| Date |
Speaker |
Title, Abstract |
| September 25 |
Du Pei (QGM Aarhus and Caltech) |
Can one hear the shape of a drum? |
|
Much like harmonics of musical instruments, spectra of quantum
systems contain wealth of interesting information. In this talk, I will
introduce new invariants of three- and four-manifolds using BPS spectra of
quantum field theories. While most of them are completely novel, some of
the new invariants categorify well-known old invariants such as the WRT
invariant of 3-manifolds and the Donaldson invariant of 4-manifolds. This
talk is based on arXiv:1701.06567 and ongoing work with Sergei Gukov, Pavel
Putrov and Cumrun Vafa. |
| October 16 |
Laura Schaposnik (UIC) |
On Cayley and Langlands type correspondences for Higgs bundles. |
|
The Hitchin fibration is a natural tool through which one can understand
the moduli space of Higgs bundles and its interesting subspaces (branes). After reviewing
the type of questions and methods considered in the area, we shall dedicate this talk to the
study of certain branes which lie completely inside the singular fibres of the Hitchin fibrations.
Through Cayley and Langlands type correspondences, we shall provide a geometric description of these
objects, and consider the implications of our methods in the context of representation theory,
Langlands duality, and within a more generic study of symmetries on moduli spaces. |
| November 6 |
Natalie Paquette (Caltech) |
Dual boundary conditions in 3d N=2 QFTs |
|
We will study half-BPS boundary conditions in 3d N=2 field theories that
preserve 2d (0,2) supersymmetry on the boundary. We will construct
simple boundary conditions and study their local operator content using
a quantity called the half-index. Using the half-index as a guide, we
study the actions of a variety of 3d dualities on the boundary
conditions, including level-rank duality, mirror symmetry, and
Seiberg-like duality. Identifying the dual pairs of boundary conditions,
in turn, helps lead to the construction of duality interfaces. This talk
is based on work in progress with T. Dimofte and D. Gaiotto.
|
November 7
Tuesday, 2pm, 02/104 |
Arnav Tripathy (Harvard) |
Special cycles and BPS jumping loci |
|
I'll sketch an attempt to bring the theory of special cycles, a deep part of
number theory, into the domain of supersymmetric string compactifications. I'll describe a
construction based on jumping loci for BPS state counts -- a separate phenomenon from the
better-known wall-crossing! -- and explain in what cases these jumping loci generalize some
parts of the theory of special cycles. Finally, I'll conclude with a host of physical and
mathematical conjectures raised by this line of investigation. |
| November 20 |
Minhyong Kim (Oxford) |
Gauge theory in arithmetic geometry I |
|
Number-theorists have been implicitly using gauge theory for perhaps 350 years,
and explicitly for about 50 years. However, they did not use the terminology at all. I will
review some of this story and explain why it's a good idea do to so now. In particular, we will
describe some of the ideas of Diophantine gauge theory and arithmetic Chern-Simons theory. |
November 21
Tuesday, 2pm, 02/104 |
Minhyong Kim (Oxford) |
Gauge theory in arithmetic geometry II |
|
This is a continuation of the previous lecture |
| December 4 |
Lukas Hahn (Heidelberg) |
Super Riemann surfaces and their moduli |
|
Abstract forthcoming |
| January 15 |
Helge Ruddat (Mainz) |
Tropical construction of
Lagrangian submanifolds |
|
Homological mirror symmetry suggests that complex submanifolds of a
Calabi-Yau manifold match Lagrangian submanifolds of the mirror dual
Calabi-Yau. In practice, a maximal degeneration needs to be chosen and
then the submanifolds are identified by a duality of their degeneration
data which is tropical geometry. Cheuk Yu Mak and I carry out this
construction for lines on the quintic threefold which become spherical
Lagrangians in the quintic mirror. Our construction applies more
generally for Calabi-Yau threefolds in the Batyrev construction and
probably even more generally at some point in the future. Quite
surprisingly, many exotic Lagrangian threefolds can be constructed this
way, many for the first time in a compact symplectic 6-manifold. |
| February 19 |
Guglielmo Lockhart (Amsterdam) |
Universal features of 6d self-dual string CFTs |
|
BPS strings are the fundamental objects on the tensor branch of 6d \((1,0)\) SCFTs. They can
be thought of as the instantons of the 6d gauge group, and are the building blocks for computing the
‘instanton piece’ of the \(\mathbb R^4\times T^2\) partition function of the parent 6d SCFT. The goal of this talk is to
rephrase their properties from the point of view of a worldsheet \(\mathcal N=(0,4)\) NLSM. This reveals that, despite
their superficial differences, self-dual strings of arbitrary 6d SCFTs share many universal features.
Along the way, this leads to a better understanding of the flavor symmetry of the parent 6d SCFTs.
Moreover, the constraints from modularity and these universal features are strong enough that one
can fix the elliptic genus of one self-dual string for a wide variety of SCFTs. |
| February 26 |
Jan Swoboda (München) |
The Higgs bundle moduli space and its asymptotic geometry |
|
The Theorem of Narasimhan and Seshadri states a correspondence between the moduli space
of stable holomorphic vector bundles over a Riemann surface \(X\) and that of irreducible unitary connections
of constant central curvature. This is one instance of a much more general correspondence due to Kobayashi
and Hitchin. Higgs bundles come into play when the compact Lie group \(\operatorname{SU}(r)\) is replaced by
\(\operatorname{SL}(r,\mathbb C)\). A suitable generalization of the constant central curvature connections in the
former case is found in the solutions to Hitchin's self-duality equations. Due to the noncompactness of
the Higgs bundle moduli space, a set of new questions revolving around its ``geometry at infinity'' arises.
In this talk I will focus on the asymptotics of the natural \(L^2\)-metric \(G_{L^2}\) on the moduli space
\(\mathcal M\) of rank-\(2\) Higgs bundles. I will show that on the regular part of the Hitchin fibration
\((A,\Phi)\mapsto\det\Phi\) this metric is well-approximated by the semiflat metric \(G_{\operatorname{sf}}\)
coming from the completely integrable system on \(\mathcal M\). This also reveals the asymptotically
conic structure of \(G_{L^2}\), with (generic) fibres of the above fibration being asymptotically flat
tori. This result confirms some aspects of a more general conjectural picture made by Gaiotto, Moore
and Neitzke. Its proof is based on a detailed understanding of the ends structure of \(\mathcal M\).
The analytic methods used here in addition yield a complete asymptotic expansion of the difference
\(G_{L^2}-G_{\operatorname{sf}}\) between the two metrics, with leading order term having polynomial
decay and a rather explicit description.
The results presented here are from recent joint work with Rafe Mazzeo, Hartmut Weiß and Frederik Witt.
|
| Date |
Speaker |
Title, Abstract |
| April 10 |
Ingmar Saberi (Heidelberg) |
Holographic lattice field theories |
|
Recent developments in tensor network models (which are, roughly speaking, quantum circuits
designed to produce analogues of the ground state in a conformal field theory) have led to
speculation that such networks provide a natural discretization of the AdS/CFT correspondence.
This raises many questions: just to begin, is there any sort of lattice field theory model
underlying this connection? And how much of the usual AdS/CFT dictionary really makes sense
in a discrete setting? I'll describe some recent work that proposes a setting in which such
questions can perhaps be addressed: a discrete spacetime whose bulk isometries nevertheless
match its boundary conformal symmetries. Many of the first steps in the AdS/CFT dictionary
carry over without much alteration to lattice field theories in this background, and one
can even consider natural analogues of BTZ black hole geometries.
|
Tuesday, April 11
2 p.m.s.t. |
Michael Gekhtman (Notre Dame) |
Higher pentagram maps via cluster mutations and networks on surfaces
|
|
The pentagram map that associates to a projective polygon a new one formed by
intersections of short diagonals was introduced by R. Schwartz and was shown to
be integrable by V. Ovsienko, R. Schwartz and S. Tabachnikov. M. Glick
demonstrated that the pentagram map can be put into the framework of the theory
of cluster algebras, a new and rapidly developing area with many exciting
connections to diverse fields of mathematics. In this talk I will explain that
one possible family of higher-dimensional generalizations of the pentagram map
is a family of discrete integrable systems intrinsic to a certain class of
cluster algebras that are related to weighted directed networks on a torus and a
cylinder. After presenting necessary background information on Poisson geometry
of cluster algebras, I will show how all ingredients necessary for integrability
- Poisson brackets, integrals of motion - can be recovered from combinatorics of
a network. The talk is based on a joint project with M. Shapiro, S. Tabachnikov
and A. Vainshtein.
|
| May 29 |
Jon Keating (Bristol) |
The Riemann hypothesis
and physics — a perspective |
|
I will give an overview of some connections, mostly
speculative, between the Riemann Hypothesis, random matrix theory, and
quantum chaos. |
| July 3 |
Alex Turzillo (Caltech) |
Spin TQFT and fermionic gapped phases |
|
I will discuss state sum constructions of two-dimensional Spin TQFTs and
their G-equivariant generalizations. These models are related to tensor network descriptions
of ground states of fermionic topological matter systems. We will revisit the classification
of fermionic short range entangled phases enriched by a finite symmetry G and derive a group
law for their stacking. |
| July 10 |
Markus Banagl (Heidelberg) |
Intersection homology and
the conifold transition |
|
Abstract forthcoming |
| Date |
Speaker |
Title, Abstract |
| November 28 |
John Alexander Cruz Morales (Bonn) |
On Stokes matrices for Frobenius manifolds |
|
In this talk we will discuss how to compute the Stokes matrices for some semisimple Frobenius manifolds
by using the so-called monodromy identity. In addition, we want to discuss the case when we get integral
matrices and their relations with mirror symmetry. This is part of an ongoing project with M. Smirnov
and previous joint work with Marius van der Put.
|
| December 5 |
Adam Alcolado (McGill) |
Extended Frobenius Manifolds |
|
Frobenius manifolds, introduced by Dubrovin, are objects which know about many different things in
mathematics, for example, the enumeration of rational curves, or the list of platonic solids. We
will introduce a generalization of Frobenius manifolds which know about real (or open) enumerative
geometry. What else do these extended Frobenius Manifolds know?
|
| December 12 |
Pietro Longhi (Uppsala) |
Probing the geometry of BPS states with spectral networks |
|
In presence of defects the Hilbert space of a quantum field theory can change in
interesting ways. Surface defects in 4d N=2 theories introduce a class of 2d-4d
BPS states, which the original 4d theory does not possess. For theories of class
S, spectral networks count 2d-4d BPS states, and through the 2d-4d wall-crossing
phenomenon the 4d BPS spectrum can be obtained. Adopting this physical viewpoint
on spectral networks, I will illustrate some recent and ongoing developments
based on this framework, with applications to the study of 2d (2,2) BPS spectra,
and of 4d N=2 BPS monodromies.
|
| Tuesday, December 13 |
Michael Bleher (Heidelberg) |
Supersymmetric Boundary Conditions in
N=4 Super Yang-Mills Theory |
|
In the last decade it was realized that supersymmetric boundary conditions in super Yang-Mills
theories can provide invaluable insight into several areas of current mathematical research.
Motivated especially by their appearance in a categorification of knot invariants, I will give
a short overview on half-BPS boundary conditions in 4d N=4 SYM theory. Of particular interest
is the Nahm-pole boundary condition and the main goal will be to review the corresponding moduli
space of supersymmetric vacua.
|
| January 30 |
Chris Elliott (IHÉS) |
Algebraic Structures for Kapustin-Witten
Twisted Gauge Theories |
|
Topological twisting is a technique for producing topological field theories from supersymmetric
field theories -- one exciting application is Kapustin and Witten's 2006 discovery that the categories
appearing in the geometric Langlands conjecture can be obtained as topological twists of N=4 supersymmetric
gauge theories, and that these two categories are interchanged by S-duality. There are, however, several
incompatibilities between Kapustin and Witten's construction and the geometric representation theory
literature. First, their techniques do not produce the right algebraic structures on the moduli
spaces appearing in geometric Langlands, and secondly, their construction doesn't explain the singular
support conditions Arinkin and Gaitsgory introduced in order to make the geometric Langlands
correspondence possible. In this talk I'll explain joint work with Philsang Yoo addressing both
of these issues: how to understand topological twisting in (derived) algebraic geometry, and
how to interpret singular support conditions as arising from the choice of a vacuum state.
|
Wednesday, February 1
SR B, 2 p.m.s.t. |
Chris Elliott (IHÉS) |
An Introduction to the Batalin-Vilkovisky Quantization Formalism |
|
Abstract forthcoming
|
| February 13 |
Steven Sivek (Bonn) |
The augmentation category of a Legendrian knot |
|
Given a Legendrian knot in R3, Shende, Treumann, and Zaslow defined a
category of constructible sheaves on the plane with singular support controlled by the
front projection of the knot. This category turns out to be equivalent to a unital
A∞ category, called the augmentation category, which is defined in terms of a
holomorphic curve invariant (Legendrian contact homology) of the knot.
In this talk I will describe the construction of these categories and outline a proof that
they are equivalent. This is joint work with Lenny Ng, Dan Rutherford, Vivek Shende,
and Eric Zaslow.
|
| February 20 |
Ben Knudsen (Harvard) |
A local-to-global approach to configuration spaces |
|
I will describe how ideas borrowed from functorial field theory,
the theory of chiral algebras, and BV theory may be profitably adapted to
the purely topological problem of calculating Betti numbers of
configuration spaces. These methods lead to improvements of classical
results, a wealth of computations, and a new and combinatorial proof of
homological stability.
|
| February 27 |
Owen Gwilliam (Bonn) |
Chiral differential
operators and the curved beta-gamma system |
|
Chiral differential operators (CDOs) are a vertex algebra analog
of the associative algebra of differential operators. They were originally
introduced by mathematicians using just sheaf theory and vertex algebraic
machinery. Later, Witten explained how CDOs on a complex manifold X arise
as the perturbative part of the curved beta-gamma system with target X. I
will describe recent work with Gorbounov and Williams in which we construct
the BV quantization of this theory and use a combination of factorization
algebras and formal geometry to recover CDOs. At the end, I hope to discuss
how the techniques we developed apply to a broad class of nonlinear sigma
models, including source manifolds of higher dimension.
|
| March 20 |
Michele Cirafici (I.S.T. Lisboa) |
Framed BPS quivers and line defects |
|
I will discuss a certain class of line defects in four dimensional supersymmetric theories
with N=2. I will show that many properties of these
operators can be rephrased in terms of quiver representation theory. In
particular one can study BPS invariants of a new kind, the so-called framed BPS
states, which correspond to bound states of ordinary BPS states with the defect.
I will discuss how these invariants arise from framed quivers. Time permitting I
will also discuss a formalism to study these quantities based on cluster
algebras.
|
| Date |
Speaker |
Title, Abstract |
| April 25 |
Zhentao Lu (Oxford) |
Quantum sheaf cohomology on Grassmannians |
|
I will give a brief introduction to quantum sheaf cohomology and correlation functions. I will
talk about the computation of the classical cohomology ring \(\sum H^q(X, \wedge^p E^*)\) for a
vector bundle \(E\) over the Grassmannian \(X\), also known as the polymology. Then I will talk
about the conjectural quantum sheaf cohomology derived from the Coulomb branch argument of
the physics theory of gauged linear sigma models. |
| May 2 |
Seung-Joo Lee (Virginia Tech) |
Witten Index for Noncompact Dynamics |
|
Many of the gauged dynamics motivated by string theory come with gapless asymptotic directions. In this
talk, we focus on d=1 GLSM's of such kind and their Witten indices, having in mind of the associated D-brane
bound state problems. Upon illustrating by examples that twisted partition functions can be misleading,
we proceed to explore how physical Witten indices can sometimes be embedded therein. There arise
further subtleties when gapless continuum sectors come from a gauge multiplet, as in non-primitive quiver
or pure Yang-Mills theories. For such theories, the twisted partition functions tend to involve fractional
expressions. We point out that these are tied to the notion of rational invariant in the wall-crossing
formulae, offering a general mechanism of extracting the Witten index directly from the twisted partition
function.
|
May 12
Thursday, 2pm, SR 11 |
Satoshi Nawata (Caltech/Aarhus) |
Various formulations of knot homology |
|
This lecture is preceded by a talk entitled
"Knot Homology from String Theory" in the
Oberseminar Conformal field theory
on Wednesday May 11, 2pm, Philosophenweg 12 |
| May 23 |
N.N. |
T.B.A. |
| June 13 |
Nicolò Piazzalunga (Trieste/Madrid) |
Real Topological String Theory |
June 23
Thursday, 2pm
RZ Statistik 02.104 |
Matt Young (Hong Kong) |
Algebra and geometry of orientifold Donaldson-Thomas theory |
|
I will give an overview of the orientifold Donaldson-Thomas theory of quivers. Roughly speaking,
orientifold Donaldson-Thomas theory is a virtual counting theory for principal G-bundles
in three dimensional Calabi-Yau categories, where G is an orthogonal or symplectic group.
This theory is best formulated in terms of geometrically defined representations of cohomological
Hall algebras. I will explain how this set-up leads naturally to a categorification of the orientifold
wall-crossing formula appearing in the string theory literature, a proof of the orientifold variant of
the integrality conjecture of Kontsevich-Soibelman and a geometric interpretation of orientifold
Donaldson-Thomas invariants. Partially based on joint work with Hans Franzen (Bonn).
|
| July 11 |
N.N. |
T.B.A. |
| July 25 |
Mauricio Romo (IAS) |
Complex Chern-Simons and cluster algebras |
|
I'll describe the cluster partition function (CPF), a computational tool that uses elements from
algebra and representation theory to describe the partition function of Chern-Simons theory with
gauge group SL(N,C) given some prescribed boundary conditions. The CPF allows for a perturbative
expansion that can be used to compute invariants of a large class of hyperbolic knots. I'll comment on
its relation with M-theory and further applications such as insertion of line operators, if time allows.
|
| Date |
Speaker |
Title |
| October 20 |
Thomas Reichelt (Heidelberg) |
Semi-infinite Hodge structures. An introduction |
| November 2 |
N.N. |
T.B.A. |
| November 16 |
Dominik Wrazidlo (Heidelberg) |
Positive Topological Quantum Field Theories and Fold Maps |
| December 7 |
Dmytro Shklyarov (Chemnitz) |
On an interplay between Hodge theoretic and categorical invariants of singularities |
| December 14 |
Cornelius Schmidt-Colinet (München) |
Conformal perturbation defects between 2d CFTs |
|
We consider some examples and properties of interfaces implementing
renormalisation group flows between two-dimensional conformal field theories, including general
perturbative results for entropy and transmissivity, and the explicit constructions for flows
between coset models.
|
January 11
INF 227 (KIP) SB1.107 |
Renato Vianna (Cambridge) |
Infinitely many monotone Lagrangian tori in del Pezzo surfaces |
|
In 2014, we showed how the Chekanov torus arises as a fiber of an
almost toric fibration and how this perspective enables us to describe an
infinite range of monotone Lagrangian tori. More precisely, for any Markov
triple of integers \((a,b,c)\) -- satisfying \(a^2+b^2+c^2=3abc\) -- we get a
monotone Lagrangian torus \(T(a^2,b^2,c^2)\) in \({\mathbb C} P^2\). Using
neck-stretching techniques we are able to get enough information on the
count of Maslov index \(2\) pseudo-holomorphic disks that allow us to show
that for \((d,e,f)\) a Markov triple distinct from \((a,b,c)\), \(T(d^2,e^2,f^2)\)
is not Hamiltonian isotopic to \(T(a^2,b^2,c^2)\).
In this talk we will describe how to get almost toric fibrations for all
del Pezzo surfaces, in particular for \({{\mathbb C} P}^2{\#{k{\overline{{{\mathbb C}P}^2}}}}\)
for \(4\le k\le 8\), where there is no toric fibrations (with monotone symplectic
form). From there, we will be able to construct infinitely many monotone
Lagrangian tori. Some Markov like equations appear. They are the same as the
ones appearing in the work of Haking-Porokhorov regarding degeneration
of surfaces to weighted projective spaces.
|
| January 18 |
N.N. |
T.B.A. |
| January 25 |
Penka Georgieva (Jussieu) |
Real Gromov-Witten theory in all genera |
|
We construct positive-genus analogues of Welschingers invariants for many real
symplectic manifolds, including the odd-dimensional projective spaces and the quintic threefold.
Our approach to the orientability problem is based entirely on the topology of real bundle pairs
over symmetric surfaces. This allows us to endow the uncompactified moduli spaces of real maps
from symmetric surfaces of all topological types with natural orientations and to verify that
they extend across the codimension-one boundaries of these spaces. In reasonably regular cases,
these invariants can be used to obtain lower bounds for counts of real curves of arbitrary genus.
Joint work with A. Zinger.
|
| February 1 |
N.N. |
T.B.A. |
