Mirror Symmetry Learning Seminar

• Fall 2015: we covered the basics of QFT. See qft-notes.tex.
• Winter 2016: we covered the basics of string theory and supersymmetry. See stringy-notes.tex.
• Summer 2016: we will cover the basics of mirror symmetry. See mirrorsym-notes.tex.

Summer 2016 Schedule

Unless otherwise announced on this page, the seminar will run twice a week:

• Tuesdays 12:30pm - 2:00pm in M3 3103.
• Fridays 12:00pm - 1:30pm in M3 2134.

Details about upcoming talks will be added incrementally as time progresses. For now, the plan is:

• Week 1 (Friday May 6th): Mathematical Preliminaries I; 1.1.1, 1.1.2, 1.2.1 in notes.
  • We will review sheaf cohomology and Poincare duality. We will also introduce Morse homology, the finite-dimensional version of Floer homology, which shows up in the statement of homological mirror symmetry.
• Week 2 (Monday May 9th): Mathematical Preliminaries II; section 1.2 in notes.
  • We will review Poincare and Serre duality, in particular emphasizing the connection between Poincare duality and Thom classes. Then we review Chern classes, and use this connection to explore properties of the top Chern class, i.e. the Euler class. Finally we introduce and practice using two big theorems: generalized Gauss--Bonnet and Hirzebruch--Riemann--Roch. A lot of the calculations today will be useful later on.
• Week 2 (Friday May 13th): Physical Preliminaries I; sections 2.1 and 2.2 in notes.
• Week 3 (Monday May 16th): Mathematical Preliminaries III; section 1.2 in notes.
  • A continuation of Mathematical Preliminaries II. We'll finish up the discussion of characteristic classes, introduce Hirzebruch--Riemann--Roch, and do some example calculations and usages, e.g. using it to prove the generalized Gauss--Bonnet formula.
• Week 3 (Friday May 20th): canceled due to GAP.
• Week 4 (Tuesday May 24rd): Mathematical Preliminaries IV; sections 1.1.3 and 1.2.5 in notes.
  • We introduce equivariant cohomology, which is a cohomology theory that captures the group action on a space, and the localization principle from a mathematical perspective. Our main goal is to state and justify the Atiyah--Bott localization formula, which will be our main tool for integrating equivariant cohomology classes. These tools form the basis for Gromov--Witten theory and have many other applications in mathematical physics whenever there is a gauge group.
• Week 4 (Friday May 27th): canceled
• Week 5 (Tuesday May 31st): Mathematical Preliminaries V; sections 1.3.1 to 1.3.3 in notes.
  • I'll use an hour or so to review Kahler and Calabi--Yau geometry. In particular we need to become really cozy with Hodge diamonds, especially for Calabi--Yau 3-folds. Then we'll look at the overall structure of the Calabi--Yau moduli space and take a small peek at mirror symmetry.
• Week 5 (Friday June 3rd): canceled
• Week 6 (Tuesday June 7th): Physical Preliminaries II; section 2.2 in notes.
  • I'll briefly review what Anton did last time (i.e. structure of supersymmetric quantum mechanical theories) and continue with the d=1 case. Mainly for d=1 we'll see our first sigma model. From it, using perturbative techniques, we'll "derive" Hodge theory. Then we"ll see what instantons are and how they give non-perturbative corrections to the perturbative techniques (which are locally correct but not globally correct), and using them we"ll "derive" Morse theory.
• Week 6 (Friday June 10th): Physical Preliminaries III; section 2.3 in notes.
  • We covered the d=2 free scalar boson, and compactified the target to obtain the sigma model on the circle. By computing the partition function of the latter theory, we saw T-duality.
• Week 7 (Tuesday June 14th): Physical Preliminaries IV; section 2.4 in notes.
  • I'll cover the superspace formalism, in particular the theory of a single chiral superfield, which will reduce in d=0 and d=1 to the Landau--Ginzburg theories and sigma models we have seen. This formalism allows us to directly construct a huge family of d=2 supersymmetric theories and investigate their symmetries (and symmetry breaking). We'll write down the general form of d=2 nonlinear sigma models, and d=2 Landau--Ginzburg models, and see some hints of the LG/CY correspondence between these two classes of theories.
• Week 7 (Friday June 17th): canceled due to convocation.
• Week 8 (Tuesday June 21st): Calabi-Yau moduli and Mirror Symmetry for the Quintic; sections 1.3.4 and 1.3.5 in notes.
  • We'll see the story of mirror symmetry in the simplest case: the quintic hypersurface in P^4. We'll first have to sketch the overall structure of the Calabi-Yau moduli space, in particular how it locally decomposes as a sum of complex moduli and Kahler moduli. Then we'll define some relevant Hodge-theoretic structures on the space of complex moduli. The quintic hypersurface will provide nice concrete examples of these structures. (We don't have the machinery yet to look at the Kahler moduli; that will have to wait until we define Gromov-Witten invariants.) Finally, we'll see what mirror symmetry has to say about these structures. One of them, the Yukawa coupling, has a very interesting connection to Gromov-Witten invariants, especially the ones counting rational curves in the Calabi-Yau manifold.
• Week 8 (Friday June 24th): Introduction to Gromov-Witten theory I; section 3.1 in notes.
  • We'll start on the long journey of exploring the Kahler moduli space (i.e. the "A-model variation of Hodge structure"). The first step is to learn Gromov-Witten theory. The first step in that is to define generalizations of the moduli space of marked Riemann surfaces, and define, on these spaces, rational numbers called Gromov-Witten invariants. These numbers, as we'll see later on, roughly count the number of rational curves in a space.
• Week 9 (Tuesday June 28th): Toric Geometry II
• Week 9 (Friday July 1st): canceled (Canada Day)
• Week 10 (Tuesday July 5th): Introduction to Gromov-Witten theory II; section 3.2 in notes.
  • We'll see the connection between enumerative geometry and the moduli space of stable maps via Kontsevich's recursion for the number of degree d rational maps passing through 3d-1 generic points. In particular we'll define Gromov-Witten (GW) invariants and more general numbers called gravitational descendant invariants. These arise naturally when considering differential equations involving GW invariants. We will also touch on the technical issue of defining the virtual fundamental class for the moduli space of stable maps, in order to have a good notion of what it means to integrate over it.
• Week 10 (Friday July 8th): Introduction to Gromov-Witten theory III; section 3.3 in notes.
  • Gromov-Witten invariants can be nicely encoded in a structure called a quantum cohomology ring; these come in two flavors, the small and the big. These are the ordinary cohomology rings, but with a deformed "quantum product". Enumerative data, including rational curve counts, is encoded in the small quantum product, while the associativity of the big quantum product (known as the WDVV equation) encodes relations such as Kontsevich's recursion. Finally, we briefly look at the Dubrovin formalism, in preparation for Givental's approach to the proof of mirror symmetry.
• Week 11 (Tuesday July 12th): Introduction to Gromov-Witten theory III; section 3.3 in notes.
  • Discussion of superspaces and supermanifolds.
• Week 11 (Friday July 15th): canceled
• Week 12: canceled
• Week 13 (Tuesday July 26th): Introduction to Gromov-Witten theory IV; section 3.4 in notes.
  • To calculate Gromov-Witten invariants, it is convenient to move to equivariant cohomology and use localization, because the fixed point loci of the moduli space of stable maps (in genus 0) is purely combinatorial and can be encoded as graphs. We first re-familiarize ourselves with localization by applying it to projective space. Then we do the moduli space of stable maps. Then we apply this newly gained machinery to the problem of multiple covers: how do degree k covers of degree d stable maps contribute to the Gromov-Witten invariant of degree kd?
• Week 13 (Friday Aug 2nd): Complex and Kahler Moduli; section 4.1 in notes.
  • We'll define, in general, variations of Hodge structure both on complex and Kahler moduli; we have seen special cases of this for the quintic threefold. In particular we'll be able to write down the mirror map, relating a neighborhood of a large radius limit point in Kahler moduli with a maximally unipotent boundary point in complex moduli. If time permits we'll see (finally!) what it means for two Calabi--Yau threefolds to be mirror to each other.
• Week 14: canceled