arXiv subjects

Tathagata Nayak

Publications and source records attributed to Tathagata Nayak.

3 recordsLinked to original sources

arXiv2026

Reversible Higgs Bundles

We study reversible Higgs bundles on a compact Riemann surface $Σ$ of genus $g\geq 2$, that is, $G$-Higgs bundles $(E,Φ)$ with $(E,Φ)\cong(E,-Φ)$. For a complex reductive group $G$ realized as a self-adjoint subgroup of $GL(n,\mathbb{C})$, we show that if $(E,Φ)$ corresponds to a representation $ρ$ of $π_1(Σ)$, then $(E,-Φ)$ corresponds to $(\overlineρ^{\,t})^{-1}$. As a consequence, the moduli space of reversible Higgs bundles is the fixed locus of an involution of the character variety and carries the structure of a real affine algebraic set. For a connected semisimple group and every topological type, we prove that the reversible locus in the smooth moduli space is a $(B,A,A)$-brane. We then describe reversibility through spectral data: in even rank, for a smooth spectral curve, reversible Higgs bundles correspond to line bundles invariant under the involution $η\mapsto-η$; in odd rank, under natural genericity assumptions for which the spectral curve is nodal, we give a gluing criterion together with an explicit polystability condition. Finally, for $SL(2n,\mathbb{C})$ we construct a relative Fourier--Mukai transform over the regular locus of the Hitchin base and prove that it maps the structure sheaf of the reversible locus to an object whose support is exactly the regular degree-zero $PSp(2n,\mathbb{C})$-locus in the Langlands dual $PGL(2n,\mathbb{C})$-moduli space. This support is of type $(B,B,B)$.

math.AG↗

arXiv2025

Functorial Free Group from Anosov Representations on Bundles

Let $ρ: Γ\to G$ be an Anosov representation, with $Γ$ a word hyperbolic group and $G$ a semisimple Lie group. Previous works (Guichard--Wienhard, Kapovich--Leeb--Porti, and Carvajales--Stecker) constructed an open domain of discontinuity $Ω_ρ\subset G/H$, where $H$ is a parabolic or symmetric subgroup. In this paper, we extend the properly discontinuous $Γ$-action via $ρ$ to the space of connections on the pullbacks of the tangent bundle over $Ω_ρ$. When $Ω_ρ$ is a complex curve, we show that the $Γ$-action is properly discontinuous on the union of Higgs bundle structures associated with the $(1,0)$ part of the complexified pullback bundles. We further construct a free abelian group $F^{ab}$ generated by these holomorphic line bundles and induce a topoogical structure on it, so that $ρ(Γ)$ acts properly discontinuously on $F^{ab} \setminus \{\mathrm{id}\}$. This free abelian group is well-defined up to isomorphism over the character variety of Zariski dense Anosov representations. Finally, we endow the space of Anosov representations with a categorical structure compatible with $Ω_ρ$ and construct a natural functor to the category of abelian groups.

math.GT↗

arXiv2025

On Character Variety of Anosov Representations

Let $Γ$ be the fundamental group of a $k$-punctured, $k \geq 0$, closed connected orientable surface of genus $g \geq 2$. We show that the character variety of the $(Q^+, Q^-)$-Anosov irreducible representations, resp. the character variety of the $(P^+, P^-)$-Anosov Zariski dense representations of $Γ$ into $\SL(n , \C)$, $n \geq 2$, is a complex manifold of complex dimension \hbox{$(2g+k-2)(n^2-1)$}. For $Γ=π_1(Σ_g)$, we also show that these character varieties are holomorphic symplectic manifolds.

math.GT↗