arXiv subjects

Jin-Cheng Guu

Publications and source records attributed to Jin-Cheng Guu.

6 recordsLinked to original sources

arXiv2026

A Disproof of Santharoubane's Conjecture on Presentations of Generic Skein Algebras

Let $\Sigma$ be a compact connected oriented surface of genus at least $3$ with at most one boundary component. Santharoubane associated to certain presentations of the mapping class group modulo its center a finitely presented algebra equipped with a canonical surjection onto the generic Kauffman bracket skein algebra of $\Sigma$, and conjectured that a suitable choice yields an algebra isomorphic to the skein algebra. We show that every algebra arising from this construction admits an augmentation character, whereas the generic skein algebra of $\Sigma$ admits no unital character over $\mathbb Q(A)$. The latter obstruction follows from the intersection-one Dehn-twist identity together with a $4$-holed-sphere skein relation. Consequently, the conjectured isomorphism does not hold as stated.

math.GT↗

arXiv2025

Equivalence of field theories: Crane-Yetter and the shadow

This work solves a 28-year conjecture by showing that two major invariants of smooth 4-manifolds, the shadow model (motivated by statistical mechanics [Tur91]) and the simplicial Crane-Yetter model (motivated by topological quantum field theory [CY93]), are in fact equal. These invariants, both of which degenerate to the 3D Witten-Reshetikhin-Turaev model in a special case, had been open for years to clarify their relationship. Despite the seeming difference in their origins and formal constructions, we prove their equivalence. Along the way, we sketch a dictionary between the two models, provide a brief survey of the shadow construction à la Turaev, and suggest once again that the semisimple models have reached their limits.

math-ph↗

arXiv2025

Skein Construction of Balanced Tensor Products

The theory of tensor categories has found applications across various fields, including representation theory, quantum field theory (conformal in 2 dimensions, and topological in 3 and 4 dimensions), quantum invariants of low-dimensional objects, topological phases of matter, and topological quantum computation. In essence, it is a categorification of the classical theory of algebras and modules. In this analogy, the Deligne tensor product $\boxtimes$ is to the linear tensor $\otimes_{\mathbb{C}}$ as the balanced tensor product $\boxtimes_C$ is to the tensor over algebra $\otimes_A$, where $\mathbb{C}$ is a field, $A$ is a $\mathbb{C}$-algebra, and $C$ is a tensor category. Before this work, several algebraic constructions for balanced tensor products were known, including categories of modules, internal Hom spaces, and generalized categorical centers. In this paper, we introduce a topological construction based on skein theory that offers a better mix of algebra and topology. This approach not only works for products of multiple module categories, but also provides the missing key to proving that the Turaev-Viro state sum model naturally arises from the 3-functor in the classification of fully extended field theories. Building on this result, we establish this long-anticipated proof in an upcoming work.

math-ph↗

arXiv2023

On the algebra generated by $\overlineμ, \overline{\partial}, \partial, μ$

In this note, we determine the structure of the associative algebra generated by the differential operators $\overlineμ, \overline{\partial}, \partial, μ$ that act on complex-valued differential forms of almost complex manifolds. This is done by showing it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential $[d,-]$, as well as its cohomology with respect to all its inner differentials.

math.DG↗

arXiv2021

Explicit Factorization of a Categorical Center

Given a braided fusion category $C$, it is well known that the natural map $C \boxtimes C^{bop} \to Z(C)$ from the square of $C$ to the (Drinfeld) categorical center $Z(C)$ is an equivalence if and only if $C$ is modular. However, it is not clear how to construct the inverse and the natural isomorphisms. In this work, we provide an explicit construction using insights from a specific quantum field theory, and explore how the equivalence fails for the degenerate cases.

math.QA↗

arXiv2021

Categorical Center of Higher Genera and 4D Factorization Homology

Many quantum invariants of knots and 3-manifolds (e.g. Jones polynomials) are special cases of the Witten-Reshetikhin-Turaev 3D TQFT. The latter is in turn a part of a larger theory - the Crane-Yetter 4D TQFT. In this work, we compute the Crane-Yetter theory for all (smooth and oriented) surfaces with at least one puncture. The results in general are constructed and called the categorical center of higher genera.

math.QA↗