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3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve

by Mathew Bullimore, Daniel Zhang

This Submission thread is now published as SciPost Phys. 13, 005 (2022)

Submission summary

As Contributors: Daniel Zhang
Arxiv Link: (pdf)
Date accepted: 2022-06-01
Date submitted: 2021-10-04 18:01
Submitted by: Zhang, Daniel
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


This paper studies $3d$ $\mathcal{N}=4$ supersymmetric gauge theories on an elliptic curve, with the aim to provide a physical realisation of recent constructions in equivariant elliptic cohomology of symplectic resolutions. We first study the Berry connection for supersymmetric ground states in the presence of mass parameters and flat connections for flavour symmetries, which results in a natural construction of the equivariant elliptic cohomology variety of the Higgs branch. We then investigate supersymmetric boundary conditions and show from an analysis of boundary 't Hooft anomalies that their boundary amplitudes represent equivariant elliptic cohomology classes. We analyse two distinguished classes of boundary conditions known as exceptional Dirichlet and enriched Neumann, which are exchanged under mirror symmetry. We show that the boundary amplitudes of the latter reproduce elliptic stable envelopes introduced by Aganagic-Okounkov, and relate boundary amplitudes of the mirror symmetry interface to the mother function in equivariant elliptic cohomology. Finally, we consider correlation functions of Janus interfaces for varying mass parameters, recovering the chamber R-matrices of elliptic integrable systems.

Published as SciPost Phys. 13, 005 (2022)

Submission & Refereeing History

Published as SciPost Phys. 13, 005 (2022)

Reports on this Submission

Anonymous Report 2 on 2022-5-4 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2109.10907v1, delivered 2022-05-04, doi: 10.21468/SciPost.Report.5024


I highly recommend the paper:
3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
by Mathew Bullimore, Daniel Zhang.
for publication in your Journal.

The authors flesh out the physics underlying a construction of elliptic stable envelopes by Aganagic and Okounkov. Elliptic stable envelopes provide a basis in elliptic cohomology of Higgs and Coulomb branches of 3d N=4 theories. They play an important role in understanding the action of quantum symmetries of these gauge theories, and in establishing precise mathematical predictions of 3d mirror symmetry.

While it may not contain directly new results, and is moreover largely limited to the simplest case of abelian gauge theories, it does describe the relevant physics in clear and insightful ways which have not appeared elsewhere (not counting the paper by Nekrasov and Dedushenko that appeared simultaneously). It will become a standard reference in this important area at the intersection of mathematics and physics.

I do not see need for any modifications, the paper can be published as is.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2021-12-20 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2109.10907v1, delivered 2021-12-19, doi: 10.21468/SciPost.Report.4068


1-Interesting new results on 3d N=4 (and 3d N=2) supersymmetric field theories on an elliptic curve, their boundary conditions, and their mathematical interpretation as elliptic cohomology classes.
2-Clearly written.
3-Very detailed.


1-There is no introduction to the mathematical literature on elliptic cohomology for a physicists audience. For the non-expect in that part of maths (like myself), it is simply stated that elliptic cohomology classes is defined in terms of sections of lines bundles on a certain associated variety , but further "intuition" might have been appreciated. On the other hand, it is understandable, given the subject, that one cannot cover all the "basics".


The authors study in much details the physics of 3d N=4 supersymmetric (softly broken to N=2 susy) NLSM, or rather their gauge theory UV completions. The paper include a review of all the relevant field theory results, including a number of things that clarify previous discussions.

The main motivation of the paper is to provide a SUSY QFT realisation of the mathematics of elliptic cohomology of some variety X (with a number of assumptions that simplify matter consideably). They do this by engineering X as a Higgs branch of a 3d N=4 theory, and by by showing how the space of vacua of the theory compactified on an elliptic curve can be identified with the elliptic cohomology variety of X. They further show that boundary conditions in this setup naturally engineer elliptic cohomology classes (sections of line bundles on the elliptic cohomology variety), and in particular that Neumann boundary condition can be identified with the so-called stable enveloppes.

The paper is clearly written and I would strongly recommend it for publication as it is.

Requested changes

Here are just a small list of likely typos:
1-In eq.(2.1) and similarly on page 8 and elsewhere, it seems the symbol "Z" is used to denote the maximal torus of a group (or the Cartan subalgebra of its Lie algebra)? This is confusing since this is not the center of G.
2-Two lines above (2.22), "m" should be "x".
3-p 25 and 26: "principle bundle" should be "principal bundle".
4-middle of p26: "can intersection"-->"can intersect".

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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