# Geometri- och fysikseminarium

Gränslandet mellan teoretisk fysik och matematik har sett flera stora framsteg över de senaste 20 åren, med gaugeteori, strängteori, topologi för fyrmångfalder, Floerhomologi och så vidare. Dessa seminarier har för avsikt att öka interaktionen mellan de två fälten. Både matematiker och fysiker kommer föreläsa i seminariumserien, och förhoppningsvis kan utövare av bägge ämnena förstå föreläsningarna.

Organiseras gemensamt av Matematiska Institutionen och Institutionen för fysik och astronomi.

Organisatörer: Tobias Ekholm och Maxim Zabzine

## Nineteenth UPPSALA GEOMETRY AND PHYSICS SEMINAR, June 4, 2019:

**A mirror symmetry correspondence for Landau-Ginzburg models**

**Speaker:** Arkady Vaintrob

**Department:** Oregon U

**Time:** 2019-06-04 13:30 - 14:30

**Location: **Å4001, Ångströmlaboratoriet

There are several known constructions (generally called Landau-Ginzburg models) of quantum invariants associated to a quasi-homogeneous polynomial W with an isolated singularity at the origin. These invariants play a prominent role in various mirror symmetry correspondences connecting LG models with other kinds of quantum invariants. If the polynomial W is invertible (i.e. when the number of monomials in W is equal to the number of variables), then the dual polynomial W' with the transposed matrix of exponents also has an isolated singularity and we can talk about relations between LG models for W and W'. Correspondences of this type were first considered by Berglund and Huebsch in the early 1990s, but their mathematical understanding was developed only relatively recently. I will present a mirror symmetry theorem connecting a LG B-model of W and a LG A-model of W' based respectively on Saito's theory of primitive forms and a cohomological field theory for W' constructed in my earlier work with Polishchuk using categories of matrix factorizations.

**Classical Freeness of CFTs **

**Speaker:** Reimundo Heluani

**Department: **IMPA, Rio de Janeiro

**Time: ** 2019-06-04 15:00 - 16:00

**Location: ** Å4001, Ångströmlaboratoriet

To any vertex operator algebra/CFT V one can attach a classical field theory limit P (roughly the Poisson algebra of local observables) and a classical mechanics limit C (roughly the Poisson algebra of zero modes). On the other hand to any classical mechanics system, say a Poisson algebra C with Hamiltonian H, one can attach a classical field theory JC "freely generated by C". When C is the classical mechanics limit of a CFT V, there is a canonical surjection JC --> P. We explore the question of when this is an isomorphism providing the first known examples and counterexamples. This is joint work with J. van Ekeren.

## Eighteenth UPPSALA GEOMETRY AND PHYSICS SEMINAR, October 3, 2018:

**An introduction to the BV-BFV formalism**

**Speaker:** Alberto Cattaneo

**Department:** Zurich U

**Time:** 2018-10-03 13:30 - 14:30

**Location: **Å2001, Ångströmlaboratoriet

The BV-BFV formalism unifies the BV formalism (which deals with the problem of fixing the gauge of field theories on closed manifolds) with the BFV formalism (which yields a cohomological resolution of the reduced phase space of a classical field theory). I will explain how this formalism arises and how it can be quantized.

**From Conformal Field Theory to Geometry **

**Speaker:** Luis Fernando Alday

**Department: **Oxford

**Time: ** 2018-10-03 15:00 - 16:00

**Location: ** Å2001, Ångströmlaboratoriet

The AdS/CFT correspondence maps correlators of local operators in a conformal field theory to scattering amplitudes in a gravitational/string theory on curved space-time. The study of such amplitudes is incredibly hard and has mostly been done in a certain classical limit. We show how modern analytic bootstrap techniques allows us to go much beyond that.

## SEVENTEENTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, MAY 24, 2018:

**Bosonization on a lattice in higher dimensions**

**Speaker:** Anton Kapustin

**Department:** Caltech, USA

**Time:** 2018-05-24 13:30 –14:30

**Location: **Häggsalen, Ångströmlaboratoriet

I describe a generalization of the Jordan-Wigner transformation for lattice systems of arbitrary dimension. It maps any fermionic system on a D-dimensional lattice to a (D-1)-form Z_2 gauge theory. The map preserves locality of the Hamiltonian.

## Sixteenth UPPSALA GEOMETRY AND PHYSICS SEMINAR, april 10, 2018:

**Geometry of Riemann-Hilbert correspondence**

**Speaker:** Yan Soibelman

**Department:** Kansas State University

**Time:** 2018-04-10 13:30–14:30

**Location: **Häggsalen, Ångströmlaboratoriet

Conventional Riemann-Hilbert correspondence relates differential equations and constructible sheaves. We propose to replace the latter by an appropriate Fukaya category. Based on this idea one can study the RH-correspondence not only for differential but also for q-difference and elliptic difference equations. Arising categories can be described, at least in dimension one, in different ways, e.g. in terms of constructible sheaves on “skeleta” which are not necessarily Lagrangian. If time permits I am going to discuss the corresponding “non-abelian Hodge theory” and its relationship to periodic monopoles.

## FIFTEENTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, DECEMBER 6, 2017:

**A generalisation of the AGT-correspondence for non-Lagrangian class S theories**

**Speaker:** Joerg Teschner

**Department:** Hamburg U & DESY

**Time:** 2017-12-06 13:30–14:30

**Location: **Å80101, Ångströmlaboratoriet

An interesting family of non-Lagrangian four-dimensional N=2, d=4 supersymmetric quantum field theories called T_N-theories is predicted by string theory. The geometric engineering of these theories gives a prediction for the partition functions of these theories by topological vertex methods. Joint work in progress with I. Coman-Lohi and Elli Pomoni will clarify the relation of these partition function to conformal blocks in Toda CFT, and to the quantisation of moduli spaces of flat SL(N) connections on the three-punctured sphere.

**Lagrangian fibrations on the projective plane and classification**

**Speaker:** Georgios Dimitroglou Rizell

**Department: **Uppsala University

**Time:** 2017-12-06 15:00–16:00

**Location:** Å80101, Ångströmlaboratoriet

The mirror of the projective plane in the sense of homological mirror symmetry consists of Landau-Ginzburg models built from Lagrangian torus fibrations on complements of elliptic curves. In ongoing work with T. Ekholm and D. Tonkonog we extend the construction to the case of certain singular torus fibrations. We also discuss recent classification results for Lagrangians in this setting.

## fourteenth UPPSALA GEOMETRY AND PHYSICS SEMINAR, april 20, 2017:

**Quantum invariants of singularities and matrix factorizations**

**Speaker:** Arkady Vaintrob

**Department: **University of Oregon, USA

**Time:** 2017-04-20 13:30–14:30

**Location: **Häggsalen, Ångströmlaboratoriet

I will discuss a cohomological field theory related to a quasihomogeneous singularity W with a group G of its diagonal symmetries (a Landau-Ginzburg A-model). The state space of this theory is the equivariant Milnor ring of W and its correlators are analogs of the Gromov-Witten invariants for the non-commutative space associated with the pair (W,G). In the case of simple singularities of type A they control the intersection theory on the moduli space of higher spin curves. The construction is based on categories of equivariant matrix factorizations.

**Kähler-Einstein metrics emerging from free fermions**

**Speaker:** Robert Berman

**Department: **Chalmers, Sweden

**Time: ** 2017-04-20 15:00–16:00

**Location: ** Häggsalen, Ångströmlaboratoriet

In this talk I will describe a statistical mechanical approach to the construction of Kähler-Einstein metrics on a complex algebraic variety X, i.e. Kähler solutions to Einstein's vacuum field equations. The microscopic theory is given by a canonical fermion gas on X whose one-particle states are pluricanonical holomorphic sections on X. The convergence problem in the case of a positive cosmological constant (i.e. the case when X is a Fano manifold) is still open and turns out to be closely connected to the Yau-Tian-Donaldson conjecture in Kähler geometry. Some intriguing relations to the AdS/CFT correspondence will also be pointed out.

## THIrteenth UPPSALA GEOMETRY AND PHYSICS SEMINAR, January 23, 2017:

**Dirac geometry and Poisson homogeneous spaces**

**Speaker:** Henrique Bursztyn

**Department:** IMPA, Rio de Janeiro

**Time:** 2017-01-23 13:30–14:30

**Location:** Polhemsalen, Ångströmlaboratoriet

We will give a brief introduction to Dirac structures, which are geometric objects extending Poisson as well as presymplectic structures. They arise naturally as the geometric framework for mechanical systems with constraints, but their recent applications are far reaching, including e.g. generalized complex geometry and the theory of moment maps. In this talk, I will explain how Dirac structures can be used as a tool to solve the problem of integrating Poisson homogeneous spaces.

**Teichmüller TQFT and traces of unitary operators**

**Speaker:** Rinat Kashaev

**Department: **Geneva Univ

**Time:** 2017-01-23 15:00–16:00

**Location:** Polhemsalen, Ångströmlaboratoriet

The Teichmüller TQFT induces unitary projective representations of punctured surfaces in infinite dimensional Hilbert spaces. The partition functions of the mapping tori of pseudo Anosov elements are finite in this theory provided a version of the volume conjecture for Teichmüller TQFT is satisfied. That means that the traces of the corresponding unitary operators and of all their powers can be defined even if these are not trace class operators.

## TWELFTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, April 20, 2016:

**Arithmetic of Calabi-Yau manifolds I**

**Speaker:** Xenia de la Ossa

**Department:** Oxford U

**Time:** 2016-04-20 13:30–14:30

**Location:** Å2005, Ångströmlaboratoriet

**Arithmetic of Calabi-Yau manifolds II**

**Speaker: **Philip Candelas

**Department:** Oxford U

**Time: **2016-04-20 15:00–16:00

**Location:** Å2005, Ångströmlaboratoriet

Calabi-Yau manifolds have many remarkable properties owing to their relation to supersymmetry and to string theory. In these two lectures we will give a self-contained introduction, aimed at a mixed audience of physicists and mathematicians, to the arithmetic of these manifolds, the aim of the lectures is to explore whether there are questions of common interest, in this context, to physicists, number theorists and geometers.

It is well known that there are certain classical enumerative problems to do with Calabi-Yau manifolds, of which the simplest is counting the number of holomorphically embedded lines, that are solved by manipulations involving the periods of the (mirror) manifold. These calculations are usually considered to belong to the province of algebraic geometry. The main quantities of interest in the arithmetic context are the numbers, N_{k}(*φ)*, of points of the manifold considered as a manifold over the field with p^{k} elements. We shall be concerned with the computation of these numbers and their dependence on the parameters, collectively denoted by *φ*. The first surprise, for a physicist, is that the N_{k}(*φ)* are also given by expressions that involve the periods of the manifold.

The N_{k}(*φ)* are encoded into the local ζ-function

about which much is known in virtue of the Weil conjectures. A topic we will stress is that, for one parameter families of manifolds, there are values of the parameter ϕ for which the manifold becomes singular and, for these values, the ζ-function degenerates and exhibits modular behaviour. A natural question for a physicist, and whose answer would be of considerable interest in mathematics, is whether the appearance of a modular group can be understood in physical terms. The topics of the talks will be:

1. A brief introduction to Calabi–Yau manifolds and their moduli. Elements of the theory of finite fields and p-adic numbers. The numbers, N_{r}(*φ)*, of F_{p}-rational points of a Calabi–Yau manifold, and their relation to the periods.

2. The form of the ζ-function for one parameter families, degenerations and modular properties. If time permits, also the relation of the ζ-functions for pairs of mirror manifolds and the role of the large complex structure limit.

## ELEVENTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, NOVEMBER 25, 2015:

**Spherical T-duality**

**Speaker:** Mathai Varghese

**Department:** Adelaide U

**Time:** 2015-11-25 13:30–14:30

**Location:** Polhemsalen, Ångströmlaboratoriet

Spherical T-duality is related to M-theory and was introduced in recent joint work with Bouwknegt and Evslin. I will begin by briefly reviewing the case of principal SU(2)-bundles with degree 7 flux, and then focus on the non-principal case for most of the talk, ending with the relation to SUGRA/M-theory.

**Topological field theory on singular spaces: quantizing moduli of microlocal sheaves**

**Speaker: **Vivek Shende

**Department:** UC Berkeley

**Time:** 2015-11-25 15:00–16:00

**Location:** Polhemsalen, Ångströmlaboratoriet

It is expected that the Fukaya category of a Weinstein manifold admits an interpretation as microlocal sheaves on a singular Lagrangian skeleton. One thus expects many moduli spaces of interest in mathematics and physics to arise as moduli of such sheaves; examples include cluster varieties and the augmentation variety of Legendrian contact homology. When such moduli spaces arise in field theoretic contexts, it is natural to want to “do quantum mechanics on the moduli space”. In this talk, we use stratified factorization homology to construct a singular topological quantum field theory on skeleta of dimensions at most 3, generalizing the analogous construction on smooth spaces by Ben Zvi, Brochier, and Jordan.

## TENTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, MARCH 18, 2014:

**Derived representation schemes
Speaker:** Giovanni Felder

**Department:**ETH, Zurich

**Time:**2015-03-18 13:30–14:30

**Location:**Polhemsalen, Ångströmlaboratoriet

Representation schemes parametrize representations of associative algebras on a given vector space. I will review a derived version of this theory, due to Berest, Khachatryan and Ramadoss, and present simple examples, such as the algebra of polynomials in two variables, featuring phenomena that are visible in computer experiments, and only partly understood mathematically. I will explain the relation with instanton partition functions of N=2 supersymmetric gauge theory on R^4xS^1 and with constant term identities. (Based on joint work with Y. Berest, A. Patotski, A. Ramadoss and T. Willwacher)

**Higher Spins & Strings
Speaker: **Matthias Gaberdiel

**Department:**ETH, Zurich

**Time:**2015-03-18 15:00–16:00

**Location:**Polhemsalen, Ångströmlaboratoriet

The conjectured relation between higher spin theories on anti de-Sitter (AdS) spaces and weakly coupled conformal field theories is reviewed. I shall then outline the evidence in favour of a concrete duality of this kind, relating a specific higher spin theory on AdS3 to a family of 2d minimal model CFTs. Finally, I shall explain how this relation fits into the framework of the familiar stringy AdS/CFT correspondence.

## NINTH UPPSALA GEOMETRY AND PHYSICS SEMINAR (COLLOQUIA), NOVEMBER 6, 2014:

**Math of quantum fields
Speaker:** Maxim Zabzine

**Department:**Uppsala University

**Time:**2014-11-06 13:30–14:30

**Location:**Polhemsalen, Ångströmlaboratoriet

Starting with the simple problem of counting partitions I will illustrate how the quantum theory can explain some non-trivial mathematical facts. Then I will give a non-expert overview of recent advances in quantum field theory and their relations to interesting geometrical invariants. I will also outline some fundamental mathematical problems which are posed by quantum field theory.

**Holomorphic curves, topological string, and low-dimensional topology
Speaker:** Tobias Ekholm

**Department:**Uppsala University

**Time:**2014-11-06 15:00–16:00

**Location:**Polhemsalen, Ångströmlaboratoriet

We discuss relations between holomorphic curve theories, topological string, and gauge theories in low-dimensional topology. We focus in particular on the case of knots and links in the three-sphere, where these different viewpoints complement each other and illustrate a number of central dualities. We also explain how this case leads to problems at the forefront in both symplectic geometry: open Gromov-Witten theory and (higher genus) Legendrian Symplectic Field Theory, and in topological string: new notions of mirror symmetry.

## EIGHTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, MAY 22, 2014:

**Knots, the augmentation variety, and topological strings
Speaker:** Lenhard Ng

**Department:**Duke University

**Time:**2014-05-22 13:30–14:30

**Location:**Polhemsalen, Ångströmlaboratoriet

Recently a striking connection has been made between the fields of symplectic topology and string theory. On the mathematical side, there is an invariant of knots called knot contact homology, which counts certain holomorphic curves in a 6-dimensional symplectic manifold. This leads to another knot invariant, a complex algebraic variety called the augmentation variety. Independently, studying topological strings on the resolved conifold yields another variety related to the HOMFLY polynomial of a knot. These two varieties are now conjectured to agree. I will discuss this conjecture and some possible consequences for both sides. This talk reports on joint work with Mina Aganagic, Tobias Ekholm, and Cumrun Vafa.

**Flat connections from 2-dimensional Quantum Field Theory
Speaker:** Anton Alekseev

**Department:**University of Geneva

**Time:**2014-05-22 15:00–16:00

**Location:**Polhemsalen, Ångströmlaboratoriet

Flat connections play an important role both in Physics and in Mathematics. In this talk, we shall give two examples of how flat connections come up in 2-dimensional Quantum Field Theories. The first example is the famous Knizhnik-Zamolodchikov connection which governs conformal blocks of the Wess-Zumino-Witten (WZW) model of Conformal Field Theory. The second example is the Torossian connection which was recently discovered in Deformation Quantization. It expresses quantum equations of motion of the 2-dimensional BF-theory (one of the simplest topological gauge theories). Both Knizhnik-Zamolodchikov and Torossian connections are intimately related to 3-dimensional Topological Field Theory, and they give rise to interesting solutions of the pentagon equation (so-called Drinfeld associators).

## SEVENTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, NOV. 5, 2013:

**SYZ Mirror symmetry for hypersurfaces**

**Speaker:** Mohammed Abouzaid

**Department:** Simons center, USA

**Time:** 2013-11-05 13:30–14:30

**Location:** Å80101, Ångströmlaboratoriet

I will discuss the construction of an SYZ mirror for hypersurfaces in a toric variety of dimension n. The mirror will in general be a Landau-Ginzburg model defined on an open subset of toric variety of dimension n+1. The precise construction will reveal a surprising twist on the expectation that the mirror of the resolution of surface singularities is given by the family of their smoothings. This is joint work with Auroux and Katzarkov.

**Advances in the theory of knot polynomials**

**Speaker:** Miranda Cheng

**Department:** Paris

**Time:** 2013-11-05 15:00–16:00

**Location:** Å80101, Ångströmlaboratoriet

The term “moonshine”, first introduced in the context of Monstrous Moonshine, describes an unexpected relation between modular functions and the representation theory of finite groups. I will give an overview of the recent developments in the area, focussing on the “umbral moonshine” phenomenon which (conjecturally) relates mock modular forms and various finite groups arising naturally from the study of the Niemeier lattices, and its possible relation to K3 surfaces and string theory. This talk is mostly based on joint work with John Duncan and Jeff Harvey.

## SIXTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, APRIL 10, 2013:

**Symplectic Khovanov cohomology**

**Speaker:** Ivan Smith

**Department:** Cambridge

**Time:** 2013-04-10 13:30–14:30

**Location:** Polhemsalen, Ångströmlaboratoriet

Symplectic Khovanov cohomology is a Floer-theoretic invariant of oriented links in the three-sphere, conjecturally isomorphic to its combinatorial sibling. I will outline a partial proof of that conjecture in characteristic zero. The key ingredient is a formality theorem for the Fukaya categories of certain symplectic manifolds arising in Lie theory; the proof of formality is motivated in part by ideas from homological mirror symmetry. This talk reports on joint work with Mohammed Abouzaid.

**Advances in the theory of knot polynomials**

**Speaker:** Alexei Morozov

**Department:** ITEP, Moscow

**Time: **2013-04-10 15:00–16:00

**Location: **Polhemsalen, Ångströmlaboratoriet

Knot polynomials are Wilson-loop averages in topological Chern-Simons theory and its generalisations. They depend on a variety of variables and satisfy rich set of equations – only few of which are already known. Generating functions of knot polynomials belong to the class of Hurwitz partition functions with non-trivial integrability properties. We review technical approaches, which allow to study these relations and connect knot theory to other branches of theoretical physics.

## FIFTH GEOMETRY AND PHYSICS SEMINAR, OCTOBER 3, 2012:

**McKay correspondence and the partition functions of 4d N=2 gauge theories
Speaker:** Vasily Pestun

**Department:**IAS, Princeton

**Time:**Wednesday, October 3, 2012 13:30–14:30

**Location:**Polhemsalen, Ångströmlaboratoriet

**Type:**Theoretical Physics

To any discrete subgroup of SU(2), or an affine ADE quiver, equipped with a certain data, there is an associated 4d N=2 gauge theory partition function, describing equivariant cohomology of the moduli space of self-dual connections on a four-manifold and defined combinatorially on the space of multi-colored partitions. For all such theories, the asymptotics of this partition function is found explicitly in terms of the periods of certain algebraic integrable system associated to the moduli of holomorphic ADE bundles on elliptic curves.

**The geometric construction of the Reshetikhin-Turaev Topological Quantum Field Theory
Speaker:** Jørgen Ellegaard Andersen

**Department:**Aarhus University

**Time:**Wednesday, October 3, 2012 15:00–16:00

**Location:**Polhemsalen, Ångströmlaboratoriet

In this talk, we will discuss the geometric construction of the Reshetikhin-Turaev Topological Quantum Field Theory using the geometric quantization of the moduli spaces of flat connections on two dimensional surfaces. We will then discuss various results on the large level asymptotics of these theories.

## FOURTH UPPSALA GEOMETRY AND PHYSICS SEMINAR, MARCH 26, 2012:

**Recent progress on four-dimensional symplectic embedding problems
Speaker:** Michael Hutchings

**Department:**UC Berkeley

**Time:**2012-03-26 13:30–14:30

**Location:**Häggsalen, Ångströmlaboratoriet

We discuss recent results on the problem of when one four-dimensional symplectic manifold (usually with boundary) can be symplectically embedded into another. For example, Dusa McDuff proved a number-theoretic criterion for when one four-dimensional ellipsoid can be symplectically embedded into another. Numerical invariants called “ECH capacities” give general obstructions to four-dimensional symplectic embeddings which turn out to be sharp in the case of ellipsoids.

**String-theory applications of integrable systems
Speaker:** Konstantin Zarembo

**Department:**Nordita

**Time:**2012-03-26 15:00–16:00

**Location:**Häggsalen, Ångströmlaboratoriet

Integrability plays an important role in many areas of physics, yielding exact results for systems where interactions may be arbitrary strong and difficult to handle by any other means. One example is string theory, where integrability has led to important insights into the nature of gauge/string dualities.

## THIRD UPPSALA GEOMETRY AND PHYSICS SEMINAR, SEPTEMBER 6, 2011:

**A piece of 21st century mathematics that didn't make it into 20th century physics**

**Speaker:** Sergei Gukov

**Department:** Caltech

**Time:** 2011-09-06 15:00–17:00

**Location:** Å80101, Ångströmlaboratoriet

This lecture will be about the physics of new knot invariants invented by Khovanov, Rozansky, Ozsvath, Szabo, Rasmussen, and other mathematicians circa 2000. The key ingredients of the proposed physical framework involve standard tools from gauge theory and string theory. It leads to a wealth of generalizations and comes with a few surprising features.

**Rigidity and Flexibility in Symplectic Geometry**

**Speaker:** Yakov Eliashberg

**Department:** Stanford

**Time:** 2011-09-06 12:30–14:30

**Location:** Å80101, Ångströmlaboratoriet

Symplectic topology was born in the 1980s on the borderline between the worlds of rigid and flexible mathematics. In the talk I will describe some recent advances on both sides of the border. On the flexible side, I will discuss a surprising h-principle for Legendrian knots of dimension >1, proven by my student Max Murphy, as well as its consequences for symplectic topology of Stein manifolds. On the rigid side, I will discuss effective techniques for computing symplectic invariants of Stein manifolds via Legendrian surgery. The flexible side of the story is joint work with K.

## SECOND UPPSALA GEOMETRY AND PHYSICS SEMINAR, DECEMBER 16, 2010:

**Derived brackets in generalized geometry
Speaker:** Ezra Getzler

**Department:**Northwestern Univ, Chicago

**Time:**2010-12-16 15:00–16:00

**Location:**Polhemsalen, Ångströmlaboratoriet

The derived bracket of Koszul is an algebraic construction which associates to a differential on a graded Lie algebra a “derived” Lie algebra. This turns out to generalize the way in which the Poisson bracket is associated to a Poisson tensor on a manifold.

**Quantum theory and enumerative problems
Speaker:** Marcos Mariño

**Department:**University of Geneva

**Time:**2010-12-16 13:30–14:30

**Location:**Polhemsalen, Ångströmlaboratoriet

**Type:**Theoretical Physics

In quantum theory we are very often interested in counting objects. For example, to understand perturbation theory in quantum field theory we have to count graphs by using combinatorial techniques, while in string theory we want to count curves in manifolds, by using algebraic geometry. In this talk I will explore some of these enumerative problems, starting with simple examples in quantum theory and ending with string theory. I will also use Chern-Simons theory and its mathematical incarnation (the LMO invariant) to discuss the interplay between counting problems and the 1/N expansion.

## FIRST UPPSALA GEOMETRY AND PHYSICS SEMINAR, MAY 6, 2010:

**Geometry and string duality
Speaker:** Chris Hull

**Department:**Imperial College

**Time:**2010-05-06 15:00–16:00

**Location:**Häggsalen, Ångströmlaboratoriet

Generalized geometry provides a natural framework for studying d-dimensional manifolds equipped with a metric and B-field. In this approach the tangent bundle is “doubled” to T+T* and there is a natural action of O(d,d) on the geometry. The group O(d,d) also arises naturally in string theory for backgrounds that are d-torus bundles. Dimensional reduction on the torus fibres gives a truncation to an effective theory with an O(d,d) symmetry.

**Generalized holomorphic bundles
Speaker:** Nigel Hitchin

**Department:**Oxford

**Time:**2010-05-06 13:30–14:30

**Location:**Häggsalen, Ångströmlaboratoriet

On a manifold with a generalized complex structure there is a natural notion of generalized holomorphic bundle introduced by Gualtieri. In the case of a symplectic structure this is just a bundle with flat connection, but for an ordinary complex structure, viewed from the generalized point of view, it becomes an interesting holomorphic object. We shall discuss this case, and also the case for a holomorphic Poisson structure, considering in particular the role of B-field transformations.