Geometry and Physics Seminars

Resurgence and topological strings

Speaker: Marcos Mariño
Department: Geneva U.
Time: 2023-05-03 13:30 – 14:30
Location: Häggsalen, Ångströmlaboratoriet

Abstract

The theory of resurgence provides a precise and unified mathematical formulation of the non-perturbative sectors of a physical theory, based solely on its perturbative expansion. In recent years it has been applied to topological quantum field theories and topological strings, leading to many insights and results. For example, it has been found that, in many cases, the Stokes constants appearing in resurgent analysis are related to BPS invariants. In this talk I will first review the relevant aspects of the resurgence program, and then I will review recent progress on its implementation in topological string theory. In particular, I will present exact results on multi-instanton amplitudes for the topological string on arbitrary Calabi-Yau manifolds, and applications of these results to asymptotic problems in the theory of Gromov-Witten invariants.

The geometry of black hole entropy functions

Speaker: James Sparks
Department: Oxford U.
Time: 2023-05-03 15:00 – 16:00
Location: Häggsalen, Ångströmlaboratoriet

Abstract

One of the major successes of string theory has been a precise microstate counting of black hole entropy, at least for certain classes of supersymmetric black holes in asymptotically flat spacetimes. Recently a growing body of similar work has been developed for black holes in asymptotically Anti-de Sitter (AdS) spacetimes, where the microstate counting uses the AdS/CFT correspondence and a dual field theory description. After introducing the general topic, I will describe a certain Riemannian geometry associated to such black holes in AdS. This involves a new type of PDE for a Kahler metric. While solving this PDE is in general a hard problem, I explain how solutions are critical points of an "entropy function", which when evaluated on a solution is precisely the associated black hole entropy. Geometric techniques then allow one to compute the entropy for various families of black holes, without solving the Einstein equations explicitly. I will conclude by commenting on the relationship between this entropy function and the dual field theory.

Black Holes, arithmetic and modularity for families of Calabi-Yau manifolds, part I

Speaker: Xenia de la Ossa
Department: Oxford
Time: 2023-02-01 15:15 – 16:15
Location: Polhemsalen, Ångströmlaboratoriet

Black Holes, arithmetic and modularity for families of Calabi-Yau manifolds, part II

Speaker: Philip Candelas
Department: Oxford
Time: 2023-02-01 16:30 – 17:30
Location: Polhemsalen, Ångströmlaboratoriet

Abstract

The main goal of these two talks is to explore some questions of common interest for physicists, number theorists and geometers, in the context of the arithmetic of Calabi-Yau 3-folds. There are many such relations, however we will focus on the rich structure of black hole solutions of type II superstrings on a Calabi-Yau manifolds. We will give a self contained introduction aimed at a mixed audience of physicists and mathematicians. The main quantities of interest in the arithmetic context are the numbers of points of the manifold, considered as a variety over a finite field. A mathematician is interested in the computation of these numbers and their dependence on the moduli of the variety. The surprise for a physicist is that the numbers of points over a finite field are also given by expressions that involve the periods of a manifold. These periods determine many aspects of the physical theory, as for example the kinetic terms of the effective Lagrangian as well as the Yukawa couplings, but also properties of black hole solutions. For a mathematician, the number of points determine the zeta function, about which much is known in virtue of the Weil conjectures. We discuss a number of interesting topics related to the zeta function, the corresponding L-function, and the appearance of modularity for one parameter families of Calabi-Yau manifolds. We will focus on an example for which the quartic numerator of the zeta function of a one parameter family factorises into two quadrics at special values of the parameter, which satisfies an algebraic equation with coefficients in Q (so independent of any particular prime), and for which the underlying manifold is smooth. The significance of these factorisations in physics is that they are due to the existence of black hole attractor points in the sense of type II supergravity and are related to a splitting of the Hodge structure and that at these special values of the parameter. For a mathematician these factorisations of the Hodge structure are related to the famous Hodge Conjecture. Modular groups and modular forms arise in relation to these attractor points. To our knowledge, the rank two attractor points that were found by the application of these number theoretic techniques, provide the first explicit examples of such attractor points for Calabi-Yau manifolds. Time permitting, we will describe this scenario also for the mirror manifold in type IIA supergravity.

Geometry and Fundamental Lessons for Quantum Gravity

Speaker: Cumrun Vafa
Department: Harvard
Time: 2020-01-29 13:15 – 14:15
Location: Häggsalen, Ångströmlaboratoriet

Abstract

I review the power of geometric ideas in the context of string theory to teach us about consistency conditions for a quantum gravitational theory. For a large class of cases we can use known mathematical and geometrical facts to rule out putative quantum gravitational systems which naively look perfectly consistent. I discuss some of the principles that have emerged in this study which distinguish good quantum systems belonging to the “string landscape” from the inconsistent ones belonging to the “string swampland”.

“Zed-hat”

Speaker: Sergei Gukov
Department: Caltech
Time: 2020-01-29 14:30 – 15:30
Location: Häggsalen, Ångströmlaboratoriet

Abstract

The goal of the talk will be to introduce a class of functions that originate from physics, answer a question in topology, can be computed via methods more common in the theory of dynamical systems, and in the end turn out to enjoy beautiful modular properties of the type first observed by Ramanujan.

CY 3-folds over finite fields, Blackhole attractors, and D-brane masses

Speaker: Albrecht Klemm
Department: Bonn U
Time: 2020-01-29 16:00 – 17:00
Location: Häggsalen, Ångströmlaboratoriet

Abstract

The integer coefficients in the numerator of the Hasse-Weil Zeta function for one parameter Calabi-Yau 3-folds are expected to be Hecke eigenvalues of Siegel modular forms. For rigid CY 3-folds as well as at conifold — and rank two attractor points of non rigid Calabi-Yau this numerator contains actors of lower degree whose coefficients are determined by the Hecke eigenvalues of weight two or four modular cusp forms of $\Gamma_0(N)$. We show that the Hecke L-function at integer arguments or more generally the periods of these modular forms give the $D$-brane masses as well as the value and the curvature of the Weil-Peterssen metric at these points. The coefficients of the connection matrix from the integer symplectic basis to a Frobenius basis at the conifold and at a rank two attractor point are given by the periods and the quasi periods of these modular forms.

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

Abstract

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 Hübsch 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

Abstract

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.

An introduction to the BV-BFV formalism

Speaker: Alberto Cattaneo
Department: Zürich U
Time: 2018-10-03 13:30 – 14:30
Location: Å2001, Ångströmlaboratoriet

Abstract

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

Abstract

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.

Bosonization on a lattice in higher dimensions

Speaker: Anton Kapustin
Department: Caltech, USA
Time: 2018-05-24 13:30 – 14:30
Location: Häggsalen, Ångströmlaboratoriet

Abstract

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 Z2 gauge theory. The map preserves locality of the Hamiltonian.

Geometry of Riemann-Hilbert correspondence

Speaker: Yan Soibelman
Department: Kansas State University
Time: 2018-04-10 13:30 – 14:30
Location: Häggsalen, Ångströmlaboratoriet

Abstract

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.

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

Speaker: Jörg Teschner
Department: Hamburg U & DESY
Time: 2017-12-06 13:30 – 14:30
Location: Å80101, Ångströmlaboratoriet

Abstract

An interesting family of non-Lagrangian four-dimensional N=2, d=4 supersymmetric quantum field theories called TN-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

Abstract

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.

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

Abstract

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

Abstract

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.

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

Abstract

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

Abstract

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.

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

Abstract

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, Nk(φ), of points of the manifold considered as a manifold over the field with pk 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 Nk(φ) are also given by expressions that involve the periods of the manifold.

The Nk(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, Nr(φ), of Fp-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.

Spherical T-duality

Speaker: Mathai Varghese
Department: Adelaide U
Time: 2015-11-25 13:30 – 14:30
Location: Polhemsalen, Ångströmlaboratoriet

Abstract

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

Abstract

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.

Derived representation schemes

Speaker: Giovanni Felder
Department: ETH, Zurich
Time: 2015-03-18 13:30 – 14:30
Location: Polhemsalen, Ångströmlaboratoriet

Abstract

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 R4xS1 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

Abstract

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.

Math of quantum fields

Speaker: Maxim Zabzine
Department: Uppsala University
Time: 2014-11-06 13:30 – 14:30
Location: Polhemsalen, Ångströmlaboratoriet

Abstract

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

Abstract

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.

Knots, the augmentation variety, and topological strings

Speaker: Lenhard Ng
Department: Duke University
Time: 2014-05-22 13:30 – 14:30
Location: Polhemsalen, Ångströmlaboratoriet

Abstract

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

Abstract

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).

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

Abstract

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.

Symplectic Khovanov cohomology

Speaker: Ivan Smith
Department: Cambridge
Time: 2013-04-10 13:30 – 14:30
Location: Polhemsalen, Ångströmlaboratoriet

Abstract

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

Abstract

Knot polynomials are Wilson-loop averages in topological Chern–Simons theory and its generalisations. They depend on a variety of variables and satisfy a 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.

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

Abstract

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

Abstract

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.

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

Abstract

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

Abstract

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.

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

Abstract

This lecture will be about the physics of new knot invariants invented by Khovanov, Rozansky, Ozsváth, 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

Abstract

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.

Derived brackets in generalized geometry

Speaker: Ezra Getzler
Department: Northwestern Univ, Chicago
Time: 2010-12-16 15:00 – 16:00
Location: Polhemsalen, Ångströmlaboratoriet

Abstract

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

Abstract

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.

Geometry and string duality

Speaker: Chris Hull
Department: Imperial College
Time: 2010-05-06 15:00 – 16:00
Location: Häggsalen, Ångströmlaboratoriet

Abstract

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

Abstract

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.