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Seven-dimensional super Yang-Mills at negative coupling
by Joseph A. Minahan, Usman Naseer, Charles Thull
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Submission summary
| Authors (as registered SciPost users): | Usman Naseer · Charles Thull |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2208.01115v2 (pdf) |
| Date submitted: | 2022-08-17 22:25 |
| Submitted by: | Naseer, Usman |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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Abstract
We consider the partition function for Euclidean $SU(N)$ super Yang-Mills on a squashed seven-sphere. We show that the localization locus of the partition function has instanton membrane solutions wrapping the six "fixed" three-spheres on the $\mathbb{S}^7$. The ADHM variables of these instantons are fields living on the membrane world volume. We compute their contribution by localizing the resulting three-dimensional supersymmetric field theory. In the round-sphere limit the individual instanton contributions are singular, but the singularities cancel when adding the contributions of all six three-spheres. The full partition function on the ${\mathbb S}^7$ is well-defined even when the square of the effective Yang-Mills coupling is negative. We show for an $SU(2)$ gauge theory in this regime that the bare negative tension of the instanton membranes is canceled off by contributions from the instanton partition function, indicating the existence of tensionless membranes. We provide evidence that this phase is distinct from the usual weakly coupled super Yang-Mills and, in fact, is gravitational.
Current status:
Reports on this Submission
Report #2 by Anonymous (Referee 1) on 2022-9-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2208.01115v2, delivered 2022-09-23, doi: 10.21468/SciPost.Report.5754
Strengths
1-Uses useful approximations to give significant evidence to the physically important conjecture that the seven-dimensional super Yang-Mills theory has a parameter region where the low-energy theory is a supergravity theory.
2-The cancellation of poles at the one-instanton level is highly non-trivial and demonstrates the consistency of the results in the paper.
Weaknesses
1-The analysis is based on an approximate solution to the saddle point equation (2.13) of a matrix integral.
2-The cancellation of poles is done only at the one-instanton level. It is desirable to do the same for higher instanton numbers.
Report
The work under review performs the supersymmetric localization computation of the partition function of the seven-dimensional super Yang-Mills theory with 16 supercharges on a squashed seven-sphere $\mathbb{S}^7$. The authors apply useful approximations to give significant evidence to the physically important conjecture that the seven-dimensional super Yang-Mills theory has a parameter region where the low-energy theory is a supergravity theory. They demonstrate the cancellation of poles at the one-instanton level, which is highly non-trivial and demonstrates the consistency of the results in the paper.
The results and the conclusions of the paper are interesting and important.
The paper is written clearly, and provide adequate references in the literature.
I recommend the paper for publication in the journal.
Requested changes
1-In the manipulation in (2.10), $\sum_i \sigma_i$ is set to zero. So the authors seem to considering $SU(N)$ as the gauge group. On the other hand the measure factor $\prod_{i=1}^N {\rm d}\sigma_i$ in (2.9) would be appropriate for $U(N)$ but not for $SU(N)$. Some clarifying comments seem required.
2-Above (3.9), they write that "the two eigenvalues are widely separated". Because the gauge group is $SU(2)$, actually there is only one eigenvalue $\sigma_{12}$. $\sigma_1$ and $\sigma_2$ do not exist independently.
3-Below (2.25), "equals equals" should be "equals''.
4-Below (2.31), "expressed" should be "suppressed".
5-Above (3.1) "Localizing" should be "localizing".
6-Above (4.1), "given given" should be "given".
Report #1 by Anonymous (Referee 2) on 2022-8-23 (Invited Report)
- Cite as: Anonymous, Report on arXiv:2208.01115v2, delivered 2022-08-23, doi: 10.21468/SciPost.Report.5571
Report
This paper studies the 7d SYM on the round and squashed $S^7$ using localization technique and obtains exact formulae for partition function. In particular, it discusses in detail the theory for negative value of YM coupling squared. Naively the contribution of instanton gets enhanced in such a phase, but it is shown that the enhancement is canceled by Nekrasov partition function. This paper also gives some evicences that the theory at negative coupling-squared can be described by a 7d supergravity, which definitely deserves further study.
Some detail regarding the computation of membrane instanton contribution is presented in the appendix.
Overall the paper is very interesting and clearly written, so I recommend it for publication in SciPost.
Requested changes
Here are some (possible) typos and suggestions.
1- (page 6) just after equation (2.5) there is an equation "$\imath_vF=F_V$" which does not make sense.
2- (page 15) It will be nice to explain the condition or the limit in which the weak equality "$\approx$" in equation (3.10) holds. Also, the equality "$=$" in equation (3.12) should be replaced with "$\approx$"
3- (page 15) In the equation right after (3.13), $q=e^{-\frac{2\pi\delta\sigma}{\omega_2}}$, replace $\omega_2$ by $\omega_1$
4- (page 28) In equation (B.20), the product in the denominator of RHS is over $i=0,\cdots,n_r$ such that $j\ne x$. It should be $i\ne x$.
5- (page 29) The range of integer variables $n,m$ in the Figure 2(a) should be the same as those in (B.19)