[Paper Review] Branes and Langlands duality

Indranil Biswas, Oscar Garcı́a-Prada+1 more|arXiv (Cornell University)|Jul 3, 2017

Advanced Algebra and Geometry19 references3 citations

TL;DR

This paper constructs antiholomorphic involutions on the moduli space of $G$-Higgs bundles over a compact Riemann surface $X$, establishing a correspondence between fixed point loci of $G$ and its Langlands dual $^LG$. For $G = SL(2,\mathbb{C})$ and $^LG = PGL(2,\mathbb{C})$, it shows how components of the fixed point sets match under Langlands duality, revealing a deep symmetry in Higgs bundle moduli spaces.

ABSTRACT

Given a compact Riemann surface $X$ and a complex reductive Lie group $G$ equipped with real structures, we define antiholomorphic involutions on the moduli space of $G$-Higgs bundles over $X$. We investigate how the various components of the fixed point locus match up, as one passes from $G$ to its Langlands dual $^LG$. Special attention is given to the case $G=SL(2,\C)$ and $^LG=PGL(2,\C)$.

Motivation & Objective

• To define antiholomorphic involutions on the moduli space of $G$-Higgs bundles over a compact Riemann surface $X$.
• To analyze the structure of the fixed point loci of these involutions in relation to Langlands duality.
• To investigate the correspondence between components of fixed point loci when passing from $G$ to its Langlands dual $^LG$.
• To provide a detailed study of the case $G = SL(2,\mathbb{C})$ and $^LG = PGL(2,\mathbb{C})$ as a key example of this duality.

Proposed method

• The construction relies on real structures on the complex reductive Lie group $G$, which induce antiholomorphic involutions on the moduli space of $G$-Higgs bundles.
• The moduli space of $G$-Higgs bundles is equipped with a natural complex structure, and the involutions are defined using the real structures on $G$.
• The fixed point loci of these involutions are studied using the geometry of Higgs bundles and the action of the Galois group associated with the real structure.
• The analysis involves comparing the topological and geometric properties of fixed point components for $G$ and $^LG$, particularly focusing on connected components and their invariants.
• The case $G = SL(2,\mathbb{C})$ is analyzed in detail, where the Langlands dual $^LG = PGL(2,\mathbb{C})$ is considered, and the matching of fixed point components is explicitly described.
• The paper uses tools from algebraic geometry, representation theory, and Higgs bundle theory to relate the moduli spaces of $G$ and $^LG$ via duality.

Experimental results

Research questions

• RQ1 How do antiholomorphic involutions on the moduli space of $G$-Higgs bundles arise from real structures on $G$?
• RQ2 What is the structure of the fixed point locus of these involutions for a given $G$?
• RQ3 How do the components of the fixed point locus of $G$ relate to those of its Langlands dual $^LG$?
• RQ4 In the case $G = SL(2,\mathbb{C})$, how do the fixed point components of $G$ and $^LG = PGL(2,\mathbb{C})$ match under Langlands duality?
• RQ5 What geometric and topological invariants are preserved or transformed under this duality in the context of Higgs bundles?

Key findings

• Antiholomorphic involutions are constructed on the moduli space of $G$-Higgs bundles using real structures on $G$, providing a geometric realization of duality.
• The fixed point locus of the involution on $G$-Higgs bundles corresponds to real forms of the Higgs bundle structure, reflecting the Galois action associated with the real structure.
• For $G = SL(2,\mathbb{C})$, the fixed point components of the involution are shown to match precisely with those of $^LG = PGL(2,\mathbb{C})$ under Langlands duality.
• The correspondence between components of the fixed point loci of $G$ and $^LG$ is established via a duality in the topological classification of Higgs bundles, particularly in terms of the degree and stability conditions.
• The analysis reveals that the Langlands dual pair $SL(2,\mathbb{C})$ and $PGL(2,\mathbb{C})$ exhibit a symmetric behavior in their fixed point loci under the antiholomorphic involutions.
• The results demonstrate a precise geometric manifestation of Langlands duality in the context of Higgs bundles, linking real forms and duality through the moduli space structure.