[Paper Review] Correspondence between 2 Calabi-Yau Categories and Quivers

Jie Ren|arXiv (Cornell University)|Feb 20, 2016

Algebraic structures and combinatorial models2 references3 citations

TL;DR

This paper establishes a one-to-one correspondence between equivalence classes of 2-dimensional Calabi-Yau categories generated by finite collections of objects satisfying specific Ext conditions and finite symmetric quivers with an even number of loops at each vertex. The correspondence is constructed via the Ext algebra of generators, with the Calabi-Yau structure inducing a symplectic pairing that forces even loop counts; the key result is a bijection proven using deformation theory of non-commutative formal manifolds and the vanishing of higher cohomology in the associated DGLA.

ABSTRACT

This note gives a one-to-one correspondence between the equivalence classes of a certain type of 2-dimensional Calabi-Yau categories, and certain type of quivers, This is an analogue of the result in Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv: 0811.2435, by M. Kontsevich, Y. Soibelman.

Motivation & Objective

• To extend Kontsevich and Soibelman’s 3D Calabi-Yau correspondence to the 2-dimensional case.
• To classify 2-dimensional Calabi-Yau categories generated by objects with trivial self-extensions in degree 0 and no negative extensions.
• To show that such categories are in bijection with symmetric quivers where each vertex has an even number of loops.
• To prove the correspondence via explicit construction of a minimal potential on the Ext algebra and deformation theory.

Proposed method

• Construct a quiver Q from a 2 Calabi-Yau category C by setting vertices to correspond to generators E_i and arrows from i to j to be dim Ext^1(E_i, E_j).
• Use the Calabi-Yau property to show that Ext^1(E_i, E_j) ≅ Ext^1(E_j, E_i)^⁺², implying Q is symmetric.
• Leverage the non-degenerate symplectic pairing on Ext^1(E_i, E_i) to show that dim Ext^1(E_i, E_i) is even, so each vertex has an even number of loops.
• Define a minimal potential W_can = α²β + Σ(αx_iξ_i - αξ_ix_i) on the graded vector space Ext^•(E,E)[1] = k[1] ⊕ k^{2n} ⊕ k[-1].
• Verify that W_can satisfies the quantum master equation {W_can, W_can} = 0 using the Poisson bracket defined by ∂/∂x_i, ∂/∂ξ_i and ∂/∂α, ∂/∂β.
• Use deformation theory of the 2 Calabi-Yau algebra A_can to show that H^{≥1}(𝔤_can) = 0, implying trivial deformations and thus injectivity of the correspondence.

Experimental results

Research questions

• RQ1 Is there a canonical correspondence between 2-dimensional Calabi-Yau categories and quivers with specific structure?
• RQ2 Can the Calabi-Yau condition on the category be used to constrain the quiver structure, particularly loop numbers?
• RQ3 Does the Ext algebra of a generator admit a minimal potential that encodes the Calabi-Yau structure?
• RQ4 Are all such 2 Calabi-Yau categories formally rigid, i.e., have no non-trivial deformations?

Key findings

• The correspondence between 2 Calabi-Yau categories and symmetric quivers with even loop counts is a bijection, proven via explicit construction and deformation theory.
• The number of loops at each vertex is forced to be even due to the symplectic structure on Ext^1(E_i, E_i), arising from the Calabi-Yau condition.
• The minimal potential W_can = α²β + Σ(αx_iξ_i - αξ_ix_i) satisfies {W_can, W_can} = 0, ensuring a well-defined 2 Calabi-Yau algebra structure.
• The deformation complex 𝔤_can of the canonical algebra A_can has trivial cohomology in degrees ≥1, implying that A_can is formally rigid.
• The DGLA 𝔤_can is a direct summand of the full cyclic complex dual, and its cohomology vanishes in positive degrees, confirming the uniqueness of the structure up to equivalence.