### Geometry and Physics: Mirror symmetry, Hodge theory, and related topics – University of Miami, January 25-30, 2016

Speakers:

Mohammed Abouzaid

Enrico Arbarello

Denis Auroux

Arnaud Beauville

Herb Clemens

Mirjam Cvetic

Emanuel Diaconescu

Alexander Efimov

Kenji Fukaya

Dennis Gaitsgory

Phillip Griffiths

Dmitry Kaledin

Mikhail Kapranov

Sheldon Katz

Maxim Kontsevich

Bong Lian

Eyal Markman

Greg Moore

John Morgan

David Morrison

Dmitri Orlov

Paul Seidel

Carlos Simpson

Yan Soibelman

Cumrun Vafa

Ilia Zharkov

A conference on the occasion of Ron Donagi’s 60th birthday

**Organizers:** D. Auroux, E. Gasparim, L. Katzarkov, M. Kontsevich, E. Lupercio, T. Pantev, M. Teicher

This conference on Geometry and Physics: Mirror Symmetry, Hodge Theory, and related topics, took place at the University of Miami (Coral Gables, FL) on **January 25-30, 2016**.

This event was partially supported by the Simons Collaboration on Homological Mirror Symmetry, and by an NSF FRG grant on Wall-Crossing in Geometry and Physics. It was informally dedicated to Ron Donagi’s 60th birthday.

**Practical information**

**Venue: ** The conference takes place at the University of Miami Coral Gables campus. Monday 1/25 through Wednesday 1/27, the talks are in the **Field House** at BankCenter United; Thursday 1/28 through Saturday 1/30, the talks are in the **Storer Auditorium** in the Aresty Building of the School of Business Administration.

**Accommodation:** Reservations have been made for participants who requested them in one of the following two hotels:

- Holiday Inn Coral Gables/University of Miami, located right next to the University of Miami campus. The hotel address is: 1350 South Dixie Highway, Coral Gables FL 33146; phone: 1-305-667-5611
- Marriott Miami Dadeland, 9090 S. Dadeland Boulevard, Miami FL 33156; phone: 1-305-670-1035. Metrorail runs every 5 minutes between the hotel (Dadeland South Station) and the U. Miami campus (University Station) (3 stops, 6 minutes).

You should have received an e-mail from the U. Miami staff confirming the details of your hotel reservation.

**Registration fee:** there is a nominal registration fee of $35 per participant ($30 for graduate students), to cover the cost of refreshments (waived for speakers). The conference banquet fee for those attending the banquet is $45 per participant.

**Airport:** Miami International Airport is about 7 miles from campus. The most convenient way to reach the hotel or the campus is to take a taxi. Participants expecting reimbursement: please keep all your original boarding passes since they may be needed for reimbursement.

### Schedule

Monday January 25 (BankCenter United Fieldhouse)

9:30-10:30 | A. Beauville: The Chow ring of hyperkähler manifolds (Slides) (Video) |

11:00-12:00 | E. Arbarello: Brill-Noether-Petri curves on K3 surfaces (Slides) |

2:00-3:00 | C. Vafa: tt* geometry (Video) |

3:15-4:15 | C. Simpson: The sphere at infinity of the character variety (Video) |

4:30-5:30 | Y. Soibelman: Symplectic geometry of Riemann-Hilbert correspondences (Slides) (Video) |

Tuesday January 26 (BankCenter United Fieldhouse)

9:30-10:30 | D. Morrison: Elliptic Calabi-Yau torsors (Video) |

11:00-12:00 | D. Gaitsgory: Arthur parameters in local geometric Langlands (Video) |

2:00-3:00 | D. Orlov: Noncommutative varieties and their geometric realizations (Video) |

3:15-4:15 | P. Seidel: Differentiation with respect to the Kähler parameter (Video) |

4:30-5:30 | M. Kapranov: Perverse Schobers on surfaces and Fukaya categories with coefficients (Video) |

Conference Dinner |

Wednesday January 27 (BankCenter United Fieldhouse)

9:30-10:30 | M. Kontsevich: Multiplication kernel (Video) |

11:00-12:00 | K. Fukaya: Lagrangian correspondences and A_{∞} functors (Video) |

2:00-3:00 | T. Pantev: Self-duality of reducible Hitchin fibers (Video) |

3:15-4:15 | J. Morgan: The Poincare duality bordism (Video) |

4:30-5:30 | D. Kaledin: A simple construction of polynomial Witt vectors (Video) |

Cancelled | G. Moore: Some math predictions from the semiclassical formulation of BPS states (Notes) (First Annual United Airlines Lecture; cancelled due to weather) |

7:00pm | Public Lecture by Yuri Tschinkel (Wilder Auditorium, Knight Physics Building) Interested participants should RSVP for this lecture at www.as.miami.edu/tschinkel |

Thursday January 28 (Storer Auditorium, School of Business Administration)

9:00-10:00 | P. Griffiths: Moduli and Hodge theory (Video) |

10:15-11:15 | S. Katz: The Calabi-Yau integrable system and gauge theories with defects (Video) |

11:30-12:30 | H. Clemens: A global algebro-geometric model for heterotic – F-theory duality (Video) |

Free afternoon |

Friday January 29 (Storer Auditorium, School of Business Administration)

9:30-10:30 | D. Auroux: Towards HMS for hypersurfaces in (C*)^{n} and toric varieties (Video) |

11:00-12:00 | E. Diaconescu: Vertical DT invariants and modular forms (Video) |

2:00-3:00 | M. Abouzaid: Lagrangian Floer theory, revisited (Video) |

3:15-4:15 | B. Lian: Hunting for large complex structure limits (Video) |

4:30-5:30 | M. Cvetic: Higher rank abelian and discrete symmetries in F-theory (Video) (Slides) |

Saturday January 30 (Storer Auditorium, School of Business Administration)

9:00-10:00 | I. Zharkov: Phase tropical hypersurfaces and their skeleta (Video) |

10:15-11:15 | A. Efimov: Categorical formal punctured neighborhood of infinity (Video) |

11:30-12:30 | E. Markman: On the Hodge conjecture for self products of K3 surfaces with a quadratic real multiplication field (Video) |

### Titles and Abstracts

E. Arbarello: Brill-Noether-Petri curves on K3 surfaces

**Abstract:** In ’86 Lazarsfeld proved that a general hyperplane section of a rank one K3 surface is Brill-Noether-Petri (BNP) general, i.e. the map $ H^0(L)\otimes H^0(\omega_C L^{-1})\to H^0(\omega_C)$ is injective for all line bundles $L$ on $C$. In ’97, Wahl showed that the non-surjectivity of the Gaussian map $\nu_1:\wedge^2H^0(\omega_C)\to H^0(\omega^3_C)$ is a necessary condition for a curve $C$ to be a hyperplane section of a K3 surface (or a limit of such). He conjectured that, for a BNP general curve of genus $g\geq 12$, this condition should also be sufficient. We prove this conjecture (jointly with A. Bruno and E. Sernesi). In a separate work (with A. Bruno, G. Farkas, and G. Saccà) we produce concrete examples of BNP general curves, of any given genus $g$, as degree $3g$ plane curves with eight $g$-tuple points and one $(g-1)$-tuple point and no other singularities (the Du Val curves). As a consequence we show the existence of BNP general curves defined over $\Bbb{Q}$, answering a question raised by Harris and Morrison in ’98.

A. Beauville: The Chow ring of hyperkähler manifolds

**Abstract:** This is a survey of the progress obtained in the last 10 years on the Chow ring of projective hyperkähler manifolds. Starting from the case of K3 surfaces, I will explain why one may hope that this Chow ring admits a natural grading. While this conjecture seems inaccessible at the moment, it implies some concrete consequences on intersections of divisors, which have been verified in an increasing number of cases.

H. Clemens: A global algebro-geometric model for heterotic – F-theory duality

**Abstract:** Heterotic string theory postulates a 10-dimensional universe with a Calabi-Yau threefold of diameter 10^{-33} cm in the 6 real directions skew to space-time. F-theory packages the physical information attached to the CY-threefold in geometric information about a family of CY-fourfolds that specializes to a union of two Fano fourfolds meeting transversely along the (heterotic) CY-threefold. We will discuss a global model for this ‘duality’ that incorporates the desired SU(5) → SU(3) × SU(2) × U(1) symmetry breaking and desired numerical invariants. A critical role in achieving the latter is played by the requirment that the elliptically fibered threefold and fourfold admit only bisections (rather than a section). Joint work with David Morrison, Tony Pantev, Stuart Raby and Sakura Schafer-Nameki.

M. Cvetic: Higher rank abelian and discrete symmetries in F-theory

E. Diaconescu: Vertical DT invariants and modular forms

**Abstract:** An explicit formula will be derived for the generating function all DT invariants counting vertical two dimensional sheaves on K3 fibrations. The final expressions will be shown to satisfy strong modularity properties. In particular a new construction of vector valued modular forms will emerge from the geometric framework, exhibiting some of the features of a Hecke tranform. This is joint work with Vincent Bouchard, Thomas Creutzig, Charles Doran, Callum Quigley and Artan Sheshmani.

A. Efimov: Categorical formal punctured neighborhood of infinity

**Abstract:** Given a smooth algebraic variety *X* over a field of characteristic zero, one can choose its smooth compactification, and take the formal punctured neighborhood of the infinity locus. The result does not depend on the choice of a compactification. In particular, we get a triangulated DG category of perfect complexes on this formal punctured neighborhood, which depends only on *X.*

According to Mohammed Abouzaid, a similar construction occurs in the framework of Fukaya categories, and this motivates a question if we can do it in purely categorical terms.

I will explain that the answer is “yes”: for any small DG category one can define the corresponding “formal punctured neighborhood of infinity”. When applied to the derived category of coherent sheaves on *X*, one gets the category described above. Our construction is closely related with (an algebraic version of) Calkin algebras.

P. Griffiths: Moduli and Hodge theory

**Abstract:** In this talk we shall relate two of the major areas of algebraic geometry, namely moduli and Hodge theory. In the classical cases of curves, abelian varieties and K3 surfaces this relationship is much studied and well known. For surfaces of general type one has the KSBA construction of the moduli space $\cal{M}$ whose canonical compactification is a projective variety. Although the singularity structure of surfaces $X_0$ corresponding to boundary points of $\cal{M}$ is understood, this is not the case for the global structure of $X_0$. In this talk we shall discuss how Hodge theory may be used as a guide to help understand the boundary structure of $\cal{M}$ in a first non-classical case of regular algebraic surfaces $X$ with $K_X^2 = 2$ and $p_g(X) = 2$. This represents joint work in progress with Mark Green, Radu Laza and Colleen Robles.

D. Kaledin: A simple construction of polynomial Witt vectors

**Abstract:** It is known that proper non-commutative generalization of the Witt vectors functor is a functor W(M/R) of two variables, a ring R and a bimodule M. It is a polynomial functor with respect to M. Even when R is a field, the theory is interesting, but unfortunately, all known definitions use high technology (topology and/or algebraic K-theory). I will give an overview of the subject, including motivations; then I will present a very simple direct construction of W(M/k), where k is a perfect field. The construction only uses basic properties of cyclic groups.

M. Kapranov: Perverse Schobers on surfaces and Fukaya categories with coefficients

**Abstract:** Perverse Schobers (conjectural “perverse sheaves of triangulated categories”) can be seen as natural coefficient data for forming Fukaya categories in the same way as sheaves serve as coefficient data for forming cohomology. In the talk I will explain how to make this proposal rigorous for the case of Riemann surfaces. The key tool is provided by the relative Waldhausen spaces of a spherical functor and the remarkable property of periodicity of orthogonals which they possess. Joint work with T. Dyckerhoff, V. Schechtman and Y. Soibelman.

S. Katz: The Calabi-Yau integrable system and gauge theories with defects

**Abstract:** I explore the geometry of the Calabi-Yau integrable system as the Calabi-Yau threefold degenerates to contain curves of ADE singularities, and relate to gauge theories on the curves with defects at isolated points. The mathematical questions addressed arise from F-theory and type IIA string theory.

This talk is based on work in progress with L. Anderson, J. Heckman, and L. Schaposnik.

M. Kontsevich: Multiplication kernel

**Abstract:** A classical finite-dimensional algebraic integrable system can be interpreted as a commutative associative algebra in the symmetric monoidal category of symplectic varieties and Lagrangian correspondences. I’ll talk about a quantum version making sense uniformly in zero and positive characteristic.

B. Lian: Hunting for large complex structure limits

**Abstract:** A large complex structure (LCS) limit in the complex structure moduli space of Calabi-Yau manifolds is supposed to be fundamental, because this is one place where we can test statements about mirror symmetry. Yet, we don’t seem to have many cases in which we understand such a limit. In this talk, we will discuss a new test case — hypersurfaces in G/P. This is based on recent joint work with S. Bloch, A. Huang, D. Srinivas, S.-T. Yau, and X. Zhu, and on work in progress with L. Fu.

E. Markman: On the Hodge conjecture for self products of K3 surfaces with a quadratic real multiplication field

**Abstract:** Let E be the algebra of rational Hodge endomorphisms of the transcendental lattice of a K3 surface S. Then E is a field, which is either totally real or a quadratic extension of a totally real field, by a result of Zarhin. In the latter case, in which S is said to have complex multiplication, E is known to be spanned by rational Hodge isometries. A recent result of Buskin, extending work of Mukai, states that rational Hodge isometries are algebraic. The Hodge conjecture for S×S follows. We discuss work in progress extending the Mukai-Buskin theorem for rational Hodge similarity endomorphisms, which would imply the Hodge conjecture for S×S in the case $E=\Bbb{Q}[\sqrt{D}]$, $D>0$.

G. Moore: Some math predictions from the semiclassical formulation of BPS states

**Abstract:** The semiclassical formulation of BPS states of four-dimensional N=2 field theories (with Lagrangian formulations) involve the $L^2$ kernels of Dirac-like operators on monopole moduli spaces. We explain the geometrical implications of the no-exotics property and wall-crossing behavior of BPS states. This talk is largely based on http://arxiv.org/pdf/1512.08923.

D. Morrison: Elliptic Calabi-Yau torsors

**Abstract:** Every elliptic fibration $\pi: X \to B$ with a rational section determines an elliptic curve $E$ defined over the function field $K=K(B)$ of the base; if two elliptic fibrations determine the same elliptic curve, then they are birationally equivalent. As a group-scheme over $K$, $E$ may admit “torsors”, i.e., projective curves $C$ over $K$ with a transitive action $E \times C \to C$ having trivial stabilizers. The question we will address is: if $X$ is an elliptically fibered Calabi-Yau variety, and $C/K$ is a torsor for the associated elliptic curve $E/K$, when does $C/K$ have a birational model which is itself a Calabi-Yau variety? The question is an important one to answer for application to F-theory.

I will discuss various partial results concerning this question, some old and some new. I will also briefly describe the application to F-theory.

D. Orlov: Noncommutative varieties and their geometric realizations

**Abstract:** In this talk we are going to discuss a question of geometric realizations for noncommutative varieties. We are mostly going to consider special noncommutative varieties that have a full exceptional collection.

C. Simpson: The sphere at infinity of the character variety

**Abstract:** After a short detour through the Barcelona countryside, we’ll get back to a topic from my talk in a previous January in Miami. In a geometric avatar of the P=W conjecture, we can formulate the hypothesis that the dual boundary complex of a character variety of a compact orbicurve is a sphere, and that this sphere corresponds to the sphere at infinity in the base of the Hitchin fibration. We’ll report on some recent progress: a proof that the dual complex is homotopically a sphere, for rank 2 connections on $\Bbb{P}^1$ with *n* singular points. The strategy is to introduce some algebraic Fenchel-Nielsen coordinates, with care being taken along the lower-dimensional strata.

Y. Soibelman: Symplectic geometry of Riemann-Hilbert correspondences

**Abstract:** The main purpose of my talk is to discuss a conjecture about RH-correspondence for holonomic q-D-modules (which are certain finite type modules over the multi-dimensional quantum torus). Differently from the case of RH-correspondence for usual D-modules, the “Betti side” of the proposed RH-correspondence is a mixture of constructible and coherent sheaves. The unifying picture for the RH-correspondence in both, D-modules and q-D-modules stories, can be achieved via appropriate Fukaya categories. The talk is based on a project “Holomorphic Floer quantization”, joint with Maxim Kontsevich.

C. Vafa: tt* geometry

**Abstract:** I review tt* geometry and explain its recent extensions from 2d QFT’s to 3d and 4d supersymmetric theories and its relation to hyper-holomorphic connections. Moreover, for 4d N=2 QFT’s, tt* geometry leads to a twistorial extension of topological strings.

I. Zharkov: Phase tropical hypersurfaces and their skeleta

**Abstract:** To a smooth complex hypersurface in $(\Bbb{C}*)^N$ we associate a phase tropical hypersurface, which is the fibration over the tropical limit by coamoebas (tori over the facets). I will show that the phase tropical hypersurface is homeomorphic to the complex one and give strategies to construct its skeleta which are PL combinatorial objects.

### List of participants

Mohammed Abouzaid

Efim Abrikosov

Enrico Arbarello

Hülya Argüz

Denis Auroux

Matthew Ballard

Vladimir Baranovsky

Rodrigo Barbosa

Lawrence Barrott

Arnaud Beauville

Enrique Becerra

Fabian Belmonte

Oren Ben-Bassat

Netanel Blaier

Pierrick Bousseau

Paul Bressler

Juan Manuel Burgos Mieres

Philip Candelas*

Catherine Cannizzo

Nuno Cardoso

Juan Carlos Castro

Patrick Clarke

Herb Clemens

Mirjam Cvetic

Emanuel Diaconescu

Colin Diemer

Jimmy Dillies

Ron Donagi

Philippe Durand

Alexander Efimov

Kuan-Ying Fang

Teng Fei

Kenji Fukaya

Dennis Gaitsgory

Sergey Galkin

Sheel Ganatra

Honghao Gao

Yuan Gao

Peng Gao

Hugo Garcia-Compean

Alberto Garcia-Raboso

Elizabeth Gasparim

Michail Gerapetritis

Eduardo Gonzalez

Lino Grama

Mark Green*

Phillip Griffiths

Roberta Guadagni

Aldo Guzmán Sáenz

Andrew Hanlon

Jeff Hicks

Jianting Huang

Matei Ionita

Natalia Iyudu

Xin Jin

Dmitry Kaledin

Mikhail Kapranov

Yusuf Baris Kartal

Ilya Karzhemanov

Alexander Kasprzyk

Sheldon Katz

Ludmil Katzarkov

Tyler Kelly

Maxim Kontsevich

Can Kozcaz

Oleksandr Kravets

Amy Ksir

Nilay Kumar

Nikon Kurnosov

Jonathan Lai

Enka Lakuriqi

Oleg Lazarev

Heather Lee

Yanki Lekili

Jun Li

Bong Lian

Dahisy Lima

Timothy Logvinenko

Zhentao Lu

Kevin Luk

Valery Lunts

Ernesto Lupercio

Timothy Magee

Svetlana Makarova

Eyal Markman

Daniel Maya

Greg Moore

John Morgan

David Morrison

Benedict Morrissey

Saeed Moshfeghyeganeh

Jacob Mostovoy

Cristian Mihai Munteanu

Dmitri Orlov

Ignacio Hermelindo Otero

Pranav Pandit

Tony Pantev

Seoree Park

Mauro Porta

Victor Przyjalkowski

Surya Raghavendran

Mohammad Reza Rahmati

Jie Ren

Larry Richardson

Helge Ruddat

Jan-David Salchow

Paul Seidel

Artan Sheshmani

Bernd Siebert

Kyler Siegel

Carlos Simpson

Yan Soibelman

Theodore Spaide

Bruno Suzuki

Zack Sylvan

Hiro Lee Tanaka

Mina Teicher

Junwu Tu

Francisco Javier Turrubiates

Cumrun Vafa

Umut Varolgunes

Sara Venkatesh

Juan Jose Villarreal Montoya

Xinli Xiao

Miguel Xicotencatl

Ivan Yakovlev

Jingyu Zhao

Ilia Zharkov

Peng Zhou

Yuecheng Zhu