common boundary, on which the doubled Dirac operator is invertible. It is a pseudodifferential operator of order zero (an operator written in an oscillatory-integral calculus that generalizes differential operators and boundary singular integrals) whose principal symbol — its leading part — coincides with the positive spectral projection of the boundary operator; boundary conditions defined by any projection sharing that symbol are well posed, with an index given by the relative index against the Calderón projection. This framework, developed in the monograph of Booß-Bavnbek and Wojciechowski and reviewed by Booß-Bavnbek and Lesch, constructs Green-kernel boundary projections for arbitrary smooth boundaries, but it does not assert exact equality between such a projection and the APS spectral projector; the exact equality computed for the ball is a statement about a special geometry. What remains genuinely open is whether the physics identity — an exact operator equality between the APS spectral projector of B∂B_{\partial} and the boundary restriction of a fermion propagator — holds for general domains, for precisely which propagator, and how it deforms when the boundary operator acquires zero modes or when the projector must actually be computed on a non-spherical boundary.

Problem Statement

Establish the integral (covariant) form of APS spectral boundary conditions beyond the spherical bag. The single overarching objective is to replace the nonlocal spectral definition of these boundary conditions by an exact, covariant Green-kernel condition that is valid for general domains; three linked questions serve this objective, the first being load-bearing for the other two.

(1) The universal identity. For a massless Dirac operator in a bag of arbitrary shape and dimension (both the classical even-dimensional APS setting and the truncated odd-dimensional setting for a spatial bag evolving in time), with B∂B_{\partial} the self-adjoint tangential boundary operator and P\mathcal{P} its APS spectral projector, state precisely — and then prove or refute — the kernel identity

P(x^,y^)=12 δ∂B(x^−y^)−i (γ^n^x) S(x^,y^),x^,y^∈∂B.\mathcal{P}(\hat{x},\hat{y})=\frac{1}{2}\,\delta_{\partial B}(\hat{x}-\hat{y})-\mathrm{i}\,(\hat{\gamma}\hat{n}_x)\,\mathcal{S}(\hat{x},\hat{y}),\qquad \hat{x},\hat{y}\in\partial B.

Here δ∂B\delta_{\partial B} is the delta-function (unit) kernel on the boundary, n^x\hat{n}_x the outward unit normal at x^\hat{x}, and γ^\hat{\gamma} the gamma matrix along the normal. A convention must also be fixed for the left-hand side: in the source, the object P\mathcal{P} appearing in the displayed identity is the chirally blocked full-spinor projector — the block operator that annihilates the positive-eigenvalue upper components and the negative-eigenvalue lower components of B∂B_{\partial} and commutes with γ5\gamma_5 — whereas the constructive gloss in the Background projects onto a single spectral sign; the two readings are related by P↦I−P\mathcal{P}\mapsto\mathcal{I}-\mathcal{P} together with the sign of the i(γ^n^x)S\mathrm{i}(\hat{\gamma}\hat{n}_x)\mathcal{S} term, and stating precisely which object and sign convention is intended forms part of question (1). The statement must also specify which Green function S\mathcal{S} makes the identity an exact operator identity in general: the free ambient propagator, the Green function of the interior problem with spectral boundary conditions, or the Green function of the doubled (invertible-double) problem — three notions that are not manifestly equal in general: for the ball, the free ambient propagator and the interior spectral-boundary-condition Green function coincide on the boundary, while the two-mode model actually uses the Green function of the doubled (full-line) problem. If the identity fails for general shapes, the required answer characterizes the correction terms (for example, as lower-order pseudodifferential or smoothing terms relative to the Calderón projection of elliptic boundary theory) and identifies the geometries in which the exact identity holds.

(2) Boundary zero modes. Extend the projector and the identity to boundaries on which B∂B_{\partial} has zero eigenvalues: fix the convention for the zero-eigenvalue subspace, state how the kernel formula is modified, and describe the behavior of the projector when boundary modes cross zero.

(3) Constructive kernels. Provide constructive projector kernels — closed-form, algorithmic, or numerical with controlled error — for non-spherical boundaries, including nontrivial tangential gauge fields and curvature, sufficient to impose spectral boundary conditions in practice and to reproduce the spherical result as a special case.

The core open item — the one the source explicitly leaves unproved — is (1); (2) extends the identity to the zero modes whose possibility the source acknowledges but excludes from its own computation, and (3) answers the constructive direction noted by the source and by later applications, in a form consistent with the identity established in (1).

References

  1. A. A. Abrikosov (jr.) and A. Wipf, "The integral form of APS boundary conditions in the bag model", J. Phys. A: Math. Theor. 40 (2007) 5163, DOI 10.1088/1751-8113/40/19/016, arXiv:hep-th/0611121, https://arxiv.org/abs/hep-th/0611121
  2. A. A. Abrikosov (jr.), "Fermionic Green function in the 3D-ball with chirally invariant boundary conditions", in: Proceedings of the 15th International Seminar on High Energy Physics (Quarks 2008), Sergiev Posad, Russia, May 23–29, 2008, INSPIRE record 1372925 (no DOI or arXiv ID), https://inspirehep.net/literature/1372925
  3. A. A. Abrikosov (jr.), "Modified spectral boundary conditions in the bag model", J. Phys. A: Math. Gen. 39 (2006) 6109–6115, DOI 10.1088/0305-4470/39/21/S01, arXiv:hep-th/0512311, https://arxiv.org/abs/hep-th/0512311
  4. M. F. Atiyah, V. K. Patodi and I. M. Singer, "Spectral asymmetry and Riemannian geometry. I", Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69, DOI 10.1017/s0305004100049410, https://doi.org/10.1017/s0305004100049410
  5. T. Birkandan and M. Hortaçsu, "Dirac equation in the background of the Nutku helicoid metric", J. Math. Phys. 48 (2007) 092301, DOI 10.1063/1.2785124, arXiv:0706.2543, https://arxiv.org/abs/0706.2543
  6. V. E. Ambruş and E. Winstanley, "Massless rotating fermions inside a cylinder", AIP Conf. Proc. 1694 (2015) 020011, DOI 10.1063/1.4937237, arXiv:1502.03073, https://arxiv.org/abs/1502.03073
  7. L. Manzo, "Worldline approach for spinor fields in manifolds with boundaries", J. High Energy Phys. 2024 (2024) 144, DOI 10.1007/JHEP06(2024)144, arXiv:2403.00218, https://arxiv.org/abs/2403.00218
  8. B. Booß-Bavnbek and K. P. Wojciechowski, "Elliptic Boundary Problems for Dirac Operators", Birkhäuser, Boston, 1993, DOI 10.1007/978-1-4612-0337-7, https://doi.org/10.1007/978-1-4612-0337-7
  9. B. Booß-Bavnbek and M. Lesch, "The invertible double of elliptic operators", Lett. Math. Phys. 87 (2009) 19–46, DOI 10.1007/s11005-009-0292-9, arXiv:0803.4047, https://arxiv.org/abs/0803.4047
  10. T. Kimura and S. Sharma, "Dirac Operators, APS Boundary Conditions, and Spectral Flow on a Finite Warped Cylinder", arXiv:2603.23275 [math-ph] (2026), https://arxiv.org/abs/2603.23275