Twisted argyle quivers and Higgs bundles

Considering a compact Riemann surface of genus greater or equal than two, a Higgs bundle is a pair composed of a holomorphic bundle over the Riemann surface, joint with an auxiliar vector field, so-called Higgs field. This theory started around thirty years ago, with Hitchin’s work, when he reduced the self-duality equations from dimension four to dimension two, and so, studied those equations over Riemann surfaces. Hitchin baptized those fields as Higgs fields because in the context of physics and gauge theory, they describe similar particles to those described by the Higgs bosson. Later, Simpson used the name Higgs bundle for a holomorphic bundle together with a Higgs field. Today, Higgs bundles are the subject of research in several areas such as non-abelian Hodge theory, Langlands, mirror symmetry, integrable systems, quantum field theory (QFT), among others. The main purposes here are to introduce these objects, and to present a brief but complete construction of the moduli space of Higgs bundles.

Categories of Representations of Coalgebroids

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We compute the monodromy of the Hitchin fibration for the moduli space of $L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any $n$, on a compact Riemann surface of genus $g>1$. We require the line bundle $L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy group is generated by Picard-Lefschetz transformations associated to vanishing cycles of singular spectral curves. We construct such vanishing cycles explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is a {\em skew-symmetric vanishing lattice} in the sense of Janssen. Using the classification of vanishing lattices over $\mathbb{Z}$, we completely determine the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image of the restriction map from the cohomology of the moduli space of Higgs bundles to the cohomology of a non-singular fibre of the Hitchin fibration.

Holomorphic chains on a Riemann surface arise naturally as fixed points of the natural C*-action on the moduli space of Higgs bundles. In this paper we associate a new quiver bundle to the Hom-complex of two chains, and prove that stability of the chains implies stability of this new quiver bundle. Our approach uses the Hitchin-Kobayashi correspondence for quiver bundles. Moreover, we use our result to give a new proof of a key lemma on chains (due to \'Alvarez-C\'onsul, Garc\'ia-Prada and Schmitt), which has been important in the study of Higgs bundle moduli; this proof relies on stability and thus avoids the direct use of the chain vortex equations.

Narasimhan–Ramanan branes, introduced by the authors in a previous paper, consist of a family of (BBB)-branes inside the moduli space of Higgs bundles, and a family of complex Lagrangian subvarieties. It was conjectured that these complex Lagrangian subvarieties support the (BAA)-branes that are mirror dual to the Narasimhan–Ramanan (BBB)-branes. In this paper, we show that the support of these branes intersects nontrivially the locus of wobbly Higgs bundles.

The appearance of the Bethe Ansatz equation for the Nonlinear Schrodinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrodinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrodinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.

We answer the questions, "What does Chern-Simons theory assign to a point?" and "What kind of mathematical object does Chern-Simons theory assign to a point?" Our answer to the first question is representations of the based loop group. More precisely, we identify a certain class of projective unitary representations of the based loop group [Formula: see text] We define the fusion product of such representations, and we prove that, modulo certain conjectures, the Drinfel'd center of that representation category of [Formula: see text] is equivalent to the category of positive energy representations of the free loop group [Formula: see text]† The abovementioned conjectures are known to hold when the gauge group is abelian or of type [Formula: see text] Our answer to the second question is bicommutant categories. The latter are higher categorical analogs of von Neumann algebras: They are tensor categories that are equivalent to their bicommutant inside [Formula: see text], the category of bimodules over a hyperfinite [Formula: see text] factor. We prove that, modulo certain conjectures, the category of representations of the based loop group is a bicommutant category. The relevant conjectures are known to hold when the gauge group is abelian or of type [Formula: see text].

Resolutions of non-regular Ricci-flat Kähler cones

Using the L^2 norm of the Higgs field as a Morse function, we study the moduli spaces of U(p,q)-Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p,q). A key step is the identification of the function's local minima as moduli spaces of holomorphic triples. In a companion paper "Moduli spaces of holomorphic triples over compact Riemann surfaces" (math.AG/0211428) we prove that these moduli spaces of triples are non-empty and irreducible. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple PU(p,q)-representations. The topological invariants of the flat bundles are used to label subspaces. These invariants are bounded by a Milnor-Wood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding subspace is non-empty and connected. If the coprimality condition does not hold, our results apply to the closure of the moduli space of irreducible representations. This paper, and its companion mentioned above, form a substantially revised version of math.AG/0206012.

Let be an arbitrary non‐compact hyperbolic Riemann surface, that is, not the complex plane or punctured plane. Let be , or . We show that, on a Higgs bundle in the ‐Hitchin section over , there always exists a harmonic metric satisfying certain nice properties. Moreover, when the Higgs field has bounded spectral curve with respect to the complete hyperbolic metric on , such a harmonic metric is unique. As a result, we obtain a new proof of the existence and uniqueness of a harmonic metric for Higgs bundles in the ‐Hitchin section over a compact hyperbolic Riemann surface. Moreover, we show an existence result of harmonic metrics for a more general family of Higgs bundles that admit a full holomorphic filtration.

Let [Formula: see text] be an abelian model category (in the sense of Hovey). For a large class of quivers, we describe associated abelian model structures on categories of quiver representations with values in [Formula: see text]. This is based on recent work of Holm and Jørgensen on cotorsion pairs in categories of quiver representations. An application on Ding projective and Ding injective representations of quivers over Ding–Chen rings is given.

For a perfect base field k , we investigate arithmetic aspects of moduli spaces of quiver representations over k : we study actions of the absolute Galois group of k on the k ¯ -valued points of moduli spaces of quiver representations over k and we provide a modular interpretation of the fixed-point set using quiver representations over division algebras, which we reinterpret using moduli spaces of twisted quiver representations (we show that those spaces provide different k -forms of the initial moduli space of quiver representations). Finally, we obtain that stable k ¯ -representations of a quiver are definable over a certain central division algebra over their field of moduli.

Representations of the fundamental group of a surface in [formula omitted] and holomorphic triples

We prove the Topological Mirror Symmetry Conjecture by Hausel–Thaddeus for smooth moduli spaces of Higgs bundles of type SL_n and PGL_n. More precisely, we establish an equality of stringy Hodge numbers for certain pairs of algebraic orbifolds generically fibred into dual abelian varieties. Our proof utilises p-adic integration relative to the fibres, and interprets canonical gerbes present on these moduli spaces as characters on the Hitchin fibres using Tate duality. Furthermore, we prove for d prime to n, that the number of rank n Higgs bundles of degree d over a fixed curve defined over a finite field, is independent of d. This proves a conjecture by Mozgovoy–Schiffmann in the coprime case.