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2019

Enno Kessler, Artan Sheshmani, and Shing-Tung Yau. 11/13/2019. “Super J-holomorphic Curves: Construction of the Moduli Space.” arXiv:1911.05607. Publisher's VersionAbstract*Let M be a super Riemann surface with holomorphic distribution D and N a symplectic manifold with compatible almost complex structure J. We call a map Φ:M→N a super J-holomorphic curve if its differential maps the almost complex structure on D to J. Such a super J-holomorphic curve is a critical point for the superconformal action and satisfies a super differential equation of first order. Using component fields of this super differential equation and a transversality argument we construct the moduli space of super J-holomorphic curves as a smooth subsupermanifold of the space of maps M→N.*

Artan Sheshmani and Shing-Tung Yau. 11/4/2019. “Higher rank flag sheaves on surfaces and Vafa-Witten invariants.” arXiv:1911.00124. Publisher's VersionAbstract*We study moduli space of holomorphic triples ϕ: E1→E2, composed of torsion-free sheaves Ei,i=1,2 and a holomorphic mophism between them, over a smooth complex projective surface S. The triples are equipped with Schmitt stability condition [Alg Rep Th. 6. 1. pp 1-32, 2003]. We observe that when Schmitt stability parameter q(m) becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute deformation-obstruction theory in some cases. We further generalize our construction by gluing triple moduli spaces, and extend the earlier work of Gholampour-Sheshmani-Yau [arXiv:1701.08899] where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of chains**where ϕi are injective morphisms and rk(Ei)≥1,∀i. There is a connection, by wallcrossing in the master space in the sense of Mochizuki, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of threefold given by a line bundle on the surface, X:=Tot(L→S). The latter, when L=K_S, provides the means to compute the contribution of higher rank flag sheaves to partition function of Vafa-Witten theory on X.*

*E1→E2→⋯→En,*

Amin Gholampour and Artan Sheshmani. 9/6/2019. “Donaldson-Thomas invariants, linear systems and punctual Hilbert schemes.” arXiv:1909.02679. Publisher's VersionAbstract*We study certain DT invariants arising from stable coherent sheaves in a nonsingular projective threefold supported on the members of a linear system of a fixed line bundle. When the canonical bundle of the threefold satisfies certain positivity conditions, we relate the DT invariants to Carlsson-Okounkov formulas for the "twisted Euler's number" of the punctual Hilbert schemes of nonsingular surfaces, and conclude they have a modular property.*

Dennis Borisov, Ludmil Katzarkov, and Artan Sheshmani. 8/8/2019. “Shifted symplectic structures on derived Quot-stacks.” arXiv:1908.03021. Publisher's VersionAbstract*It is shown that derived Quot-stacks can be mapped into moduli functors of perfect complexes in a formally etale way. In the case of moduli of sheaves on Calabi-Yau manifolds this implies existence of shifted symplectic structures on derived Quot-stacks.*

Dennis Borisov, Artan Sheshmani, and Shing-Tung Yau. 8/1/2019. “Global Shifted Potentials for moduli spaces of sheaves on CY4.” arXiv:1908.00651. Publisher's VersionAbstract*It is shown that any derived scheme over ℂ equipped with a −2-shifted symplectic structure, and having a Hausdorff space of classical points, admits a globally defined Lagrangian distribution as a dg ℂ^∞-manifold. The main tool for proving this theorem is a strictification result for Lagrangian distribution. It is shown that existence of a global Lagrangian distribution allows us to realize the moduli space of sheaves on Calabi-Yau fourfolds as a derived critical locus of a potential of degree −1 on the moduli space of Spin(7) instantons.*

Artan Sheshmani. 7/1/2019. “Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories.” Notices of International Congress of Chinese Mathematicians (2019), 7, 2, Pp. 25-31.Abstract iccm_07_02_a03.pdf

*This article provides the summary of arXiv:1701.08899 and arXiv:1701.08902 where the authors studied the enumerative geometry of “nested Hilbert schemes” of points and curves on algebraic surfaces and their connections to threefold theories, and in particular relevant Donaldson-Thomas, Vafa-Witten and Seiberg-Witten theories.*

Amin Gholampour, Artan Sheshmani, and Shing-Tung Yau. 2/19/2019. “Localized Donaldson-Thomas theory of surfaces.” American Journal of Mathematics (February 2020), 142, 1. Publisher's VersionAbstract*Let S be a projective simply connected complex surface and L be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of L. The moduli space admits a C∗-action induced by scaling the fibers of L. We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on S. We define the localized Donaldson-Thomas invariants of L by virtual localization in the case that L twisted by the anti-canonical bundle of S admits a nonzero global section. When pg(S)>0, in combination with Mochizuki's formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When L is the canonical bundle of S, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.*

2018

Duiliu Emanuel Diaconescu, Artan Sheshmani, and Shing-Tung Yau. 10/22/2018. “Atiyah class and sheaf counting on local Calabi Yau fourfolds.” arXiv: 1810.09382. Publisher's VersionAbstract*We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface using results in arXiv:1701.08899 and arXiv:1701.08902 and . We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.*

Amin Gholampour, Artan Sheshmani, and Shing-Tung Yau. 7/18/2018. “Nested Hilbert schemes on surfaces: Virtual fundamental class”. Publisher's VersionAbstract*We construct natural virtual fundamental classes for nested Hilbert schemes on a nonsingular projective surface S. This allows us to define new invariants of S that recover some of the known important cases such as Poincaré invariants of Dürr-Kabanov-Okonek and the stable pair invariants of Kool-Thomas. In certain cases, we can express these invariants in terms of integrals over the products of Hilbert scheme of points on S, and relate them to the vertex operator formulas found by Carlsson-Okounkov. The virtual fundamental classes of the nested Hilbert schemes play a crucial role in the Donaldson-Thomas theory of local-surface-threefolds that we study in [GSY17b] (arXiv:1807.05697).*

Melissa Liu and Artan Sheshmani. 6/16/2018. “Stacky GKM graphs and orbifold Gromov-Witten theory *(2018)*”. Publisher's VersionAbstract*Following Zong (arXiv:1604.07270), we define an algebraic GKM orbifold X to be a smooth Deligne-Mumford stack equipped with an action of an algebraic torus T, with only finitely many zero-dimensional and one-dimensional orbits. The 1-skeleton of X is the union of its zero-dimensional and one-dimensional T-orbits; its formal neighborhood X^ in X determines a decorated graph, called the stacky GKM graph of X. The T-equivariant orbifold Gromov-Witten (GW) invariants of X can be computed by localization and depend only on the stacky GKM graph of X with the T-action. We also introduce abstract stacky GKM graphs and define their formal equivariant orbifold GW invariants. Formal equivariant orbifold GW invariants of the stacky GKM graph of an algebraic GKM orbifold X are refinements of T-equivariant orbifold GW invariants of X.*

Amin Gholampour and Artan Sheshmani. 2/21/2018. “Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms.” Advances in Mathematics , 326, Pp. 79-107. Publisher's VersionAbstract*Motivated by the S-duality conjecture, we study the Donaldson-Thomas invariants of the 2 dimensional Gieseker stable sheaves on a threefold. These sheaves are supported on the fibers of a nonsingular threefold X fibered over a nonsingular curve. In the case where X is a K3 fibration, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the K3 fiber and the Noether-Lefschetz numbers of the fibration. We prove that a certain generating function of these invariants is a vector modular form of weight -3/2 as predicted in S-duality.*

2017

Sergey Gukov, Chiu-Chu Melissa Liu, Artan Sheshmani, and Shing-Tung Yau. 10/30/2017. “On topological approach to local theory of surfaces in Calabi-Yau threefolds.” Advances in Theoretical and Mathematical Physics , 21, 2017 no 7, Pp. 1679-1728. Publisher's VersionAbstract*We study the web of dualities relating various enumerative invariants, notably Gromov-Witten invariants and invariants that arise in topological gauge theory. In particular, we study Donaldson-Thomas gauge theory and its reductions to D=4 and D=2 which are relevant to the local theory of surfaces in Calabi-Yau threefolds.*

Amin Gholampour and Artan Sheshmani. 6/20/2017. “Intersection numbers on the relative Hilbert schemes of points on surfaces,” Asian Journal of Mathematics, 21, 3, Pp. 531-542. Publisher's VersionAbstract*We study certain top intersection products on the Hilbert scheme of points on a nonsingular surface relative to an effective smooth divisor. We find a formula relating these numbers to the corresponding intersection numbers on the non-relative Hilbert schemes. In particular, we obtain a relative version of the explicit formula found by Carlsson-Okounkov for the Euler class of the twisted tangent bundle of the Hilbert schemes.*

Artan Sheshmani and Chiu-Chu Melissa Liu. 6/14/2017. “Equivariant Gromov-Witten Invariants of Algebraic GKM Manifolds.” Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 13, 48, Pp. 1-21. Publisher's VersionAbstract*An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use virtual localization to express equivariant Gromov-Witten invariants of any algebraic GKM manifold (which is not necessarily compact) in terms of Hodge integrals over moduli stacks of stable curves and the GKM graph of the GKM manifold.*

Amin Gholampour, Artan Sheshmani, and Yukinobu Toda. 3/14/2017. “Stable pairs on nodal K3 fibrations.” International Mathematics Research Notices (IMRN), Vol. 2017, No. 00, Pp. 1-50. Publisher's VersionAbstract*We study Pandharipande-Thomas's stable pair theory on K3 fibrations over curves with possibly nodal fibers. We describe stable pair invariants of the fiberwise irreducible curve classes in terms of Kawai-Yoshioka's formula for the Euler characteristics of moduli spaces of stable pairs on K3 surfaces and Noether-Lefschetz numbers of the fibration. Moreover, we investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the K3fibration. In the case that the K3 fibration is a projective Calabi-Yau threefold, by means of wall-crossing techniques, we write the stable pair invariants in terms of the generalized Donaldson-Thomas invariants of 2-dimensional Gieseker semistable sheaves supported on the fibers.*

2016

Artan Sheshmani. 12/1/2016. “Higher rank stable pairs and virtual localization.” Communications in Analysis and Geometry, 24, 1, Pp. 139-193. Publisher's VersionAbstract*We define and compute higher rank analogs of Pandharipande-Thomas stable pair invariants in primitive classes for K3 surfaces. Higher rank stable pair invariants for Calabi-Yau threefolds have been defined by Sheshmani \cite{shesh1,shesh2} using moduli of pairs of the form $Ø^n\into \F$ for $\F$ purely one-dimensional and computed via wall-crossing techniques. These invariants may be thought of as virtually counting embedded curves decorated with a (n−1)-dimensional linear system. We treat invariants counting pairs $Ø^n\into \E$ on a $\K3$ surface for $\E$ an arbitrary stable sheaf of a fixed numerical type ("coherent systems" in the language of \cite{KY}) whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of $\K3$ surfaces is treated by \cite{MPT}; there they prove the KKV conjecture in primitive classes by showing the resulting partition functions are governed by quasimodular forms. We prove a "higher" KKV conjecture by showing that our higher rank partition functions are modular forms.*

Artan Sheshmani. 12/1/2016. “Wall-crossing and invariants of higher rank stable pairs.” Illinois Journal of Mathematics, 59, 1, Pp. 55-83. Publisher's VersionAbstract*We introduce a higher rank analog of the Joyce-Song theory of stable pairs. Given a nonsingular projective Calabi-Yau threefold X, we define the higher rank Joyce-Song pairs given by OrX(−n)→F where F is a pure coherent sheaf with one dimensional support, r>1 and n≫0 is a fixed integer. We equip the higher rank pairs with a Joyce-Song stability condition and compute their associated invariants using the wallcrossing techniques in the category of weakly semistable objects.*

Vincent Bouchard, Thomas Creutzig, Emanuel Diaconescu, Charles Doran, Callum Quigley, and Artan Sheshmani. 11/14/2016. “Vertical D4-D2-D0 bound states on K3 fibrations and modularity.” Communications in Mathematical Physics, 350, 3, Pp. 1069–1121. Publisher's VersionAbstract*An explicit formula is derived for the generating function of vertical D4-D2-D0 bound states on smooth K3 fibered Calabi-Yau threefolds, generalizing previous results of Gholampour and Sheshmani. It is also shown that this formula satisfies strong modularity properties, as predicted by string theory. This leads to a new construction of vector valued modular forms which exhibits some of the features of a generalized Hecke transform.*

Artan Sheshmani. 9/1/2016. “Weighted Euler characteristic of the moduli space of higher rank Joyce-Song pairs,” European Journal of Mathematics. Publisher's VersionAbstract*The invariants of rank 2 Joyce-Song semistable pairs over a Calabi-Yau threefold were computed in arXiv:1101.2252, using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman. Such wall-crossing computations often depend on the combinatorial properties of certain elements of a Hall-algebra (these are the stack functions defined by Joyce). These combinatorial computations become immediately complicated and hard to carry out, when studying higher rank stable pairs with rank>2. The main purpose of this article is to introduce an independent approach to computation of rank 2 stable pair invariants, without applying the wallcrossing formula and rather by stratifying their corresponding moduli space and directly computing the weighted Euler characteristics of the strata. This approach may similarly be used to avoid complex combinatorial wallcrossing calculations in rank>2 cases.*

2014

Amin Gholampour, Artan Sheshmani, and R. P. Thomas. 10/1/2014. “Counting curves on surfaces in Calabi-Yau 3-folds.” Mathematische Annalen, 360, 1, Pp. 67-78. Publisher's VersionAbstract*Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of 2-dimensional torsion sheaves, enumerating pairs Z⊂H in a Calabi-Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a 1-dimensional subscheme of it. The associated sheaf is the ideal sheaf of Z⊂H, pushed forward to X and considered as a certain Joyce-Song pair in the derived category of X. We express these invariants in terms of the MNOP invariants of X.*

In gauge theory, the prominent examples of enumerative invariants are Donaldson polynomials and Seiberg-Witten invariants, which help to distinguish different smooth structures on 4 dimensional manifolds. In recent years, other 4-manifold invariants have been introduced by changing the gauge theory (the PDE’s and the “counting” problem) or, by changing the dimension, similar gauge theory invariants were defined on higher-dimensional manifolds. Notable examples include Donaldson-Thomas (DT) invariants for six-dimensional, Calabi-Yau, manifolds. The study of enumerative geometry (counting of algebraic subspaces) of complex surfaces and threefolds has proven to be deeply related to physical structures, e.g. around Gopakumar-Vafa invariants (GV); Gromov-Witten invariants, DT, as well as Pandharipande-Thomas (PT) invariants; and their “motivic lifts". On the other hand, physical dualities in Gauge and String theory, such as Montonen-Olive duality and heterotic/Type II duality have also been a rich source of spectacular predictions about these counting invariants. An example of this is the modularity properties of GW or DT invariants which is proved mathematically in some cases, as suggested by the heterotic/Type II duality.

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