Mathematical physics

Mathematical physics, including mathematics, is a research area where novel mathematical techniques are invented to tackle problems in physics, and where novel mathematical ideas find an elegant physical realization. Historically, it would have been impossible to distinguish between theoretical physics and pure mathematics. Often spectacular advances were seen with the concurrent development of new ideas and fields in both mathematics and physics. Here one might note Newton's invention of modern calculus to advance the understanding of mechanics and gravitation. In the twentieth century, quantum theory was developed almost simultaneously with a variety of mathematical fields, including linear algebra, the spectral theory of operators and functional analysis. This fruitful partnership continues today with, for example, the discovery of remarkable connections between gauge theories and string theories from physics and geometry and topology in mathematics.

Displaying 445 - 456 of 887

Twisted 3d supersymmetric gauge theories and the enumerative geometry of quasi-maps

Andrea Ferrari
March 18, 2019

PIRSA:19030107
Mathematical physics
Topological Holography Course - Lecture 5

Kevin Costello Perimeter Institute for Theoretical Physics
March 15, 2019

PIRSA:19030091
Mathematical physics
On one example of a chiral Lie group

Fyodor Malikov University of Southern California
March 01, 2019

PIRSA:19030096
Mathematical physics
Fusion Hall algebra and shuffle conjectures

Anton Mellit University of Vienna
March 01, 2019

PIRSA:19030095
Mathematical physics
TBA

Alexander Braverman University of Toronto
March 01, 2019

PIRSA:19030094
Mathematical physics
Connecting affine Yangians with W-algebras

Andrei Negut Massachusetts Institute of Technology (MIT) - Center for Extreme Quantum Information Theory (xQIT)
February 28, 2019

PIRSA:19020066
Mathematical physics
COHA and AGT for Spiked Instantons

Miroslav Rapcak Qminers
February 28, 2019

PIRSA:19020067
Mathematical physics
Glueing W-algebras

Thomas Creutzig University of Alberta
February 28, 2019

PIRSA:19020068
Mathematical physics
Partition functions and the McKay correspondence

Balazs Szendroi University of Oxford
February 28, 2019

PIRSA:19020065
Mathematical physics
An old-fashioned view of BPS-algebras

Natalie Paquette University of Washington
February 27, 2019

PIRSA:19020064
Mathematical physics
Perverse sheaves, perverse schobers and physical "theories"

Mikhail Kapranov University of Tokyo
February 27, 2019

PIRSA:19020063
Mathematical physics
Comultiplications on cohomological Hall algebras and vertex algebras

Gufang Zhao University of Melbourne
February 27, 2019

PIRSA:19020062
Mathematical physics

Twisted 3d supersymmetric gauge theories and the enumerative geometry of quasi-maps

Andrea Ferrari
March 18, 2019

PIRSA:19030107
Mathematical physics
I discuss a geometric interpretation of the twisted indexes of 3d (softly broken) $\cN=4$ gauge theories on $S^1 \times \Sigma$ where $\Sigma$ is a closed genus $g$ Riemann surface, mainly focussing on quivers with unitary gauge groups. The path integral localises to a moduli space of solutions to generalised vortex equations on $\Sigma$, which can be understood algebraically as quasi-maps to the Higgs branch. I demonstrate that the twisted indexes computed in previous work reproduce the virtual Euler characteristic of the moduli spaces of twisted quasi-maps. I investigate 3d $\cN=4$ mirror symmetry in this context, which implies an equality of enumerative invariants associated to mirror pairs of Higgs branches under the exchange of equivariant and degree counting parameters. I will conclude with some remarks about how holomorphic Morse theory can be used to access the spaces of supersymmetric ground states in limits where $\cN=4$ supersymmetry is fully restored. These spaces of ground states may be related to the spaces of conformal blocks for the VOAs introduced by Costello and Gaiotto.
Topological Holography Course - Lecture 5

Kevin Costello Perimeter Institute for Theoretical Physics
March 15, 2019

PIRSA:19030091
Mathematical physics
On one example of a chiral Lie group

Fyodor Malikov University of Southern California
March 01, 2019

PIRSA:19030096
Mathematical physics
We quantize the Khesin-Zakharevich Poisson-Lie group of pseudo-differential symbols. This is a joint work with A.Linshaw
Fusion Hall algebra and shuffle conjectures

Anton Mellit University of Vienna
March 01, 2019

PIRSA:19030095
Mathematical physics
The classical Hall algebra of the category of representations of one-loop quiver is isomorphic to the ring of symmetric functions, and Hall-Littlewood polynomials arise naturally as the images of objects. I will talk about a second "fusion" product on this algebra, whose structure constants are given by counting of bundles with nilpotent endomorphisms on P^1 with restrictions at 0, 1 and infinity. The two products together make up a structure closely related to the elliptic Hall algebra. In the situations when bundles can be explicitly enumerated, I will explain how this leads to q,t-identities conjectured by combinatorists, such as the shuffle conjecture and its generalizations. This is a joint project with Erik Carlsson.
TBA

Alexander Braverman University of Toronto
March 01, 2019

PIRSA:19030094
Mathematical physics
Connecting affine Yangians with W-algebras

Andrei Negut Massachusetts Institute of Technology (MIT) - Center for Extreme Quantum Information Theory (xQIT)
February 28, 2019

PIRSA:19020066
Mathematical physics
Alday-Gaiotto-Tachikawa connect instanton counts in gauge theory with conformal blocks for W-algebras. We realize this mathematically by relating q-deformed W-algebras with the affine q-Yangians that control gauge theory, thus offering an affine, q-deformed generalization of the well-known Brundan-Kleshchev construction in type A
COHA and AGT for Spiked Instantons

Miroslav Rapcak Qminers
February 28, 2019

PIRSA:19020067
Mathematical physics
The well-known AGT correspondence relates $\mathcal{W}_N$-algebras and supersymmetric gauge theories on $\mathbb{C}^2$. Embedding $\mathbb{C}^2$ as a coordinate plane inside $\mathbb{C}^3$, one can associate the COHA to $\mathbb{C}^3$ and derive the corresponding $\mathcal{W}_N$ as a truncation of its Drinfeld double. Building up on Zhao's talk, I will discuss a generalization of this story, where $\mathbb{C}^2$ is replaced by a more general divisor inside $\mathbb{C}^3$ with three smooth components supported on the three coordinate planes. Truncations of the Drinfeld double lead to a three-parameter family of algebras $\mathcal{W}_{L,M,N}$ determining the vertex algebras associated to Nekrasov's spiked instantons. Many interesting questions emerge when considering a general Calabi-Yau three-fold instead of $\mathbb{C}^3$. I will discuss a class of vertex algebras conjecturally arising from divisors inside more general toric Calabi-Yau three-folds.
Glueing W-algebras

Thomas Creutzig University of Alberta
February 28, 2019

PIRSA:19020068
Mathematical physics
S-duality predicts rather surprising isomorphisms of extensions of W-algebras. The aim of this talk is to present some explanation. Firstly, I will explain the concept of glueing W-algebras along certain categories of modules and then I will introduce what we call a W-algebra translation functor.
Partition functions and the McKay correspondence

Balazs Szendroi University of Oxford
February 28, 2019

PIRSA:19020065
Mathematical physics
I will explain some results on certain sheaf-theoretic partition functions defined on Calabi-Yau orbifolds, and their connection to the McKay correspondence, the representation theory of affine Lie algebras, and cohomological Hall algebras. Based on joint work with Gyenge and Nemethi, respectively Davison and Ongaro
An old-fashioned view of BPS-algebras

Natalie Paquette University of Washington
February 27, 2019

PIRSA:19020064
Mathematical physics
The notion of the algebra of BPS states goes back to work of Harvey and Moore in the late 90's. Explicit computations in perturbative heterotic string theory point to an algebraic structure isomorphic to a Generalized Kac-Moody (GKM) algebra in that context; at the same time, rather mysteriously, denominators of GKMs furnish signed counts of BPS states in certain supersymmetric string vacua. The most famous example of this is the reciprocal of the Igusa cusp form of weight 10, which governs some BPS state counts related to string theory on the product of a nonsingular K3 surface and an elliptic curve E. Other denominator formulas, more generally, produce interesting examples of Siegel automorphic forms, and have enumerative geometric interpretations. I will review recent progress understanding the role of GKMs in some special string vacua related to Monstrous and Conway moonshine modules (based on work with Persson & Volpato, and Harrison & Volpato), as well as a new class of putative GKM denominators obtained from a large class of quotients of K3xE (based on work with Volpato & Zimet).
Perverse sheaves, perverse schobers and physical "theories"

Mikhail Kapranov University of Tokyo
February 27, 2019

PIRSA:19020063
Mathematical physics
The mathematical concept of sheaves is a tool for > describing global structures via local data. Its generalization, the > concept of perverse sheaves, which appeared originally in the study of > linear PDE, turned out to be remarkably useful in many diverse areas > of mathematics. I will review these concepts as well as a more recent conjectural categorical generalization, called perverse schobers. > One reason for the interest in such structures is the remarkable > parallelism between: > > (1) The purely mathematical classification theory of perverse sheaves > on a complex plane with several singular points > (Gelfand-MacPherson-Vilonen). > > (2) The "infrared" analysis of > 2d supersymmetric theories (Gaiotto-Moore-Witten). > > I will explain this parallelism which suggests that the infrared > analysis should be formulated in terms of a perverse schober. This is > based on work in progress with Y. Soibelman and L. Soukhanov.
Comultiplications on cohomological Hall algebras and vertex algebras

Gufang Zhao University of Melbourne
February 27, 2019

PIRSA:19020062
Mathematical physics
In this talk we will explore a factorization structure of the cohmological Hall algebra (COHA) of a quiver, and the occurrence of the same structure from Beilinson-Drinfeld Grassmannians. In particular, in collaboration with Mirkovic and Yang, we identified a Drinfeld-type comultiplication on the COHA with the factorizable line bundle on the zastava space. I will discuss one aspect of a recent joint work with Rapcak, Soibelman, and Yang, which can be reviewed as a construction of a vertex algebra from the standard comultiplication on the double COHA of the Jordan quiver. The standard comultiplication differs from the Drinfeld comultiplication by an S-operator. The relevant vertex algebra is the VOA at the corner of Rapcak-Gaiotto.

Title	Speaker(s)	Date	Talk Type	Info link
Twisted 3d supersymmetric gauge theories and the enumerative geometry of quasi-maps	Andrea Ferrari	2019-03-18	Scientific Series	View details
Topological Holography Course - Lecture 5	Kevin Costello	2019-03-15	Course	View details
On one example of a chiral Lie group	Fyodor Malikov	2019-03-01	Conference	View details
Fusion Hall algebra and shuffle conjectures	Anton Mellit	2019-03-01	Conference	View details
TBA	Alexander Braverman	2019-03-01	Conference	View details
Connecting affine Yangians with W-algebras	Andrei Negut	2019-02-28	Conference	View details
COHA and AGT for Spiked Instantons	Miroslav Rapcak	2019-02-28	Conference	View details
Glueing W-algebras	Thomas Creutzig	2019-02-28	Conference	View details
Partition functions and the McKay correspondence	Balazs Szendroi	2019-02-28	Conference	View details
An old-fashioned view of BPS-algebras	Natalie Paquette	2019-02-27	Conference	View details
Perverse sheaves, perverse schobers and physical "theories"	Mikhail Kapranov	2019-02-27	Conference	View details
Comultiplications on cohomological Hall algebras and vertex algebras	Gufang Zhao	2019-02-27	Conference	View details

Mathematical physics

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