# Matrix factorizations Vienna May 13-17, 2013

Registration is from 9:30 to 10:00 on Monday, May 13.

# Program

 Monday, May 13 Tuesday, May 14 Wednesday, May 15 Thursday, May 16 Friday, May 17 10:00 D. Favero A. Takahashi N. Addington H. Uehara V. Lunts COFFEE 11:30 M. Ballard S. Hosono O. Schnürer P. Sosna A. Preygel LUNCH 2:00 L. Hille H-C. G. von Bothmer W. Lowen H. Lenzing E. Scheidegger 3:15 H. Krause N. Carqueville E. Segal J. Knapp P. Clarke 4:30 I. Dolgachev W. Donovan Soccer / Party D. Pomerleano E. Sharpe

# Abstracts

### D. Favero - Homological Projective Duality via Matrix Factorizations

Homological projective duality, due to Kuznetsov, provides an elegant framework for understanding the relationship between the derived category of an algebraic variety and its projective dual. It has provided many of the more deeply geometric examples of relationships between derived categories. I will discuss how homological projective duality can be naturally treated in the context of Landau-Ginzburg models and algorithmically approached for geometric invariant theory quotients.

### M. Ballard - Kernels for equivariant factorizations

We discuss a factorization construction for the internal Hom between equivariant factorizations in the homotopy category of dg-categories.

### L. Hille - Rational Surfaces, Spherical Twists and the Auslander Algebra of k[T]/T^t

In a recent work with Markus Perling we have constructed a tilting bundle on any rational surface. Moreover, the endomorphism algebra can be chosen to be quasi-hereditary. This establishes a close connection between certain subcategories generated by line bundles and the so-called good modules over the endomorphism algebra of the tilting bundle. If one considers a partial tilting bundle, one might get even an equivalence of abelian categories. If the rational surface contains a chain of (-2)-curves one gets an equivalence with the modules over the Auslander algebra of the truncated polynomial ring. For this category we can classify all full exceptional sequences, all spherical objects, and all tilting modules in a joint work with David Ploog. Using the equivalence again we get an analogous result for certain sheaves on the rational surface. In the talk I concentrate mainly on the above classification and try to motivate further results that might generalize this equivalence.

### H. Krause - Thick subcategories and noncrossing partitions for categories of matrix factorizations of finite type

We consider the triangulated category of graded matrix factorizations for a polynomial of type ADE, as studied by Kajiura, Saito, and Takahashi (and idependently by Lenzing and de la Pena). In my talk, I'll explain a description of the lattice of thick subcategories in terms of noncrossing partitions. This is based on a K-theoretic study of representations of quivers of Dynkin type, going back to work of Ingalls and Thomas

### I. Dolgachev - Unitary representations of triangle groups and matrix factorizations

The now classical McKay correspondence assigns to a unitary finite-dimensional representation of a central extension $\Pi$ of a finite subgroup $\Gamma$ of PSU(2) an object of the category of matrix factorizations of the Klein polynomial $f$ generating the basic relation between the minimal generators of the ring of invariants $\mathbb{C}[x,y]^{\Pi}$. In this talk we will discuss a similar construction, where $\Gamma$ is replaced by a triangle subgroup of PSU(1,1) with algebra of automorphic forms generated by 3 elements and the Klein polynomial is replaced by a generator of relations between the minimal generators of this algebra. An example of such polynomial is the polynomial $f = x^2+y^3+z^7$.

### A. Takahashi - Categories of matrix factorizations for elliptic Landau-Ginzburg orbifolds

The categories of graded matrix factorizations for invertible polynomials defining simple elliptic singularities will be considered. We show that the category admits a full strongly exceptional collection and is equivalent to the derived category of coherent sheaves on a weighted projective line if and only if the dualizing element in the grading is non-zero. In particular, we shall explain how some of exceptional objects in a f.s.e.c. "decomposes" by taking a quotient of the grading group and then forms a part of a f.s.e.c. in the category of matrix factorizations graded by the quotient, which is compatible with the geometry of coverings for weighted projective lines. My talk is based on a joint paper in preparation with Hidemasa Oda and Kyoji Saito.

### S. Hosono - BPS numbers and projective geometry of Reye congruences

Based on recent collaborations with Hiromichi Takagi, I will talk about the mirror symmetry of a Calabi-Yau 3 fold of Reye congruences and its Fourier-Mukai partner. Studying the projective geometry of Reye congruences of dimension four, we have found a pair (X,Y) of two (non-birational) Calabi-Yau threefolds, which has many similarities to the well-studied Grassmann-Pfaffian Calabi-Yau threefolds. I will discuss mirror symmetry in our case and determine the BPS numbers of X and Y. Observing the Euler number of X in the table of BPS numbers, I briefly sketch our proof of the derived equivalence between X and Y.

### N. Carqueville - Orbifold completion

Motivated by defects in topological quantum field theory, one can start from any pivotal bicategory B and construct its "orbifold completion" B_orb in terms of Frobenius algebras. The completion satisfies the natural properties B \subset B_orb and (B_orb)_orb = B_orb, and it gives rise to various new equivalences and nondegeneracy results. I will explain this construction (which is joint work with Ingo Runkel) along with the necessary algebraic background, and then apply it to the bicategory of Landau-Ginzburg models. Applications include a unified perspective on ordinary equivariant matrix factorisations, a one-line proof of a cousin of the Hirzebruch-Riemann-Roch theorem, and new equivalences for simple singularities.

### W. Donovan - Mixed braid groups, deformations, and derived categories

Seidel-Thomas gave a celebrated construction of braid group actions on certain derived categories, with good examples provided by the derived categories of minimal resolutions of A_n surface singularities. I will explain joint work with Ed Segal, in which we use deformations of these examples to produce higher-dimensional varieties which carry interesting actions of some relatives of the braid group, namely pure and mixed braid groups.

### N. Addingtion - Categories of massless D-branes and del Pezzo surfaces

We study examples of Calabi-Yau 3-folds containing del Pezzo surfaces, and ask which D-branes "go massless" as the del Pezzo is contracted to a point. We approach the question via monodromy in the stringy Kaehler moduli space, using spherical functors. For del Pezzos can be written as a surface in weighted P^3, the category of massless D-branes seems to be identified with the category of matrix factorizations.

### O. Schnürer - Matrix factorizations, semi-orthogonal decompositions and motivic measures

I will report on joint work with Valery Lunts. We establish semi-orthogonal decompositions for matrix factorizations on projective space bundles and blowing-ups and use them for constructing motivic measures with values in Grothendieck rings of saturated dg categories

### W. Lowen - Hochschild cohomology with support and the Grothendieck construction

In this talk we explain how the functoriality properties of Hochschild cohomology are essentially determined by an underlying level of "grading categories". We describe some techniques for deconstructing Hochschild cohomology, based upon sheaf properties and arrow categories, and we relate this to a generalized Grothendieck construction.

### E. Segal - The Pfaffian-Grassmannian Correspondence Revisited

'The Pfaffian-Grassmannian correspondence' is a curious example of two particular Calabi-Yau threefolds which appear to have the same mirror, despite not being birational. As a consequence they should be derived equivalent, and this was proved by Borisov and Caldararu a few years ago. At around the same time Hori and Tong gave a physical derivation of the correspondence, by writing down a (non-abelian) GLSM that interpolates between the two spaces. I'll discuss joint work with Nick Addington and Will Donovan in which we give a new proof of the derived equivalence, based on the Hori-Tong construction. In particular we break the equivalence into three steps by passing through some intermediate categories of matrix factorizations.

### H. Uehara - Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks

For toric Deligne--Mumford stacks X, we can consider a cerrtain generalization of Frobenius endomorphism. For such an endomorphism on 2-dimensional toric Deligne--Mumford stacks X, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on X. My talk is based on a joint work with Ryo Ohkawa.

### P. Sosna - On autoequivalences of some Calabi-Yau and hyperkähler varieties

I will explain how to construct new autoequivalences of some Calabi-Yau and hyperkähler varieties by passing from a smooth projective surface to the associated Hilbert scheme of points. This is joint work with David Ploog.

### H. Lenzing - Categories of matrix factorizations for triangle singularities

This is joint work with D. Kussin and H. Meltzer. We investigate categories of matrix factorizations of the (universally graded) triangle singularity $f=x^a + y^b + z^c$ for a tripel of integers $a,b,c$ which are greater or equal $2$. Such a category $\mathcal{T}$ always has a tilting object, formed from a complete exceptional sequence, with an endomorphism ring that is the tensor product of three path algebras of equioriented Dynkin quivers of type $A$. Moreover, $\mathcal{T}$ is fractional Calabi-Yau, and the Auslander-Reiten components are known. The results are obtained by exploiting the relationship to the category of coherent sheaves on the weighted projective line given by the weight triple $(a,b,c)$.

### J. Knapp - New Calabi-Yaus from non-abelian 2D theories

Using a supersymmetric gauge theory - the gauged linear sigma model - with non-abelian gauge groups we construct new examples of one-parameter Calabi-Yau threefolds and explore their Kahler moduli spaces. In particular, we find a gauge theory description for four Pfaffian Calabi-Yaus and construct a new determinantal Calabi-Yau described by a gauged linear sigma model with gauge group U(1)xO(2).

### D. Pomerleano - SYZ mirror symmetry for toric Calabi-Yau varieties

A recent preprint of Abouzaid, Auroux, and Katzarkov constructs SYZ mirrors for certain "generalized conic bundles" over toric varieties. In this talk, after reviewing the construction, we examine their construction from the point of view of Homological Mirror Symmetry. I will describe joint work with Kazushi Ueda and Kwokwai Chan where we examine mirror symmetry for generalized conic bundles over $\mathbb{C}*$. We compute wrapped Floer homology of Lagrangian sections of the SYZ fibration and identify it with endomorphisms of certain line bundles on the mirror Calabi-Yau. We will also discuss compactly supported categories and categories of matrix factorizations as time permits.

### V. Lunts - Categorical Lefschetz fixed point theorem

I will discuss categorical versions of the Lefschetz FPT. As an example I will prove such a theorem for an endofunctor of the category of perfect dg modules over a smooth and proper dg algebra.

# Events

### Soccer

Meet at front door of the ESI at 5:10 to play Soccer (aka Football). Field is reserved from 6:00 to 8:00 pm, Wednesday, May 15 at this location:
Soccerclub Wien; Gutheil-Schoder-Gasse 9, 1100 Wien

### Party

Wine and cheese at Garnisongasse 3 (2nd floor) at 8:00, Wednesday May 15. We will watch Europa League Final at 8:45. Directions from ESI