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Generalized Toric Polygons, Tbranes, and 5d SCFTs
by Antoine Bourget, Andrés Collinucci, Sakura SchäferNameki
Submission summary
Authors (as registered SciPost users):  Antoine Bourget 
Submission information  

Preprint Link:  https://arxiv.org/abs/2301.05239v1 (pdf) 
Date submitted:  20230126 19:04 
Submitted by:  Bourget, Antoine 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
5d Superconformal Field Theories (SCFTs) are intrinsically stronglycoupled UV fixed points, whose realization hinges on string theoretic methods: they can be constructed by compactifying Mtheory on local CalabiYau threefold singularities or alternatively from the worldvolume of 5branewebs in type IIB string theory. There is a correspondence between 5branewebs and toric CalabiYau threefolds, however this breaks down when multiple 5branes are allowed to end on a single 7brane. In this paper, we extend this connection and provide a geometric realization of brane configurations including 7branes. A web with 7branes defines a socalled generalized toric polygon (GTP), which corresponds to combinatorial data that is obtained by removing vertices along external edges of a toric polygon. We identify the geometries associated to GTPs as nontoric deformations of toric CalabiYau threefolds and provide a precise, algebraic description of the geometry, when 7branes are introduced along a single edge. The key ingredients in our analysis are Tbranes in a type IIA frame, which includes D6branes. We show that performing HananyWitten moves for the 7branes on the type IIB side corresponds to switching on semisimple vacuum expectation values on the worldvolume of D6branes, which in turn uplifts to complex structure deformations of the CalabiYau geometries. We test the proposal by computing the crepant resolutions of the deformed geometries, thereby checking consistency with the expected properties of the SCFTs.
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Reports on this Submission
Report
In this paper, the authors take valuable steps towards generalizing the correspondence between 5D SCFTs and toric polygons to accommodate socalled generalized toric polygons (GTPs), which are further characterized by a choice of color for each node associated to specific choices of fivebrane boundary conditions. The key missing ingredient in this more general correspondence is the geometric interpretation of GTPs as local (but not necessarily toric) CY3s defining Mtheory backgrounds.
The authors make a compelling proposal on how to supply this missing ingredient by interpreting the local geometries associated to a certain class of GTPs as comprising local CY3 patches modified by algebraic (i.e., complex structure) deformations associated to Tbranes in the type IIA duality frame. This proposal, supported by ample technical arguments and numerous illustrative examples, appears to initiate a fruitful line of investigation with clearly defined future directions. The overarching efforts, if eventually successful, will clearly constitute significant progress towards understanding geometric engineering of 5D SCFTs in string theory.
One aspect that I find somewhat unclear is to what extent the capacity for toric polygons to be able to describe fivebranes ending on mutually nonlocal sevenbrane configurations is essential for the central result of this paper, especially since the authors confine their attention to mutually local configurations of sevenbranes (specifically, parallel stacks). Said differently, one might wonder whether or not the local geometric interpretation could have been spelled out simply using stacks of parallel sevenbranes, without reference to complete toric polygons, despite the fact that toric polygons nicely contextualize the results and clearly motivate future research.
My opinion is that this paper could perhaps benefit from an additional comment clarifying and/or emphasizing the extent to which the complete (generalized) toric polygons were necessary for obtaining the advertised results. With this minor revision (and leaving the specifics to the authors' discretion), I would enthusiastically recommend this paper for publication.
Strengths
1. First proposal of geometric description of 5d SCFTs described by 5brane webs with multiple 5branes ending on the same 7brane.
2. Passes consistency checks using known deformations.
Weaknesses
1. The Type IIA description of the deformation as 6brane recombination is not sufficiently explained.
Report
The authors propose an Mtheoretic geometric description of 5d SCFT's corresponding to 5brane webs in Type IIB string theory
in which groups of multiple external 5branes are constrained to end on common 7branes.
The geometry in this case is not a toric CalbiYau, and the corresponding grid diagram is referred to as a Generalized Toric Polygon (GTP).
More specifically they consider a subclass of such models, in which the constraints apply to just one edge of the polygon.
Their proposal involves a deformation of the toric CY which is motivated by an intermediate description of the system in Type IIA string theory,
that involves a combination of singular geometry and 6branes wrapping collapsed 2cycles.
There are two steps leading to their proposal.
In the first step, they argue that the GTP itself, namely the constraints imposed on some of the external 5branes,
corresponds in the Type IIA picture to nilpotent VEVs for the Higgs field on the 6brane worldvolume.
Such VEVs do not separate the 6branes, and therefore do not correspond to a modification of the geometry in the Mtheory picture.
The second step is the deformation itself.
They propose an algebraic expression for the deformation, and argue that from the 5braneweb viewpoint it corresponds
to a HananyWitten type move that releases the 7branes that have multiple 5branes ending on them.
The intermediate Type IIA picture of this step, which is what is supposed to provide the main motivation for this proposal,
is that some of the 6branes combine and move away from the singularity, as they say below equation (3.45).
I did not really understand this process.
Is there an intuitive explanation?
Their proposal does pass consistency checks using some known deformations of the T_N theories.
I therefore believe it is correct, but I think a simple explanation of the 6brane recombination process should be possible.
Two minor comments about section 5.2 are that
a. It would be helpful to clarify that the "matter" they discuss
corresponds to massive (and nongauge invariant) BPS states on the Coulomb branch, since one might
mistakingly assume that the discussion is about (gaugeinvariant) BPS operators of the 5d SCFT.
b. In equation (5.14) one should probably also include the SU(2) charges, i.e. the last four states are really two SU(2) doublets.
There is also a small typo in equation (4.9): In the second equation the power of Z should be 2.
Requested changes
1. A simple explanation of the 6brane recombination.
2. Minor clarifications in section 5.2 (see report).
Report
The paper studies constructions of 5d superconformal field theories (SCFTs) via brane webs in IIB string theory and via singular CalabiYau threefolds in Mtheory. One of the unsolved questions has been to identify the Mtheory CalabiYau threefold corresponding to 5branes webs with 7branes, where there are multiple 5branes ending on a single 7branes. The presence of these 7branes makes the dual Mtheory CalabiYau threefold nontoric. Therefore the standard duality between webs and toric polygons does not apply. The authors address this question in a restricted but still large class of examples, highlighting also the general procedure. In particular, the paper explicitly identifies the CalabiYau geometries corresponding to the dual generalized toric polygons (GPT). GPTs were previously introduced as dual diagram of the certain IIB brane webs with 7branes, but they did not have a full geometric description in Mtheory. Key for the understanding of the Mtheory geometry is the intermediate step of the dualization procedure between IIB and Mtheory. This is the IIA construction, which is given by D6branes intersecting along genus0 curves and forming a 5d linear quiver theory. The paper shows that, when in the web multiple 5branes end on a 7brane, a nilpotent Tbrane background for the D6brane configuration in IIA has been activated. Finally semisimple deformations of the Tbrane configuration, which are dual to HananyWitten transitions in the IIB web, lead to complex structure deformations of the CalabiYau geometry upon uplift to Mtheory. This last step is key for having an explicit geometry in Mtheory. The authors finally propose some checks of this proposal, and they apply them to some explicit examples. In particular, the main test consists in resolving the resulting singular CalabiYau geometry, and in matching properties derived from these resolutions with the ones computed from the dual brane webs. These include flavor symmetries and/or charges of BPS states. The examples studied in this paper successfully pass the proposed tests.
The paper provide interesting new results and methodologies to address the question mentioned above, therefore I recommend it for publication to SciPost, when the following minor points about section 2 are addressed, answered or justified:
The idea of having a general section about the technical Tbrane procedure is very useful. On the other hand sometimes the used terminology can be a bit too technical in the mathematical sense, and it can sound a bit cryptic to some physics readers. The general procedure does become clearer when looking at the explicit examples though.
 To be more concrete the sentence after (2.7) " ... are monic polynomials ...". This sentence together with footnote 2 sounds a bit obscure. It might be more helpful to just show the general intuitive form of the SNF(T) matrix, or expand the discussion making it more explicit or manifest.
 in (2.10) the partition notation on the T matrices has been introduced without explanation (only mentioned in introduction). Again, this becomes clear later on, but when reading this first, one might get confused on what this notation might mean here.
 I do not understand how (2.14)  (2.17) reveals the structure of the deformation (2.11), in a different guise, or what "reveals the same structure in a different guise" refers to.