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Modified instanton sum and 4-group structure in 4d $\mathcal{N}=1$ $SU(M)$ SYM from holography
by Marwan Najjar
Submission summary
| Authors (as registered SciPost users): | Marwan Najjar |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2503.17108v1 (pdf) |
| Date submitted: | Sept. 1, 2025, 11:05 a.m. |
| Submitted by: | Marwan Najjar |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the decomposition of the holographic 4d $\mathcal{N}=1$ $SU(M)$ gauge theory within the Klebanov-Strassler set-up. In particular, we propose a consistent framework for defining a modified instanton sum and a 4-group structure for the SYM theory, derived from its $AdS/CFT$ construction. To achieve this, we analyse symmetry topological operators associated with continuous $(-1)$-form symmetries, derive the corresponding 5-dimensional Symmetry Topological Field Theory (SymTFT), and impose specific discrete gaugings.
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- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
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Report
This paper revisits the Klebanov-Strassler setup in Type IIB as holographic realization of 4d N=1 SYM theory. The emphasis is on (-1)-form symmetries, which are related to modified instanton sums and decomposition in SYM. The author studies the Type IIB origin of these (-1)-form symmetries from various viewpoints, including symmetry operators from branes and SymTFT. This work completes a nice circle of ideas related to the realization of 4d N=1 SYM via geometric engineering in M-theory.
The paper is recommended for publication after a minor revision to address the points listed below.
The author explains how fluxbranes realize Page charges. The latter are closed, but can be gauge not-invariant. The localized gauge fields on the fluxbranes are important in ensuring gauge invariance. The author proposes to set the localized gauge fields to zero, but I am not sure whether this is a consistent operation. I agree with the author that, with the localized gauge fields turned on, the fluxbrane might describe a non-invertible operator in the bulk. Such an operator, however, could still describe an invertible symmetry in the QFT. This depends on the boundary conditions. I think it would be beneficial to clarify this point.
Around eqs (3.19), (3.20), (3.21) the objects $F_4^{(9)}$, $F_4^{(7)}$, $H_4^{(7)}$ are defined indirectly, equating their derivatives to some expressions in square brackets. I think it would be useful to add a comment on why these combinations in square brackets are closed. Moreover, in defining $F_4^{(9)}$, $F_4^{(7)}$, $H_4^{(7)}$ from $dF_4^{(9)}$, $dF_4^{(7)}$, $dH_4^{(7)}$ some choices have to be made. Can anything in the subsequent discussion depend on such choices?
Eq (3.35) introduces the map $\phi_{(1)}^*$, but does not quite explain what it is.
I believe the identification in (3.48) might require some further explanation . Let us think of $\theta_{\rm YM}$ as a background 0-form gauge field coupled to the Chern-Weil current ${\rm Tr}(FF)$. In this sense, it seems natural to extend $\theta_{\rm YM}$ (regarded as background) to the bulk of the SymTFT, where it becomes a dynamical gauge field with field strength $F_1^{(3)}$. Now, it seems to me that (3.48) promotes the current ${\rm Tr}(FF)$ itself to another gauge field in the bulk, with field strength $F_4^{(7)}$. In my opinion, this step is not entirely obvious, and it would be useful to motivate it in greater detail.
I am not sure I understand (3.59). $F_{p+2}$ is replaced by $2\pi N dA_{p+1}$. Does this mean that we are restricting the fluxes of $F_{p+2}$ to be multiples of $N$? I believe that it would be beneficial to explain this point further.
In (3.63) we find the notation $(\widetilde \delta (F_{p+2}))_{p+1}$, but a general definition is not provided (one example is given in (3.64)). It would be useful to define this notation in general, as it is not standard.
Around (3.78), the author's goal is to diagnose the higher-group structure involving $\mathbb Z_N^{[1,e]}$ and $\mathbb Z_{p}^{[3]}$. It is stated that, to do so, we have to gauge $\mathbb Z_N^{[1,e]}$ and $\mathbb Z_{p}^{[3]}$. I would say, what we need is to couple the system to non-dynamical background fields for $\mathbb Z_N^{[1,e]}$ and $\mathbb Z_{p}^{[3]}$. The key relations are (3.88), (3.89) for $b_2$, $a_3$, $a_4$, but in these relations $b_2$, $a_3$, $a_4$ need not be dynamical. It might be that this is merely an issue of terminology. It would be beneficial if the author could comment on this point.
The paper is recommended for publication after a minor revision to address the points listed below.
The author explains how fluxbranes realize Page charges. The latter are closed, but can be gauge not-invariant. The localized gauge fields on the fluxbranes are important in ensuring gauge invariance. The author proposes to set the localized gauge fields to zero, but I am not sure whether this is a consistent operation. I agree with the author that, with the localized gauge fields turned on, the fluxbrane might describe a non-invertible operator in the bulk. Such an operator, however, could still describe an invertible symmetry in the QFT. This depends on the boundary conditions. I think it would be beneficial to clarify this point.
Around eqs (3.19), (3.20), (3.21) the objects $F_4^{(9)}$, $F_4^{(7)}$, $H_4^{(7)}$ are defined indirectly, equating their derivatives to some expressions in square brackets. I think it would be useful to add a comment on why these combinations in square brackets are closed. Moreover, in defining $F_4^{(9)}$, $F_4^{(7)}$, $H_4^{(7)}$ from $dF_4^{(9)}$, $dF_4^{(7)}$, $dH_4^{(7)}$ some choices have to be made. Can anything in the subsequent discussion depend on such choices?
Eq (3.35) introduces the map $\phi_{(1)}^*$, but does not quite explain what it is.
I believe the identification in (3.48) might require some further explanation . Let us think of $\theta_{\rm YM}$ as a background 0-form gauge field coupled to the Chern-Weil current ${\rm Tr}(FF)$. In this sense, it seems natural to extend $\theta_{\rm YM}$ (regarded as background) to the bulk of the SymTFT, where it becomes a dynamical gauge field with field strength $F_1^{(3)}$. Now, it seems to me that (3.48) promotes the current ${\rm Tr}(FF)$ itself to another gauge field in the bulk, with field strength $F_4^{(7)}$. In my opinion, this step is not entirely obvious, and it would be useful to motivate it in greater detail.
I am not sure I understand (3.59). $F_{p+2}$ is replaced by $2\pi N dA_{p+1}$. Does this mean that we are restricting the fluxes of $F_{p+2}$ to be multiples of $N$? I believe that it would be beneficial to explain this point further.
In (3.63) we find the notation $(\widetilde \delta (F_{p+2}))_{p+1}$, but a general definition is not provided (one example is given in (3.64)). It would be useful to define this notation in general, as it is not standard.
Around (3.78), the author's goal is to diagnose the higher-group structure involving $\mathbb Z_N^{[1,e]}$ and $\mathbb Z_{p}^{[3]}$. It is stated that, to do so, we have to gauge $\mathbb Z_N^{[1,e]}$ and $\mathbb Z_{p}^{[3]}$. I would say, what we need is to couple the system to non-dynamical background fields for $\mathbb Z_N^{[1,e]}$ and $\mathbb Z_{p}^{[3]}$. The key relations are (3.88), (3.89) for $b_2$, $a_3$, $a_4$, but in these relations $b_2$, $a_3$, $a_4$ need not be dynamical. It might be that this is merely an issue of terminology. It would be beneficial if the author could comment on this point.
Recommendation
Ask for minor revision