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Instanton Density Operator in Lattice QCD from Higher Category Theory
by JingYuan Chen
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Authors (as registered SciPost users):  JingYuan Chen 
Submission information  

Preprint Link:  scipost_202407_00023v1 (pdf) 
Date submitted:  20240714 16:02 
Submitted by:  Chen, JingYuan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
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Approach:  Theoretical 
Abstract
A natural definition for instanton density operator in lattice QCD has been long desired. We show this problem is, and has to be, resolved by higher category theory. The problem is resolved by refining at a conceptual level the YangMills 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 generalizationthrough the lens of higher category theoryof 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, hence the language of categories becomes 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 WessZuminoWitten term, 3d skyrmion density operator and 4d hedgehog defect for lattice $S^3$ (pion vacua) nonlinear sigma model, and the 3d ChernSimons term, 4d instanton density operator and 5d Yang monopole defect for lattice $SU(N)$ YangMills theory. 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 nonlinear sigma model or gauge theory in the continuum, realizing the desired topological operators on the lattice.
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