Long, C., et al., 2021. Non-Holomorphic cycles and Non-BPS black Branes. arXiv:2104.06420. Publisher's VersionAbstract
We study extremal non-BPS black holes and strings arising in M-theory compactifications on Calabi-Yau threefolds, obtained by wrapping M2 branes on non-holomorphic 2-cycles and M5 branes on non-holomorphic 4-cycles. Using the attractor mechanism we compute the black hole mass and black string tension, leading to a conjectural formula for the asymptotic volumes of connected, locally volume-minimizing representatives of non-holomorphic, even-dimensional homology classes in the threefold, without knowledge of an explicit metric. In the case of divisors we find examples where the volume of the representative corresponding to the black string is less than the volume of the minimal piecewise-holomorphic representative, predicting recombination for those homology classes and leading to stable, non-BPS strings. We also compute the central charges of non-BPS strings in F-theory via a near-horizon AdS3 limit in 6d which, upon compactification on a circle, account for the asymptotic entropy of extremal non-supersymmetric 5d black holes (i.e., the asymptotic count of non-holomorphic minimal 2-cycles).
Sheshmani, A. & You, Y., 2021. Categorical Representation Learning: Morphism is all you need. Machine Learning: Science and Technology. Publisher's VersionAbstract
We provide a construction for categorical representation learning and introduce the foundations of ''categorifier". The central theme in representation learning is the idea of everything to vector. Every object in a dataset C can be represented as a vector in ℝ^n by an encoding map E: Obj(C)→ℝ^n. More importantly, every morphism can be represented as a matrix E: Hom(C)→ℝ^n_n. The encoding map E is generally modeled by a deep neural network. The goal of representation learning is to design appropriate tasks on the dataset to train the encoding map (assuming that an encoding is optimal if it universally optimizes the performance on various tasks). However, the latter is still a set-theoretic approach. The goal of the current article is to promote the representation learning to a new level via a category-theoretic approach. As a proof of concept, we provide an example of a text translator equipped with our technology, showing that our categorical learning model outperforms the current deep learning models by 17 times. The content of the current article is part of the recent US patent proposal (patent application number: 63110906).
Saadat, M., et al., 2021. Hyrdodynamic advantage of in-line schooling. Bioinspiration & Biomimetics , pp. 1-21. Publisher's VersionAbstract
Fish benefit energetically when swimming in groups, which is reflected in lower tail-beat frequencies for maintaining a given speed. Recent studies further show that fish save the most energy when swimming behind their neighbor such that both the leader and the follower benefit. However, the mechanisms underlying such hydrodynamic advantages have thus far not been established conclusively. The long-standing drafting hypothesis – reduction of drag forces by judicious positioning in regions of reduced oncoming flow– fails to explain advantages of in-line schooling described in this work. We present an alternate hypothesis for the hydrodynamic benefits of in-line swimming based on enhancement of propulsive thrust. Specifically, we show that an idealized school consisting of in-line pitching foils gains hydrodynamic benefits via two mechanisms that are rooted in the undulatory jet leaving the leading foil and impinging on the trailing foil: (i) leading-edge suction on the trailer foil, and (ii) added-mass push on the leader foil. Our results demonstrate that the savings in power can reach as high as 70% for a school swimming in a compact arrangement. Informed by these findings, we designed a modification of the tail propulsor that yielded power savings of up to 56% in a self-propelled autonomous swimming robot. Our findings provide insights into hydrodynamic advantages of fish schooling, and also enable bioinspired designs for significantly more efficient propulsion systems that can harvest some of their energy left in the flow.
Kessler, E., Sheshmani, A. & Yau, S.-T., 2021. Super J-holomorphic Curves: Construction of the Moduli Space. Mathematischel Annalen. 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.
Liu, M. & Sheshmani, A., 2021. Stacky GKM graphs and orbifold Gromov-Witten theory. Asian Journal of Mathematics , 24 (5) , pp. 48. 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.
Kessler, E., Sheshmani, A. & Yau, S.-T., 2020. Super quantum cohomology I: Super stable maps of genus zero with Neveu-Schwarz punctures. arXiv:2010.15634. Publisher's VersionAbstract
In this article we define stable supercurves and super stable maps of genus zero via labeled trees. We prove that the moduli space of stable supercurves and super stable maps of fixed tree type are quotient superorbifolds. To this end, we prove a slice theorem for the action of super Lie groups on supermanifolds with finite isotropy groups and discuss superorbifolds. Furthermore, we propose a generalization of Gromov convergence to super stable maps such that the restriction to fixed tree type yields the quotient topology from the superorbifold and the reduction is compact. This would, possibly, lead to the notions of super Gromov-Witten invariants and small super quantum cohomology to be discussed in sequels.
McBreen, M., Sheshmani, A. & Yau, S.-T., 2020. Elliptic stable envelopes and hypertoric loop spaces. arXiv:2010.00670. Publisher's VersionAbstract
This paper relates the elliptic stable envelopes of a hypertoric variety X with the K-theoretic stable envelopes of the loop hypertoric space, ℒ˜X. It thus points to a possible categorification of elliptic stable envelopes.
Borisov, D., Sheshmani, A. & Yau, S.-T., 2020. Global shifted potentials for moduli stacks of sheaves on Calabi-Yau four-folds II (the stable locus). arXiv:2007.13194. Publisher's VersionAbstract
It is shown that there are globally defined Lagrangian distributions on the stable loci of derived Quot-stacks of coherent sheaves on Calabi-Yau four-folds. Dividing by these distributions produces perfectly obstructed smooth stacks with globally defined −1-shifted potentials, whose derived critical loci give back the stable loci of smooth stacks of sheaves in global Darboux form.
McBreen, M., Sheshmani, A. & Yau, S.-T., 2020. Twisted Quasimaps and Symplectic Duality for Hypertoric Spaces. arXiv:2004.04508. Publisher's VersionAbstract
We study moduli spaces of twisted quasimaps to a hypertoric variety X, arising as the Higgs branch of an abelian supersymmetric gauge theory in three dimensions. These parametrise general quiver representations whose building blocks are maps between rank one sheaves on ℙ^1, subject to a stability condition, associated to the quiver, involving both the sheaves and the maps. We show that the singular cohomology of these moduli spaces is naturally identified with the Ext group of a pair of holonomic modules over the `quantized loop space' of X, which we view as a Higgs branch for a related theory with infinitely many matter fields. We construct the coulomb branch of this theory, and find that it is a periodic analogue of the coulomb branch associated to X. Using the formalism of symplectic duality, we derive an expression for the generating function of twisted quasimap invariants in terms of the character of a certain tilting module on the periodic coulomb branch. We give a closed formula for this generating function when X arises as the abelianisation of the N-step flag quiver.
Gholampour, A. & Sheshmani, A., 2020. Donaldson-Thomas invariants, linear systems and punctual Hilbert schemes. Mathematical Research Letters (MRL), Accepted. 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.
Diaconescu, D.E., Sheshmani, A. & Yau, S.-T., 2020. Atiyah class and sheaf counting on local Calabi Yau fourfolds. Advances in Mathematics , 368 (2020) , pp. 1-54. 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.
Gholampour, A., Sheshmani, A. & Yau, S.-T., 2020. Nested Hilbert schemes on surfaces: Virtual fundamental class. Advances in Mathematics , 365 ( 13). 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).
Gholampour, A., Sheshmani, A. & Yau, S.-T., 2020. Localized Donaldson-Thomas theory of surfaces. American Journal of Mathematics , 142 (2). 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.
Sheshmani, A. & Yau, S.-T., 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.
Borisov, D., Katzarkov, L. & Sheshmani, A., 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.
Borisov, D., Sheshmani, A. & Yau, S.-T., 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.
Sheshmani, A., 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

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.

Gholampour, A. & Sheshmani, A., 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.
Gukov, S., et al., 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.
Gholampour, A. & Sheshmani, A., 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.