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Gravitational Path Integral and String Theory

Ashoke Sen

Oct 19, 2026arXiv:2610.27073v1
hep-thgr-qc
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Scorecard· 16/16
6.5/10 impact

Authoritative, coherent synthesis of a live debate on UV sensitivity of gravitational path integral for BPS indices, with clarifications and a small new appendix, but mostly previously published and confined to formal hep-th.

Abstract

Gravitational path integral has had many applications in recent years. Using the example of supersymmetric index, which provides one of the most controlled tests of gravitational path integral, we argue that even in this controlled case, naive gravitational path integral cannot produce consistent results beyond low orders. However one can go further using stringy inputs, e.g. the choice of UV cut-off and the choice of correct path integration variable, without having to do full fledged string calculation. We also discuss to what extent we are led to string scale UV cut-off even if we begin with a supergravity theory without an a priori bias towards string theory. This is a review of mostly published results, but we clarify several details.

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Paper type: Theoretical review/perspective in hep-th by Ashoke Sen. It is mostly a synthesis of published results, with a new appendix and some clarifications. Its argumentative thrust is partly position-like.

Core Contribution

The paper asks how far the gravitational path integral can be trusted without a UV completion. It uses the supersymmetric index of BPS black holes in 4D N≥2 supergravity as the test case. The central claim is that the one-loop logarithmic correction to the index depends on the UV cutoff, through factors of g_s^γ. It also depends on how zero modes are measured and which dual field (2-form vs axion) is the integration variable. Consistency then requires a string-scale cutoff (γ=0). Two tests are used:

  • agreement with microscopic counting for N=8 and N=4 pure-NSNS black holes;
  • g_s-independence of the index for pure-RR black holes, which needs no microscopic input.

    • A further step asks whether a pure supergravity theory, with no string input, can be led to the string-scale cutoff. Sen lays out a set of UV-completion criteria:

  • a string-frame action of the form e^{-2φ}L;
  • a uniform cutoff;
  • quantized charge lattices that are independent of the moduli;
  • a rule for choosing the fundamental field in a dual pair, based on the weak-coupling power of e^{-αφ};
  • g_s-independence of the index when the modulus is not fixed by attractors.

    • Under these criteria, N=6 and N=8 supergravity force γ=0. N=4 is agnostic. N=2 needs extra stringy input, namely the −χ_CY/24 ln g_s E_4 term, which comes from a ten-dimensional Euler density.

    The paper also yields concrete predictions for N=6 and N=2 microscopic degeneracies. These are S = S_cl − 2 ln(λ_Qλ_P) for N=6, and the N=2 formula (6.9). The appendix proves that for general N=2 supergravity the scalars stay constant when their asymptotic values are set to attractor values. This generalizes earlier STU/N=4/N=8 results and corrects a minor point in the literature.

    Methodological Rigor

    The calculation uses heat-kernel regularization, scaling arguments for zero-mode measures, and a duality-frame analysis. It is internally coherent and physically well motivated. The treatment of zero-mode measures (translations, rotations, goldstinos, 2-form zero modes) is careful and clear.

    Caveats:

  • Several steps are scaling arguments that treat the kinetic operator as "pretending diagonal". They track only powers of λ_S and g_s, not O(1) terms.
  • The UV-completion criteria are partly chosen so that the desired answer emerges. Criterion 7, index independence of g_s, is physically sound. The N=2 case needs ad hoc infrared reinterpretation of an RR-sector divergence.
  • Dropping RR zero modes is justified by the field-strength formulation of string field theory.
  • The check against microscopic data is limited to N=8 and N=4. The N=6 and N=2 results are predictions rather than tests.
  • The critique of the Planck-scale localization results (ref [17]) is only sketched here. The technical argument is deferred to ref [18], and the "wrong answer" claim rests on the footnote about gauge zero modes.
  • The paper is a review, so the proofs and many details are in the cited work.

    • Potential Impact

    Its main audience is the black-hole microstate counting, supergravity localization, and string-theory communities. It sharpens a live controversy. This is whether supergravity localization or the gravitational path integral can give exact or quantitative results without stringy input, as pursued by Iliesiu–Murthy–Turiaci and others. The message that even controlled quantities need UV input shapes expectations for gravitational path integral applications such as replica wormholes, JT gravity and near-extremal black holes. It also points toward string field theory in black-hole backgrounds, and toward combining localization with worldsheet methods.

    The practical takeaway is a clear hierarchy of what can be computed from low-energy physics. Logarithmic terms are accessible with a string-scale cutoff. Full one-loop corrections require string theory. The results have no direct application outside fundamental theory. The perspective could influence how people interpret gravitational-path-integral claims, but it will probably not change the practice of the broader community.

    Timeliness

    The topic is timely, with active debate on gravitational path integrals, supersymmetric index factorization, and the Euclidean saddles of the BPS index. The paper consolidates a scattered series of recent results by Sen and collaborators into one narrative with clarified conventions, for example the fixed-charge-lattice convention and the 2-form vs axion integration variable. That makes it a useful reference.

    Strengths

  • A clear, pedagogical derivation of the log corrections with explicit g_s and λ_S dependence.
  • A nontrivial cross-check across N=8, N=4, N=6 and N=2 theories.
  • A conceptual framework for when a string-scale cutoff is forced, beyond string-theory input.
  • Concrete predictions for unknown microscopic results.
  • The appendix generalizes the constant-scalar attractor solution.

    • Limitations

  • It is a review, so there is limited new technical content.
  • Only logarithmic terms are addressed. The paper does not fully compute one-loop effects, treat non-supersymmetric holes, or handle full string-theory backgrounds with RR flux.
  • The criteria for a UV completion are assumptions, and N=2 requires extra input.
  • Some arguments are heuristic and at the level of scaling.
  • The conclusion that "naive gravitational path integral fails beyond low orders" is likely to be disputed by the localization community, since the disagreement is not independently resolved here.
  • Much of the work is tied to a narrow class of supersymmetric black holes.

    • Overall

    The paper is a high-quality, authoritative synthesis from a leading figure. It will be cited as a reference for the UV-sensitivity argument and for the stringy-cutoff conclusion. Its novelty is moderate because the results are mostly published. Its audience is limited to formal high-energy theory.

    Rating:6.5/ 10
    Significance 6.5Rigor 6.5Novelty 5.5Clarity 7.5

    Generated Oct 21, 2026

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