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Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model
by Belal Nazzal, Anton Nedelin, Shlomo S. Razamat
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Submission summary
| Authors (as registered SciPost users): | Anton Nedelin · Shlomo Razamat |
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| Preprint Link: | https://arxiv.org/abs/2106.08335v3 (pdf) |
| Date accepted: | 2022-01-06 |
| Date submitted: | 2021-11-05 10:36 |
| Submitted by: | Nedelin, Anton |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
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| Approach: | Theoretical |
Abstract
We consider supersymmetric surface defects in compactifications of the $6d$ minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a punctured Riemann surface. For the case of $N=1$ such defects are introduced into the supersymmetric index computations by an action of the $BC_1\,(\sim A_1\sim C_1)$ van Diejen model. We (re)derive this fact using three different field theoretic descriptions of the four dimensional models. The three field theoretic descriptions are naturally associated with algebras $A_{N=1}$, $C_{N=1}$, and $(A_1)^{N=1}$. The indices of these $4d$ theories give rise to three different Kernel functions for the $BC_1$ van Diejen model. We then consider the generalizations with $N>1$. The operators introducing defects into the index computations are certain $A_{N}$, $C_N$, and $(A_1)^{N}$ generalizations of the van Diejen model. The three different generalizations are directly related to three different effective gauge theory descriptions one can obtain by compactifying the minimal $(D_{N+3},D_{N+3})$ conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the $A_N$ case, and derive various properties these operators have to satisfy as a consequence of $4d$ dualities following from the geometric setup. In some cases we are able to verify these properties which in turn serve as checks of said dualities. As a by-product of our constructions we also discuss a simple Lagrangian description of a theory corresponding to compactification on a sphere with three maximal punctures of the minimal $(D_5,D_5)$ conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.
Author comments upon resubmission
List of changes
1. Question: In eq. 2.5 s is used as the power of q (power of derivative) and as the
number of punctures in Cg,s , maybe use a different letter.
Answer: We fixed this by changing s → m in the power of q.
2. Question: The discussion leading to eqs. 3.19 and 3.21, is quite hard to follow, perhaps because of the notation and should be improved. For example the $h_i^{(a)}$ and $h_a$ in 3.19 are not clearly defined. It is also not obvious why $h^{(a)}_ih^{-1/4}_a$ (with a referring to either x or y punctures) is the label of the operator, which before was labelled by the charges of the moment map used to close the z puncture.
Answer: Thank you for noticing that $h^{(a)}$ variables where not defined explicitly. We now added explicit explanation in the sentences just before eq. 3.18. Regarding $h^{(a)}_ih^{-1/4}_a$ label of the index. It is indeed the charge of the puncture we close. It is not independent of the charges of the punctures we act on ($h_i^{(a)}$) so we choose to express everything in terms of one single set of charges. To be more clear we added explanation about this relation after eq.3.18.
3. Question: After eq. 3.21 authors say that if $h_i=u^{-6}w^{-12}$, eq. 3.13 is recovered, but there is no $h_i$ in eq. 3.21, perhaps the condition is $h^{(a)}_ih^{-1/4}_a=u^{-6}w^{-12}$?
Answer: To be more precise here the condition should be $a=u$ and choosing $h_i^{(u)}=u^{-6}w^{-12}$. We added this specification after eq. 3.21 (3.22 in the previous version)
4. Question: To be more precise here the condition should be $a=u$ and choosing $h_i^{(u)}=u^{-6}w^{-12}$. We added this specification after eq. 3.21 (3.22 in the previous version)
Answer: To be more precise here the condition should be $a=u$ and choosing $h_i^{(u)}=u^{-6}w^{-12}$. We added this specification after eq. 3.21 (3.22 in the previous version)
5. Question: Eq. 2.7 states the kernel property. Here a generic difference operator can act either on the puncture $u$ or $z$ of $I(z,u)$ which is not defined but from fig. 3 it seems to refer to the index of a generic surface with two max puncture and a closed minimal puncture, perhaps to be more consistent with their previous notation should refer to $I(z,u)$ as $I[C_{g,s}(u,z),F]$.
Answer: Kernel property is true for any index of any theory obtained compactifying on a surface with two or more maximal punctures, i.e. having at least two global symmetries of certain type. Closure of the minimal puncture we show on Fig. 3 is used to derive this property. But the property itself is only right part of this Figure. Hence we called this general index ${\cal I}(u,z)$ with $u$ and $z$ referring to two global symmetries (or equivalently punctures) we can act on. However we agree that the notation proposed by the referee is even better and we performed suggested modification in the corresponding equation.
6. Question: Eq. 3.35 states the kernel property for trinions. The difference operator is labelled by $z$ and $u^{12}w^6$. From the notation used before this would suggest that the puncture $z$ is closed by the vev of an operator with charge $u^{12}w^6$?, but then we would get a tube as in the case later in 3.38 while the kernel property should hold with the $z$ puncture open as well.
Answer: The difference operators we get are indeed obtained by closing minimal punctures with space-dependent vevs of the moment maps. However the trinions we act on in eq.3.36 have nothing to do with closing procedure. Instead of trinions there can be tube theories or any other theory corresponding to compactification on a surface with at least two maximal punctures. So there is no need to close $\SU(2)_z$ puncture of the trinion we act on. We specify trinions as kernel functions since they together with the tubes are the simplest examples and building blocks for all other kernel functions.
Published as SciPost Phys. 12, 140 (2022)