Open problems in the spectral action for noncommutative spaces with boundary: gauge-field boundary conditions, Lorentzian extension, and applications

In noncommutative geometry the bosonic action is computed, not postulated, as the spectral action: the trace of a cutoff function of the Dirac operator. Chamseddine and Connes (J. Geom. Phys. 61 (2011) 317-332) evaluated the spectral action for the noncommutative space of the Standard Model on a compact Riemannian manifold with boundary and showed that the self-adjointness boundary conditions of the Dirac operator force the correct gravitational boundary (Gibbons-Hawking-type) and Higgs boundary terms, with the right signs and coefficients, provided the normal component of the gauge field vanishes on the boundary. The paper left three directions open, and all three remain unresolved in the subsequent literature. First, the gauge-field boundary conditions are imposed as natural assumptions: the observation that their second parts 'arise as integrability condition derived from the boundary conditions of the Dirac operator' was never proven or classified. Second, the construction is intrinsically Euclidean (compact resolvent): a Lorentzian-signature spectral action with boundary, including the fate of the boundary terms under Wick rotation, has not been constructed; Lorentzian spectral actions exist so far only without boundary. Third, concrete applications of the boundary formulas (explicit geometries, boundary contributions to observables) were deferred and only mathematical reformulations have appeared since.

Background

In Connes' noncommutative geometry, a geometric space is described spectrally by a spectral triple $(\mathcal{A},\mathcal{H},D)$: an involutive algebra $\mathcal{A}$ of 'coordinates' represented on a Hilbert space $\mathcal{H}$ of 'spinors', together with a self-adjoint Dirac operator $D$ (the first-order square root of the Laplacian acting on spinor fields, carrying the spin and gauge couplings of the geometry) with compact resolvent (meaning $(D-i)^{-1}$ is a compact operator, which forces a discrete spectrum). The commutative prototype is the algebra of smooth functions on a compact Riemannian spin manifold; 'almost-commutative' triples, products of such a manifold with a finite-dimensional internal space, encode gauge theories, with the gauge and Higgs fields arising as inner fluctuations $D\to D+A$ of the Dirac operator by one-forms built from $\mathcal{A}$.

The spectral action principle states that the bosonic action is a function of the spectrum of $D$ alone: $S=\mathrm{Tr}\,f(D/\Lambda)$, the trace of a positive even cutoff function evaluated at energy scale $\Lambda$. Its large-$\Lambda$ expansion is governed by the heat-kernel coefficients $a_n$ of $D^2$ (the local invariants appearing in the small-$t$ expansion $\mathrm{Tr}\,e^{-tD^2}\sim\sum_n a_n t^{(n-d)/2}$), so all relative coefficients of the resulting gravitational and Yang-Mills-Higgs terms are fixed, with no free parameters beyond a few moments of $f$. For the almost-commutative triple of the Standard Model this yields the Einstein-Hilbert action together with the full Standard Model bosonic action.

On a compact manifold $M$ with smooth boundary $\partial M$, two new phenomena appear. First, in general relativity the Einstein-Hilbert action $\int_M\sqrt{g}R$ alone is not variationally well-posed for fixed boundary metric: one must add the Gibbons-Hawking term $\int_{\partial M}\sqrt{h}\,K$, where $h$ is the metric induced on the boundary and $K$ is the trace of the extrinsic curvature (the second fundamental form, measuring how the boundary curves inside $M$). Whether the spectral action reproduces this boundary term with the correct sign and coefficient is a stringent test, since the spectral action leaves no freedom to tune it. Second, the Dirac operator on a manifold with boundary is not essentially self-adjoint (it admits many self-adjoint extensions rather than a unique one, so a boundary condition must be chosen): hermiticity of its quadratic form requires a local projector boundary condition $\Pi_-\Psi\big|_{\partial M}=0$ with $\Pi_-=(1-\chi)/2$ and $\chi=\gamma^n\gamma^5$ ($\gamma^n$ the Clifford contraction with the inward unit normal, $\gamma^5$ the chirality matrix), a condition of the local 'chiral bag' type (so named after the MIT bag model of quark confinement, where such projector conditions on spinors at a boundary first appeared) studied in the heat-kernel literature, whose square imposes mixed Dirichlet-Neumann-type conditions on $D^2$. With such conditions, the heat-kernel coefficients acquire, besides the usual curvature integrals, boundary integrals polynomial in $K$, the second fundamental form $K_{ab}$, and the normal component of the connection.

Chamseddine and Connes (announced in Phys. Rev. Lett. 99 (2007) 071302, developed in J. Geom. Phys. 61 (2011) 317-332) evaluated the spectral action for the noncommutative space of the Standard Model, a product of a compact Riemannian four-manifold with boundary with the finite Standard-Model space, under exactly these conditions. They found that the spectral action automatically contains $\int_M\sqrt{g}R+2\int_{\partial M}\sqrt{h}K$ (after fixing the gravitational constant) with the sign and coefficient needed for consistency of the Hamiltonian, together with the correct boundary partner of the Higgs kinetic term, and extended the computation to a dilaton background. The one input that did not follow from the spectral data was the treatment of the gauge fields: to keep gauge fields out of the boundary terms they imposed, as an assumption, the vanishing of the normal component of the gauge field on the boundary ($B_n\big|_{\partial M}=0$ for the background in Section 4 of the paper, equivalently $A_n\big|_{\partial M}=0$ for the fluctuation in Section 5, where $n$ denotes the inward normal direction).

Three directions were left open at that point. (i) The structural origin of the gauge-field boundary conditions: after imposing a gauge condition, the spectral action's boundary terms need only part of the two admissible boundary-condition pairs, and the authors observed, without proof, that the remaining parts 'arise as integrability condition derived from the boundary conditions of the Dirac operator'. (ii) The Lorentzian question: the spectral action is intrinsically Euclidean, since compactness of the resolvent is what makes the trace well defined, and the passage to physical Lorentzian signature via Wick rotation (the analytic continuation of the time coordinate relating Euclidean and Lorentzian metrics), with the fate of the boundary terms and of self-adjointness, was not addressed. (iii) Applications: the paper closes the dilaton section with 'Practical applications of these results will be dealt with in the future.'

Problem Statement

The problem is to complete the spectral action for noncommutative spaces with boundary, in the setting of Chamseddine and Connes (J. Geom. Phys. 61 (2011) 317-332), along the three directions that paper leaves open.

(1) Gauge-field boundary conditions. In the source computation, after the gauge condition $\nabla_\mu A^{\alpha\mu}=0$, either of two pairs of boundary conditions may be imposed on the gauge fluctuation $A$ (indices: $n$ the inward normal direction, $a,b$ tangential directions, $\nabla_n$ the normal covariant derivative, $K_{ab}$ the second fundamental form of the boundary, and $\alpha$ a gauge-group index):

$$A_n\big|_{\partial M}=0,\qquad (\nabla_n\delta^{ab}-K^{ab})A_b\big|_{\partial M}=0,$$

or

$$(\nabla_n-K)A_n\big|_{\partial M}=0,\qquad A_a\big|_{\partial M}=0.$$

The boundary terms of the spectral action need only the first parts of these conditions. Prove, refute, or classify: determine, in full generality, which boundary conditions on the gauge (inner-fluctuation) fields are forced by, merely compatible with, or excluded by the self-adjointness/hermiticity boundary conditions of the Dirac operator on an almost-commutative spectral triple with boundary; in particular, whether the second parts of the pairs above arise as integrability (consistency) conditions of the Dirac boundary data, as the source observes but does not prove. The general classification is required in addition to this integrability sub-claim: settling whether the second parts are integrability conditions resolves only that special case and leaves open the general classification of which boundary conditions are forced, which are merely compatible, and which are excluded.

(2) Lorentzian extension. The construction is Euclidean, since the spectral action $\mathrm{Tr}\,f(D/\Lambda)$ requires $D$ to have compact resolvent. Extend the boundary framework to Lorentzian signature: define a spectral action with boundary for Lorentzian (almost-commutative) geometries; establish the self-adjointness properties of the boundary value problem, in the Hilbert-space sense or in the Krein-space sense (a Krein space being a Hilbert space with an indefinite inner product, the natural setting for Lorentzian spinors); and determine the fate of the boundary terms, the Gibbons-Hawking-type and Higgs boundary partners, under the rotation.

(3) Applications. Apply the boundary spectral-action formulas to concrete geometries and settings of physical interest, such as explicit boundary-value problems or cosmological and black-hole backgrounds, computing the boundary contributions to observable quantities; carry out such a computation for at least one explicitly named geometry and boundary condition, using the boundary conditions and heat-kernel coefficients.

The common objective uniting the three components is to turn the formal Euclidean results of Chamseddine and Connes into a structural (which boundary data are admissible), dynamical (Lorentzian signature), and applied (computable boundary observables) framework for the spectral action on noncommutative spaces with boundary.

References

1. Ali H. Chamseddine, Alain Connes, 'Noncommutative geometric spaces with boundary: Spectral action', Journal of Geometry and Physics 61 (2011) 317-332. DOI: 10.1016/j.geomphys.2010.10.002; arXiv:1008.3980. URL: https://doi.org/10.1016/j.geomphys.2010.10.002
2. Ali H. Chamseddine, Alain Connes, 'Quantum gravity boundary terms from the spectral action of noncommutative space', Physical Review Letters 99 (2007) 071302. DOI: 10.1103/PhysRevLett.99.071302; arXiv:0705.1786. URL: https://doi.org/10.1103/PhysRevLett.99.071302
3. Bruno Iochum, Cyril Levy, 'Spectral triples and manifolds with boundary', Journal of Functional Analysis 260 (2011) 117-134. DOI: 10.1016/j.jfa.2010.09.006; arXiv:1001.3927. URL: https://doi.org/10.1016/j.jfa.2010.09.006
4. Bruno Iochum, Cyril Levy, Dmitri Vassilevich, 'Spectral action for torsion with and without boundaries', Communications in Mathematical Physics 310 (2012) 367-382. DOI: 10.1007/s00220-011-1406-7; arXiv:1008.3630. URL: https://doi.org/10.1007/s00220-011-1406-7
5. Bruno Iochum, Cyril Levy, Dmitri Vassilevich, 'Global and local aspects of spectral actions', Journal of Physics A: Mathematical and Theoretical 45 (2012) 374020. DOI: 10.1088/1751-8113/45/37/374020; arXiv:1201.6637. URL: https://doi.org/10.1088/1751-8113/45/37/374020
6. Yong Wang, 'A Kastler-Kalau-Walze type theorem and the spectral action for perturbations of Dirac operators on manifolds with boundary', arXiv:1210.5005 (2012). URL: https://arxiv.org/abs/1210.5005
7. Jingbo Wang, 'Holograph in noncommutative geometry: Part 1', arXiv:1007.2976 (2010). URL: https://arxiv.org/abs/1007.2976
8. Ubertino Battisti, Joerg Seiler, 'Boundary value problems with Atiyah-Patodi-Singer type conditions and spectral triples', Journal of Noncommutative Geometry 11 (2017) 887-917. DOI: 10.4171/jncg/11-3-4; arXiv:1503.02897. URL: https://doi.org/10.4171/jncg/11-3-4
9. Maxim A. Kurkov, Fedele Lizzi, 'Clifford structures in noncommutative geometry and the extended scalar sector', Physical Review D 97 (2018) 085024. DOI: 10.1103/PhysRevD.97.085024; arXiv:1801.00260. URL: https://doi.org/10.1103/PhysRevD.97.085024
10. Maxim Kurkov, 'Parity anomalies on 4D manifolds with boundaries', EPJ Web of Conferences 191 (2018) 06013. DOI: 10.1051/epjconf/201819106013. URL: https://doi.org/10.1051/epjconf/201819106013
11. Michał Eckstein, Bruno Iochum, 'Spectral Action in Noncommutative Geometry', SpringerBriefs in Mathematical Physics, Springer (2018). DOI: 10.1007/978-3-319-94788-4; arXiv:1902.05306. URL: https://doi.org/10.1007/978-3-319-94788-4
12. Christian Bär, Alexander Strohmaier, 'An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary', American Journal of Mathematics 141 (2019) 1421-1455. DOI: 10.1353/ajm.2019.0037; arXiv:1506.00959. URL: https://doi.org/10.1353/ajm.2019.0037
13. Nguyen Viet Dang, Michał Wrochna, 'Complex powers of the wave operator and the spectral action on Lorentzian scattering spaces', Journal of the European Mathematical Society 27 (2025) 971-1054 (published online 16 December 2023). DOI: 10.4171/jems/1389; arXiv:2012.00712. URL: https://doi.org/10.4171/jems/1389
14. Ruben Zeitoun, 'Feynman spectral action of the wave operator on asymptotically de Sitter spaces', arXiv:2511.17359 (2025). URL: https://arxiv.org/abs/2511.17359
15. Francesco D'Andrea, Maxim A. Kurkov, Fedele Lizzi, 'Wick rotation and fermion doubling in noncommutative geometry', Physical Review D 94 (2016) 025030. DOI: 10.1103/PhysRevD.94.025030; arXiv:1605.03231. URL: https://doi.org/10.1103/PhysRevD.94.025030
16. Agostino Devastato, Shane Farnsworth, Fedele Lizzi, Pierre Martinetti, 'Lorentz signature and twisted spectral triples', Journal of High Energy Physics 2018 (2018) 089. DOI: 10.1007/JHEP03(2018)089; arXiv:1710.04965. URL: https://doi.org/10.1007/JHEP03(2018)089
17. Agostino Devastato, Manuele Filaci, Pierre Martinetti, Devashish Singh, 'Actions for twisted spectral triple and the transition from the Euclidean to the Lorentzian', arXiv:2002.11700 (2020). URL: https://arxiv.org/abs/2002.11700
18. Ali H. Chamseddine, 'Hearing the Shape of the Universe: A Personal Journey in Noncommutative Geometry', arXiv:2511.05909 (2025). URL: https://arxiv.org/abs/2511.05909