Introductory Talk Fall 2018 Student Symplectic Seminar

Rough notes: introductory talk Fall 2018 Student Symplectic Seminar

September 25, 2018

Abstract Mirror symmetry is a rapidly-growing area linking symplectic and algebraic geometry, which also incorporates ideas from many other fields including differential geometry, homological algebra, category theory and physics. I will give a brief survey of some of the main conjectures in this field, with a view of introducing and motivating the topics which will be discussed in the seminar.

Main reference: Nick Sheridan’s youtube lecture “Lectures on Homological mirror symmetry” (ﬁrst lecture)

1 Introduction

1.1 Physics origin

Like many subjects in modern geometry and topology, many of the foundational ideas in mirror symmetry came to mathematics from physics, and more specifically from string theory and quantum field theory. We consider a quantum field theory built from the data of maps from a Riemann surface Σ to a Calabi-Yau 3-fold. You get probabilities by considering the partition function R eiS. There are two particular theories called A-model and B-model. Now physicists realized that there are certain pairs of Calabi-Yaus X,X∨ such that the A-model of X agrees with the B-model of X∨, and vice-versa.

1.2 Basic math picture

I want to begin with some definitions, which we will revisit throughout the quarter. Let (Xn, J, ω) be Kaehler (i.e. simultaneously a complex and symplectic manifold). Then Xn is said to be Calabi-Yau if there exists a nowhere-vanishing section θ ∈ Hn,0(X; C) (i.e. a holomorphic volume form). There are many equivalent definitions. In particular, if X is simply-connected, then it is Calabi-Yau if and only if it admits a Ricci-flat Kaehler metric. Meta mirror-symmetry: There should exist pairs of “spaces” (X, ω, J), (X∨, ω∨,J ∨) such that the A and B quantities are interchanged, i.e. A(X, ω) ↔ B(X∨,J ∨) and B(X, ω) ↔ A(X∨,J ∨). Traditionally, X and X∨ are Calabi-Yau -although apparently this requirement is not seen as so crucial to the current mathematical understanding of mirror symmetry. In fact, mirror symmetry is expected to hold in families. More precisely, we expect isomorphisms ∨ Msymp(X) ↔ Mcpx(X ) and vice versa. And we expect that, under these isomorphisms, the previous equivalences hold in families.

1 A slightly enhanced version of this implies that Hq(ΛpTX) = Hp,q(X∨). If X,X∨ are Calabi-Yau, we n−p,q q p also have H (X) = H (Λ TX), simply by sending v1, . . . , vp 7→ iv1,...,vp θ. This is the mirror in mirror symmetry. I will now brieﬂy run through certain versions of mirror symmetry that are being studied by mathe- maticians. Many of the things I write will seem very mysterious: it is our goal to understand them! I also want to say that -although there are some fairly precise conjectures which have been formulated and which we will discuss, I personally ﬁnd it easier to think of mirror symmetry as a broad philosophy which says that certain objects occurring in symplectic geometry should be related to certain objects in algebraic geometry.

2 Mathematical theories

2.1 Closed string (Hodge theoretic) mirror symmetry

Historically, this was the ﬁrst version of mirror symmetry, and it is particularly close to physics. In this version, we are working with Calabi-Yaus -in particular with Calabi-Yau 3-folds. On the A side, we have things called Gromov-Witten invariants. These are counts of pseudo-holomorphic curves weighted by symplectic area, and -despite the “holomorphic”, they turn out to be symplectic invariants. On the B side, we have quantities as periods of diﬀerential forms (i.e. the integral of a holomorphic form along a holomorphic submanifold) and variations of Hodge structures (i.e. how the Dolbeault decomposition varies as we change the complex structure).

5 5 4 Example 2.1 (Candelas-de la Ossa-Green-Parkes). Considered Q3 = {x0 + ··· + x4 = 0} ⊂ CP . This is a K3 surface called the “quintic 3-fold”. They predicted nd := #{degreedrational curves} in Q3. E.g. n1 = 2875.

The mirror is a certain resolution of a Z/5 action on the quintic 3-fold (see Gross’ paper). The nd arise as coeﬃcients in a certain power series called the Yukawa coupling, which is a triple-pairing on H1,1. There is a mirror Yukawa coupling on H2,1(X∨) whose coeﬃcients depends on periods.

2.2 Open string (homological) mirror symmetry

Physicists also want to consider theories involving maps from open Riemann surfaces Σ → X. These Riemann surfaces need to have boundary in “branes” i.e. Lagrangian submanifolds of X. Now the A-side objects are complex submanifolds with a holomorphic line bundle. The B-side objects are special Lagrangians with a flat U(1)-bundle. (!! holomorphic objects ↔ flat objects is an old story -see Riemann-Hilbert correspondence!!). These spaces turns out to be very complicated. In his 1994 ICM address, Kontsevich proposed that one could enlarge these spaces to make them more tractable. This leads to homological mirror symmetry. On the A-side, our enlargement is called the Fukaya category Fuk(X). It contains all Lagrangians submanifolds of X, equipped say with a flat connection (+maybe other stuff). On the B-side, the enlargement is the derived category of coherent sheaves.

Conjecture 2.2 (Homological Mirror Symmetry). Given X,X∨ mirror pairs, there are (derived . . . ) ∨ ∨ equivalences of categories Coh(X) Fuk(X ) and Fuk(X) Coh(X ).

2 2.3 Geometric mirror symmetry

How to explain the existence of mirrors? How to build a mirror? The Strom SYZ conjecture; Gross- Siebert is an AG version

Conjecture 2.3 (Strominger-Yau-Zaslow). If X and X∨ are mirror Calabi-Yaus, then there exist fibrations f : X → B and f˜ : X∨ → B whose general fibers are special Lagrangian. These fibrations are ∨ 1 1 ∨ dual, in the sense that Xb = H (Xb; R/Z) and Xb = H (Xb ; R/Z) whenever both fibers are smooth.

We will see later in the seminar what this means in some simple examples. I should say that this conjecture is likely false without modifications. Gross-Siebert give an algebro-geometric generalization, and in particular a method to construct mirrors. This involves tropical geometry, log geometry, other fancy stuff . . . We will spend time trying to understand the basics of their program. What is the status of these conjectures? First of all, the general expectation is that geometric mirror symmetry → open string MS → closed string MS. Ganatra-Perutz-Sheridan prove a lot for the second implication. My understanding is that there is reasonable prospect of proving HMS for varieties built from Gross-Siebert in the next fifteen years.

3 Our plan

Our plan: understand what’s going on in simple cases. We’ll focus on the elliptic curve; then we’ll discuss the SYZ conjecture and the Gross-Siebert program. This will hopefully motivate us to learn some algebraic and symplectic geometry!