The spectral Einstein functional for the Dirac operator with torsion

Jin Hong; Yong Wang

REVIEW 3 major objections 5 minor 3 cited by

The spectral Einstein functional for the Dirac operator with torsion

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Explicit local formula found for the spectral Einstein functional with torsion.

desk verdict Main Einstein functional formula is plausible and internally consistent, but (3.4) has a normalization typo, and the torsion sector wants a cross-check. read the letter →

arxiv 2412.08028 v4 pith:JXYVCA63 submitted 2024-12-11 math.DG math-phmath.MP

Jin Hong , Yong Wang This is my paper · ORCID

classification math.DGmath-phmath.MP MSC 53C2758J40

keywords DiracoperatorwithtorsionspectralEinsteinfunctionalnoncommutativeresiduespinmanifoldsymbolexpansionLichnerowiczformula

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper aims to compute, on a closed even-dimensional spin manifold, the spectral Einstein functional built from the Dirac operator with skew torsion $D_T$: the noncommutative residue of $c(v)(c(w)D_T + D_T c(w))D_T^{1-n}$. It claims that this functional is exactly a local integral whose integrand is a combination of the Einstein tensor contracted with $v,w$, a torsion-norm term, a torsion cross term, a covariant-divergence term of the torsion, and a term involving the Levi-Civita derivative of $w$. If the computation is right, spectral geometry with torsion supplies a concrete, computable analogue of the gravitational action that reduces to the torsion-free case when $T=0$.

What carries the argument

The central machinery is the noncommutative residue, a trace over the unit cosphere bundle of the $-n$-order symbol of a pseudodifferential operator; the spectral Einstein functional is defined through it. The computation is carried by the asymptotic symbol expansion of $\Delta_{T,E}^{-m}$ with $\Delta_{T,E}=D_T^2$, imported from references [1] and [13], together with the torsion Lichnerowicz formula $D_T^2 = -g^{ab}(\nabla_{\partial_a}\nabla_{\partial_b}-\nabla_{\nabla^L_{\partial_a}\partial_b}) + E$, where $E=\frac{3}{2}dT+\frac{1}{4}s-\frac{3}{4}\|T\|^2$. The Clifford trace identities in the appendix reduce products of Clifford symbols to metric contractions, and the composition formula for symbols organizes the product $c(v)D_T c(w)D_T D_T^{-n}$ into six symbol-order terms that are integrated over the unit sphere.

What would settle it

On a flat $2m$-dimensional torus with constant skew-symmetric torsion $T$ and parallel unit vector fields $v,w$, the claimed formula reduces to $\frac{2^m 2\pi^m}{\Gamma(m)}\int_M \bigl(-\frac{9}{2}\|T\|^2 g(v,w)+\frac{9}{2}\sum_{j,l}T(v,e_j,e_l)T(w,e_j,e_l)\bigr)$. An independent Fourier-mode calculation of $\mathrm{Wres}(c(v)(c(w)D_T+D_T c(w))D_T^{-n+1})$ on this torus would either reproduce that number or contradict Theorem 3.3.

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Extended reading notes

Core claim

On an $n=2m$ dimensional closed spin manifold with torsion, the paper's Theorem 3.3 asserts $A_{D_T} = \mathrm{Wres}(c(v)c(w)D_T^{-n}) = -\frac{2^m 2\pi^m}{\Gamma(m)}\int_M g(v,w)\,d\mathrm{Vol}_M$ and $B_{D_T} = \mathrm{Wres}(c(v)(c(w)D_T + D_T c(w))D_T^{-n+1}) = \frac{2^m 2\pi^m}{\Gamma(m)}\int_M \bigl(-\frac{1}{6}G(v,w) - \frac{9}{2}\|T\|^2 g(v,w) + \frac{9}{2}\sum_{j,l} T(v,e_j,e_l)T(w,e_j,e_l) + \frac{3}{2}\sum_a \nabla_{e_a}(T)(e_a,v,w) + 3\sum_j T(v,\nabla^L_{e_j}w,e_j)\bigr)\,d\mathrm{Vol}_M$, where $G(v,w)=\mathrm{Ric}(v,w)-\frac{1}{2}s\,g(v,w)$. The proof splits $B_{D_T}$ into $B_1=\mathrm{Wres}(c(v)c(w)D_T^{-n+2})$ and $B_2=\mathrm{Wres}(c(v)D_T c(w)D_T D_T^{-n})$, computes each $-2m$-order symbol using the pseudodifferential composition formula and Clifford trace identities, and sums all contributions to reach the displayed formula.

Load-bearing premise

The load-bearing premise is that the symbol expansion of $(D_T^2)^{-m}$ and the torsion Lichnerowicz formula with $E=\frac{3}{2}dT+\frac{1}{4}s-\frac{3}{4}\|T\|^2$, both imported from prior references without re-derivation, are correct; if either is wrong, the final integrand changes.

Editorial extensions

If this is right

The functional is local: the residue depends only on the metric, the torsion tensor, and its first covariant derivatives at each point, not on global topological data.
Setting $T=0$ recovers the torsion-free spectral Einstein functional, so $B_{D_T}$ specializes to a multiple of $\int_M G(v,w)\,d\mathrm{Vol}_M$ in the ordinary Dirac case.
On a flat manifold with constant torsion and parallel $v,w$, only the torsion-norm and torsion-cross terms survive, so the formula predicts a purely torsion-driven residue.
The same computation yields the metric functional $A_{D_T}$ as a universal multiple of $\int_M g(v,w)\,d\mathrm{Vol}_M$, independent of the torsion.
Definition 3.7 turns the computed expression into a candidate spectral Einstein functional with torsion on every closed even-dimensional spin manifold.

Reading between the lines

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

A direct test the paper does not run: compute both sides on a flat torus with constant torsion via Fourier modes; if the numbers disagree, the imported symbol expansion is the first place to recheck.
The derivative term $\sum_a \nabla_{e_a}(T)(e_a,v,w)$ and the connection term $\sum_j T(v,\nabla^L_{e_j}w,e_j)$ are not obviously symmetric in $v,w$, so $B_{D_T}$ may be an asymmetric bilinear form; the paper does not discuss this.
The same symbol-expansion strategy should extend to the Hodge–Dirac operator with torsion or to manifolds with boundary, where the analogous spectral Einstein functional is not yet computed.
Because the torsion enters through three visibly different channels (pointwise norm, cross term, and derivative terms), the formula suggests that spectral measurements of the residue could in principle separate intrinsic torsion from its gradient, an inversion problem the paper leaves untouched.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the paper likely gets the right closed formula for the spectral Einstein functional with torsion in even dimensions. The B1+B2 decomposition is coherent, the Clifford trace appendix is detailed, and the T=0 limit reproduces the standard -1/6 G(v,w) term. But the companion metric functional (3.4) has a normalization error: it states -2m 2π^m/Γ(m), whereas the paper's own conventions (tr[id]=2^m, unnormalized sphere measure) give -2^m 2π^m/Γ(m). For m=2 they agree; for m≠2 they don't. The stress-test note's claim that the leading symbol yields (3.4) exactly is mistaken. This looks like a typo (2m instead of 2^m), and it does not threaten the B_DT result.

What's new: the explicit even-dimensional no-boundary formula for B_DT with torsion. It's a natural extension of the program in [1,13,23], and the new content is the assembly. The assembly is done carefully: the m-dependent terms cancel in the final sum, which is a good sign, and the known zero-torsion limit checks out.

Soft spots: besides the (3.4) typo, the torsion part of B_DT rests entirely on imported Lemma 2.1 and the torsion Lichnerowicz formula in (2.2)-(2.3). No independent check is reported, not even the 4D case or a flat-torsion limit. A wrong coefficient in those imports would change every torsion term while leaving T=0 intact. The paper also has many typos and minor notational slips that don't affect the math but make verification harder.

Who it's for: people working on spectral actions and noncommutative residues with torsion. It's a solid technical contribution once the metric functional typo is fixed and a cross-check is added. I'd send it to a referee—the main result deserves scrutiny—but I'd tell the referee to verify (3.4) and ask the authors to check a known limit.

Referee Report

3 major / 5 minor

Summary. The paper computes the Wodzicki-residue functionals A_DT and B_DT for the Dirac operator with torsion D_T on a closed even-dimensional spin manifold. Theorem 3.3 states that A_DT = -2^m (2π^m/Γ(m)) ∫_M g(v,w) dVol and that B_DT = Wres(c(v)(c(w)D_T + D_T c(w))D_T^{-n+1}) is given by an explicit local integral involving the Einstein tensor G(v,w), the torsion norm squared, the bilinear term Σ T(v,e_j,e_l)T(w,e_j,e_l), the covariant derivative ∇T, and a term T(v,∇^L w,·). The proof splits B_DT into B1 = Wres(c(v)c(w)D_T^{-n+2}) and B2 = Wres(c(v)D_T c(w)D_T D_T^{-n}), relies on the symbol expansion of Δ_{T,E}^{-m} imported from [1] and on the torsion Lichnerowicz formula (2.2)–(2.3) from [13], and supplies an appendix of Clifford trace identities.

Significance. If Theorem 3.3 is correct, the paper gives a complete local description of the spectral Einstein functional for Dirac operators with torsion in arbitrary even dimension, generalizing the torsion-free result of [1] and matching the expected Einstein term when T=0. The computation is systematic and includes machine-checkable-style trace identities; there are no fitted parameters and no circular assumption of the final formula. The main risk is that every torsion-dependent coefficient in Theorem 3.3 is obtained by substituting (2.3), (2.7), (2.8) into the imported Lemma 2.1, so an incorrect sign or normalization in those inputs would change the final formula while preserving the torsion-free limit.

major comments (3)

[§2, Lemmas 2.1–2.2 and Theorem 3.3] The torsion coefficients in Theorem 3.3 all flow from Lemma 2.1, imported from [1], and from the torsion Lichnerowicz formula (2.2)–(2.3), imported from [13]. The substitution of (2.3), (2.7), and (2.8) into Lemma 2.1 is stated as Lemma 2.2 without a detailed check. Because a wrong sign or factor in T_a, T_ab, or E would change every torsion term in the final formula while leaving the T=0 limit unchanged, this is load-bearing. I request either a full derivation of Lemma 2.2 from the definitions of D_T and Δ_{T,E}, or an independent cross-check, for example comparison with the 4-dimensional computation in [13] and a flat-torsion test.
[§3, Part II-3-H, Eq. (3.58)] The dT contribution to B2 is discarded using the identity Σ_{i<j<k<t} tr(c(v)c(e_f)c(w)c(e_f)c(e_i)c(e_j)c(e_k)c(e_t)) = 0, stated in (3.58) without proof. This identity is not an immediate consequence of Lemma A.1 for general m, since products of six Clifford generators can have nonvanishing trace when m>2. Because this is the only place where the dT part of E enters the computation, a proof or a correct derivation is needed before the final formula can be regarded as fully verified.
[§3, Lemma 3.4 and Eq. (2.4)] The subprincipal symbol σ0(D_T) is imported from [13] with no proof, and its connection-term sign is not reconciled with the convention in (2.4), where ∇_{∂_a} = ∂_a + (1/4) Σ ⟨∇^L_{∂_a} e_s,e_t⟩ c(e_s)c(e_t). The curvature contribution to σ0(AB) in Eq. (3.26), and hence the metric part of BDT, depends on this sign. The authors should state explicitly how Lemma 3.4 follows from the conventions of Section 2, or derive σ0(D_T) directly.

minor comments (5)

[Theorem 3.3 and Eqs. (3.4), (3.5)] The displayed normalization `-2m 2π^m / Γ(m)` is ambiguous in the text; if it is intended to be `-2^m 2π^m / Γ(m)`, the superscripts should be typeset clearly, since the same expression later appears as `2 m 2πm / Γ(m)`.
[§3, Part II, before Eq. (3.25)] The text says one needs to compute ∫ tr[σ_{-2m}(P1 D_T^{-2m})], but the quantity is AB D_T^{-2m}, not P1 D_T^{-2m}; this is a typo that should be corrected.
[§3, Eq. (3.29)] The term 3/2 Σ ∂T/∂x_j c(v)c(dx_j)c(ew)c(e_f)c(e_α)c(e_β) appears to contain a typo: `c(ew)` should presumably be `c(w)`, with the ∂w term handled separately in the following term.
[Throughout] There are several typographical issues, including "there form" for "three-form" and the title "Dirac operator wit h torsion"; the paper should be carefully proofread.
[§3, Lemmas 3.4–3.6] Lemmas 3.4 and 3.5 are stated without proof or precise reference to [13]; since Lemma 3.6 depends on their signs and normalizations, a short derivation or explicit pointer would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 3.3 is a symbol computation from imported but independent lemmas, not an assumption of its own conclusion.

full rationale

The claimed spectral Einstein functional B_DT is computed, not assumed. The derivation chain is: define the Dirac operator with torsion; import the Lichnerowicz formula (2.2)-(2.3) from [13] and Lemma 2.1 from [1] for the homogeneous symbols of Delta_{T,E}^{-m}; specialize those symbols to the torsion case in Lemma 2.2; then evaluate the noncommutative residues in Parts I and II using the symbol calculus and Clifford trace identities proved in the Appendix. The target formula (3.5) appears only as the output of this computation, and Definition 3.7 names the resulting integral after the theorem has been established. There is no fitted parameter, no data subset, no uniqueness claim, and no renaming of an empirical pattern. The citation to [13] is partially self-citational because Yong Wang is an author of both papers, and the torsion Lichnerowicz formula together with Lemma 3.4 (symbols of D_T) are load-bearing for the torsion-dependent coefficients. However, these are parameter-free background lemmas with stated assumptions, and nothing in the present paper reduces Theorem 3.3 to a restatement of them; the full trace and symbol computations are carried out here. Under the rules, such external or prior-work citations constitute real evidence rather than circularity. The skeptic's concern about the correctness of the imported torsion coefficients is a correctness-risk issue, not a circularity issue.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters or invented entities. The central formula rests on imported symbol expansions and an imported Lichnerowicz formula; the trace identities are standard and proved in the appendix but not machine-checked.

assumptions (4)

domain assumption Lemma 2.1 of [1]: exact symbol expansions (2.9)-(2.11) for Δ_{T,E}^{-m}.
Invoked as Lemma 2.1 and used to produce Lemma 2.2; no proof is given in this paper.
domain assumption Torsion Lichnerowicz formula: D_T^2 = -g^{ab}(∇_a∇_b - ∇^L_{∂_a}∂_b) + E with E = 3/2 dT + 1/4 s - 3/4 ||T||^2.
Taken from [13] and used in Eq. (2.2)-(2.3); the torsion coefficients enter every symbol formula.
standard math Clifford trace identities in Lemma A.1, including traces of products of up to eight Clifford generators.
Proved in the appendix from c(X)c(Y)+c(Y)c(X)=-2g(X,Y) and cyclicity of trace; used throughout the residue computation.
domain assumption M is a compact even-dimensional spin manifold without boundary, dimension n=2m>=3, and T is a totally antisymmetric 3-form defining a metric connection.
This is the stated setting of the paper in the abstract and Section 2.

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Pith. "Pith review of The spectral Einstein functional for the Dirac operator with torsion." pith.science (2026). https://pith.science/paper/JXYVCA63

@misc{pith2026241208028,
  author       = {Pith},
  title        = {Pith review of: The spectral Einstein functional for the Dirac operator with torsion},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JXYVCA63}},
  note         = {Machine review of arXiv:2412.08028}
}

read the original abstract

In this paper, we compute the spectral Einstein functional associated with the Dirac operator with torsion on even-dimensional spin manifolds without boundary.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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The spectral Einstein functional of the nonminimal de Rham-Hodge operator on even-dimensional manifolds equals an explicit constant times the integral of Ric(u,v) minus (s/2)g(u,v).

Reference graph

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23 extracted references · 22 canonical work pages · cited by 3 Pith papers

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

Dabrowski., A

L. Dabrowski., A. Sitarz., P. Zalecki.: Spectral Metric and Einstein Functionals. Advances in Mathematics, 427, 10 9128, (2023)

work page 2023

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W ang, Y

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work page 2023

[3] [2]

Connes.: Quantized calculus and applications

A. Connes.: Quantized calculus and applications. XIth I nternational Congress of Mathematical Physics(Paris,199 4), In- ternat Press, Cambridge, MA, 15-36, (1995)

work page 1995

[4] [3]

Connes.: The action functinal in Noncommutative geom etry

A. Connes.: The action functinal in Noncommutative geom etry. Comm. Math. Phys. 117, 673-683, (1988)

work page 1988

[5] [4]

Connes, J

A. Connes, J. Lott.: Particle models and Non-commutativ e geometry. Nucl. Phys. B Proc.Supp 18B, 29-47, (1990)

work page 1990

[6] [5]

Connes., A H

A. Connes., A H. Chamseddine.: Inner ﬂuctuations of the s pectral action. J. Geom. Phys. 2007, 57(1): 1-21

work page 2007

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Kastler.: The Dirac Operator and Gravitation

D. Kastler.: The Dirac Operator and Gravitation. Comm. M ath. Phys. 166, 633-643, (1995)

work page 1995

[8] [7]

Kalau and M

W. Kalau and M. W alze.: Gravity, Noncommutative geometr y and the W odzicki residue. J. Geom. Physics. 16, 327- 344,(1995)

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