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As Contributors: | Joseph Hayling · Rodolfo Panerai · Constantinos Papageorgakis |

Arxiv Link: | https://arxiv.org/abs/1803.06177v3 |

Date accepted: | 2018-06-15 |

Date submitted: | 2018-06-07 |

Submitted by: | Papageorgakis, Constantinos |

Submitted to: | SciPost Physics |

Domain(s): | Theoretical |

Subject area: | High-Energy Physics - Theory |

A formula was recently proposed for the perturbative partition function of certain $\mathcal N=1$ gauge theories on the round four-sphere, using an analytic-continuation argument in the number of dimensions. These partition functions are not currently accessible via the usual supersymmetric-localisation technique. We provide a natural refinement of this result to the case of the ellipsoid. We then use it to write down the perturbative partition function of an $\mathcal N=1$ toroidal-quiver theory (a double orbifold of $\mathcal N=4$ super Yang-Mills) and show that, in the deconstruction limit, it reproduces the zero-winding contributions to the BPS partition function of (1,1) Little String Theory wrapping an emergent torus. We therefore successfully test both the expressions for the $\mathcal N=1$ partition functions, as well as the relationship between the toroidal-quiver theory and Little String Theory through dimensional deconstruction.

Published
as
SciPost Phys. **4**, 042
(2018)

Dear Editor,

We thank all referees for a careful reading of the manuscript along with the associated comments and suggestions. We have implemented a list of changes to our submission based on these.

Sincerely,

J. Hayling, R. Panerai and C. Papageorgakis

We order our response by referee report:

Referee 1:

We first reply to the points raised in the main report below:

1 - We believe that the phrasing in this instance is sufficiently clear; the qualifier “exact (to all orders in the coupling)” refers to the localisation of supersymmetric gauge theories where such a concept exists (and not to the localisation technique in full generality). We have not implemented any changes associated with this point.

2 - We feel that the phrasing “1-loop contribution” is misleading because: 1) it would imply that this calculation is the result of bona fide localisation calculation in 4D (see also point 8 below) and 2) even in that event the 1-loop contributions refer to the functional determinants of the free limit for the deformed theory. The qualifier “vector- and chiral-multiplet contributions” is more appropriate, in our opinion, as it captures the full contributions to the end result from these multiplets in the zero-instanton sector. We have not implemented any changes associated with this point.

3 - We thank the referee for raising this point, as the statement in the paper was misleading. We have stressed both above Eq. (9) as well as above Eq. (19) that the quantities presented are for the integrand of the perturbative partition function. There was also a notation clash towards the end of the letter, where the integrated LST partition function Eq. (29) and the integrand Eq. (31) were denoted with the same symbol. We have changed the former to resolve this ambiguity and stressed that Eq. (31) is to be integrated over $\lambda$.

4- We believe that it is clearly stated in the text that the two results match up to this overall factor. This is a purely geometric contribution that is not dropped for the sole purpose of ensuring the matching of the two calculations presented in our work, but often dropped in the topological strings literature as it contains no dynamical information. We have clarified this point in Footnote 11.

5- This paragraph was not meant to demonstrate tests of our proposal, but illustrate how this is a generalisation of known results. We have changed the phrasing slightly to clarify this.

6 - This sentence is indeed referring to SYM + KK + S-dual spectra. The S-dual spectra are needed in order to reproduce the winding sectors of LST as in [14], the deconstruction of which we are reviewing at this stage without restricting to the 0-winding sector. We have not implemented any changes associated with this point.

7 - We interpret this comment to refer to $b$ and $\hat b$ taking values in $\mathbb Z$ instead of $\mathbb N$, in which case we are not double counting as this is reproducing all the KK modes of the higher-dimensional theory (positive and negative), as e.g. in the original paper of Arkani-Hamed, Cohen and Georgi below equation (2.8). We have not implemented any changes associated with this point.

8 - There is no known way to localise these $\mathcal N=1$ quiver theories on $S^4$, and therefore no associated 4D localisation locus. The integration variables are inherited from the 2D/3D calculation of [10], where such a locus exists, via analytic continuation. This is also why we chose not to use the 1-loop nomenclature in point 2 above. We have not implemented any changes associated with this point.

9 - The number of derivatives that needs to be taken depends on the quantity that is probed, as can be easily found in our references [3] and [9]. We find that providing all these details in the opening paragraph of this letter would detract from introducing the topic to a general audience. The interested reader can find more information in the references provided. We have not implemented any changes associated with this point.

For the final point in this section of the report:

There is no need to take the large-rank limit on each node for our arguments to go through (large K). Our claim is not that the instantons are suppressed but that we are only reproducing the zero-winding sector of LST. Therefore the non-zero-winding sectors are very much there but we do not have a four-dimensional derivation for them. We have not implemented any changes associated with this point.

Moving on to the Requested Changes section:

1 - Addressed as above.

2 - We have included a new footnote 3 addressing this point.

3 - Since this is a letter submission, we decided not to include this calculation as it would significantly increase the length of the paper and they can be reproduced using standard technology from the literature. We have not implemented any changes associated with this point.

4 - Eq. (17) presents a mathematical identity that holds for any $\tau$, provided that $\Im \tau>0$. We specify $\tau$ only when we apply this to the case in hand, namely from Eq. (19) onwards. We have not implemented any changes associated with this point. However, we have clarified the ten-dimensional geometry at the beginning of that paragraph and the notation used for the susy projections should now be evident. This also follows the conventions of [10].

We hope that we have sufficiently addressed the referee’s concerns.

Referee 2:

This point was raised and addressed in the response to the first referee (point 3; main report). We include it here once again for ease of navigation:

We thank the referee for raising this point, as the statement in the paper was misleading. We have stressed both above Eq. (9) as well as above Eq. (19) that the quantities presented are for the integrand of the perturbative partition function. There was also a notation clash towards the end of the letter, where the integrated LST partition function Eq. (29) and the integrand Eq. (31) were denoted with the same symbol. We have changed the former to resolve this ambiguity and stressed that Eq. (31) is to be integrated over $\lambda$.

Referee 3:

No changes requested.

Resubmission 1803.06177v3 (7 June 2018)

Submission 1803.06177v2 (3 May 2018)