# Mirrors in the Midlands, April 1-3 2019

We are excited to host the conference 'Mirrors in the Midlands' on 1-3 April 2019. Funding has been provided by the London Mathematical Society and the University of Birmingham School of Mathematics.

**Speakers:**

- Andrea Brini (Birmingham)
- Xenia de la Ossa (Oxford)
- Sara Angela Filippini (Imperial)
- Mark Gross (Cambridge)
- Tyler Kelly (Birmingham)
- Cristina Manolache (Imperial)
- Ian Strachan (Glasgow)
- Richard Thomas (Imperial)
- Alexander Veselov (Loughborough)

**Organisers:**

- Andrea Brini
- Tyler Kelly

**Where:**

The conference will be held in the Watson Building at the University of Birmingham in Lecture Rooms B and C. Watson Building is an 8 minute walk from the University [UNI] National Rail Station.

**Registration and Support:**

Please register using this Google form for our conference. We have some funding to support students and early career postdocs to attend. The registration deadline is 27 February 2019.

## Schedule:

### Monday April 1st 2019 in Lecture Room C, Watson Building

- 1:00pm-2:00pm: Richard Thomas
- 2:00pm-2:30pm: Coffee Break
- 2:30pm-3:30pm: Alexander Veselov
- 3:30pm-4:00pm: Tea Break
- 4:00pm-5:00pm: Andrea Brini
- 5:00pm-6:00pm: Reception

### Tuesday April 2nd 2019 in Lecture Room B, Watson Building

- 10:00am-11:00am: Ian Strachan
- 11:00am-11:30am: Coffee Break
- 11:30am-12:30pm: Cristina Manolache
- 12:30pm-2:30pm: Lunch
- 2:30pm-3:30pm: Sara Angela Filippini
- 3:30pm-4:00pm: Coffee Break
- 4:00pm-5:00pm: Tyler Kelly
- 5:00pm-6:00pm: Reception
- 7:00pm-9:00pm: Conference Dinner

### Wednesday April 3rd 2019 in Lecture Room C, Watson Building

- 10:00am-11:00am: Mark Gross
- 11:00am-11:30am: Coffee Break
- 11:30am-12:30pm: Xenia de la Ossa
- 12:30pm Farewells.

## Titles and Abstracts

### Andrea Brini:

*Chern-Simons theory and the higher genus B-model*

Abstract: I'll give an update on the relation of quantum invariants of knots and 3-manifolds with higher genus mirror symmetry and the topological recursion. For 3-manifolds, I'll give a general realisation of quantum invariants of spherical space forms via the Eynard-Orantin recursion on spectral curves of the relativistic Toda system on ADE root latices. For knots, I'll outline a conjecturally general correspondence between the coloured HOMFLY polynomial and the topological recursion, which rests on the identification of a natural group of piecewise linear transformations on both sides of the correspondence.

### Xenia de la Ossa:

*On the deformation theory of Heterotic structures*

Abstract: A heterotic $G$-structure is the geometry associated to heterotic compactifications leading to supersymmetric effective field theories. It amounts to the geometry of instantons on manifolds with a G-structure with an extra condition which involves the B field. In this talk, I will give an introduction to the study of the moduli space of those structures, illustrating with the case where $G=G_2$.

I will explain how these systems can be recast in terms of a nilpotent operator which acts on the space of forms on the manifold with values on a certain bundle $Q$ whose sections correspond to the gauge symmetries of the heterotic structure. Using these opertaros, I will show how to describe the infinitesimal moduli space in terms of the first cohomology of the nilpotent operator. Time permitting, I will specialise to the case $ G = SU(3)$ where we have been able to write a Kahler metric on moduli, as well the efforts to understand higher order deformations and the global structure of the full moduli space. In particular, we will see that the moduli space is described by an $L_3$ algebra obtained by deforming an appropriate superpotential.

A motivation behind these efforts to discover the mathematical structure of the quantum moduli spaces of these superstring theories, apart from the expectation of finding interesting new mathematics, is the well known fact that these structures contain a wealth of information about the physical behaviour of the low energy effective field theories.

### Sara Angela Filippini:

*Orbital degeneracy loci and applications*

Abstract: We consider a generalization of degeneracy loci of morphisms between vector bundles modelled on orbit closures of algebraic groups in their linear representations. In a preferred class of examples we gain some control over their canonical sheaf. We show how these techniques can be applied to construct Calabi-Yau and Fano varieties of dimension three and four. This is joint work with Vladimiro Benedetti, Laurent Manivel, and Fabio Tanturri.

### Mark Gross:

*Open FJRW Mirror Symmetry*

Abstract: I will talk about joint work with Tyler Kelly and Ran Tessler. We propose a form of mirror symmetry for open FJRW theory associated to Fermat polynomials x_1^{r_1}+x_2^{r_2}. I will attempt to give some hint as to what the invariants we calculate are and what the shape of mirror symmetry is in this case.

### Tyler Kelly:

*Hypergeometric Decompositions of Symmetric Quartic K3 Pencils*

Abstract: We will consider two different types of hypergeometric decompositions of special data associated to five pencils of quartic surfaces. We see that over the complex numbers, the middle cohomology of these five pencils yield hypergeometric Picard-Fuchs equations. Using these parameters, we then consider the same pencils over finite fields, decomposing their rational point counts using the finite-field hypergeometric functions with the same parameters as above. This is joint work with Charles Doran, Adriana Salerno, Steven Sperber, John Voight, and Ursula Whitcher.

### Cristina Manolache:

*Beyond the perfect world*

Abstract: Morphisms with perfect obstruction theories are the starting point of modern enumerative geometry. These are unfortunately rare. I will elaborate on the localized cosection construction of Kiem and Li to give a construction in which we deal with a three term obstruction theory.

### Ian Strachan:

*Modular Frobenius Manifolds*

Abstract: Frobenius manifolds with modular symmetry are discussed from a number of different points of view. Examples of such manifolds include:

- g=1 Hurwitz spaces;
- Jacobi group orbit spaces;
- Tensor products of Coxeter group orbit spaces;

as well from the quantum cohomology of certain manifolds. In this talk the properties of such manifolds will be discussed, namely:

- progress towards a classification of such manifolds;
- the development of elliptic V-systems for the corresponding almost dual Frobenius manifolds.

### Richard Thomas:

*Vafa-Witten invariants for projective surfaces*

Abstract*: *I will say something about how to define VW invariants, refined VW invariants, and how to calculate them via degeneracy loci.

### Alexander Veselov:

*Markov triples, quantum cohomology and integrability*

Abstract: Markov triples are the integer solutions of the celebrated Markov equation $x^2+y^2+z^2=3xyz.$ These triples were originated in number theory in Markov’s 1880 work, but since then they surprisingly appeared in many areas of mathematics, including hyperbolic geometry, theory of Teichmueller spaces and, most recently, in algebraic geometry. I will talk about one particular remarkable appearance of Markov triples, discovered by Boris Dubrovin in relation with Frobenius manifolds, enumerative geometry and quantum cohomology. Due to Kontsevich and Manin the corresponding structure is described by a solution of a particular Painleve-VI equation, which raised a natural question whether this solution is “special” in some precise sense. I will discuss this question from the point of view of the theory of two-valued groups, following recent joint work with V.M. Buchstaber.