arXiv subjects

Gorapada Bera

Publications and source records attributed to Gorapada Bera.

7 recordsLinked to original sources

arXiv2025

Deformations of asymptotically cylindrical associative submanifolds

This article develops the deformation theory of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds, laying the foundation for the gluing of ACyl associative submanifolds in twisted connected sum $G_2$-manifolds presented by the author in [Ber22]. We study the moduli space of ACyl associative submanifolds with a fixed asymptotic holomorphic curve and a fixed rate, as well as the moduli space where this asymptotic data is allowed to vary, each endowed with a natural topology. We also express their virtual dimensions in terms of data on the asymptotic cross-sections.

math.DG↗

arXiv2025

Dirac operators twisted by ramified Euclidean line bundles

This article is concerned with the analysis of Dirac operators $D$ twisted by ramified Euclidean line bundles $(Z,\mathfrak{l})$-motivated by their relation with harmonic $\mathbf{Z}/2\mathbf{Z}$ spinors, which have appeared in various context in gauge theory and calibrated geometry. The closed extensions of $D$ are described in terms of the Gelfand-Robbin quotient $\check{\mathbf{H}}$. Assuming that the branching locus $Z$ is a closed cooriented codimension two submanifold, a geometric realisation of $\check{\mathbf{H}}$ is constructed. This, in turn, leads to an $L^2$ regularity theory.

math.DG↗

arXiv2024

Uniqueness in the local Donaldson-Scaduto conjecture

The local Donaldson-Scaduto conjecture predicts the existence and uniqueness of a special Lagrangian pair of pants with three asymptotically cylindrical ends in the Calabi-Yau 3-fold $X \times \mathbb{R}^2$, where $X$ is an ALE hyperk\"ahler 4-manifold of $A_2$-type. The existence of this special Lagrangian has previously been proved. In this paper, we prove a uniqueness theorem, showing that no other special Lagrangian pair of pants satisfies this conjecture.

math.DG↗

arXiv2024

Remarks on $\mathrm{Sp}(1)$-Seiberg-Witten equation over $3$-manifolds

We prove that the $\mathrm{Sp}(1)$-Seiberg-Witten equation over a closed hyperbolic $3$-manifold ${\mathbb H}^3/\Gamma$ always admits a canonical irreducible solution induced by the hyperbolic metric. We also prove that the Zariski tangent space of the moduli space at this canonical solution is same as the Zariski tangent space of the moduli space of locally conformally flat structures at the hyperbolic metric. This space is again same as the space of trace-free Codazzi tensors and carries an injection to $H^1(\Gamma,\mathbb R^{1,3})$, the first group cohomology of the $\Gamma$-module $\mathbb R^{1,3}$. In particular, if $H^1(\Gamma,\mathbb R^{1,3})=0$ then the canonical irreducible solution is infinitesimally rigid. We also prove that the $\mathrm{Sp}(1)$-Seiberg-Witten equation over $S^1\times \Sigma$ has no irreducible solutions and the moduli space of reducible solutions is same as the moduli space of flat $\mathrm{SU}(2)$-connections.

math.DG↗

arXiv2023

Growth of spinors in the generalized Seiberg-Witten equations on $\mathbb R^4$ and $\mathbb R^3$

The classical Seiberg-Witten equations in dimensions three and four admit a natural generalization within a unified framework known as the generalized Seiberg-Witten (GSW) equations, which encompasses many important equations in gauge theory. This article proves that the averaged $L^2$-norm of any spinor with non-constant pointwise norm in the GSW equations on $\mathbb R^4$ and $\mathbb R^3$, measured over large-radius spheres, grows faster than a power of the radius, under a suitable curvature decay assumption. Separately, it is shown that if the Yang-Mills-Higgs energy of any solution of these equations is finite, then the pointwise norm of the spinor in it must converge to a non-negative constant at infinity. These two behaviors cannot occur simultaneously unless the spinor has constant pointwise norm. This work may be seen as partial generalization of results obtained by Taubes[Tau17a], and Nagy and Oliveira [NO19] for the Kapustin-Witten equations.

math.DG↗

arXiv2023

Deformations and desingularizations of conically singular associative submanifolds

The proposals of Joyce [Joy18], and Doan and Walpuski [DW19] on counting closed associative submanifolds of $G_2$-manifolds depend on various conjectural transitions. This article contributes to the study of transitions arising from the degenerations of associative submanifolds into conically singular (CS) associative submanifolds. First, we study the moduli space of CS associative submanifolds with isolated singularities modeled on associative cones in $\mathbb R^7$, establishing transversality results in both fixed and one-parameter family of co-closed $G_2 $-structures. We prove that for a generic co-closed $G_2$-structure (or a generic path thereof) there are no CS associative submanifolds having singularities modeled on cones with stability-index greater than $0$ (or $1$, respectively). We establish that associative cones whose links are null-torsion holomorphic curves in $S^6$ have stability-index greater than $4$, and all special Lagrangian cones in $\mathbb C^3$ have stability-index greater than or equal to $1$ with equality only for the Harvey-Lawson $T^2$-cone and a transverse pair of planes. Next, we study the desingularizations of CS associative submanifolds in a one-parameter family of co-closed $G_2$-structures. Consequently, we derive desingularization results relating the above transitions for CS associative submanifolds with a Harvey-Lawson $T^2$-cone singularity and for associative submanifolds with a transverse self-intersection.

math.DG↗

arXiv2022

Associative submanifolds in twisted connected sum $G_2$-manifolds

We introduce a method to construct closed rigid associative submanifolds in twisted connected sum $G_2$-manifolds. More precisely, we prove a gluing theorem of asymptotically cylindrical (ACyl) associative submanifolds in ACyl $G_2$-manifolds under a transverse intersection hypothesis. This is analogous to the gluing theorem for $G_2$-instantons introduced in [SW15]. We rephrase the hypothesis in the special cases where the ACyl associative submanifolds are obtained from holomorphic curves or special Lagrangians in ACyl Calabi-Yau $3$-folds, in terms of algebraic-geometric conditions and topological conditions, respectively. This yields many rigid associative submanifolds with new topological types $S^3$, $\mathbf R\mathbf P^3$ or $\mathbf R\mathbf P^3\#\mathbf R\mathbf P^3$.

math.DG↗