Special Lecture Series on Donaldson-Thomas and Gromov-Witten Theories

From March 8 to April 19, the Center of Mathematical Sciences and Applications will be hosting a special lecture series on Donaldson-Thomas and Gromov-Witten Theories. Artan Sheshmani (QGM Aarhus and CMSA Harvard) will give eight talks on the topic on Wednesdays and Fridays from 9:00-10:30 am, which will be recorded and promptly available on CMSA’s Youtube Channel.

The topic of the talks are as follows:

 

Date Title Abstract
3-3-2017 Modularity of DT invariants on smooth K3 fibrations I

Video

Motivated by S-duality modularity conjectures in string theory, we study the Donaldson-Thomas invariants of 2-dimensional sheaves inside a nonsingular threefold X. Our main case of study is when X is given by a smooth surface fibration over a curve with the canonical bundle of X pulled back from the base curve. We study the Donaldson-Thomas invariants, as defined by Richard Thomas, of the 2-dimensional Gieseker stable sheaves in X supported on the fibers. In case whereXis aK3 fibration, analogous to the Gromov-Witten theory formula established by Maulik-Pandharipande, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the fiber and the Noether-Lefschetz numbers of the fibration, and prove that the invariants have modular properties. We are intending to go through these lectures with a relatively slow speed, that is: Theorem-Proof style, (so we will be less hand-wavy!) and we intend to establish some of the required background, e.g Noether-Lefschetz theory, vector valuedmodular forms etc.
 3-8-2017 Modularity of DT invariants on smooth K3 fibrations II

Video

Motivated by S-duality modularity conjectures in string theory, we study the Donaldson-Thomas invariants of 2-dimensional sheaves inside a nonsingular threefold X. Our main case of study is when X is given by a smooth surface fibration over a curve with the canonical bundle of X pulled back from the base curve. We study the Donaldson-Thomas invariants, as defined by Richard Thomas, of the 2-dimensional Gieseker stable sheaves in X supported on the fibers. In case whereXis aK3 fibration, analogous to the Gromov-Witten theory formula established by Maulik-Pandharipande, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the fiber and the Noether-Lefschetz numbers of the fibration, and prove that the invariants have modular properties. We are intending to go through these lectures with a relatively slow speed, that is: Theorem-Proof style, (so we will be less hand-wavy!) and we intend to establish some of the required background, e.g Noether-Lefschetz theory, vector valued modular forms etc.
3-10-2017 Conifold Transitions and modularity of DT invariants on Nodal fibrations

Video

Following lectures I and II we continue the discussion on the moduli space of shaves with two dimensional support on K3-fibered threefolds, which can admit finitely many nodal (rational double point singularity at worst) fibers. We will use the conifold transitions and degeneration techniques in this case to relate the geometry of our moduli space and its enumerative invariants to the ones studied in lectures I, II over smooth K3-fibrations.
 4-5-2017 Stable pair PT invariants on smooth fibrations I

Video

We study Pandharipande-Thomas’s stable pair theory on smooth 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.
 4-7-2017 Stable pair PT invariants on smooth fibrations II

Video

 

We study Pandharipande-Thomas’s stable pair theory on smooth 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.
 4-12-2017 Stable pair PT invariants on nodal fibrations: perverse sheaves, Wallcrossings, and an analog of fiberwise T-duality

Video

Following lecture 4, we continue the study of stable pair invariants of K3-fibered threefolds., We investigate the relation of these invariants with the perverse (non-commutative) stable pair invariants of the K3-fibration. In the case that the 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.
 4-14-2017 DT versus MNOP invariants and S_duality conjecture on general complete intersections

Video

Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs ZH in a Calabi–Yau threefold X. Here H is a member of a sufficiently positive linear system and Z is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of ZH, 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.
 4-19-2017 Proof of S-duality conjecture on quintic threefold I

Video

I will talk about an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. We use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity. More precisely, we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve an algebraic-geometric proof of S-duality modularity conjecture.
 4-28-2017 Proof of S-duality conjecture on Quintic threefold II I will talk about an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. We use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over the relative Hilbert scheme of points on surfaces and then prove modularity. More precisely, we have proven that the generating series, associated to the top intersection numbers of the Hilbert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson, where they used representation theory and the machinery of vertex operators to prove this statement for absolute Hilbert schemes. These intersection numbers eventually, together with the generating series of Noether-Lefschetz numbers as I will explain, will provide the ingredients to achieve an algebraic-geometric proof of S-duality modularity conjecture.

Comments are closed.