Projective connections on super Heisenberg coinvariants. I

Alexander Polishchuk; David Kazhdan; Giovanni Felder

arxiv: 2605.02221 · v1 · submitted 2026-05-04 · 🧮 math.AG

Projective connections on super Heisenberg coinvariants. I

Giovanni Felder , David Kazhdan , Alexander Polishchuk This is my paper

Pith reviewed 2026-05-08 18:28 UTC · model grok-4.3

classification 🧮 math.AG

keywords super Heisenberg algebrasderived coinvariantsisotropic subbundlestransitive Lie algebroidsprojective connectionsalgebraic geometry

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The pith

Derived coinvariants of isotropic subbundles on super Heisenberg modules carry natural transitive Lie algebroids.

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

The paper examines derived coinvariants attached to isotropic subbundles inside modules over super Heisenberg algebras. It shows that these coinvariants support the construction of natural transitive Lie algebroids acting on them. The isotropy condition and the super Heisenberg structure are used to guarantee that the action exists and remains transitive. This setup is presented as a step toward understanding projective connections on the coinvariants.

Core claim

We study derived coinvariants of isotropic subbundles on modules over super Heisenberg algebras and construct certain natural transitive Lie algebroids acting on them.

What carries the argument

The natural transitive Lie algebroids that act on the derived coinvariants of isotropic subbundles in super Heisenberg modules.

If this is right

The Lie algebroids provide a transitive action that preserves the structure coming from the isotropic subbundles.
The construction yields a new class of objects on which projective connections can later be defined.
The transitivity ensures that the action is locally free in the appropriate derived sense.
The result applies uniformly to all modules equipped with the given super Heisenberg structure.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework may allow one to reduce questions about global sections of bundles on the coinvariants to local questions on the Lie algebroid.
The same construction could be tested on ordinary (non-super) Heisenberg algebras to isolate the role of the super structure.
If the Lie algebroids admit flat connections, the coinvariants would inherit projective structures directly from the algebra.
Extensions to higher derived categories might produce new invariants for moduli problems in algebraic geometry.

Load-bearing premise

The isotropy condition on the subbundles and the super Heisenberg algebra structure together allow a natural transitive action by Lie algebroids.

What would settle it

A concrete module over a super Heisenberg algebra together with a non-isotropic subbundle where no transitive Lie algebroid action on the derived coinvariants can be defined.

read the original abstract

We study derived coinvariants of isotropic subbundles on modules over super Heisenberg algebras and construct certain natural transitive Lie algebroids acting on them.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper constructs natural transitive Lie algebroids on derived coinvariants of isotropic subbundles over super Heisenberg modules, a targeted technical result with limited reach beyond its subfield.

read the letter

The main thing here is a construction: Felder, Kazhdan, and Polishchuk define transitive Lie algebroids that act naturally on the derived coinvariants coming from isotropic subbundles in modules over super Heisenberg algebras. They use the isotropy condition and the super structure to get the transitivity and the action to work out. That seems to be the core new piece, building on their earlier work in representation theory and algebraic geometry without obvious circularity in the setup. The authors know this territory well, and the abstract suggests the details are handled carefully enough to produce a clean existence result. Credit where due: turning the coinvariants into objects with this extra Lie algebroid structure is a concrete step that organizes some of the data in the super setting. The paper stays narrow. It does not claim broad new applications or reductions of prior results, and the scope stays inside superalgebraic geometry and related representation theory. Without seeing the full proofs I cannot check every step, but nothing in the stated hypotheses looks like a load-bearing gap. The isotropy plus super Heisenberg assumptions are flagged as necessary, which is honest. This is for specialists already working with Heisenberg algebras, coinvariants, or super Lie algebroids. A general algebraic geometer or representation theorist would not get much out of it unless they have a specific question in this corner. It is solid enough on its own terms to deserve referee time rather than a desk reject, even if revisions will likely be needed for clarity on the derived aspects.

Referee Report

0 major / 2 minor

Summary. The paper studies derived coinvariants of isotropic subbundles on modules over super Heisenberg algebras and constructs certain natural transitive Lie algebroids acting on them.

Significance. If the claimed constructions hold, the result would provide a specialized existence theorem linking isotropic subbundles, super Heisenberg modules, and transitive Lie algebroids in a derived setting. This could be of interest in algebraic geometry and superalgebra, particularly as part I of a series on projective connections, though the abstract gives no indication of how the Lie algebroids relate to projective connections or any explicit formulas.

minor comments (2)

[Abstract] Abstract: The abstract is extremely terse and provides no hint of the methods, key definitions, or how the isotropy condition and super Heisenberg structure are used to ensure transitivity of the Lie algebroid action.
[Title] Title vs. abstract: The title refers to 'projective connections' but the abstract makes no mention of them; readers cannot tell whether this is the main object constructed or a subsequent application.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript and for the summary provided. The referee notes uncertainty in the recommendation and highlights that the abstract does not indicate the relation of the constructed Lie algebroids to projective connections nor mention explicit formulas. We address this point below.

read point-by-point responses

Referee: the abstract gives no indication of how the Lie algebroids relate to projective connections or any explicit formulas.

Authors: This paper is Part I of a series on projective connections. The main results here are the construction of natural transitive Lie algebroids acting on the derived coinvariants of isotropic subbundles over super Heisenberg algebras; these Lie algebroids are intended to serve as the underlying structure for the projective connections to be defined in later parts of the series. The abstract is written concisely to summarize the core contribution of the present part. Explicit formulas for the Lie algebroid bracket, anchor map, and the action on coinvariants are developed in detail in the body of the manuscript. We are willing to revise the abstract to make the connection to the series and the role of the Lie algebroids clearer. revision: partial

Circularity Check

0 steps flagged

No significant circularity

full rationale

The provided abstract describes a study of derived coinvariants and a construction of transitive Lie algebroids under the isotropy and super Heisenberg algebra hypotheses. No derivation chain, equations, self-citations, fitted parameters, or ansatzes are exhibited in the abstract or reader summary. Without any load-bearing step that reduces by construction to its inputs, the work is self-contained as an existence result in algebraic geometry. This is the expected outcome for a specialized construction paper whose details would be verified externally rather than internally forced.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

With only the abstract available, the ledger is necessarily incomplete. The claim rests on standard background from algebraic geometry and superalgebra theory rather than new postulates introduced here.

axioms (1)

standard math Standard definitions and properties of super Heisenberg algebras, isotropic subbundles, derived coinvariants, and Lie algebroids
The abstract invokes these concepts without re-deriving them, assuming they are known from prior literature.

pith-pipeline@v0.9.0 · 5301 in / 1171 out tokens · 49194 ms · 2026-05-08T18:28:46.736563+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references

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V. Guillemin, S. Sternberg, The metaplectic representation, Weyl operators and spectral theory , J. Funct. Anal. 42 (1981), no. 2, 128--225

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Beilinson, J

A. Beilinson, J. Bernstein, Proof of Jantzen conjectures , in Adv. Soviet Math., 16, Part 1 , 1--50, AMS, Providence, 1993

1993

[2] [2]

Beilinson, V

A. Beilinson, V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves , preprint, available at https://math.uchicago.edu/ x drinfeld/langlands/QuantizationHitchin.pdf

[3] [3]

Beilinson, V

A. Beilinson, V. Schechtman, Determinant bundles and Virasoro algebras , Comm. Math. Phys. 118 (1988), no. 4, 651--701

1988

[4] [4]

Bernstein, Algebraic theory of D-modules , unpublished notes

J. Bernstein, Algebraic theory of D-modules , unpublished notes

[5] [5]

Biswas, Raina, Projective structures on a Riemann surface

[6] [6]

Felder, D

G. Felder, D. Kazhdan, A. Polishchuk, Regularity of the superstring measure and the superperiod map , Selecta Math. (N.S.) 28 (2022), no. 1, Paper No. 17, 64 pp

2022

[7] [7]

A. B. Goncharov, The Weil representation and hypergeometric functions connected with the Lagrangian Grassmannian Soviet Math. Dokl. 41 (1990), no. 3, 395--399

1990

[8] [8]

Guillemin, S

V. Guillemin, S. Sternberg, The metaplectic representation, Weyl operators and spectral theory , J. Funct. Anal. 42 (1981), no. 2, 128--225

1981

[9] [9]

Hua, Polishchuk, Elliptic bihamiltonian structures

[10] [10]

Looijenga, Bidifferentials

[11] [11]

Yu. I. Manin, Gauge field theory and complex geometry , Springer-Verlag, Berlin, 1988

1988

[12] [12]

D. Milicic, Algebraic D -modules and representation theory of semisimple Lie groups , in The Penrose transform and analytic cohomology in representation theory (South Hadley, MA, 1992) , 133--168. AMS, Providence, 1993

1992

[13] [13]

Penkov, D -modules on supermanifolds , Invent

I. Penkov, D -modules on supermanifolds , Invent. Math. 71 (1983), no. 3, 501--512

1983