The Komar charges of type II supergravity and supersymmetric localization

Gabriele Barbagallo; Tom\'as Ort\'in

REVIEW

The Komar charges of type II supergravity and supersymmetric localization

Gabriele Barbagallo, Tomás Ortín

High Energy Physics - Theory · Submitted 7 Oct 2026

Pith's verdict: Careful new Komar charge computations for type II supergravity, but the headline vanishing theorem rests on one asserted, unproved differential equation — send it to review and demand the derivation.

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arXiv ID2610.10045 (version 3)

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CitationGabriele Barbagallo, Tom\'as Ort\'in. "The Komar charges of type II supergravity and supersymmetric localization." arXiv:2610.10045 (2026). https://pith.science/paper/ODWBIMQL

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Editor's note

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Two things to know. The charge derivations in Sections 2–5 are careful, new, and worth having. But the vanishing theorem (6.14) sits on a single asserted equation: (6.10) is introduced with "Using the KSEs it can be shown that" and never proved, even though the introduction admits this differential equation for dK^(1) is not available in the literature. The rest of the chain — Hodge dualization, momentum-map identifications (6.9), duality constraints — is written out. The proof of the theorem is effectively the proof of (6.10). That is a genuine missing link, and the reader's CONDITIONAL verdict is right.

Credit where it's due. This is the first generalized Komar charge computation for ten-dimensional IIA, massive IIA, IIB, and the democratic formulation, and the Noether-Wald charges, momentum-map equations, and Page-charge rewritings are worked through with the key steps displayed. I did not find internal inconsistencies in Sections 2–5, and the unified result — the same form K = −e^{−2φ}⋆dk + P_B ∧ e^{−2φ}⋆H + ½ P∧⋆G in all four formulations — is useful. Identifying momentum maps with spinor bilinears (6.9) is a clean step. The citation pattern is fine; heavy citation of the program's own prior work is natural when you are extending that program. The authors are also honest about what they cannot claim: the equivariant localization connection stays motivational, and they say plainly they do not know why the vanishing property should hold in general.

On the gap itself: the first-order equations quoted from Ref. [32] give the Killing property and the momentum-map equations, but not dK^(1). Deriving (6.10) means differentiating the Killing spinor equations and doing Fierz-type rearrangements, exactly the kind of calculation where a wrong sign or coefficient changes which combination of G∧⋆Ω and Ω∧⋆G is matched, and the charge would fail to vanish by a boundary term. So this is not a cosmetic omission. It is also not evidence the theorem is wrong — the structure of (6.10) is precisely what the charge needs — but plausibility is not a proof.

Separately: the manuscript needs a proofreading pass ("The, we show" in the abstract, a doubled "study", garbled k=0 subscripts in (2.79) and (3.41)). Nothing load-bearing there.

For whom: people doing black p-brane thermodynamics and anyone connecting Komar charges to supersymmetric localization. It deserves a serious referee, with the report conditional on an appendix deriving (6.10). I would cite the charge formulas now; I would wait for the appendix before relying on the vanishing theorem.

Referee report

Significance

The Komar-charge computations are a solid, detailed contribution: the momentum-map formalism with compensating gauge transformations is applied consistently, the Noether identities are checked, the Page-charge and potential rewritings are explicit, and the democratic formulation reproduces the separate IIA/massive-IIA/IIB results after the duality constraints are imposed. No free parameters are introduced: the momentum-map identifications are fixed by supersymmetry rather than fitted. If the vanishing theorem holds, it extends the observation of Ref. [31] to ten dimensions and provides a sharp structural property of supersymmetric solutions that can be tested on explicit solutions (modulo the divergence caveat the authors note for brane solutions) and that may ultimately bear on equivariant localization. The paper is candid about its limitations, including the ad hoc status of the compensating-gauge-transformation construction of δξ (footnote 12) and the statement in the introduction that the key differential equation is not available in the literature.

Major objections (2)

[Sec. 6, Eq. (6.10)] The vanishing theorem Eq. (6.14) rests entirely on the differential identity Eq. (6.10), which is introduced with the sentence 'Using the KSEs it can be shown that' and is not derived. The introduction (p. 4) explicitly states that this differential equation 'does not seem to be directly available in the literature [32]', so the identity is not a quoted result but an unproved assertion. Because the passage from (6.10) to (6.13) is just the Hodge-duality identity (6.12), and (6.13) is identified with K[K^(1)] through (6.9) and (6.7), the claim (6.14) is equivalent to (6.10): a sign or coefficient error in (6.10) would change the relative coefficient of the G∧⋆Ω and Ω∧⋆G terms and would leave a nonzero boundary term in (6.14). The authors should supply the derivation from the Killing spinor equations (6.2a)–(6.2b), including the Fierz rearrangements, the chirality conventions (6.1), the handling of the mass parameter G^(0)=m for massive IIA, and the restriction to the IIB sector with G^(5)=⋆G^(5).
[Sec. 6, Eqs. (6.10) and (6.13)] As printed, the equations are not well formed. With the contraction operator defined in Eq. (6.11), the term ı_{G^(n)}⋆Ω^(n+2) is defined only when 8−n>n and has degree 8−2n, while dK^(1) is a 2-form, so (6.10) is not a homogeneous identity as written and some rank projection or a different contraction convention must be intended. Likewise, in (6.13) the term shown as '\tildeK^(1)H' has degree 4 as an ordinary wedge product, whereas ⋆dK^(1) has degree 8; presumably the intended term is \tildeK^(1)∧⋆H, the Hodge dual of ı_{\tildeK^(1)}H via (6.12), but the displayed formula lacks the star. The authors should define the contractions and specify which homogeneous components are used at each step; this is part of the missing derivation of (6.10), but it is separately relevant because (6.10)–(6.13) are not checkable as typeset.

Minor objections (4)

[Abstract] The abstract ends mid-sentence ('...the momentum maps of the different fields of the theory are') and contains the typo 'The, we show that'; it should be completed and proofread.
[Section 1] There are several typos: 'I general' should be 'In general'; footnotes 1 and 2 are garbled ('See Ref. [6] a comprehensive approach.'); 'objects that naturally occur equivariant localization approach' is missing words; 'we have chosen to study study first' has a duplicated word; and the sentence 'some of the objects ... seem to be related to the are related to the generalized Komar charge' is broken.
[Title] Given that the introduction states that the relation between the equivariant localization approach and the generalized Komar charge 'has not yet been made precise', the title's phrasing 'and supersymmetric localization' promises more than the paper delivers; the title or the framing should be adjusted.
[Sections 3.1.3 and 7] Section 3.1.3 contains the grammatical slip 'Furthermore Proceeding in exactly in the same way', and Section 7's 'Thus form should immediately allow us' should read 'This form'.

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The paper

The paper claims

For supersymmetric solutions of the ten-dimensional type II supergravities, the generalized Komar charge of the supersymmetric Killing vector vanishes identically.

The reading

This paper computes the generalized Komar charge — the conserved 8-form associated with a spacetime symmetry in a gravitational theory, the analogue of the Komar mass of general relativity — for the ten-dimensional $\mathcal{N}=2$ supergravity theories: IIA, Romans' massive IIA, and IIB. In each theory the charge is on-shell closed for any Killing vector and can be rewritten in terms of Page charges and horizon-closed potentials, which is the form needed for Smarr formulas and generalized zeroth laws. The central result concerns supersymmetric solutions: when the Killing vector is the supersymmetric Killing vector $K^{(1)}$ built from a Killing spinor, and when the momentum maps are identified with spinor bilinears by Eq. (6.9), the generalized Komar charge vanishes identically, $K[K^{(1)}] = 0$. The proof passes through a new differential equation for $K^{(1)}$, Eq. (6.10), which the authors say follows from the Killing spinor equations but do not derive; Hodge-dualizing it and applying duality constraints turns it into the vanishing of the charge. The authors do not yet see a general reason for this vanishing, and they note that if the link to equivariant localization is confirmed, the identity would constrain the Euclidean action of supersymmetric solutions.

What carries the argument

The machinery is the generalized Komar charge: an 8-form $K[k]$ defined for any Killing vector $k$ as the difference between the Noether-Wald charge and a term built from the on-shell Lagrangian. In manifestly gauge-invariant form it reads $K[k] = -e^{-2\phi}\star(e^a \wedge e^b)P_{k\,ab} + P^{(1)}_{B\,k}\wedge(e^{-2\phi}\star H) + \tfrac{1}{2}P_k\wedge\star G$, where $P_{k\,ab}$, $P^{(1)}_{B\,k}$, and $P_k$ are the Lorentz, Kalb-Ramond, and RR momentum maps. The paper rewrites this charge as a combination of Page charges $\tilde{H}$ and $\star\tilde{G}$ with potentials $\Phi$ closed on Killing horizons, preparing it for black-hole thermodynamics. The vanishing theorem rests on Eq. (6.10), a first-order differential equation for the supersymmetric Killing vector whose Hodge dual, combined with Eq. (6.7) and momentum-map identification (6.9), is term by term the charge $K[K^{(1)}]$.

Extended reading notes

Core claim

The central claim is Eq. (6.14): for every supersymmetric solution of the $\mathcal{N}=2$, $d=10$ IIA, massive IIA, and IIB theories, the on-shell closed generalized Komar charge associated with the supersymmetric Killing vector vanishes identically, $K[K^{(1)}] = 0$, provided the momentum maps are fixed by Eq. (6.9) as specific Killing spinor bilinears. With that choice, the Killing spinor equations imply the momentum-map equations, so $K^{(1)}$ is a symmetry of all fields. The proof uses a first-order differential equation for $K^{(1)}$, Eq. (6.10), which the authors state is not directly available in the literature and is asserted to follow from the Killing spinor equations. After Hodge dualization and imposition of the electric-magnetic duality constraints, that equation becomes exactly the equality of the generalized Komar charge with zero. The same conclusion holds in all four formulations constructed in the paper (massless IIA, massive IIA, IIB, and the democratic formulation), which all reduce to the same expression (5.33) once the duality constraints are imposed.

Load-bearing premise

The entire vanishing result rests on Eq. (6.10), a differential equation for the supersymmetric Killing vector that the paper records as following from the Killing spinor equations but does not derive; if that equation is false, or silently assumes particular Fierz identities, spinor chiralities, or duality constraints, the conclusion $K[K^{(1)}] = 0$ collapses.

Editorial extensions

If this is right

Each of the charges (2.73), (3.42), (4.26), and (5.33) is on-shell closed for any Killing vector, so every type II theory carries a conserved 8-form charge for each spacetime symmetry.
Every charge can be rewritten in Page-charge form with potentials closed on Killing horizons, the input required for deriving Smarr formulas and restricted zeroth laws.
For supersymmetric solutions, $K[K^{(1)}] = 0$ gives a new route to proving saturation of BPS bounds in ten-dimensional type II supergravity.
In massive IIA, the KR momentum map becomes exact on horizons, so the KR charge and the anomalous $\Phi^{(0)}$ term contribute only total derivatives; the massive theory behaves like a Stückelberg system whose would-be charges drop out.
In the democratic formulation, imposing duality constraints and keeping only even- or odd-rank RR fields recovers exactly the IIA and IIB charges, confirming consistency of the separate derivations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Eq. (6.10) is asserted rather than derived: a stand-alone derivation from the Killing spinor equations, or a supersymmetric solution violating it, would settle whether $K[K^{(1)}] = 0$ is a theorem or an additional assumption.
If the vanishing charge corresponds to equivariant localization, the on-shell Euclidean action of a supersymmetric solution should be determined by boundary data at the fixed points of $K^{(1)}$; the identity in this paper would then be the integration-constant constraint.
The same construction is testable in other supergravity theories: wherever a supersymmetric Killing vector obeys an analogue of Eq. (6.10) with spinor-bilinear momentum maps, the generalized Komar charge should vanish.
Since generalized Komar integrals typically diverge for brane solutions, the Page-charge rewriting may give the finite combination needed to assign conserved charges to black $p$-branes and derive their Smarr relations.

Original abstract

We compute the on-shell closed, generalized Komar charges associated to an arbitrary Killing vector of the two 10-dimensional $\mathcal{N}=2$ supergravity theories which are the low-energy effective field theories of the type~IIA and IIB superstrings. We study different rewritings of this charge in terms of Page charges and forms with special properties that are used in the derivation of thermodynamical relations in black $p$-brane spacetimes. The, we show that, when the Killing vector is the supersymmetric Killing vector built as a bilinear of a Killing spinor, the momentum maps of the different fields of the theory are

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We compute the on-shell closed, generalized Komar charges associated to an arbitrary Killing vector of the two 10-dimensional $\mathcal{N}=2$ supergravity theories which are the low-energy effective field theories of the type~IIA and IIB superstrings. We study different rewritings of this charge in terms of Page charges and forms with special properties that are used in the derivation of thermodynamical relations in black $p$-brane spacetimes. The, we show that, when the Killing vector is the supersymmetric Killing vector built as a bilinear of a Killing spinor, the momentum maps of the different fields of the theory are

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We compute the on-shell closed, generalized Komar charges associated to an arbitrary Killing vector of the two 10-dimensional $\mathcal{N}=2$ supergravity theories which are the low-energy effective field theories of the type~IIA and IIB superstrings.	contribution 0.97	0.95	0.17	asserted 1.00
s1	We study different rewritings of this charge in terms of Page charges and forms with special properties that are used in the derivation of thermodynamical relations in black $p$-brane spacetimes.	contribution 0.90	0.85	0.07	asserted 0.95
s2	The, we show that, when the Killing vector is the supersymmetric Killing vector built as a bilinear of a Killing spinor, the momentum maps of the different fields of the theory are	result 0.92	0.87	0.28	asserted 0.79

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References

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Local supersymmetry implies momentum map/Gauntlett-Pakis equations and vanishing generalized Komar forms
hep-th 2026-10 accept novelty 6.0 of 10

The local supersymmetry algebra implies all first-order equations satisfied by Killing spinor bilinears and the vanishing of the generalized Komar charge.
On the quantum statistical relation and the covariant phase space Hamiltonian
hep-th 2026-10 accept novelty 6.0 of 10

A closed expression for the covariant phase space Hamiltonian in gauge theories is derived and used to express the Euclidean action as a boundary integral, yielding the quantum statistical relation for black holes.

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