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Swarnava Mukhopadhyay

Publications and source records attributed to Swarnava Mukhopadhyay.

At least 19 recordsLinked to original sources
arXiv2026

Can one hear the shape of a lattice random walk?

We construct distinct high-dimensional mean-zero finite range lattice random walks having pairwise-equal return probabilities for all step counts. The same examples provide pairwise-distinct shapes of discretizations of the standard Laplacian with pairwise-equal density state functions. The main contribution is a reconstruction theorem: to a colored trivalent graph one associates a quantum Clebsch--Gordan polytope, and this association is a full functor, in particular from the polytope one can uniquely recover the original graph. These polytopes appear as moment polytopes of toric degenerations of character varieties (moduli spaces of rank-2 bundles on curves), yielding a combinatorial non-abelian Torelli theorem. In symplectic geometry, the reconstruction theorem implies that monotone Lagrangian tori on odd character varieties associated with these degenerations are pairwise non-Hamiltonian isotopic. These results, and some of the applications, arose from the study of mirror symmetry for moduli spaces of vector bundles, and of the related Laurent phenomenon for mutations of graph potentials.

math.PR↗
arXiv2025

Applications of the Liouville symplectic form on the cotangent bundle of a loop group

Let $G$ be a semisimple, simply connected, affine algebraic group defined over $\mathbb C$. Consider the Liouville symplectic structure on the total space $T^*G((t))$ of the cotangent bundle of the loop group $G((t))$, where $t$ is a formal parameter. We show that the Liouville symplectic structure on $T^*G((t))$ induces the symplectic structures on the moduli stack of framed principal Higgs $G$-bundles on a compact connected Riemann surface $X$ and also on the moduli spaces of framed $G$-connections on $X$. Similar symplectic structures -- on the moduli stack of framed principal Higgs $G$-bundles, with finite order framing, and also framed connections on $X$, with finite order framing -- were constructed earlier by various authors. Our results show that they all have a common origin.

math.AG↗
arXiv2024

Torelli theorem for moduli stacks of vector bundles and principal G-bundles

Given any irreducible smooth complex projective curve $X$, of genus at least $2$, consider the moduli stack of vector bundles on $X$ of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve $X$ and the rank of the vector bundles. The case of trivial determinant, rank $2$ and genus $2$ is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is $\mathbb{P}^3$ thus independently of the curve). We also prove a Torelli theorem for moduli stacks of principal $G$-bundles on a curve of genus at least $3$, where $G$ is any non-abelian reductive group.

math.AG↗
arXiv2024

Torsors on moduli spaces of principal $G$-bundles

Let $G$ be a semisimple complex algebraic group with a simple Lie algebra $\mathfrak{g}$, and let $\mathcal{M}^0_{G}$ denote the moduli stack of topologically trivial stable $G$-bundles on a smooth projective curve $C$. Fix a theta characteristic $κ$ on $C$ which is even in case $\dim{\mathfrak{g}}$ is odd. We show that there is a nonempty Zariski open substack ${\mathcal U}_κ$ of $\mathcal{M}^0_{G}$ such that $H^i(C,\, \text{ad}(E_G)\otimesκ) \,=\, 0$, $i\,=\, 1,\, 2$, for all $E_G\,\in\, {\mathcal U}_κ$. It is shown that any such $E_G$ has a canonical connection. It is also shown that the tangent bundle $T{U}_κ$ has a natural splitting, where $U_κ$ is the restriction of $\mathcal{U}_κ$ to the semi-stable locus. We also produce an isomorphism between two naturally occurring $Ω^1_{{M}^{rs}_{G}}$--torsors on the moduli space of regularly stable ${M}^{rs}_{G}$.

math.AG↗
arXiv2023

Motivic factorisation of KZ local systems and deformations of representation and fusion rings

Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbb{C}$. The KZ connection is a connection on the constant bundle associated to a set of $n$ finite dimensional irreducible representations of $\mathfrak{g}$ and a nonzero $κ\in \mathbb{C}$, over the configuration space of $n$-distinct points on the affine line. Via the work of Schechtman--Varchenko and Looijenga, when $κ$ is a rational number the associated local systems can be seen to be realisations of naturally defined motivic local systems. We prove a basic factorisation for the nearby cycles of these motivic local systems as some of the $n$ points coalesce. This leads to the construction of a family (parametrised by $κ$) of deformations over $\mathbb{Z}[t]$ of the representation ring of $\mathfrak{g}$--we call these enriched representation rings--which allows one to compute the ranks of the Hodge filtration of the associated variations of mixed Hodge structure; in turn, this has applications to both the local and global monodromy of the KZ connection. In the case of $\mathfrak{sl}_n$ we give an explicit algorithm for computing all products in the enriched representation rings, which we use to prove that if $1/κ$ is an integer then the global monodromy is finite and scalar. We also prove a similar factorisation result for motivic local systems associated to conformal blocks in genus $0$; this leads to the construction of a family of deformations of the fusion rings. Computations in these rings have potential applications to finiteness of global monodromy. Several open problems and conjectures are formulated. These include questions about motivic BGG-type resolutions and the relationship between the Hodge filtration and the filtration by conformal blocks at varying levels.

math.AG↗
arXiv2023

Graph potentials and topological quantum field theories

We introduce graph potentials, which are Laurent polynomials associated to (colored) trivalent graphs. We show that the birational type of the graph potential only depends on the homotopy type of the colored graph, and use this to define a topological quantum field theory. A similar construction was recently introduced independently by Kontsevich--Odesskii under the name of multiplicative kernels. We end our paper by giving an efficient computational method to compute its partition function. This is the first paper in a series, and we give a survey of the applications of graph potentials in the other parts.

math.AG↗
arXiv2023

Geometrization of the TUY/WZW/KZ connection

Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. As an application, we give a geometric construction of the Knizhnik-Zamolodchikov connection on the trivial bundle over the configuration space of points in the projective line whose typical fiber is the space of invariants of tensor product of representations.

math.AG↗
arXiv2023

Generalized theta functions, strange duality, and odd orthogonal bundles on curves

This paper studies spaces of generalized theta functions for odd orthogonal bundles with nontrivial Stiefel-Whitney class and the associated space of twisted spin bundles. In particular, we prove a Verlinde type formula and a dimension equality that was conjectured by Oxbury-Wilson. Modifying Hitchin's argument, we also show that the bundle of generalized theta functions for twisted spin bundles over the moduli space of curves admits a flat projective connection. We furthermore address the issue of strange duality for odd orthogonal bundles, and we demonstrate that the naive conjecture fails in general. A consequence of this is the reducibility of the projective representations of spin mapping class groups arising from the Hitchin connection for these moduli spaces. Finally, we answer a question of Nakanishi-Tsuchiya about rank-level duality for conformal blocks on the pointed projective line with spin weights.

math.AG↗
arXiv2023

A parabolic analog of a theorem of Beilinson and Schechtman

For a simple, simply connected, complex group G, we prove an explicit formula to compute the Atiyah class of parabolic determinant of cohomology line bundle on the moduli space of parabolic $G$-bundles. This generalizes an earlier result of Beilinson-Schechtman.

math.AG↗
arXiv2022

Graph potentials and symplectic geometry of moduli spaces of vector bundles

We give the first examples of Fano manifolds with multiple optimal tori, i.e.~we construct monotone Lagrangian tori $L$, such that the weighted number of holomorphic Maslov index two discs with boundary on $L$ equals the upper bound given by the symplectic invariant $\limsup_n ([m_0(L)^n]_{x^0})^{1/n}$, where $m_0(L)$ is the Floer potential. To every trivalent graph $γ$ of genus $g$ we associate an optimal torus $L_γ$ on the celebrated symplectic Fano manifold $\mathcal{N}_g$ (of complex dimension $3g-3$) with $\mathrm{T}_{\mathcal{N}_g} = 8g-8$), given by the character variety of rank 2 on a genus $g$ surface with prescribed odd monodromy at a puncture, We moreover show that all pairs $(\mathcal{N}_g,L_γ)$ are pairwise non-isotopic. In particular, we confirm a form of mirror symmetry between the A-model of the pairs $(\mathcal{N}_g,L_γ)$ (and also spaces $\mathcal{N}_g$ standalone) and B-model of graph potentials, a family of Laurent polynomials we introduced in earlier work. A crucial input from outside of symplectic geometry is an analysis of Manon's toric degenerations of algebro-geometric models $\mathrm{M}_C(2,\mathcal{L})$ for the spaces $\mathcal{N}_g$, as moduli spaces of stable rank $2$ bundles on an algebraic curve with a fixed determinant, constructed using conformal field theory.

math.AG↗
arXiv2022

Decompositions of moduli spaces of vector bundles and graph potentials

We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for, and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau-Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Munoz's decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We will explain how these decompositions can be seen as evidence for the conjectural semiorthogonal decomposition.

math.AG↗
arXiv2022

Crossed modular categories and the Verlinde formula for twisted conformal blocks

In this paper, we give a Verlinde formula for computing the ranks of the bundles of twisted conformal blocks associated with a simple Lie algebra equipped with an action of a finite group $Γ$ and a positive integral level $\ell$ under the assumption that "$Γ$ preserves a Borel". As a motivation for this Verlinde formula, we prove a categorical Verlinde formula which computes the fusion coefficients for any $Γ$-crossed modular fusion category as defined by Turaev. To relate these two versions of the Verlinde formula, we formulate the notion of a $Γ$-crossed modular functor and show that it is very closely related to the notion of a $Γ$-crossed modular fusion category. We compute the Atiyah algebra and prove (with same assumptions) that the bundles of $Γ$-twisted conformal blocks associated with a twisted affine Lie algebra define a $Γ$-crossed modular functor. Along the way, we prove equivalence between a $Γ$-crossed modular functor and its topological analogue. We then apply these results to derive the Verlinde formula for twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.

math.AG↗
arXiv2021

Fundamental group of moduli of principal bundles on curves

Let $X$ be a compact connected Riemann surface of genus at least two, and let ${G}$ be a connected semisimple affine algebraic group defined over $\mathbb C$. For any $δ\in π_1({G})$, we prove that the moduli space of semistable principal ${G}$--bundles over $X$ of topological type $δ$ is simply connected. In contrast, the fundamental group of the moduli stack of principal ${G}$--bundles over $X$ of topological type $δ$ is shown to be isomorphic to $H^1(X, π_1({G}))$.

math.AG↗
arXiv2018

Spectral data for spin Higgs bundles

In this paper we determine the spectral data parametrizing Higgs bundles in a generic fiber of the Hitchin map for the case where the structure group is the special Clifford group with fixed Clifford norm. These are spin and "twisted" spin Higgs bundles. The method used relates variations in spectral data with respect to the Hecke transformations for orthogonal bundles introduced by Abe. The explicit description also recovers a result from the geometric Langlands program which states that the fibers of the Hitchin map are the dual abelian varieties to the corresponding fibers of the moduli spaces of projective orthogonal Higgs bundles (in the even case) and projective symplectic Higgs bundles (in the odd case).

math.AG↗
arXiv2018

Admissible subcategories in derived categories of moduli of vector bundles on curves

We show that the Poincaré bundle gives a fully faithful embedding from the derived category of a curve of sufficiently high genus into the derived category of its moduli space of bundles of rank $r$ with fixed determinant of degree 1. Moreover we show that a twist of the embedding, together with 2 exceptional line bundles, gives the start of a semi-orthogonal decomposition. This generalises results of Narasimhan and Fonarev-Kuznetsov, who embedded the derived category of a single copy of the curve, for rank 2.

math.AG↗
arXiv2018

Examples violating Golyshev's canonical strip hypotheses

We give the first examples of smooth Fano and Calabi-Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarised varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.

math.AG↗
arXiv2017

Hyperplane arrangements and tensor product invariants

In the first part of this paper, we consider, in the context of an arbitrary hyperplane arrangement, the map between compactly supported cohomology to the usual cohomology of a local system. A formula (i.e., an explicit algebraic de Rham representative) for a generalized version of this map is obtained. These results are applied in the second part to invariant theory: Schechtman and Varchenko connect invariant theoretic objects to the cohomology of local systems on complements of hyperplane arrangements. The first part of this paper is then used, following and completing arguments of Looijenga, to determine the image of invariants in cohomology. In suitable cases (e.g., corresponding to positive integral levels), the space of invariants is shown to acquire a mixed Hodge structure over a cyclotomic field. We investigate the Hodge filtration on the space of invariants, and characterize the subspace of conformal blocks in Hodge theoretic terms.

math.AG↗