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arxiv: 2409.00219 · v2 · pith:NAEYK2Q2new · submitted 2024-08-30 · 🧮 math.CT · math.AT· math.QA

Higher categories of push-pull spans, II: Matrix factorizations

Lorenzo Riva This is my paper

Pith reviewed 2026-05-23 21:22 UTC · model grok-4.3

classification 🧮 math.CT math.ATmath.QA
keywords matrix factorizationsRozansky-Witten modelshigher categoriestopological field theoriesfunctorial field theorypush-pull spansaffine modelssymmetric monoidal categories
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The pith

A functor connects the 2-category of matrix factorizations for affine Rozansky-Witten models to the homotopy 2-category of the (∞,3)-category 𝒞RW, allowing calculation of the associated TFTs.

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

This paper constructs a functor from the 2-category of matrix factorizations modeling affine Rozansky-Witten theories into the homotopy 2-category of 𝒞RW, the symmetric monoidal (∞,3)-category of commutative Rozansky-Witten models built in the companion paper. It then computes the topological field theories that arise from applying this functor. A sympathetic reader would care because the construction supplies a concrete algebraic-to-higher-categorical bridge for these models and advances their placement inside a framework intended to approximate the 3-category of Kapustin and Rozansky.

Core claim

The central claim is that there exists a functor from the 2-category of matrix factorizations associated to affine Rozansky-Witten models into the homotopy 2-category of 𝒞RW, and that the topological field theories associated to these models can be calculated by passing through this functor.

What carries the argument

The functor from the 2-category of matrix factorizations to the homotopy 2-category of 𝒞RW, which transfers the algebraic data into the higher-categorical setting.

If this is right

  • The topological field theories of the affine models are obtained by applying the functor and then evaluating the resulting objects in 𝒞RW.
  • The symmetric monoidal structure on 𝒞RW induces a corresponding structure on the images of the matrix factorizations.
  • This supplies an explicit approximation, in the affine case, to the 3-category of Kapustin and Rozansky.

Where Pith is reading between the lines

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

  • The functor might serve as a template for defining similar maps from other algebraic models of field theories into higher categories.
  • If the functor preserves additional structure, it could be used to extract numerical invariants of three-manifolds directly from matrix factorizations.

Load-bearing premise

The (∞,3)-category 𝒞RW constructed in the companion paper must possess the symmetric monoidal and homotopy properties needed for the functor to land in its homotopy 2-category.

What would settle it

A concrete affine matrix factorization whose image under the functor fails to respect the composition or monoidal product in the homotopy 2-category of 𝒞RW would show the claimed functor does not exist.

read the original abstract

This is the second part of a project aimed at formalizing Rozansky-Witten models in the functorial field theory framework. In the first part we constructed a symmetric monoidal $(\infty, 3)$-category $\mathscr{CRW}$ of commutative Rozansky-Witten models with the goal of approximating the $3$-category of Kapustin and Rozansky. In this paper we extend work of Brunner, Carqueville, Fragkos, and Roggenkamp on the affine Rozansky-Witten models: we exhibit a functor connecting their $2$-category of matrix factorizations with the homotopy $2$-category of $\mathscr{CRW}$, and calculate the associated TFTs.

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.

Referee Report

1 major / 1 minor

Summary. This paper is the second part of a project formalizing Rozansky-Witten models in the functorial field theory framework. Building on the symmetric monoidal (∞,3)-category 𝒞RW from the companion paper (part I), it exhibits a functor from the 2-category of matrix factorizations (as studied by Brunner–Carqueville–Fragkos–Roggenkamp) to the homotopy 2-category of 𝒞RW and calculates the associated TFTs for affine Rozansky-Witten models.

Significance. If the functor exists and the TFT calculations are valid, the work connects concrete algebraic data from matrix factorizations to a higher-categorical model approximating the Kapustin–Rozansky 3-category. The explicit functor construction and TFT computations are strengths that could support further calculations in this area.

major comments (1)
  1. [Introduction and main construction] The existence of the functor and the TFT calculations rest on the symmetric monoidal structure and homotopy 2-category of 𝒞RW as constructed in part I. The manuscript supplies neither an independent definition of 𝒞RW nor a verification that the required properties survive passage to the homotopy 2-category (Introduction and main construction sections). This assumption is load-bearing for the central claim.
minor comments (1)
  1. [Abstract] The abstract refers to 'the associated TFTs' without indicating which specific models or invariants are computed; a short clarification would aid readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. The manuscript is the second part of a project, and we address the concern about reliance on part I below.

read point-by-point responses

  1. Referee: [Introduction and main construction] The existence of the functor and the TFT calculations rest on the symmetric monoidal structure and homotopy 2-category of 𝒞RW as constructed in part I. The manuscript supplies neither an independent definition of 𝒞RW nor a verification that the required properties survive passage to the homotopy 2-category (Introduction and main construction sections). This assumption is load-bearing for the central claim.

    Authors: This paper is the second installment of the project and is explicitly framed as such, with the symmetric monoidal (∞,3)-category 𝒞RW constructed in the companion paper (part I). The homotopy 2-category is the standard 2-truncation of this (∞,3)-category, and the symmetric monoidal structure descends to it by the general theory of symmetric monoidal ∞-categories (as verified in part I, sections 3 and 4). The current work applies these structures to construct the functor and compute the TFTs. To make the dependence fully explicit without duplicating part I, we will add a concise summary paragraph in the introduction of the revised version, recalling the relevant definitions and citing the precise statements from part I on which the functor and TFT calculations rely. revision: partial

Circularity Check

0 steps flagged

No circularity; new functor constructed atop prior category

full rationale

The central result is the explicit construction of a functor from the 2-category of matrix factorizations (Brunner–Carqueville–Fragkos–Roggenkamp) into the homotopy 2-category of the symmetric monoidal (∞,3)-category 𝒞RW, together with the associated TFT calculations. 𝒞RW itself is taken from the companion paper (part I), but the present text supplies an independent mapping and does not redefine any quantity in terms of the functor or TFTs it claims to produce. No equation, definition, or step reduces the claimed functor or TFTs to the inputs by construction, nor does any self-citation serve as the sole justification for a uniqueness or ansatz that would force the result. This is ordinary dependence on prior work rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the prior construction of 𝒞RW and on standard facts from higher category theory and matrix factorization literature; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Existence and symmetric monoidal structure of the (∞,3)-category 𝒞RW constructed in part I.
    Invoked to define the target of the functor.

pith-pipeline@v0.9.0 · 5631 in / 1198 out tokens · 15885 ms · 2026-05-23T21:22:02.707371+00:00 · methodology

discussion (0)

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Reference graph

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