A Formula for Connected Bosonic $n$-Point Functions for the BKP Hierarchy

Jian Zhou; Xuhui Zhang

REVIEW 3 major objections 5 minor 30 references

A Formula for Connected Bosonic $n$-Point Functions for the BKP Hierarchy

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The connected bosonic n-point functions of a BKP tau-function can be computed by embedding the BKP hierarchy into the KP hierarchy, and the resulting formula provably agrees with the existing BKP formula.

desk verdict The paper's central equivalence theorem is not self-consistent as printed: the n=1 case contradicts the authors' own (3.25), and the z1...zn factor vanishes between (4.2) and (4.8). read the letter →

arxiv 2506.00451 v1 pith:ZJL3ZIWN submitted 2025-05-31 math-ph math.MP

Xuhui Zhang , Jian Zhou This is my paper

classification math-phmath.MP MSC 37K10

keywords KPhierarchyBKPconnectedn-pointfunctionstau-functionsaffinecoordinatesboson-fermioncorrespondenceSatoGrassmannianintegrablehierarchies

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 establishes an embedding route to the connected bosonic $n$-point functions of BKP tau-functions: it exploits the classical identity $\tau^{KP}(t_1,0,t_3,0,\ldots)=\tau^{BKP}(t_1,t_3,\ldots)^2$ and rewrites the known KP formula for connected $n$-point functions in terms of BKP affine coordinates. The formula that comes out is visibly different from the one obtained earlier by the type-B boson-fermion correspondence, and the paper's main result is a proof that the two formulas are equivalent. A reader should care because the embedding method does not need the type-B boson-fermion correspondence at all; it uses only the KP machinery plus a coordinate conversion, and it is intended to transfer to other reductions of KP, where such equivalence checks are expected to be equally technical.

What carries the argument

The machinery has three parts. The first is the embedding relation $\tau^{KP}(t_1,0,t_3,0,\ldots)=\tau^{BKP}(t_1,t_3,\ldots)^2$, which makes $F^{BKP}=\frac12 F^{KP}$ after the even times are set to zero. Here the affine coordinates are the expansion coefficients specifying the point of the Sato Grassmannian carried by the tau-function. The second is an explicit conversion of affine coordinates: the paper expresses the generating series $A^{KP}(w,z)$ of the KP affine coordinates in terms of the BKP affine coordinates $a^{BKP}_{m,n}$ and the diagonal coefficients $a^{BKP}_{m,0}$ (Theorem 3.1(1)), and derives the antisymmetrization relation $A^{BKP}(w,z)=\frac14(zA^{KP}(w,-z)-wA^{KP}(z,-w))$. The third is combinatorial: the KP $n$-point formula is a cycle sum of $\hat A^{KP}$ factors, and a sign-averaging projection selects the odd time variables that BKP depends on. The equivalence of the two BKP formulas is ultimately decided by Lemma 4.1, an algebraic identity relating two products built from formal series $s$ and $t$; its proof in the appendix is a long induction, and it is the load-bearing step of the comparison.

What would settle it

Expand both sides of equation (1.10) for $n=3$ and $n=4$ with a truncated BKP affine-coordinate choice, keeping a few nonzero $a^{BKP}_{m,n}$ and $a^{BKP}_{m,0}$, and compare coefficients in the variables $z_i$ up to a fixed order. A single mismatching coefficient would refute Theorem 1.2. A cheaper check is to verify Lemma 4.1 for $k=2,3$ with generic truncated series $s$ and $t$.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.2: the new embedding-derived formula (1.3) for the connected bosonic $n$-point functions of a BKP tau-function is equivalent to the earlier type-B formula (1.8). Equivalently, Theorem 3.1(3) gives the generating series $$\sum_{i_1,\ldots,i_n\ge 1,\, i_k\text{ odd}}\frac{\partial^n $F^{{BKP}}$}{\partial t_{i_1}\cdots\partial t_{i_n}}\Big|_{t=0} $z_1^{{-i_1-1}}$\cdots $z_n^{{-i_n-1}}$$$ as a sum over $n$-cycles of products of $\hat A^{KP}$ factors, with $A^{KP}$ built from BKP affine coordinates by (1.1), up to an explicit two-point correction. The proof shows that this expression coincides with the cycle sum built from $\hat A^{BKP}$, and the final comparison is reduced to a purely algebraic identity, Lemma 4.1, for two formal series $s$ and $t$ with $s(y,x)=-s(x,y)$.

Load-bearing premise

The comparison rests entirely on Lemma 4.1, a purely algebraic identity about two formal series, whose proof in the appendix is a long sign-sensitive induction that the paper does not machine-check; if that identity is wrong, the equivalence theorem falls with it.

Editorial extensions

If this is right

The two known presentations of BKP $n$-point functions are interchangeable, so computations can use whichever form is easier to evaluate.
BKP $n$-point functions can be obtained without the type-B boson-fermion correspondence: the KP formula plus the $A^{KP}$-from-$A^{BKP}$ conversion is sufficient.
The two-point correction term $-\delta_{n,2}\frac{z_1^2+z_2^2}{2(z_1^2-z_2^2)^2}$ reappears naturally from the embedding, matching its role in KdV correlator formulas.
The same embedding strategy should produce $n$-point formulas for other KP reductions, and the paper states that checking these against the known formulas is a natural but technically heavy next step.

Reading between the lines

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

Because Lemma 4.1 is not machine-checked, a cheap way to test the paper's central equivalence is to verify the identity for small $k$ with random truncated series; the paper does not report such a check.
If the equivalence holds generally, the relation $A^{BKP}(w,z)=\frac14(zA^{KP}(w,-z)-wA^{KP}(z,-w))$ may be a model for a general embedding dictionary between the affine coordinates of a KP reduction and those of KP itself.
One could test the same technique on the CKP or DKP reductions: if analogous antisymmetrization relations hold, the embedding approach would give a uniform derivation of $n$-point functions across all KP reductions, not just BKP.

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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

Main take: the central equivalence, Theorem 1.2/4.2, is not correct as stated. For n=1, the claimed identity says A_BKP(z,-z) = 1/4(A_KP(z,z)+A_KP(-z,-z)). The paper's own (3.25), plus antisymmetry of A_BKP, gives A_BKP(z,-z) = -z/4(A_KP(z,z)+A_KP(-z,-z)). Those can only agree for zero tau functions. Independently, Section 4 starts with the correct-looking identity (4.2), which includes a z1...zn factor because the Wang-Yang formula uses z^{-i} while the embedding formula uses z^{-i-1}; the restated Theorem 4.2 drops that factor. So the proof establishes a different identity from the theorem, and this failure is prior to Lemma 4.1.

What is genuinely useful: Section 3 works out the embedding of BKP into KP explicitly, giving A_KP in terms of BKP affine coordinates and a structurally new formula (3.26) for the n-point functions. This is a direct realization of Zhou's embedding proposal, and the method should transfer to other KP reductions. Lemma 4.1 is a bold combinatorial reduction; if it survives a machine check, it would be a nice result in its own right. The paper is honest about the technical difficulty and does not hide the gap behind vague statements.

Soft spots: the errors in the theorem statements are the biggest problem, but there is another layer of risk. Lemma 4.1's proof in Appendix A is long and sign-sensitive, and the OCR-corrupted text makes line-by-line verification impractical; a CAS check of the lemma and of small-n cases would settle it. The reliance on the second author's arXiv preprints for the KP formula and embedding proposal is not itself a flaw, but external confirmation would help. No data or code, so the claims rest entirely on the written proofs.

Bottom line: the paper is for specialists in integrable hierarchies and tau-function correlators. The derivation idea has value, but the central claim as printed is broken. I would not cite it in its current form. A serious referee could help the authors fix the sign and factor issues, so the paper deserves peer review rather than desk rejection, but only with the expectation of substantial revision.

Referee Report

3 major / 5 minor

Summary. The paper derives a formula for the connected bosonic n-point functions of a BKP tau-function by embedding the BKP hierarchy into the KP hierarchy. Using the relation tau_KP(t1,0,t3,0,...) = tau_BKP(t1,t3,...)^2, the authors apply Zhou's KP formula to obtain a new expression (Theorem 3.1) in terms of the affine coordinates of the BKP tau-function, converted into a KP generating series A_KP. They then claim that this formula is equivalent to the Wang--Yang formula for BKP n-point functions, a statement formalized as Theorem 1.2 and restated as Theorem 4.2. The proof of equivalence relies on a technical combinatorial lemma (Lemma 4.1) whose proof occupies Appendix A.

Significance. The embedding approach is natural and, if correct, would provide a uniform method for transferring n-point function formulas from KP to its reductions without developing a separate boson-fermion correspondence. The paper contains an explicit and internally consistent computation of the induced KP affine coordinates from BKP coordinates (Theorem 3.1(1)-(2)), which is a useful contribution. However, the central claim of the paper, the equivalence of the new formula with the Wang--Yang formula, is false as stated: it fails the one-point test. Since the main theorem is load-bearing for the paper's purpose, the present manuscript cannot be recommended for publication.

major comments (3)

[Section 4, Eqs. (4.2) and (4.8)] The claimed equivalence fails already for n=1. For n=1, the Wang--Yang sum in (2.29) equals xi(z,-z) = A_BKP(z,-z). The embedding formula (3.26) gives the generating series in z^{-i-1} as 1/4[A_KP(z,z)+A_KP(-z,-z)]; after multiplying by z to compare with the z^{-i} series in (2.29), this becomes z/4[A_KP(z,z)+A_KP(-z,-z)]. By relation (3.25), this equals A_BKP(-z,z). Since A_BKP is antisymmetric, A_BKP(z,-z) = -A_BKP(-z,z). Thus (1.10) for n=1 asserts that two opposite quantities are equal. The remark in Section 4 that A_BKP(-z,z)=1/4(zA_KP(-z,-z)+zA_KP(z,z)) and therefore the one-point functions coincide compares with the wrong object: the WY one-point function is A_BKP(z,-z), not A_BKP(-z,z).
[Section 4, Eqs. (4.2) and (4.8)] The proof in Section 4 is directed at identity (4.2), whose right-hand side contains an explicit factor z1...zn and is subsequently rewritten with factors epsilon_{sigma_i(1)} z_{sigma_i(1)} attached to each A_KP factor. The theorem as stated in (4.8), and likewise (1.10), omits this z1...zn factor. The proof therefore establishes at most a different identity from the announced theorem. This is not a cosmetic discrepancy: for n=1, (4.2) with the z1 factor gives A_BKP(-z,z), while (4.8) gives 1/4[A_KP(z,z)+A_KP(-z,-z)], neither of which equals the Wang--Yang value A_BKP(z,-z).
[Lemma 4.1 / Appendix A] Lemma 4.1 is the technical engine of the equivalence proof, and its proof in Appendix A is a long induction with many sign-sensitive rewritings, for example the manipulations leading from (A.5a) to (A.5e) and the identity (A.6). No machine-checked or computer-algebra verification is supplied, so the lemma carries residual risk. This is secondary to the two points above, but if the authors revise the theorem, the lemma and its application in (4.6) must be re-examined together with the corrected statement.

minor comments (5)

[Throughout, esp. Eqs. (1.3), (1.8), (2.23), (2.29)] The text contains corrupted symbol sequences such as '⌟⟨rro⟪⟪⟩r⟪⌟...' that make many formulas unreadable; the manuscript needs a clean typesetting pass before any resubmission.
[Abstract and keywords] There are multiple typos: 'tau-funtion', 'bosoincn-piont' in the abstract, and the keywords should be proofread.
[Section 4 heading and Theorem 3.1(1)] The heading 'The Equivalence of Two Fomulae' and the phrase 'for convinience' should be corrected; in Theorem 3.1(1), 'sesne' should be 'sense'.
[Eqs. (1.5) and (2.25)] The notation i_{x,y} is used before it is defined; it should be defined at first use, for example in Section 1.
[Section 4, definition of the map O] In the definition of the map O, the notation z_i1...z_in is ambiguous; use z_{i_1}...z_{i_n} for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the BKP n-point formula is derived from an independent KP-hierarchy result and checked against the external Wang–Yang formula; the self-citations are not circular inputs.

full rationale

The paper's central derivation is self-contained with respect to the claimed BKP result. Theorem 3.1 obtains the KP affine-coordinate generating series A_KP from BKP affine coordinates by an explicit operator computation (equations (3.8)–(3.24)), and then derives the BKP connected n-point function generating series (3.26) by applying the E-averaging procedure to the KP formula (2.23). The KP formula is cited from [Zho15a], a prior work by one of the authors, but it is a statement about KP connected n-point functions, not about BKP connected n-point functions; it does not assume the target result, so under the stated rules it counts as independent support rather than circular input. The embedding proposal [Zho15b] is also by the same author, but the actual embedding computation is carried out in this paper, not imported as a black box. The claimed equivalence with the Wang–Yang formula is tested against an external paper [WY22], and the key Lemma 4.1 is proved from scratch for arbitrary formal series s and t in Appendix A, with the BKP case obtained by specialization; there is no fitted parameter, no imported uniqueness theorem, and no renaming of the target formula as a new result. The sign and prefactor discrepancies noted by a skeptical reader (for example the missing z_1...z_n factor when comparing (4.2) with the printed (4.8), or the one-point sign issue) are potential correctness defects in the theorem as stated, but they are not instances of circular reasoning: they do not make the claimed formula coincide with an input by construction. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; the affine coordinates a_{n,m} are inputs, not fitted constants. The paper relies on standard integrable-hierarchy machinery and two prior results ([Zho15a] KP formula, [DJKM82]/[You89] BKP-to-KP embedding), with no new postulates beyond the technical Lemma 4.1.

assumptions (5)

standard math Boson-fermion correspondence for KP and BKP hierarchies and the standard fermionic Fock space formalism.
Invoked throughout Section 2 as the framework; taken from [MJD00], [JM83], [DJKM82].
domain assumption Embedding identity tau_KP(t1,0,t3,0,...)=tau_BKP(t1,t3,...)^2.
Used at eq. (3.1) and Theorem 1.1 to convert BKP log derivatives into half of KP log derivatives; cited to [DJKM82] and [You89].
domain assumption KP connected n-point function formula, [Zho15a, Theorem 5.3] (Theorem 2.1 here).
The new BKP formula is obtained by applying the projection operator E to this prior formula; the current paper does not prove it.
domain assumption Affine coordinates of BKP satisfy a_{n,m}=-a_{m,n} and a_{0,0}=0.
Assumed at eq. (2.19) and used in the coordinate conversion (3.8)-(3.12) and in Theorem 3.1(2).
standard math Glauber or Baker-Campbell-Hausdorff formula and the vanishing of the relevant commutator chains in the fermionic algebra.
Used in Section 3 to write e^{X+Y}|0> as e^{X-[X,Y]/2}|0>; also the anticommutation relations (2.2).

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Cite this review

Pith. "Pith review of A Formula for Connected Bosonic $n$-Point Functions for the BKP Hierarchy." pith.science (2026). https://pith.science/paper/ZJL3ZIWN

@misc{pith2026250600451,
  author       = {Pith},
  title        = {Pith review of: A Formula for Connected Bosonic \(n\)-Point Functions for the BKP Hierarchy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJL3ZIWN}},
  note         = {Machine review of arXiv:2506.00451}
}

abstract

We present a formula for the connected $n$-point functions of a tau-funtion of the BKP hierarchy by embedding BKP hierarchy into KP hierarchy. This formula is different from the one given by Wang and Yang. We prove that these two formulae are equivalent.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Buryak--Okounkov formula for the n -point function and a new proof of the Witten conjecture

Alexander Alexandrov, Francisco Hern \'a ndez Iglesias, and Sergey Shadrin. Buryak--Okounkov formula for the n -point function and a new proof of the Witten conjecture. International Mathematics Research Notices , 2021(18):14296--14315, 2021

work page 2021

[2] [2]

KdV solves BKP

Alexander Alexandrov. KdV solves BKP . Proceedings of the National Academy of Sciences , 118(25):e2101917118, 2021

work page 2021

[3] [3]

Correlation functions of the KdV hierarchy and applications to intersection numbers over M _ g,n

Marco Bertola, Boris Dubrovin, and Di Yang. Correlation functions of the KdV hierarchy and applications to intersection numbers over M _ g,n . Physica D: Nonlinear Phenomena , 327:30--57, 2016

work page 2016

[4] [4]

Simple Lie algebras, Drinfeld-Sokolov hierarchies, and multi-point correlation functions

Marco Bertola, Boris Dubrovin, and Di Yang. Simple Lie algebras, Drinfeld-Sokolov hierarchies, and multi-point correlation functions. Moscow Mathematical Journal , 21(2):233--270, 2021

work page 2021

[5] [5]

Integrals of -classes over double ramification cycles

Alexandr Buryak, Sergey Shadrin, Loek Spitz, and Dimitri Zvonkine. Integrals of -classes over double ramification cycles. American Journal of Mathematics , 137(3):699--737, 2015

work page 2015

[6] [6]

Geometric interpretation of Zhou’s explicit formula for the Witten--Kontsevich tau function

Ferenc Balogh and Di Yang. Geometric interpretation of Zhou’s explicit formula for the Witten--Kontsevich tau function. Letters in Mathematical Physics , 107:1837--1857, 2017

work page 2017

[7] [7]

Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method

Mattia Cafasso and Di Yang. Tau-functions for the Ablowitz--Ladik hierarchy: the matrix-resolvent method. Journal of Physics A: Mathematical and Theoretical , 55(20):204001, 2022

work page 2022

[8] [8]

Transformation groups for soliton equations: IV

Etsuro Date, Michio Jimbo, Masaki Kashiwara, and Tetsuji Miwa. Transformation groups for soliton equations: IV . A new hierarchy of soliton equations of KP -type. Physica D: Nonlinear Phenomena , 4(3):343--365, 1982

work page 1982