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Dualities and loops on squashed $S^3$

by Charles Thull

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Authors (as registered SciPost users): Charles Thull
Submission information
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Date submitted: 2023-01-24 09:32
Submitted by: Thull, Charles
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory


We consider $\mathcal{N}=4$ supersymmetric gauge theories on the squashed three-sphere with six preserved supercharges. We first discuss how Wilson and vortex loops preserve up to four of the supercharges and we find squashing independence for the expectation values of these $\frac{2}{3}$-BPS loops. We then show how the additional supersymmetries facilitate the analytic matching of partition functions and loop operator expectation values to those in the mirror dual theory, allowing one to lift all the results that were previously established on the round sphere to the squashed sphere. Additionally, on the squashed sphere with four preserved supercharges, we numerically evaluate the partition functions of ABJM and its dual super-Yang-Mills at low ranks of the gauge group. We find matching values of their partition functions, prompting us to conjecture the general equality on the squashed sphere. From the numerics we also observe the squashing dependence of the Lee-Yang zeros and of the non-perturbative corrections to the all order large $N$ expression for the ABJM partition function.

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Reports on this Submission

Anonymous Report 2 on 2023-3-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2212.06813v2, delivered 2023-03-15, doi: 10.21468/SciPost.Report.6910


This paper studies 3d theories on the squashed sphere with extended supersymmetry (more than 4 supercharges). These theories/backgrounds were constructed previously and this paper focuses on finding BPS line operators and discussing the partition functions and expectation values of the operators.

Overall this is an original and rigorous paper that should be published. I have two main concerns that I'd like the author to address. One is a matter of terminology and the other is related to the numerical calculations.

The author refers to the theories as having ${\cal N}=4$ supersymmetry. This language attempts to separate the theory from the background (geometry, masses, fluxes). Such a split is unnatural in this context and indeed the relations shown for these theories are associated to changing the squashing and matching theories with different parameters. I would recommend to find a different language that recognizes that the field content is the same as ${\cal N}=4$ theories in flat space, but there is no $SO(4)$ R-symmetry (or eight preserved supercharges).

Also in the discussion of mirror pairs, the discussion should be more relations between partition functions and observables rather than theories. This is evident in the formulas, but not in the preceding discussions.

I am not very familiar with numerical calculations of such integrals and the paper does not explain it in much detail or give appropriate references. There are no error estimates and the differences of $10^{-5}$ or $10^{-3}$ are said to be good. A match is good if the difference is of order of the expected numerical error, but no error estimates of the numerics are provided and I think that fixing this will significantly strengthen this part of the paper

Requested changes

1. A clearer language to refer to theories + backgrounds with enhanced supersymmetry.
2. Error estimates of the numerical calculations.

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

Anonymous Report 1 on 2023-3-15 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2212.06813v2, delivered 2023-03-15, doi: 10.21468/SciPost.Report.6906


The author considers $\mathcal{N}=4$ supersymmetric quantum field theories on a squashed three-sphere. As in one of the previous works by the author and collaborators, a specific choice of the mass parameters is made such that 6 out of 8 supercharges are preserved. In such a setup, the author considers line operators that preserve 2/3 of the six unbroken supercharges. The author then argues that the expectation values of those operators can be expressed via the ones on a round (i.e. non-squashed) three-sphere. The author also considers numerical evaluations of the partition functions of two specific theories: ABJM and SYM, with various values of the parameters (such that four supercharges are preserved) and finds very strong evidence of their matching, which is in agreement with the conjectured duality between these two theories.

I believe that the obtained results and the developed techniques are of interest to other researchers working on related topics, such as supersymmetric localization and dualities. The paper is in general well-written and the presentation is clear. I recommend the manuscript for publication.

Requested changes

I have just very minor suggestions:

1) I think it could be helpful to clarify the notations $A$ and $\sigma$ appearing in (3.1). In particular, it is somewhat clear that A there is not the same as $A$ in section 2 (the gauge field in the supergravity multiplet), but it might be helpful to clarify it after (3.1) anyway.

2) In the sentence above (2.15) there is a typo: missing "h" in "the".

3) It could be helpful to add clarification of the meaning of the + operation in $\vec{\eta}/b+iq/b$ in the middle line of (4.4), as $q$, unlike $\vec{\eta}$, does not seem to be a vector.

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

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