Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.17515

Chern classes of the multilayer fractional quantum Hall bundle on Riemann surfaces

María Abad Aldonza·Florent Dupont
View original source ↗Read source document ↗Export metadata ↓

Abstract

The so-called multilayer wave functions were introduced in the study of the fractional Quantum Hall effect by Halperin and others. They are defined with the help of a symmetric matrix $K$ in $M^k(\mathbb{N})$, which encodes the couplings between the $k$ layers where particles live. We study the multilayer quantum states in the case where each layer is a Riemann surface of genus $g$. These states form a vector bundle over the Jacobian variety of the Riemann surface, or the space of Aharonov-Bohm fluxes in physics terminology. Burban-Klevtsov have determined the rank and first Chern class of this bundle when $g=1$, and Klevtsov-Zvonkine computed the Chern character of this bundle in the single layer case for any genus. We generalize the latter approach to the multilayer case and compute the Chern character of the multilayer bundle for any genus, and any possible number of non-localized quasi-holes, under the assumption that the bilinear form associated to $K-I$ is non negative. The key tools are the Grothendieck-Riemann-Roch formula, Berezin integration and Wick's formula for exterior algebras. We show that when all quasi-holes are localized, the Chern character is compatible with the bundle being projectively flat. Furthermore, for those configurations, the conductance becomes independent of the genus and is equal to the sum of the coefficients of the inverse matrix $K^{-1}$, proving two conjectures by Keski-Vakkuri and Wen. We also find the relation linking $K$, the genus of the surface, the magnetic field and the number of non-localized quasi holes in each layer, which was studied under the name of ''shift formula''. Finally, we study conditions on the matrix $K$ under which states having only localized quasi-holes maximize the total particle number, as well as the asymptotics for large magnetic fields in this scenario.

Explore related subjects

math.AGcond-mat.str-elmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

María Abad Aldonza, Florent Dupont. 2025-09-22. Chern classes of the multilayer fractional quantum Hall bundle on Riemann surfaces. https://arxiv.org/abs/2509.17515

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Lawson--Deligne Classes and Applications

We construct the integral Lawson--Deligne map of weight $q=n-p-k-1$ on smooth complex projective $n$-folds using filtered currents. It lifts the Friedlander--Mazur cycle class, recovers the reduced generalized Abel--Jacobi invariant on homologically trivial classes, and is compatible with algebraic correspondences. A Picard--Fuchs separation argument applied to the conic and van Geemen normal functions on the mirror quintic determines explicit regulator subspaces modulo the full rational period group. For prescribed elliptic moduli and a suitable mirror-quintic fiber, the subspace generated by their $a$- and $b$-loop products has dimension twice the $\Q$-dimension of the period-monomial space. Moduli $i\sqrt{\ell_j}$ for distinct primes $\ell_j$ give $2^{k+1}$ independent images on varieties of dimension $p+k+2$; one repeated imaginary quadratic modulus gives dimension four for every $k\geq1$. Compatibility with known projective-bundle and blow-up decompositions yields independent exceptional subspaces on smooth rational varieties. We also compare the higher Chow composite with the Bloch--KLM regulator after lowering the Hodge filtration. The KLM representative reduces to a cut-current class, and equality with the Lawson composite is proved in degree zero and for constant-unit decomposable classes. The general positive-degree comparison is reduced to an explicit filtered-realization condition.

math.AG↗
arXiv2026

Complete quasimaps to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$

We introduce a moduli space of ``complete quasimaps'' to $\mathsf{Bl}_{\mathbb{P}^s}(\mathbb{P}^r)$. The construction, following previous work for curves on projective spaces, essentially proceeds by blowing up Ciocan-Fontanine--Kim's space of quasimaps at loci where sections of line bundles are linearly dependent. We conjecture that tautological intersection numbers on these moduli spaces give enumerative counts of curves of fixed complex structure on $X$ subject to general incidence conditions, in contrast with traditional compactifications of the moduli spaces of maps. A result of Farkas guarantees that these spaces are pure of expected dimension. The conjecture is proven in dimension 2, where the main input is a Brill-Noether theorem for general curves on toric surfaces.

math.AG↗