Duiliu Emanuel Diaconescu, Artan Sheshmani, and Shing-Tung Yau. Submitted. “Atiyah class and sheaf counting on local Calabi Yau fourfolds (2018)”. 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.
Melissa Liu and Artan Sheshmani. Submitted. “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, Artan Sheshmani, and Shing-Tung Yau. 2/19/2019. “Localized Donaldson-Thomas theory of surfaces (2018).” American Journal of Mathematics (To Appear). 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.
Artan Sheshmani. 1/12/2019. “Hilbert Schemes, Donaldson-Thomas theory, Vafa-Witten and Seiberg-Witten theories.” Notices of International Congress of Chinese Mathematicians (To Appear), TBA, Pp. TBA. survey-iccm-2.pdf
Amin Gholampour, Artan Sheshmani, and Shing-Tung Yau. 7/18/2018. “Nested Hilbert schemes on surfaces: Virtual fundamental class .” Journal of Differential Geometry (To Appear). 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).
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.
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.
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.
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.
Amin Gholampour and Artan Sheshmani. 8/31/2013. “Generalized Donaldson-Thomas Invariants of 2-Dimensional sheaves on local P^2.” Advances in Theoretical and Mathematical Physics, 19, 3, Pp. 673 – 699.Abstract
Let X be the total space of the canonical bundle of P^2. We study the generalized Donaldson-Thomas invariants, defined in the work of Joyce-Song, of the moduli spaces of the 2-dimensional Gieseker semistable sheaves on X with first Chern class equal to k times the class of the zero section of X. When k=1, 2 or 3, and semistability implies stability, we express the invariants in terms of known modular forms. We prove a combinatorial formula for the invariants when k=2 in the presence of the strictly semistable sheaves, and verify the BPS integrality conjecture of Joyce-Song in some cases.
8/1/2012. “An introduction to the theory of Higher rank stable pairs and Virtual localization (2012).” In Proceedings of Symposia in Pure Mathematics, 85th ed., 2012: Pp. 455-465. Publisher's VersionAbstract
This article is an attempt to briefly introduce some of the results from arXiv:1011.6342 on development of a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for highly frozen triples given by the data O^r-->F for r>1 where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory. Instead, we build a zero-dimensional virtual fundamental class by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. We briefly include the results of calculations in this enumerative theory for local P^1 using the Graber-Pandharipande virtual localization technique. We emphasize that in this article we merely include the statement of our theorems and illustrate the final result of some of the computations. The proofs and more detailed calculations in arXiv:1011.6342 have appeared elsewhere.
Artan Sheshmani. 5/1/2011. “Towards studying the higher rank theory of stable pairs (PhD Thesis).” University of Illinois at Urbana Champaign. Publisher's VersionAbstract
This thesis is composed of two parts. In the first part we introduce a higher rank analog of the Pandharipande-Thomas theory of stable pairs on a Calabi-Yau threefold X. More precisely, we develop a moduli theory for frozen triples given by the data O_X(−n)^r-->F where F is a sheaf of pure dimension 1. The moduli space of such objects does not naturally determine an enumerative theory: that is, it does not naturally possess a perfect symmetric obstruction theory. Instead, we build a zero-dimensional virtual fundamental class by hand, by truncating a deformation-obstruction theory coming from the moduli of objects in the derived category of X. This yields the first deformation-theoretic construction of a higher-rank enumerative theory for Calabi-Yau threefolds. We calculate this enumerative theory for local ℙ1 using the Graber-Pandharipande virtual localization technique. In the second part of the thesis we compute the Donaldson-Thomas type invariants associated to frozen triples using the wall-crossing formula of Joyce-Song and Kontsevich-Soibelman.
Banavara N. Shashikanth, Artan Sheshmani, Scott D. Kelly, and Mingjun Wei. 8/1/2010. “Hamiltonian structure and dynamics of a neutrally buoyant rigid sphere interacting with thin vortex rings.” Journal of Mathematical Fluid Mechanics, 12, 3, Pp. 335-353. Publisher's VersionAbstract
In a previous paper, we presented a (noncanonical) Hamiltonian model for the dynamic interaction of a neutrally buoyant rigid body of arbitrary smooth shape with N closed vortex filaments of arbitrary smooth shape, modeled as curves, in an infinite ideal fluid in ℝ3R3. The setting of that paper was quite general, and the model abstract enough to make explicit conclusions regarding the dynamic behavior of such systems difficult to draw. In the present paper, we examine a restricted class of such systems for which the governing equations can be realized concretely and the dynamics examined computationally. We focus, in particular, on the case in which the body is a smooth sphere. The equations of motion and Hamiltonian structure of this dynamic system, which follow from the general model, are presented. Following this, we impose the constraint of axisymmetry on the entire system and look at the case in which the rings are all circles perpendicular to a common axis of symmetry passing through the center of the sphere. This axisymmetric model, in our idealized framework, is governed by ordinary differential equations and is, relatively speaking, easily integrated numerically. Finally, we present some plots of dynamic orbits of the axisymmetric system.
Banvara N. Shashikanth, Artan Sheshmani, Scott D. Kelly, and Jerrold E. Marsden. 1/1/2008. “Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape : The case of arbitrary smooth body shape.” Theoretical and Computational Fluid Dynamics, 22, 1, Pp. 37-64. Publisher's VersionAbstract
We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.