2025/06/25 by Semyon Klevtsov, Klevtsov, Semyon, Dimitri Zvonkine +1 · 1 citation
Mathematics · Physics and Astronomy · #14C17 #81V70 #Advanced Operator Algebra Research #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum and electron transport phenomena #Strongly Correlated Electrons (cond-mat.str-el) #Topological Materials and Phenomena
paper · pdf · doi:10.48550/arxiv.2506.20363
openalex publication_date 2025/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We begin by explaining how a physical problem of studying the quantum Hall effect on a closed surface C leads, via Laughlin's approach, to a mathematical question of describing the rank and the first Chern class of a particular vector bundle on the Picard group \rm Picg(C). Then we formulate and solve the problem mathematically, proving several important conjectures made by physicists, in particular the Wen-Niu topological degeneracy conjecture and the Wen-Zee shift formula. Let C be a closed Riemann surface of genus~g and SNC its Nth symmetric power. The product C × \rm Picd(C) carries a universal line bundle. On the product CN × \rm Picd(C) we consider the product of N pull-backs of this universal line bundle and twist it by a power of the diagonal on CN. The resulting line bundle descends onto SNC × \rm Picd(C). Its push-forward (as a sheaf) to \rm Picd(C) is a vector bundle that we call Laughlin's vector bundle. We determine all the Chern characters of the Laughlin vector bundle via a Grothendieck-Riemann-Roch calculation.