Kähler quantization of vortex moduli

Abstract

We discuss the Kähler quantization of moduli spaces of vortices in line bundles over compact surfaces Σ. This furnishes a semiclassical framework for the study of quantum vortex dynamics in the Schrödinger-Chern-Simons model. We employ Deligne's approach to Quillen's metric in determinants of cohomology to construct all the quantum Hilbert spaces in this context. An alternative description of the quantum wavesections, in terms of multiparticle states of spinors on Σ itself (valued in a prequantization of a multiple of its area form), is also obtained. This viewpoint sheds light on the nature of the quantum solitonic particles that emerge from the gauge theory. We find that in some cases (where the area of Σ is small enough in relation to its genus) the dimensions of the quantum Hilbert spaces may be sensitive to the input data required by the quantization scheme, and also address the issue of relating different choices of such data geometrically.

Vortex counting and the quantum Hall effect

“…The duality (1.1) was more-or-less proved in the Abelian case of N = 1 in [12]. In particular, it was shown in [12] that the low-temperature topological quantum mechanics on both sides of the duality were equivalent: an isomorphism of the vector spaces of states JHEP02(2022)155 was given (although the question of whether this introduced an isomorphism of 'natural' Hilbert space structures was not resolved, see [12,Remark 7.3]). At the level of indices, the same result was found in [13].…”

Twisted Hilbert spaces of 3d supersymmetric gauge theories

“…With the above construction in hand, we can proceed to compute the twisted Hilbert space following arguments in [34], but including higher degree cohomology. First, a short spectral sequence argument shows that the cohomology of the symmetric product in equation (4.55) can be identified with the S m Φ -invariant part of the cohomology H 0,…”

“…We start from a holomorphic line bundle L T on C. Then there are two natural holomorphic line bundles on Sym d C × C: the pull-back p * L T and the universal line bundle L. As explained in reference [34], we can then produce a holomorphic line bundle L T on Sym d C known as the Deligne pairing. This is usually denoted by…”

“…Furthermore, reference [34] shows that the holomorphic line bundle (A.25) is isomorphic to the holomorphic line bundle denoted similarly in section 4.4, which is obtained via the identification Sym d C ∼ = C d /S d .…”

Quantum soliton scattering manifolds

“…There is a detailed understanding of quantum multiparticle states in Kähler quantisation of abelian vortices[17]. This BPS system may be a good place to carefully study some of the claims in this paper.…”