arXiv subjects
Publications and source records attributed to Xiaoyue Sun.
We propose an approach to construct three-dimensional lattice models using line defects in state integral models on shaped triangulations of 3-manifolds. The Boltzmann weights for these models satisfy a variant of the tetrahedron equation, which implies integrability under suitable assumptions on R-matrices and transfer matrices. As an explicit example, we present a solution produced by Teichmüller TQFT.
We construct a new solution to the tetrahedron equation by further pursuing the quantum cluster algebra approach in our previous works. The key ingredients include a symmetric butterfly quiver attached to the wiring diagrams for the longest element of type $A$ Weyl groups and the implementation of quantum $Y$-variables through the $q$-Weyl algebra. The solution consists of four products of quantum dilogarithms. By exploring both the coordinate and momentum representations, along with their modular double counterparts, our solution encompasses various known three-dimensional (3D) $R$-matrices. These include those obtained by Kapranov-Voevodsky (1994) utilizing the quantized coordinate ring, Bazhanov-Mangazeev-Sergeev (2010) from a quantum geometry perspective, Kuniba-Matsuike-Yoneyama (2023) linked with the quantized six-vertex model, and Inoue-Kuniba-Terashima (2023) associated with the Fock-Goncharov quiver. The 3D $R$-matrix presented in this paper offers a unified perspective on these existing solutions, coalescing them within the framework of quantum cluster algebra.
We define three families of quivers in which the braid relations of the symmetric group $S_n$ are realized by mutations and automorphisms. A sequence of eight braid moves on a reduced word for the longest element of $S_4$ yields three trivial cluster transformations with 8, 32 and 32 mutations. For each of these cluster transformations, a unitary operator representing a single braid move in a quantum mechanical system solves the tetrahedron equation. The solutions thus obtained are constructed from the noncompact quantum dilogarithm and can be identified with the partition functions of three-dimensional $\mathcal{N} = 2$ supersymmetric gauge theories on a squashed three-sphere.
Appropriate credit assignment for delay rewards is a fundamental challenge for reinforcement learning. To tackle this problem, we introduce a delay reward calibration paradigm inspired from a classification perspective. We hypothesize that well-represented state vectors share similarities with each other since they contain the same or equivalent essential information. To this end, we define an empirical sufficient distribution, where the state vectors within the distribution will lead agents to environmental reward signals in the consequent steps. Therefore, a purify-trained classifier is designed to obtain the distribution and generate the calibrated rewards. We examine the correctness of sufficient state extraction by tracking the real-time extraction and building different reward functions in environments. The results demonstrate that the classifier could generate timely and accurate calibrated rewards. Moreover, the rewards are able to make the model training process more efficient. Finally, we identify and discuss that the sufficient states extracted by our model resonate with the observations of humans.