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Disordered $\mathcal{N} = (2, 2)$ Supersymmetric Field Theories
by ChiMing Chang, Xiaoyang Shen
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Authors (as registered SciPost users):  ChiMing Chang · Xiaoyang Shen 
Submission information  

Preprint Link:  https://arxiv.org/abs/2307.08742v1 (pdf) 
Date submitted:  20231014 05:13 
Submitted by:  Shen, Xiaoyang 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We investigate a large class of $\mathcal{N} = (2, 2)$ supersymmetric field theories in two dimensions, which contains the MuruganStanfordWitten model, and can be naturally regarded as a disordered generalization of the twodimensional LandauGinzburg models. We analyze the two and fourpoint functions of chiral superfields, and extract from them the central charge, the operator spectrum, and the chaos exponent in these models. Some of the models exhibit a conformal manifold parameterized by the variances of the random couplings. We compute the Zamolodchikov metrics on the conformal manifold, and demonstrate that the chaos exponent varies nontrivally along the conformal manifolds. Finally, we introduce and perform some preliminary analysis of a disordered generalization of the gauged linear sigma models, and discuss the low energy theories as ensemble averages of CalabiYau sigma models over complex structure moduli space.
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Reports on this Submission
Anonymous Report 2 on 2024312 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2307.08742v1, delivered 20240312, doi: 10.21468/SciPost.Report.8694
Report
This paper discusses ensembleaveraging 2d SUSY CFTs, generalizing the MuruganStanfordWitten model. The standard methods used in ensemble averaged theories allows the authors to compute 2point functions, 4point functions and to extract the chaos exponent while SUSY also allows for some exact computations using localization.
The paper slightly lacks motivation. The original motivation for studying ensembleaveraged theories in this specific context was the large chaos exponent of the SYK model which hinted at a holographic dual, but the theories at hand never approach this regime. The appearance of a conformal manifold in the ensembleaveraged theory is also not new, with a nonSUSY example appearing already in [Gross,Rosenhaus].
The computations themselves are clear, exhaustive and precise. The comparison to exact SUSY results using localization is very convincing.
Requested changes
1. small typos:
(a) in eq 2.55, in the second equation it should be G_{\Phi^{(1)}} > G_{\Phi^{(2)}}
(b) on page 32, \Delta_1 and \Delta_1 > \Delta_1 and \Delta_2
2. It is not completely clear why solutions for the 2point function become nonunitary for some values of \lambda. Taking section 2.4.1 as a specific example, there doesn't seem to be anything wrong with the model 2.76 for \lambda<1/2 before the ensemble average. Is the claim that there is no CFT for \lambda<1/2, or that it is a CFT but the ansatz 2.80 is wrong? For example for \lambda=0 the ansatz is probably wrong since the Rcharge assignments would be found using just the first term in 2.76 and by cmaximization, and so the ansatz fails.
It is also not clear why the appearance of a moduli space at \lambda=1/2 leads to this nonunitarity. Is the claim that the moduli space leads to instabilities at \lambda<1/2?
Anonymous Report 1 on 202439 (Invited Report)
 Cite as: Anonymous, Report on arXiv:2307.08742v1, delivered 20240309, doi: 10.21468/SciPost.Report.8686
Strengths
1. The paper is clear and well written.
2. The authors study disorderaveraged supersymmetric field theories in two dimensions, giving an interesting generalisation of the analysis of SYK models. The main focus of the paper is on disordered LandauGinzburg models. The authors compute the central charge, the operator spectrum and the chaos exponent of these models. They use both diagrammatic and exact methods, such as localisation, with the results agreeing.
4. The authors point out that these models admit a conformal manifold parametrised by the variances of the random couplings which lead to the disorder average, and they show that the above quantities depend nontrivially on these marginal couplings.
5. The authors give a preliminary study of disordered GLSMs, which can be interpreted at low energies as ensemble averages of CalabiYau sigma models, where one averages over the complex structure moduli space.
Weaknesses
1. The disorder average for the GLSMs uses marginal couplings which are Gaussian random variables. A more natural average might come from taking the distribution implied by the canonical metric on complex structure moduli space.
2. The treatment of GLSMs is somewhat cursory, though the authors indicate that they will return to this question.
Report
This article aims to extend the techniques of disorderaveraging to a large class of (2,2) field theories in two dimensions. A particular focus is on theories that are disordered versions of Landau–Ginzburg models. Using largeN techniques and exact localisation results, the authors study the disorderaverage two and fourpoint functions of chiral fields. These give access to the central charge, the spectrum of operators, and the chaos exponent for these disordered theories.
Since these theories can have multiple chiral fields, the random couplings that appear in their superpotentials are parametrised by multiple variances. Not all of these variances can be shifted away by field redefinitions, leaving a conformal manifold of theories. By varying these variances, the authors are able to move around the conformal manifold and thus study the chaos exponent as a function of these marginal couplings.
The authors then extend their approach to gauged linear sigma models. In the IR, these models flow to Calabi–Yau sigma models, which the random couplings of the GLSM corresponding to random choices of complex structure moduli. Studying the disorderaveraged GLSMs then amounts to studying ensemble averages of Calabi–Yau sigma models.
The paper is an interesting contribution to the literature on disorderaveraged field theories, and the use of localisation to obtain exact results is particularly nice.
Some comments:
The disorder average is performed using a Gaussian distribution. It would be interesting to understand whether there is a real reason for doing this. For example, for Calabi–Yau sigma models, one might imagine that the natural way to average over the complex structure moduli space is to use the Weyl–Peterson measure on the moduli space. (A similar approach has been taken when considering disorderaveraged theories on the torus.) This would amount to requiring that the random couplings J are distributed with respect to the measure implied by the Zamolodchikov metric. This would clearly complicate the analysis, but may eventually lead to simple (and canonical) results.
It would also be interesting to see further work on GLSMs, particularly on understanding whether quantities such as Yukawa couplings have simple, disorderaveraged expressions which might help in understanding general features about both CalabiYau manifolds and, for phenomenology, the resulting particle physics models.
Requested changes
1. It would be sensible to cite 2103.15826 for results on averaging over the complex structure moduli space of the torus, and its discussion in Section 8 on averaging CalabiYau CFTs.
2. On page 32, below Equation (3.12), the authors write “\Delta_{1} and \Delta_{1}”, when I believe they mean “\Delta_{1} and \Delta_{2}”.
Author: Xiaoyang Shen on 20240422 [id 4439]
(in reply to Report 1 on 20240309)
We thank the referee for the comments and have resubmitted the paper with the following adjustments:
1. It is more natural to calculate the Zamolodchikov metric and then do the subsequent averaging according to the measure. We included a discussion on averaging using the Weil–Petersson metric and cited 2103.15826 on page 33.
2. In the updated version, we have fixed the typos in eq 2.55 and on page 32.
Author: Xiaoyang Shen on 20240422 [id 4438]
(in reply to Report 2 on 20240312)We thank the referee for the comments and have resubmitted the paper with the following adjustments:
1. We thank the referee for pointing out the typos and have fixed the corresponding typos.
2. (a) The unitary bound of lambda arises due to the constraint that the coefficients of the twopoint function should be positive. Another clue for the nonunitarity is that part of the spectrum read off from the fourpoint function becomes complex when lambda is smaller than 1/2. As the referee pointed out, the CFT before averaging should be unitary, and it remains a puzzle to us how disorder averaging generates nonunitarity. Another instance of this phenomenon is in the disordered vector model studied in 2112.09157. As discussed above figure 3, the spectrum becomes complex in some range of the parameter, while the theory before averaging does not exhibit any nonunitarity.
(b) When the lambda = 1/2, the theory becomes noncompact and has an emergent U(1) symmetry. Correspondingly we find a conserved spin1 current in the spectrum, which indicates the flat direction in the supermultiplet.