Instanton density operator in lattice QCD from higher category theory, by Jing-Yuan Chen
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SciPost Physics Home Authoring Refereeing Submit a manuscript About Instanton density operator in lattice QCD from higher category theory Jing-Yuan Chen SciPost Phys. 19, 158 (2025) · published 17 December 2025 doi: 10.21468/SciPostPhys.19.6.158 pdf BiBTeX RIS Submissions/Reports Abstract A natural definition for instanton density operator in lattice QCD has long been desired. We show this problem is, and has to be, solved by higher category theory. The problem is solved by refining at a conceptual level the Yang-Mills theory on lattice, in order to recover the homotopy information in the continuum, which would have been lost if we put the theory on lattice in the traditional way. The refinement needed is a generalization—through the lens of higher category theory—of the familiar process of Villainization that captures winding in lattice XY model and Dirac quantization in lattice Maxwell theory. The apparent difference is that Villainization is in the end described by principal bundles, hence familiar, but more general topological operators can only be captured on the lattice by more flexible structures beyond the usual group theory and fibre bundles, making the language of categories natural and necessary. The key structure we need for our particular problem is called multiplicative bundle gerbe, based upon which we can construct suitable structures to naturally define the 2d Wess-Zumino-Witten term, 3d skyrmion density operator and 4d hedgehog defect for lattice $S^3$ (pion vacua) non-linear sigma model, and the 3d Chern-Simons term, 4d instanton density operator and 5d Yang monopole defect for lattice $SU(N)$ Yang-Mills theory; moreover, the structures behind the non-linear sigma model and the Yang-Mills theory are related via an implicit Yang-Baxter equation. In a broader perspective, higher category theory enables us to rethink more systematically the relation between continuum quantum field theory and lattice quantum field theory. We sketch a proposal towards a general machinery that constructs the suitably refined lattice degrees of freedom for a given non-linear sigma model or gauge theory in the continuum, realizing the desired topological operators on the lattice. × TY - JOURPB - SciPost FoundationDO - 10.21468/SciPostPhys.19.6.158TI - Instanton density operator in lattice QCD from higher category theoryPY - 2025/12/17UR - https://scipost.org/SciPostPhys.19.6.158JF - SciPost PhysicsJA - SciPost Phys.VL - 19IS - 6SP - 158A1 - Chen, Jing-YuanAB - A natural definition for instanton density operator in lattice QCD has long been desired. We show this problem is, and has to be, solved by higher category theory. The problem is solved by refining at a conceptual level the Yang-Mills theory on lattice, in order to recover the homotopy information in the continuum, which would have been lost if we put the theory on lattice in the traditional way. The refinement needed is a generalization—through the lens of higher category theory—of the familiar process of Villainization that captures winding in lattice XY model and Dirac quantization in lattice Maxwell theory. The apparent difference is that Villainization is in the end described by principal bundles, hence familiar, but more general topological operators can only be captured on the lattice by more flexible structures beyond the usual group theory and fibre bundles, making the language of categories natural and necessary. The key structure we need for our particular problem is called multiplicative bundle gerbe, based upon which we can construct suitable structures to naturally define the 2d Wess-Zumino-Witten term, 3d skyrmion density operator and 4d hedgehog defect for lattice $S^3$ (pion vacua) non-linear sigma model, and the 3d Chern-Simons term, 4d instanton density operator and 5d Yang monopole defect for lattice $SU(N)$ Yang-Mills theory; moreover, the structures behind the non-linear sigma model and the Yang-Mills theory are related via an implicit Yang-Baxter equation. In a broader perspective, higher category theory enables us to rethink more systematically the relation between continuum quantum field theory and lattice quantum field theory. We sketch a proposal towards a general machinery that constructs the suitably refined lattice degrees of freedom for a given non-linear sigma model or gauge theory in the continuum, realizing the desired topological operators on the lattice.ER - × @Article{10.21468/SciPostPhys.19.6.158, title={{Instanton density operator in lattice QCD from higher category theory}}, author={Jing-Yuan Chen}, journal={SciPost Phys.}, volume={19}, pages={158}, year={2025}, publisher={SciPost}, doi={10.21468/SciPostPhys.19.6.158}, url={https://scipost.org/10.21468/SciPostPhys.19.6.158},} Ontology / Topics See full Ontology or Topics database.
Instantons Yang Mills Theory Author / Affiliation: mappings to Contributors and Organizations See all Organizations. 1 Jing-Yuan Chen 1 Tsinghua University [THU] Funder for the research work leading to this publication National Natural Science Foundation of China [NSFC]