Coloumb Branches reading seminar

Feb 2023

talk notes seminar

Summary:

I organized a semester-long reading seminar on Coloumb branches. I gave talks here about superalgebras, supersymmetry in quantum field theory, hypertoric geometry, and abelian coloumb branches.

Presented at:

• UC Berkeley, spring 2023

Coloumb branch reading seminar

Meeting time and location: Wednsday 2:00-4:00 pm, Evans 730
Past talks: Playlist

Schedule

Description

We’ll be learning about coloumb branches, a structure arising from a supersymmetric quantum field theory. Physically, this represents a sector of the moduli space of vacua, forming a (possibly singular) manifold. Supersymmetry endows this with all sorts of structure, like a hyperkahler metric. Historically, we understand this only thru specific examples, but those examples are rich and varied. They include:

• Elliptic fibrations (Seiberg-Witten theory)
• quiver varieties of various forms
• ALE hyperkahler spaces a.la kronheimer
• gauge theoretic moduli spaces of instatons (The Atiyah-Hitchen moduli space of \(SU(2)\) instatons, moduli spaces of ASD yang-mills connections, etc)
• springer resolutions and other critters from geometric representation theory

These examples suggest that coloumb branches are more than mere hyperkaheler manifolds. They usually connect to gauge theoretic moduli spaces. They often connect to geometric representation theory. They have subtle dualities (Mirror symmetry) which shed new light on gauge theory and representation theory. All this begs for a general, unifying desecription coloumb branches and their geometry. In this seminar, we will try to understand the couloumb branches of \(3D \; \mathcal{N}=4\) supersymmetric quantum field theories.

Resources

Coloumb branches are a active area of research which is not yet mature, meaning we don’t have any textbooks to read. Instead we’ll follow some papers in the mathematical physics literature

History:

• Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
  • The premiere work by seiberg and witten introduced Coloumb branches and coloumb branch geometry, as a tool to find the low enegry effective field theory for \(\mathcal{N}=2\) yang mills theory. This paper also lead to seiberg-witten theory, a gauge theoretic topological set of invariants dual to donaldson theory. Since this work is in 4D and \(\mathcal{N}=2\), the coloumb branch is not hyperkahler. Still, it anticipates many core ideas and developments in the field.

Physics:

• Mirror Symmetry in Three Dimensional Gauge Theories
  • Paper from the 90s, shortly after the Seiberg-Witten paper. Discusses coloumb branches and mirror symmetry in the context of ALE hyperkahler spaces, like those studied by Kronheimer.
• The Coulomb Branch of \(3D \; \mathcal{N}=4\) Theories
  • Provides a general construction of Coloumb branches for \(3D \; \mathcal{N}=4\) theories, from a physics perspective.
• Coulomb branches with complex singularities
• SUSY localization for Coulomb branch operators in omega-deformed \(3D \; \mathcal{N}=4\) gauge theories

Mathematics:

• Towards a mathematical definition of Coulomb branches of \(3D \; \mathcal{N}=4\) gauge theories I and II
  • This introduces the Braverman-Finkelberg-Nakajima (BFN) construction, a mathematically precise definition of coloumb branches. This is the spectrum of the ring of some equivariant homology (or K-theory) of a based loop group. Understanding what this possibly means, or how it relates to the physics above, is one of my goals this semster.
• The role of Coulomb branches in 2D gauge theory
  • an alternative description/construction of the spaces constructed by BFN in the above paper

Talk details

2/1

Speaker: Elliot Kienzle
Topic: Planning meeting
Notes: Day one notes
Summary: Discussed the motivation behind coloumb branches, and why we care about studying them. Discussed the future topics of the reading seminar.

2/8

Speaker: Eric Jankowski, Elliot Kienzle
Topic: Superalgebras
Notes: Lie superalgebras, Physicist’s supersymmetry algebras
Video: link
Summary: Eric introduced supersymmetric lie algebras discussing their basic properties and classification. This follwed A sketch of Lie superalgebra theory by Kac, and MATHEMATICAL FOUNDATIONS OF SUPERSYMMETRY by Caston and Fioresi. Elliot described what physicsts call a supersymmetry algebra, described in the first part of chapter 5 of Hyperkahler metrics and supersymmetry by HKLR.

2/15

Speaker: Elliot Kienzle
Topic: Spin, spinor, spinest
Notes: First ~1.5 pages
Video: link
Summary: Quick scamper through clifford algebras, spins, and spinor representations. The material can be found in a great many places, for example Spin Geometry, Lawson and Michelson. I focus on \(3D\) spinors with signature \((2,1)\), following the development in chapter 5 Hyperkahler metrics and supersymmetry. I conclude with the notion that a spinor is a square root of a vector (a general philosophy which manifests particularly nicely in 3 dimensions), and a disucssion of Bott Periodicity:

2/22

Speaker: Elliot Kienzle
Topic: Square roots in supersymmetry
Notes: pages 1.5 to end
Video: link
Summary: We’ll talk about supersymmetry in the context of quantum field theory. Philosophically, we describe various ways in which supersymmetry gives square roots of familiar objects: A spinor is the square root of a vector, a supersymmetry transform is the square root of translation, and multiple superpersymmetries give square roots of negative 1. We will roughly follow chapter 5 of Hyperkahler metrics and supersymmetry, talking about superspace, superfields, the meaning of \(\mathcal{N}=1,2,4\) supersymmetry. This is supplemented by parts of Five Lectures on Supersymmetry, by Dan Freed.

3/1

Speaker: Che Shen
Topic: Supersymmetric field theory via superspace
Notes: first 4.5 pages
Video: link
Summary: We will introduce sigma model and leverage superspace formulation to construct supersymmetric field theories with 2 or 4 supercharges, following Freed, Five Lectures on Supersymmetry. We will discuss the relationship between Kahler manifolds and SUSY field theories.

3/8

Speaker: Che Shen
Topic: Supersymmetric field theory via superspace
Notes: pages 4-end
Summary: We will talk about the relationship between supersymmetry and Kahler/hyperkahler manifolds. This goes in two ways. On the one hand, the target space of a supersymmetric sigma model with 4 or 8 supercharges must be Kahler or hyperkahler. On the other hand, for some supersymmetric gauge theories, the moduli of vacua is given by Kahler/hyperkahler quotient.

References: The main reference will be Freed, Five Lectures on Supersymmetry. If people want to read ahead, the last paragraph on P71 has a general discussion about gauge theory. The section The superspacetime \(M^{4,4}\) will also be useful.

3/21

Speaker: Swapnil Garg
Topic: Seiberg-Witten theory
Summary: We will introduce our first example of a Coulomb branch, which comes from Seiberg Witten theory. We study the low energy effective field theory limit of \(4D\), \(\mathcal{N}=2\) supersymmetric Yang mills theory. This produces a moduli space with a special Kahler structure. In particular, for gauge group \(SU(2)\), the moduli space has a fibration by Elliptic curves.

For refrence, see the origional paper by Seiberg and Witten. More pedagogically, check out these Notes by Martone.

4/5-4/12

Speaker: Jacob Erlikhman
Topic: Rozansky-Witten theory
Notes: Notes
Summary: we introduced RW theories as twisted 3d N=4 SYM sigma models and going over some basic results, we’ll investigate its reduction/compactification on the circle. This time, We will find in the process that although the theory is usually defined only on compact hyperkahler targets, we can actually extend that definition to include certain non-compact ones, like the Coulomb branch of Seiberg-Witten theory. We’ll obtain some interesting results for some simple spacetime 3-manifolds using this approach.

Refrences:

1. Rozansky, Witten “hyperkahler geometry and invariants of 3 manifolds, 1996
2. Seiberg, Witten “Gauge dynamics and compactification to three dimensions” 1996
3. Gukov et al “Rozansky-Witten geometry of Coulomb branches and logarithmic knot invariants” 2018
4. Kapustin, rozansky, saulina “rozansky Witten theory and symplectic algebraic geometry I” 2008

4/19

Speaker: Elliot Kienzle
Topic: Toric geometry
Notes: Notes
Summary: I describe Toric geometry, including a dettailed description of moment maps and the symplectic reduction. I state the DelZant correspondence, and describe the combinatorial data underlying every toric manifold. This is a prelude to Hypertoric geometry, which is necessary to understand abelian Coulomb branches.

Refrences: See De Silva’s notes on Toric Geometry

4/26

Speaker: Elliot Kienzle
Topic: Abelian Coloumb branch
Notes: Notes
Summary: I talk about the Higgs branches and Coloumb branches of \(3D, \, \mathcal{N}=4\) SUSY gauge theories with abelian gauge group \(G\) and matter \(M\). In particular, I do out the examples of \((G,M) = (U(1),\mathbb{C}^{n+1})\) and \((U(1)^n,\mathbb{C}^{n+1})\). We compute these spaces using hypertoric geometry, and a taste of quiver varieties. We see that the Higgs branch of the first equals the Coloumb branch of the second, and vice versa. This is because the theories are mirror dual. This follows section 3 of Bullimore, Dimofte, and Gaiotto. Finally, we discuss general how general coulomb branches can be described in terms of abelian gauge theories. This includes a very recent paper, which shows that all coloumb branches are hilbert schemes of hypertoric varieties, by Roger Bielawski, Lorenzo Foscolo. This is the first construction of hyperkahler metrics on general Coloumb branches.