Swarnava Mukhopadhyay

Presently I am thinking about questions related to semiorthogonal decomposition of bounded derived categories, modular functors and braided tensor categories, Verlinde type formulas for twisted conformal blocks, moduli of parahoric Bruhat-Tits torsors and their connections to conformal blocks. I am also interested in questions related to hyperplane arrangements and in particular its connection to representation theory of Lie algebras. I have also worked on applying conformal blocks divisor to the birational geometry of moduli of curves.

A brief presentation of my recent work can be found here. Please feel free to browse my papers below. My collaborators are D. Alfaya, P. Belkale, P. Belmans, I. Biswas, P. Brosnan,T. Deshpande, N. Fakhruddin, S. Galkin, A. Gibney, T. Gomez, M. Inaba, A. Komoyo, A. Paul, M. Saito, R. Wentworth and H. Zelaci.

Here is the link to the algebraic geometry preprint seminar in TIFR. Here is the link to an earlier algebraic geometry seminar in TIFR. Along with Chiara Damiolini and Johan Martens, I organized a conference on Bundles and Conformal Blocks with a twist in ICMS, Edinburgh from June 13-17, 2022. Along with Johan Martens and Richard Wentworth, I am organizing a conference on Geometry and representation theory associated to G-torsors on curves. in the Micheal Brin center for mathematics from April 21-25, 2025. I am also the organizer of the colloquium of the School of Mathematics. You can find the tentative schedule for the upcoming talks here.

1. Applications of the Liouville symplectic form on the cotangent bundle of the loop group
   with I. Biswas, Michi-Aki Inaba, Arata Komyo, Masa-Hiko Saito. Click here for the present version, 23 pages, Submitted.
   [Abstract]

   Let G be a semisimple affine algebraic group defined over C. Consider the Liouville symplectic structure on the total space T^∗G((t)) 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 Higgs G-bundles on a compact connected Riemann surface X and also on the moduli spaces of framed G-connections on X. These symplectic structures on the on the moduli stack of framed Higgs G-bundles and framed connections on X were constructed earlier. Our results show that they have a common origin. Similar results for these moduli stacks with finite order framings are also obtained.
   



2. Torelli theorem for moduli stack of vector bundles and principal G bundles
   with D. Alfaya, I. Biswas and Tomas Gomez, arXiv:2310.07666 for the present version, 18 pages, to appear in Journal of Geometry and Physics.
   [Abstract]

   Given any irreducible smooth complex projective curve X, of genus at least two, 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.
   



3. Motivic factorisation of KZ local systems and deformations of representation and fusion rings
   with P. Belkale and N. Fakhruddin, arXiv:2309.16993 for the present version, 88 pages Submitted.
   [Abstract]

   Let g be a simple Lie algebra over C. The KZ connection is a connection on the constant bundle associated to a set of n finite dimensional irreducible representations of g and a nonzero κ∈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 Z[t] of the representation ring of 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 sln 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.
   



4. Torsors on moduli spaces of principal G-bundles on curves
   with I. Biswas Click here for the present version, to appear in International Journal of Mathematics.
   [Abstract]

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



5. Combinatorial non-abelian Torelli theorem
   with P. Belmans and S. Galkin. Click here for the present version.
   [Abstract]

   We prove that a (colored) trivalent graph can be recovered from (the polar dual of) the associated quantum Clebsch-Gordan polytope and that any isomorphism between such polytopes is induced by a unique properly defined isomorphism of underlying colored graphs. We also show how graph potentials introduced by the authors relate to the theory of random walks, and we use our combinatorial Torelli theorem to construct random walks with distinct shapes but equal return probabilities for every number of steps
   



6. Graph potentials and symplectic geometry of the moduli space of vector bundles
   with P. Belmans and S. Galkin, arXiv:2206.11584, separated from arXiv:2009.05568 along with proof of conjecture on equality of periods. Click here for slides of a talk. Click here for video of a talk.
   [Abstract] For any monotone Lagrangian torus L on a Fano manifold X the weighted number of holomorphic Maslov index 2 discs with boundary on L is bounded from below by an invariant TX (introduced earlier in the Galkin–Golyshev–Iritani conjecture). If this bound is attained the torus L is said to be optimal. We give the first examples of Fano manifolds with multiple optimal tori. To every trivalent graph $\gamma$ of genus g we associate an optimal torus L on the celebrated symplectic Fano manifold Ng (of complex dimension 3g - 3 with TNg = 8g- 8), namely the character variety of rank 2 on a genus g surface with prescribed odd monodromy at a puncture, and we show that all pairs N_g; L_{\gamma} are pairwise non-isotopic. In particular, we confirm a form of mirror symmetry between the A-model of these pairs N_g; L_{\gamma} and B-model of a subclass 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 for the spaces Ng, as moduli spaces of stable rank 2 bundles on an algebraic curve with a fixed determinant, constructed using conformal field theory.

   
   



7. Graph potentials and topological quantum field theories
   with P. Belmans and S. Galkin, arXiv:2205.07244. Click here for slides of a talk. Click here for video of a talk.
   [Abstract]

   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 thus define a topological quantum field theory. We end by giving an efficient computational method to compute its partition function.
   



8. Decompositions of the moduli space of vector bundles and graph potentials
   with P. Belmans and S. Galkin, formerly part of arXiv:2009.05568 along with new material on dimension of critical loci, to appear in Forum of Mathematics, Sigma. Click here for slides of a talk. Click here for video of a talk.
   [Abstract]

   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 Muñoz’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.
   



9. Geometrization of the Hitchin/WZW/KZ connection
   with I. Biswas and R. Wentworth, arXiv:2110.00430, 29 Pages, to appear in Letters in Mathematical Physics. Click here for video of a talk by Richard Wentworth.
   [Abstract]

   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.
   



10. A parabolic analog of a theorem of Beilinson-Schechtman
    with I. Biswas and R. Wentworth, arXiv:2307.09196, 30 Pages, to appear in International Mathematics Research Notices (IMRN).
    [Abstract]

    For a simple, simply connected, complex group $G$, we prove an explict 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.
    



11. A Hitchin connection on non-ableian theta functions for parabolic G-bundles
    with I. Biswas and R. Wentworth, arXiv:2103.03792, J. Reine Angew. Math. (Crelle's Journal) 803(2023), 137–181. Click here for video of a talk.
    [Abstract]

    For a simple, simply connected, complex group G, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over families of smooth projective curves with marked points.
    



12. Appendix D: Rank-level duality of conformal blocks-A brief survey
    Book by Shrawan Kumar entitled: Conformal blocks, generalized theta functions and the Verlinde formula, New Mathematical Monograph series, Cambridge University Press, 2021, 15 Pages.
    [Abstract]

    This is a brief survey on rank-level duality results of conformal blocks on curves. This will appear as an appendix to Shawan Kumar's book.
    



13. Crossed modular categories and the Verlinde formula for twisted conformal blocks
    with T. Deshpande, arXiv:1909.10799, Camb. J. Math.11(2023), no.1, 159–297. Click here for slides of a talk.
    [Abstract]

    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 ℓ 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 twisted conformal blocks. We also explicitly describe the crossed S-matrices that appear in the Verlinde formula for twisted conformal blocks.
    



14. Spectral data for spin Higgs bundles
    with R. Wentworth, Preprint arXiv:1809.05516
    International Math. Research Notices no.6 2021, 20 Pages.
    [Abstract]

    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).
    



15. Conformal Embeddings and twisted theta functions at level one
    with H. Zelaci
    in Proceedings of the American Mathematical Society 148 (2020), 14 Pages.
    [Abstract]

    In this paper, we consider the conformal embedding of so(r) into sl(r) and study relations between level one SO(r)-theta func- tions and twisted SL(r)-theta functions coming from parahoric moduli spaces. In particular, we give another proof of a theorem by Pauly- Ramanan.
    



16. Examples violating Golyshev canonical strip hypothesis
    with P. Belmans and S. Galkin
    to appear in Experimental Mathematics 31 (2022), no. 1, 233–237, Preprint arXiv:1806.07648, 8 Pages.
    [Abstract]

    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.
    



17. Admissble subcategories in the derived category of moduli spaces of vector bundles on curves.
    with P. Belmans
    Advances in Mathematics, 351(2019), Preprint arXiv:180700216. 22 Pages.
    [Abstract]

    We show that the Poincare 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.
    



18. Topology of hyperplane arrangements and invariant theory.
    with P. Belkale and P. Brosnan.
    Michigan Mathematical Journal 68(2019), Preprint arXiv:1611.01861. 37 Pages.
    [Abstract]

    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 locals ystems on complements of hyperplane arrangements. The rest 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 (as the smallest part of the Hodge filtration).
    


19. On higher Chern classes of vector bundles of conformal blocks
    with A. Gibney. Preprint 13 Pages.
    [Abstract] Here we consider higher Chern classes of vector bundles of conformal blocks on the moduli space of stable pointed curves of genus zero, giving explicit formulas for them, and extending various results that hold for first Chern classes to them. We use these classes to form a full dimensional subcone of the Pliant cone on $\overline{M}_{0,n}$..
    


20. Fundamental groups of moduli spaces of Principal bundles over a curve .
    with I. Biswas and A. Paul.
    in Geometriae Dedicata 214(2021), Preprint arXiv:1609.06436, 12 Pages.
    [Abstract] 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 C. For any \pi_1(G), we prove that the moduli space of semistable principal G--bundles over X of topological type d is simply connected. In contrast, the fundamental group of the moduli stack of principal G--bundles over X of topological type d is shown to be isomorphic to H^1(X,\pi_1(G)).
    


21. Generalized theta functions, strange duality, and odd orthogonal bundles on curves.
    with R. Wentworth.
    Communications in Mathematical Physics, 370 (2019) no.1, Preprint arXiv:1608.04990, 51 pages.
    [Abstract]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 c onformal blocks on the pointed projective line with spin weights.
    


22. Strange duality of Verlinde spaces for G_2 and F_4.
    Mathematische Zeitschrift 283 (2016), no 1-2, arXiv:1504.03757, 13 Pages.
    [Abstract] We prove that the pull back of the canonical theta divisor for E8-bundles at level one induces a strange duality between Verlinde spaces for G2 and F4 at level one on smooth curves of genus g. We also prove a parabolic generalization in terms of conformal blocks and write down identities between conformal blocks divisors in Pic(M_{g,n}-bar)_{Q}.
    


23. Rank-level duality and conformal blocks divisors.
    Published in Adv. Math. Volume 287, 10 January 2016, arXiv:1308.0854, 23 Pages.
    [Abstract]We describe new relations among conformal block divisors in Pic(M_{0,n}-bar)_{Q}. These relations appear from various rank-level dualities of conformal blocks on P^1 with n-marked points. We also give a coordinate free description of rank-level duality maps on M_{g,n}-bar.
    


24. Non-vanishing of conformal blocks divisors on \bar-M_{0,n}.
    with P. Belkale and A. Gibney
    Transformation Groups 21(2016), no-2, arXiv:1410.2459, 25 Pages.
    [Abstract] We introduce and study the problem of finding necessary and sufficient conditions under which a conformal blocks divisor on M_{0,n}-bar is nonzero, solving the problem completely for sl2 . We give necessary nonvanishing conditions in type A, which are sufficient when theta and critical levels coincide. We also show divisors are subject to additive identities, reflecting a decomposition of the weights and level.
    


25. Vanishing and identities of conformal blocks divisors.
    with P. Belkale and A. Gibney
    Published in Algebr. Geom. 2 (2015), no. 1 arXiv:1308.4906, 29 Pages.
    [Abstract]Conformal block divisors in type A on M_{0,n}-bar are shown to satisfy new symmetries when levels and ranks are interchanged in non-standard ways. A connection with the quantum cohomology of Grassmannians r eveals that these divisors vanish above the critical level.
    


26. Rank-level duality of conformal blocks for odd orthogonal Lie algebras in genus 0.
    Trans. Amer. Math. Soc. 368 (2016), no. 9, arXiv:1211.2204, 38 Pages.
    [Abstract]Classical invariants for representations of one Lie group can often be related to invariants of some other Lie group. Physics suggests that the right objects to consider for these questions are certain refinements of classical invariants known as conformal blocks. Conformal blocks appear in algebraic geometry as spaces of global sections of line bundles on the moduli stack of parabolic bundles on a smooth curve. Rank-level duality connects a conformal block associated to one Lie algebra to a conformal block for a different Lie algebra. In this paper we prove a rank-level duality for type so(2r+1) on the pointed projective line conjectured by T. Nakanishi and A. Tsuchiya.
    


27. Conformal blocks to cohomology in genus zero.
    with P. Belkale
    Published in Ann. Inst. Fourier 64(2014). arXiv:1205.3199, 52 Pages.
    [Abstract]We give a characterization of conformal blocks in terms of the singular cohomology of suitable smooth projective varieties, in genus 0 for classical Lie algebras and G_2.
    


28. Remarks on level one conformal blocks divisors.
    Published in C. R. Math. Acad. Sci. Paris 352 (2014), no. 3, 4 Pages.
    [Abstract]We show that conformal blocks divisors of type B_r and D_r at level one are effective sums of boundary divisors of M_{0,n}-bar. We also prove that the conformal blocks divisor of type Br at level one with weights (\omega_1,...,\omega_1) scales linearly with the level.
    


29. Diagram automorphisms and rank-level duality.
    arXiv:1308.1756, 17 Pages.(part of this has appeared in my thesis Trans. Amer. Math. Soc. 368 (2016), no. 9 and also in Appedix D of the book of Shrawan Kumar).
    [Abstract] We study the effect of diagram automorphisms on rank-level duality. We create new symplectic rank-level dualities from T. Abe's symplectic rank-level duality on genus zero smooth curves with marked points and chosen coordinates. We also show that ranklevel dualities for the pair sl(r), sl(s) in genus 0 arising from representation theory can also be obtained from the parabolic strange dualities of R. Oudompheng.
    

1. Phd Thesis: Rank-level duality of conformal blocks.
   under the direction of P. Belkale.
   Click here to view. [abstract]TBD. write up something


2. Masters Paper: Factorization of conformal blocks.
   under the direction of S. Kumar.
   Click here to view. [abstract]TBD. write up something


3. Rank-level duality of conformal blocks for SL_n in genus 0 revisited.
   Unpublished. We give an alternate proof of Nakanishi-Tsuchiya's rank-level duality theorem. Notes available on request. [abstract]TBD. write up something


4. Notes on action of diagram automorphism on conformal blocks divisors
   Not for publication. Click here for a copy. [abstract]TBD. write up something