Noncommutative resolutions and CICY quotients from a non-abelian GLSM

1 - The authors provide a GLSM with a phase that at the same time exhibits highly non-trivial gauge theory dynamics - a detailed understanding of which will hopefully be achieved in future work - as well as a compelling geometric interpretation in terms of a non-commutative resolution of a nodal degeneration of the complete intersection $\mathbb{P}^5_{111223}[4,6]$.
2 - They arrive at this interpretation by combining their GLSM analysis with techniques from mirror symmetry and novel results from topological string theory, thus allowing them to overcome the limitations of the gauge theory analysis.
3 - By providing the first example of a non-commutatively resolved Calabi-Yau threefold that admits both a GLSM realization and corresponds to a $\mathbb{Z}_3$ torsional B-field, the authors open up a new pathway for explicitly understanding the corresponding shaves of non-commutating algebras beyond the case of Clifford algebras that had been studied before.

The authors study the GLSM that is associated to a $\mathbb{Z}_3$-quotient of a certain Calabi-Yau threefold $Y$ that is a complete intersection in $(\mathbb{P}^2)^3$, where the $\mathbb{Z}_3$ cyclically permutes the three $\mathbb{P}^2$ factors of the ambient space. The gauge group of the corresponding GLSM is non-abelian and a semi-direct product of $U(1)^3$ with $\mathbb{Z}_3$. In the phase where the FI-parameter $\zeta$ of the GLSM is large, the GLSM flows to a non-linear sigma model on said geometry.
The analysis of this phase is largely standard and the non-abelian gauge symmetry doesn't pose much of a problem. However, the authors observe that in the phase $\zeta\ll 0$ the dynamics of the GLSM are significantly harder to understand. While they observe that the phase appears to be hybrid like, fibered over a non-linear sigma model on $\mathbb{P}^2$, the existence of a non-compact Coulomb branch -- and of a cubic potential -- renders the dynamics of this phase inaccessible with standard techniques.

Leaving a detailed analysis of the gauge theory dynamics in this phase to future work, the authors then turn to mirror symmetry and topological string theory in order to understand the infrared fixed point.
The authors observe two points of maximally unipotent monodromy in the complex structure moduli space of the mirror of $Y$, one being mirror to the limit $\zeta\gg 0$ and the other to $\zeta\ll 0$.
By studying the integral structure of the periods of this mirror, as well as the topological string free energies, they argue that the IR theory in the phase $\zeta\ll 0$ should admit an interpretation as a non-linear sigma model on a singular degeneration $X$ of the complete intersection $\mathbb{P}_{111223}[4,6]$, with 63 isolated nodes, that supports a so-called ``fractional B-field'' -- and therefore should in turn admit an interpretation as a non-commutative resolution of $X$.
To this end they use modifications of the usual formulae for an integral period basis, as well as for the classical terms and the constant map contributions to the free energies, that have recently been proposed for topological strings on such non-commutative resolutions.
The authors fix the holomorphic ambiguity up to genus 11 and calculate the $\mathbb{Z}_3$-torsion refined Gopakumar-Vafa invariants that are encoded in the A-model topological string free energies of the non-commutative resolution of $X$ together with those of its smooth deformation.
The integrality of these invariants, together with the Castelnuovo vanishing and the correct constant map contributions provide highly non-trivial evidence that their interpretation is indeed correct.

The paper is well written and the exposition is very clear.
Moreover, only few examples of non-commutative resolutions of nodal Calabi-Yau threefolds have been discussed in the literature and the authors provide the first example that appears to admit a GLSM description and at the same time corresponds to a geometry with $\mathbb{Z}_3$-torsion.
Even though the analysis of the gauge theory dynamics in the relevant phase has so far largely been constrained to highlighting the associated difficulties, the authors make elegant use of mirror symmetry to find a convincing proposal for what the IR theory should be.
It will be interesting to see if subsequent work can elucidate how the gauge theory reproduces this proposal.

• On p.7, the D-term equations also exclude solutions where, for example, $x_1=x_2=x_3=0$ but $(y_1,y_2,y_3)\ne 0\ne (z_1,z_2,z_3)$. Is it clear that (2.11) does not admit any solutions of this form? Of course this would affect the structure of the hybrid model in the phase $\zeta\ll 0$, leading to geometric "branches" in addition to the $\mathbb{P}^2$ that is parametrized by $p_1,p_2,p_3$.
• On p.14, perhaps the authors could slightly elaborate on why solutions where additional matter fields become massless have to be discarded.
• On p.16, tegarding the statement "having such a configuration at a point at infinity in the moduli space is a feature specific to non-abelian GLSMs.": For example, in 1305.5767 it has been observed that conifold transitions can occur both at phase limits and at phase boundaries, with the two sometimes appearing for different GLSM realisations of the same geometry. This already happens for abelian GLSM and one would naively expect that such GLSMs where the transition is located at a phase limit lead to counterexamples to the statements of the authors. Is the Coulomb branch still expected to be compact in those examples?
• In the calculation of the hemisphere partition function in Section 2.5, it seems that the discrete part of the gauge group can essentially be ignored. Is it clear why this is the case?
• On p.20, below eq.(3.5) the authors refer to "The first three solutions"while, given the order of the solutions in (3.5), the reader might easily be misled to consider those the first, the third and the fifth. Perhaps the exposition can slightly be improved to avoid any potential for misunderstanding.
• On p.26, above eq.(3.26), is it clear that $c_2$ agrees for $X_{\text{def}}$, $X$ and $\widehat{X}$? How would one even define this quantity for the singular variety $X$?
• It would be useful if, at least for very low genus, some of the holomorphic ambiguities of the free energies could be provided directly in the paper.

In terms of typos we have observed the following:

• On p.1, "... turns out to be a very useful too ..."
• On p.11, "The approach of ... is to use that fact that ..." presumably should read "... is to use the fact that ..."
• On p.17, below eq.(2.61), "... but is can cancel ..."
• On p.17, in the paragraph above eq.(2.63), "... it is fairly straightforward to this read off from ..."
• On p.19, last sentence above Section 3.1, it should be either "one ... basis ... attached to each MUM point ...", or "two ... bases, one attached to each MUM point ...".
• On p.19, "$U^1,....,W^1,W^3$" should be "$U^1,....,W^1,W^2$". Also, in the following these coordinates are sometimes used with upper and sometimes with lower indices. It seems that no problem would arise if lower indices were to be used throughout.
• On p.24, in eq.(3.20) the sum over curve classes could either be over elements of $H_2$, in which case the upper limit $\infty$ should be dropped, or over natural numbers.
• On p.26, below eq.(3.25), we are under the impression that, grammatically speaking, "obstruction" instead of "obstructions" would be in order here.
• On p.28, in the sentence "If $\phi$ was an $N=1$ MUM point ..." shouldn't it read "If $\phi$ were an $N=1$ MUM point ..."?
• On p.35, last paragraph, in the sentence "The existence of the Coulomb branch at infinity ..." the word "formulation" should be dropped.
• On p.36, last paragraph of the conclusion, in the sentence "A final remark in relation ..." it seems to us that it should either by "concerns the" or "pertains to the".

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