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Supersymmetric ground states of 3d $\mathcal{N}=4$ SUSY gauge theories and Heisenberg Algebras
by Andrea E. V. Ferrari
This Submission thread is now published as
Submission summary
| Authors (as registered SciPost users): | Andrea Ferrari |
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| Preprint Link: | https://arxiv.org/abs/2205.06216v2 (pdf) |
| Date accepted: | 2022-12-23 |
| Date submitted: | 2022-11-16 18:28 |
| Submitted by: | Ferrari, Andrea |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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Abstract
We consider 3d $\mathcal{N} = 4$ theories on the geometry $\Sigma\times\mathbb{R}$, where $\Sigma$ is a closed and connected Riemann surface, from the point of view of a quantum mechanics on $\mathbb{R}$. Focussing on the elementary mirror pair in the presence of real deformation parameters, namely SQED with one hypermultiplet (SQED[1]) and the free hypermulitplet, we study the algebras of local operators in the respective quantum mechanics as well as their action on the vector space of supersymmetric ground states. We demonstrate that the algebras can be described in terms of Heisenberg algebras, and that they act in a way reminiscent of Segal-Bargmann (B-twist of the free hypermultiplet) and Nakajima (A-twist of SQED[1]) operators.
Author comments upon resubmission
List of changes
All the points I do not mention below have been addressed without requiring further explanation (please note that I removed an unnecessary equation after 4.13 and so after then the numbering is off by one with respect to the referee's comments).
Report 2:
3. I introduce the mirror of the twisted fermions in section 4.2. I added a comment in section 4.1 mentioning this, and then emphasised this fact below equation 4.27 (see also 15 below).
Report 1:
3. This comment made me realise that I was using a non-standard convention for A- and B- twist indices. I consistently changed dotted and undotted indices in 2.1 and 2.2.
5. I give concrete examples of secondary products in later sections (3.2.2 and 4.1).
7. The extra sign that caused the confusion was spurious, I thank the referee for pointing this out.
11. I actually found a similar computation in one of the references I was using, and therefore I cited it. Normalisations can be introduced at various stages (normalisation of the fields, of the brackets, etc.). I work with one that is fixed by mirror symmetry and that ensures that \varphi is the complex moment map for the topological symmetry. This ensures in particular that the monoopole operators have integer charges under this symmetry.
12. I agree that it is unhelpful to introduce the notion of perverse sheaf, which would take too long to explain in this context. It is better to talk about geometric quantisation, and I do so instead. The geometric quantisation used here reduces to the computation of standard de Rham cohomology of the target. I refer to my previous paper (cited) for further details.
15. I introduced a schematic expression in 4.38.
Finally, I have corrected a few additional typos, especially in 2.32, 2.34 and the appendices. A wrong equation was referenced after 2.45 and I corrected the discussion accordingly.
Published as SciPost Phys. 14, 063 (2023)