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Register a new userA general Cayley correspondence and higher Teichmuller spaces
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We introduce a new class of $mathfrak{sl}_2$-triples in a complex simple Lie algebra $mathfrak{g}$, which we call magical. Such an $mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $mathfrak{g}$. Using this decomposition data, we explicitly parameterize special connected components of the moduli space of Higgs bundles on a compact Riemann surface $X$ for an associated real Lie group, hence also of the corresponding character variety of representations of $pi_1X$ in the associated real Lie group. This recovers known components when the real group is split, Hermitian of tube type, or $mathrm{SO}_{p,q}$ with $1<pleq q$, and also constructs previously unknown components for the quaternionic real forms of $mathrm{E}_6$, $mathrm{E}_7$, $mathrm{E}_8$ and $mathrm{F}_4$. The classification of magical $mathfrak{sl}_2$-triples is shown to be in bijection with the set of $Theta$-positive structures in the sense of Guichard--Wienhard, thus the mentioned parameterization conjecturally detects all examples of higher Teichmuller spaces. Indeed, we discuss properties of the surface group representations obtained from these Higgs bundle components and their relation to $Theta$-positive Anosov representations, which indicate that this conjecture holds.
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Some connected components of a moduli space are mundane in the sense that they are distinguished only by obvious topological invariants or have no special characteristics. Others are more alluring and unusual either because they are not detected by primary invariants, or because they have special geometric significance, or both. In this paper we describe new examples of such `exotic components in moduli spaces of SO(p,q)-Higgs bundles on closed Riemann surfaces or, equivalently, moduli spaces of surface group representations into the Lie group SO(p,q). Furthermore, we discuss how these exotic components are related to the notion of positive Anosov representations recently developed by Guichard and Wienhard. We also provide a complete count of the connected components of these moduli spaces (except for SO(2,q), with q> 3).
We introduce $Theta$-positivity, a new notion of positivity in real semisimple Lie groups. The notion of $Theta$-positivity generalizes at the same time Lusztigs total positivity in split real Lie groups as well as well known concepts of positivity in Lie groups of Hermitian type. We show that there are two other families of Lie groups, SO(p,q) for p<q, and a family of exceptional Lie groups, which admit a $Theta$-positive structure. We describe key aspects of $Theta$-positivity and make a connection with representations of surface groups and higher Teichmuller theory.
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of algebraic geometry, dynamical systems, geometry and topology lead to the study of such objects. It comes as a natural consequence that we can employ in our study algebraic, analytic, combinatorial and dynamical perspectives. These lecture notes aim to provide an expository introduction to this subject that will emphasize the aforementioned links between different areas of mathematics. We will associate to an Abelian differential a flat surface with conical singularities such that the underlying Riemann surface is obtained from a polygon by identifying edges with one another via translation. We will focus on studying these objects in families and describe some properties of the orbit as we vary the polygon by the action of $GL_2^{+}(mathbb{R})$ on the plane.
We derive generalizations of McShanes identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank positive representations with unipotent boundary monodromy. Our identities are systematically expressed in terms of projective invariants, and we study these invariants: we establish boundedness and Fuchsian rigidity results for triple and cross ratios. We apply our identities to derive the simple spectral discreteness of unipotent-bordered positive representations, collar lemmas, and generalizations of the Thurston metric.
Let $X$ be a compact connected Riemann surface, $D, subset, X$ a reduced effective divisor, $G$ a connected complex reductive affine algebraic group and $H_x, subsetneq, G_x$ a Zariski closed subgroup for every $x, in, D$. A framed principal $G$--bundle is a pair $(E_G,, phi)$, where $E_G$ is a holomorphic principal $G$--bundle on $X$ and $phi$ assigns to each $x, in, D$ a point of the quotient space $(E_G)_x/H_x$. A framed $G$--Higgs bundle is a framed principal $G$--bundle $(E_G,, phi)$ together with a section $theta, in, H^0(X,, text{ad}(E_G)otimes K_Xotimes{mathcal O}_X(D))$ such that $theta(x)$ is compatible with the framing $phi$ for every $x, in, D$. We construct a holomorphic symplectic structure on the moduli space $mathcal{M}_{FH}(G)$ of stable framed $G$--Higgs bundles. Moreover, we prove that the natural morphism from $mathcal{M}_{FH}(G)$ to the moduli space $mathcal{M}_{H}(G)$ of $D$-twisted $G$--Higgs bundles $(E_G,, theta)$ that forgets the framing, is Poisson. These results generalize cite{BLP} where $(G,, {H_x}_{xin D})$ is taken to be $(text{GL}(r,{mathbb C}),, {text{I}_{rtimes r}}_{xin D})$. We also investigate the Hitchin system for $mathcal{M}_{FH}(G)$ and its relationship with that for $mathcal{M}_{H}(G)$.
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