arXiv subjects

Nuno M. Romão

Publications and source records attributed to Nuno M. Romão.

14 recordsLinked to original sources

arXiv2020

The geometry of the space of BPS vortex-antivortex pairs

The gauged sigma model with target $\mathbb{P}^1$, defined on a Riemann surface $Σ$, supports static solutions in which $k_+$ vortices coexist in stable equilibrium with $k_-$ antivortices. Their moduli space is a noncompact complex manifold $M_{(k_+,k_-)}(Σ)$ of dimension $k_++k_-$ which inherits a natural Kähler metric $g_{L^2}$ governing the model's low energy dynamics. This paper presents the first detailed study of $g_{L^2}$, focussing on the geometry close to the boundary divisor $D=\partial M_{(k_+,k_-)}(Σ)$. On $Σ=S^2$, rigorous estimates of $g_{L^2}$ close to $D$ are obtained which imply that $M_{(1,1)}(S^2)$ has finite volume and is geodesically incomplete. On $Σ=\mathbb{R}^2$, careful numerical analysis and a point-vortex formalism are used to conjecture asymptotic formulae for $g_{L^2}$ in the limits of small and large separation. All these results make use of a localization formula, expressing $g_{L^2}$ in terms of data at the (anti)vortex positions, which is established for general $M_{(k_+,k_-)}(Σ)$. For arbitrary compact $Σ$, a natural compactification of the space $M_{(k_+,k_-)}(Σ)$ is proposed in terms of a certain limit of gauged linear sigma models, leading to formulae for its volume and total scalar curvature. The volume formula agrees with the result established for $Vol(M_{(1,1)}(S^2))$, and allows for a detailed study of the thermodynamics of vortex-antivortex gas mixtures. It is found that the equation of state is independent of the genus of $Σ$, and that the entropy of mixing is always positive.

math.DG↗

arXiv2017

The Nahm-Schmid equations and Hypersymplectic Geometry

We explore the geometry of the Nahm-Schmid equations, a version of Nahm's equations in split signature. Our discussion ties up different aspects of their integrable nature: dimensional reduction from the Yang--Mills anti-self-duality equations, explicit solutions, Lax-pair formulation, conservation laws and spectral curves, as well as their relation to hypersymplectic geometry.

math.DG↗

arXiv2017

Kähler quantization of vortex moduli

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 follow 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.

math-ph↗

arXiv2016

A no-go theorem for nonabelionic statistics in gauged linear sigma-models

Gauged linear sigma-models at critical coupling on Riemann surfaces yield self-dual field theories, their classical vacua being described by the vortex equations. For local models with structure group ${\rm U}(r)$, we give a description of the vortex moduli spaces in terms of a fibration over symmetric products of the base surface $Σ$, which we assume to be compact. Then we show that all these fibrations induce isomorphisms of fundamental groups. A consequence is that all the moduli spaces of multivortices in this class of models have abelian fundamental groups. We give an interpretation of this fact as a no-go theorem for the realization of nonabelions through the ground states of a supersymmetric version (topological via an A-twist) of these gauged sigma-models. This analysis is based on a semi-classical approximation of the QFTs via supersymmetric quantum mechanics on their classical moduli spaces.

math-ph↗

arXiv2016

Divisor braids

We study a novel type of braid groups on a closed orientable surface $Σ$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $Σ$; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles over $Σ$. The elements of these braid groups, which we call divisor braids, have coloured strands that are allowed to intersect according to rules specified by a graph $Γ$. In situations where there is more than one strand of each colour, we show that the corresponding braid group admits a metabelian presentation as a central extension of the free Abelian group $H_1(Σ;\mathbb{Z})^{\oplus r}$, where $r$ is the number of colours, and describe its Abelian commutator. This computation relies crucially on producing a link invariant (of closed divisor braids) in the three-manifold $S^1 \times Σ$ for each graph $Γ$. We also describe the von Neumann algebras associated to these groups in terms of rings that are familiar from noncommutative geometry.

math.GT↗

arXiv2014

Pairs of pants, Pochhammer curves and $L^2$-invariants

We propose an intuitive interpretation for nontrivial $L^2$-Betti numbers of compact Riemann surfaces in terms of certain loops in embedded pairs of pants. This description uses twisted homology associated to the Hurewicz map of the surface, and it satisfies a sewing property with respect to a large class of pair-of-pants decompositions. Applications to supersymmetric quantum mechanics incorporating Aharonov-Bohm phases are briefly discussed, for both point particles and topological solitons (abelian and non-abelian vortices) in two dimensions.

math-ph↗

arXiv2013

On the curvature of vortex moduli spaces

We use algebraic topology to investigate local curvature properties of the moduli spaces of gauged vortices on a closed Riemann surface. After computing the homotopy type of the universal cover of the moduli spaces (which are symmetric powers of the surface), we prove that, for genus g>1, the holomorphic bisectional curvature of the vortex metrics cannot always be nonnegative in the multivortex case, and this property extends to all Kaehler metrics on certain symmetric powers. Our result rules out an established and natural conjecture on the geometry of the moduli spaces.

math-ph↗

arXiv2012

Moduli of vortices and Grassmann manifolds

We use the framework of Quot schemes to give a novel description of the moduli spaces of stable n-pairs, also interpreted as gauged vortices on a closed Riemann surface with target Mat(r x n, C), where n >= r. We then show that these moduli spaces embed canonically into certain Grassmann manifolds, and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are smooth at least in the local case r=n. For abelian local vortices we prove that, if a certain "quantization" condition is satisfied, the embedding can be chosen in such a way that the induced Fubini-Study structure realizes the Kaehler class of the usual L^2 metric of gauged vortices.

math.AG↗

arXiv2010

Vortices and Jacobian varieties

We investigate the geometry of the moduli space of N-vortices on line bundles over a closed Riemann surface of genus g > 1, in the little explored situation where 1 =< N < g. In the regime where the area of the surface is just large enough to accommodate N vortices (which we call the dissolving limit), we describe the relation between the geometry of the moduli space and the complex geometry of the Jacobian variety of the surface. For N = 1, we show that the metric on the moduli space converges to a natural Bergman metric on the Riemann surface. When N > 1, the vortex metric typically degenerates as the dissolving limit is approached, the degeneration occurring precisely on the critical locus of the Abel-Jacobi map at degree N. We describe consequences of this phenomenon from the point of view of multivortex dynamics.

hep-th↗

arXiv2005

Spectral curves and the mass of hyperbolic monopoles

The moduli spaces of hyperbolic monopoles are naturally fibred by the monopole mass, and this leads to a nontrivial mass dependence of the holomorphic data (spectral curves, rational maps, holomorphic spheres) associated to hyperbolic multi-monopoles. In this paper, we obtain an explicit description of this dependence for general hyperbolic monopoles of magnetic charge two. In addition, we show how to compute the monopole mass of higher charge spectral curves with tetrahedral and octahedral symmetries. Spectral curves of euclidean monopoles are recovered from our results via an infinite-mass limit.

math-ph↗

arXiv2005

Gauged vortices in a background

We discuss the statistical mechanics of a gas of gauged vortices in the canonical formalism. At critical self-coupling, and for low temperatures, it has been argued that the configuration space for vortex dynamics in each topological class of the abelian Higgs model approximately truncates to a finite-dimensional moduli space with a Kaehler structure. For the case where the vortices live on a 2-sphere, we explain how localisation formulas on the moduli spaces can be used to compute explicitly the partition function of the vortex gas interacting with a background potential. The coefficients of this analytic function provide geometrical data about the Kaehler structures, the simplest of which being their symplectic volume (computed previously by Manton using an alternative argument). We use the partition function to deduce simple results on the thermodynamics of the vortex system; in particular, the average height on the sphere is computed and provides an interesting effective picture of the ground state.

hep-th↗

arXiv2005

A combinatorial formula for homogeneous moments

We establish a combinatorial formula for homogeneous moments and give some examples where it can be put to use. An application to the statistical mechanics of interacting gauged vortices is discussed.

math.SG↗

arXiv2004

Dynamics of CP^1 lumps on a cylinder

The slow dynamics of topological solitons in the CP^1 sigma-model, known as lumps, can be approximated by the geodesic flow of the L^2 metric on certain moduli spaces of holomorphic maps. In the present work, we consider the dynamics of lumps on an infinite flat cylinder, and we show that in this case the approximation can be formulated naturally in terms of regular Kaehler metrics. We prove that these metrics are incomplete exactly in the multilump (interacting) case. The metric for two-lumps can be computed in closed form on certain totally geodesic submanifolds using elliptic integrals; particular geodesics are determined and discussed in terms of the dynamics of interacting lumps.

math-ph↗

arXiv2001

Quantum Chern-Simons Vortices on a Sphere

The quantisation of the reduced first-order dynamics of the nonrelativistic model for Chern-Simons vortices introduced by Manton is studied on a sphere of given radius. We perform geometric quantisation on the moduli space of static solutions, using a Kaehler polarisation, to construct the quantum Hilbert space. Its dimension is related to the volume of the moduli space in the usual classical limit. The angular momenta associated with the rotational SO(3) symmetry of the model are determined for both the classical and the quantum systems. The results obtained are consistent with the interpretation of the solitons in the model as interacting bosonic particles.

hep-th↗