# Henry Liu

## Enumerative Geometry Seminar (Fall 2018)

The goal of this seminar is to understand the celebrated Gromov-Witten/Donaldson-Thomas correspondence.

• Organizers: Henry Liu, Melissa Chiu-Chu Liu
• Time/date: Tuesdays 2:40-3:55pm and Wednesdays 1:10-2:25pm
• Location: Math 507 (Tues) and Math 622 (Weds)

Talks on the Gromov-Witten side will generally be on Tuesdays. Talks on the Donaldson-Thomas side will generally be on Wednesdays.

I am live-TeXing notes for this seminar, available here.

Please email me at hliu at math dot columbia dot edu if you would like to be on the mailing list.

### Plan/References

The plan is to go through the MNOP and MOOP papers, which state and prove (primary, descendant) GW/DT correspondences for (toric, CY) 3-folds.

• [MNOP1] Gromov-Witten theory and Donaldson-Thomas theory, I
• [MNOP2] Gromov-Witten theory and Donaldson-Thomas theory, II
• [MOOP] Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds

For the GW side, we need the GW vertex, which involves Hodge integrals. The 1-leg and 2-leg cases are in the following papers.

• [LLZ1] A Proof of a Conjecture of Mariño-Vafa on Hodge integrals
• [OP1] Hodge integrals and invariants of the unknot
• [LLZ2] A Formula of Two-Partition Hodge Integrals

To do GW/DT for arbitrary toric 3-folds, not just CY3s, we need the theory for local curves and $$A_n$$ resolutions.

• [BP] The local Gromov-Witten theory of curves
• [OP2] The local Donaldson-Thomas theory of curves
• [M] Gromov-Witten theory of $$A_n$$-resolutions
• [MO] Donaldson-Thomas theory of $$A_n \times \mathbb{P}^1$$

We may go through additional related topics/papers as time permits.

### Schedule

 Tues Sept 11 Melissa Liu Hurwitz numbers and the ELSV formula The ELSV formula, first proved by Ekedahl, Lando, Shapiro, and Vainshtein, relates Hurwitz numbers to Hodge integrals. In this talk, we explain what the ELSV formula is and how to prove it by virtual localization on moduli of relative stable maps to the projective line relative a point, following Graber-Vakil. Reference: Lectures on the ELSV formula Weds Sept 12 Clara Dolfen GW/DT for local CY toric surfaces In their paper "Gromov-Witten theory and Donaldson Thomas theory I" (MNOP1), Maulik et al. conjecture a correspondence between the generating functions of Gromov-Witten and Donaldson-Thomas invariants for 3-folds, and prove it in the case of local Calabi-Yau toric surfaces. In this talk, we will give a brief introduction to DT theory, and talk about the fixed points of the DT moduli space under the torus action. We will use this insight to compute the DT counts via virtual localization following MNOP1. Reference: [MNOP1] Tues Sept 18 Melissa Liu ELSV formula via relative virtual localization This is a sequel of my talk on September 11. In my talk on September 11, we defined Hodge integrals and simple Hurwitz numbers, and stated the ELSV formula (first proved by Ekedahl-Lando-Shapiro-Vainshtein) expressing simple Hurwitz numbers in terms of Hodge integrals. We then interpreted each simple Hurwitz number as the degree of a morphism from certain moduli of relative stable maps to a projective space. In this talk, we prove the ELSV formula by computing this degree via relative virtual localization. Reference: Lectures on the ELSV formula Weds Sept 19 Clara Dolfen DT for local CY toric surfaces Last week we set up the machinery needed to compute the DT invariants for toric CY 3-folds via virtual localization. In this talk, we will briefly recall the weight decomposition of the virtual tangent space and use it to derive an explicit formula for the DT counts in the case of local CY toric surfaces. Reference: [MNOP1] Tues Sept 25 Melissa Liu GW/DT correspondence for the resolved conifold The resolved conifold is the total space of $$\mathcal{O}(-1)\oplus \mathcal{O}(-1)$$ over $$\mathbb{P}^1$$. It is a toric Calabi-Yau threefold whose GW/DT invariants can be interpreted as local GW/DT invariants of a super-rigid smoothly embedded rational curve in a Calabi-Yau threefold. We will explain GW/DT correspondence for the resolved conifold, and its relation with the Gopakumar-Vafa conjecture. References: Graber and Pandharipande, Localization of virtual classes Faber and Pandharipande, Hodge integrals and Gromov-Witten theory Behrend and Bryan, Super-rigid Donaldson-Thomas invariants Weds Sept 26 Ivan Danilenko GW/DT in relative setting and with descendants This time we follow "Gromov-Witten theory and Donaldson Thomas theory II" (MNOP2), Maulik et al. The main goal of the talk is to introduce DT invariants with insertions and relative DT invariants. We will state (partly conjectural) relations with their GW counterparts. Time permitting, we'll show how to compute zero dimensional contributions in DT theory by localization. References: [MNOP2] Tues Oct 02 Melissa Liu GW/DT correspondence for the resolved conifold II This is a sequel of my talk on September 25. References: Graber and Pandharipande, Localization of virtual classes Faber and Pandharipande, Hodge integrals and Gromov-Witten theory Behrend and Bryan, Super-rigid Donaldson-Thomas invariants Weds Oct 03 Ivan Danilenko GW/DT in relative setting and with descendants II We'll define the DT rubber theory and use it to compute the equivariant vertex in the relative setting. References: [MNOP2] Tues Oct 09 Melissa Liu Introduction to Relative Gromov-Witten Theory References: Jun Li, Stable morphisms to singular schemes and relative stable morphisms Jun Li, A degeneration formula of GW-invariants Jun Li, Lecture notes on relative GW-invariants (ICTP lecture notes) Weds Oct 10 Ivan Danilenko GW/DT in relative setting and with descendants III We'll define the DT rubber theory and use it to compute the equivariant vertex in the relative setting. References: [MNOP2], [OP2] Tues Oct 16 Henry Liu The GW local curves TQFT We'll define the 2d TQFT associated to the GW partition function for local curves. If time permits we'll prove its semisimplicity. References: [BP] Bryan and Pandharipande, Curves in Calabi–Yau threefolds and topological quantum field theory Freed, Lectures on topological quantum field theory Weds Oct 17 Anton Osinenko Local DT theory of curves The local Donaldson-Thomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating Gromov-Witten theory, Donaldson-Thomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane. The quantum differential equation of the Hilbert scheme of points of the plane has a natural interpretation in the local Donaldson-Thomas theory of curves. The solution determines the 1-legged equivariant vertex. References: [OP2] Tues Oct 23 Henry Liu Local curve computations We'll prove the GW local curves TQFT is semisimple and compute its pieces: tube, caps, and pair of pants. References: [BP] Bryan and Pandharipande, Curves in Calabi–Yau threefolds and topological quantum field theory Faber and Pandharipande, Hodge integrals and Gromov-Witten theory Weds Oct 24 Shuai Wang Local DT theory of curves The local Donaldson-Thomas theory of curves is solved by localization and degeneration methods. The results complete a triangle of equivalences relating Gromov-Witten theory, Donaldson-Thomas theory, and the quantum cohomology of the Hilbert scheme of points of the plane. The quantum differential equation of the Hilbert scheme of points of the plane has a natural interpretation in the local Donaldson-Thomas theory of curves. The solution determines the 1-legged equivariant vertex. References: [OP2] Tues Oct 30 Henry Liu Cap and pants We'll first finish computing the level (-1,0) cap. Then we'll finish computing the whole local curves TQFT by computing the level (0,0) pair of pants explicitly in one (simple) case, and then in general via a reconstruction result. References: [BP] Bryan and Pandharipande, Curves in Calabi–Yau threefolds and topological quantum field theory Faber and Pandharipande, Hodge integrals and Gromov-Witten theory Weds Oct 31 Shuai Wang Local DT theory of curves We will compute the cap and pair of pants. References: [OP2] Tues Nov 06 No meeting (election day) Weds Nov 07 No meeting (we're taking a break!) Tues Nov 13 Melissa Liu The 1-leg Gromov-Witten vertex References: [LLZ1], [OP1] Weds Nov 14 Yakov Kononov Relative DT theory of $$A_n \times \mathbb{P}^1$$ The talk will be devoted to the relative Donaldson-Thomas theory of $$A_n \times \mathbb{P}^1$$. I will express the action of divisor operators in terms of the action of $$\widehat{\mathfrak{gl}}(n+1)$$ on the Fock space. References: [MO] Tues Nov 20 Melissa Liu The 2-leg Gromov-Witten vertex References: Chiu-Chu Melissa Liu, Kefeng Liu, and Jian Zhou, A formula of two-partition Hodge integrals Chiu-Chu Melissa Liu, Formulae of one-partition and two-partition Hodge integrals Weds Nov 21 No meeting (Thanksgiving) Tues Nov 27 Melissa Liu A mathematical theory of the topological vertex References: Mina Aganagic, Albrecht Klemm, Marcos Mariño, and Cumrun Vafa, The Topological Vertex Jun Li, Chiu-Chu Melissa Liu, Kefeng Liu, and Jian Zhou, A Mathematical Theory of the Topological Vertex Chiu-Chu Melissa Liu, Gromov-Witten invariants of toric Calabi-Yau threefolds Weds Nov 28 Andrei Okounkov Toric GW/DT correspondence I will describe the full GW/DT correspondence for toric 3-folds using capped vertices and then answer all your questions. References: [MOOP] Tues Dec 04 Henry Liu Capping and quantum differential equation We'll see in detail the derivation of the QDE for the capping operator, using rubber calculus. Then we'll match the operator $$M_D$$ on the GW/DT sides and use it to match capped edges. References: [MOOP], [OP2] Weds Dec 05 Henry Liu Capping and quantum differential equation Continuation of Tuesday. References: [MOOP], [OP2]