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arxiv: 2605.21159 · v1 · pith:3ZIIG55Fnew · submitted 2026-05-20 · 🧮 math.SG · math.AG· math.RT

Mirror Symmetry of the Affine Toda Systems

Xin Jin , Zhiwei Yun This is my paper

Pith reviewed 2026-05-21 01:24 UTC · model grok-4.3

classification 🧮 math.SG math.AGmath.RT
keywords homological mirror symmetryaffine Toda systemswrapped Fukaya categorycoherent sheavesregular centralizer group schemegeometric LanglandsLanglands dualitywild ramification
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The pith

Homological mirror symmetry equates the wrapped Fukaya category of the affine Toda system for G with coherent sheaves on the regular centralizer of G^vee.

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

The paper proves that for any complex reductive group G the wrapped Fukaya category of its affine Toda system is equivalent as a category to the derived category of coherent sheaves on the regular centralizer group scheme of the Langlands dual group G^vee. The equivalence is presented as realizing the geometric Langlands correspondence in the concrete setting of the projective line with the mildest form of wild ramification at zero and infinity. A reader would care because the result supplies an explicit bridge between the symplectic geometry of integrable systems and the algebraic geometry of group schemes, allowing invariants and structures to be transferred from one side to the other.

Core claim

For a complex reductive group G, the wrapped Fukaya category of the affine Toda system for G is equivalent to the category of coherent sheaves on the regular centralizer group scheme for the Langlands dual group G^vee. This equivalence supplies a geometric Langlands correspondence for the projective line with mildest wild ramification at the points 0 and infinity.

What carries the argument

The homological mirror symmetry equivalence that identifies the wrapped Fukaya category of the affine Toda system with the coherent sheaf category on the regular centralizer group scheme of the Langlands dual.

Load-bearing premise

The affine Toda system for a complex reductive group G admits a well-defined wrapped Fukaya category whose objects and morphisms can be compared directly with those in the category of coherent sheaves on the regular centralizer group scheme of G^vee.

What would settle it

An explicit computation for G equal to SL(2) in which the ranks of the two categories or their Hochschild homologies fail to agree would show the claimed equivalence is false.

Figures

Figures reproduced from arXiv: 2605.21159 by Xin Jin, Zhiwei Yun.

Figure 1
Figure 1. Figure 1: An illustration of an admissible covering, with each UJ enclosed by a dashed curve. The filled region on the left represents U ♭ J1 for some J1. The filled region on the right represents U ♯ J2 for some J2 [PITH_FULL_IMAGE:figures/full_fig_p041_1.png] view at source ↗
read the original abstract

For a complex reductive group $G$, we prove a homological mirror symmetry between the wrapped Fukaya category of the affine Toda system for $G$ and coherent sheaves on the regular centralizer group scheme for the Langlands dual group $G^\vee$. This can be interpreted as a geometric Langlands equivalence for $\mathbb{P}^1$ with mildest wild ramification at $0$ and $\infty$.

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 / 2 minor

Summary. For a complex reductive group G, the manuscript proves a homological mirror symmetry between the wrapped Fukaya category of the affine Toda system for G and coherent sheaves on the regular centralizer group scheme for the Langlands dual group G^∨. This is interpreted as a geometric Langlands equivalence for P^1 with mildest wild ramification at 0 and ∞.

Significance. If the claimed equivalence holds, the result would constitute a notable contribution to homological mirror symmetry by linking the symplectic geometry of affine Toda integrable systems to algebraic geometry via the geometric Langlands program. The direct construction for general complex reductive G, avoiding parameter fitting, is a potential strength.

major comments (1)
  1. [§3] §3: The construction of the wrapped Fukaya category for the affine Toda system is central to the main theorem, yet the Liouville structure, exactness conditions, and handling of the cylindrical ends at the ramification points are not specified with sufficient precision to verify that the category is well-defined and that its morphism spaces can be compared directly to Ext groups on the centralizer scheme.
minor comments (2)
  1. [Introduction] The abstract and introduction could include a brief comparison to existing HMS results for Toda systems or other integrable systems to clarify novelty.
  2. [Throughout] Notation for the group scheme and its dual is occasionally inconsistent between sections; a dedicated notation table would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address the major comment point by point below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3] §3: The construction of the wrapped Fukaya category for the affine Toda system is central to the main theorem, yet the Liouville structure, exactness conditions, and handling of the cylindrical ends at the ramification points are not specified with sufficient precision to verify that the category is well-defined and that its morphism spaces can be compared directly to Ext groups on the centralizer scheme.

    Authors: We agree that greater precision in §3 would strengthen the exposition and facilitate verification. The wrapped Fukaya category is defined on the affine Toda integrable system equipped with its natural Liouville structure coming from the Toda potential on the cotangent bundle of the adjoint quotient, with exactness of the relevant Lagrangians following from the fact that the symplectic form admits a primitive that vanishes at infinity along the cylindrical ends. The cylindrical ends at the ramification points 0 and ∞ are modeled on the standard wild ramification data for the geometric Langlands correspondence on P^1. In the revised manuscript we will add an expanded subsection in §3 that explicitly records the Liouville 1-form, states the exactness conditions for the generating Lagrangians, and describes the asymptotic coordinates and wrapping behavior at the ends. These additions will make the well-definedness of the category and the direct comparison of morphism spaces with Ext groups on the regular centralizer scheme fully transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The manuscript claims a direct proof of homological mirror symmetry by constructing the wrapped Fukaya category of the affine Toda system for a complex reductive group G and exhibiting an equivalence to the derived category of coherent sheaves on the regular centralizer group scheme of the Langlands dual G^∨. The abstract and stated claims define the two categories independently—one via symplectic geometry and the other via algebraic geometry—without any reduction to fitted parameters, self-referential definitions, or load-bearing self-citations that collapse the result to its own inputs. No equations or steps are presented that rename a known result or smuggle an ansatz via prior work by the same authors. The central claim remains independent and externally falsifiable against benchmarks in mirror symmetry and geometric Langlands.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions from symplectic geometry and algebraic geometry concerning the definitions and properties of wrapped Fukaya categories and regular centralizer group schemes for reductive groups.

axioms (2)
  • domain assumption The affine Toda system for a complex reductive group G admits a well-defined wrapped Fukaya category
    Invoked to make the left-hand side of the claimed equivalence meaningful.
  • domain assumption The regular centralizer group scheme for the Langlands dual G^vee is a well-defined object in algebraic geometry
    Invoked to make the right-hand side of the claimed equivalence meaningful.

pith-pipeline@v0.9.0 · 5581 in / 1383 out tokens · 44342 ms · 2026-05-21T01:24:48.686652+00:00 · methodology

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