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Exponential Networks for Linear Partitions
by Sibasish Banerjee, Mauricio Romo, Raphael Senghaas, Johannes Walcher
Submission summary
| Authors (as registered SciPost users): | Johannes Walcher |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2403.14588v2 (pdf) |
| Date submitted: | 2024-04-16 11:44 |
| Submitted by: | Walcher, Johannes |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
Previous work has given proof and evidence that BPS states in local Calabi-Yau 3-folds can be described and counted by exponential networks on the punctured plane, with the help of a suitable non-abelianization map to the mirror curve. This provides an appealing elementary depiction of moduli of special Lagrangian submanifolds, but so far only a handful of examples have been successfully worked out in detail. In this note, we exhibit an explicit correspondence between torus fixed points of the Hilbert scheme of points on $\mathbb C^2\subset\mathbb C^3$ and anomaly free exponential networks attached to the quadratically framed pair of pants. This description realizes an interesting, and seemingly novel, "age decomposition" of linear partitions. We also provide further details about the networks' perspective on the full D-brane moduli space.
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This paper is a continuation in a series of papers discussing exponential network techniques applied to counting BPS D-brnaes on Calabi-Yau threefolds.
The heart of this technique is in consideration of a mirror dual curve, then D-branes are dual to sLags, whose forms and filling numbers may be identified explicitly from solving first order differential equations numerically.
In this paper authors concentrate on the case of C3 when a non-compact D4-brane is present.
From the point of view of an effective quiver QFT this situation is captured by specific quiver framing. In this particular case the quiver setup boils down to a Hilbert scheme of points on C2.
Authors propose to filter out the BPS spectrum in this case by a specific limit for the C3 mirror curve.
The proposal is confirmed by explicit comparison of the resulting BPS spectrum with equivariant fixed points on Hilb(C2).
Speculations on the life outside the fixed point locus are also present in the text.
On the quiver theory side framing imposes significant corrections to the BPS spectra and may be involved in wall-crossing via mutations.
This is why it is important to study a proper mirror dualization of the framing elements, and this manuscript is the first step down this route.
This research is new and vibrant. And I definitely recommend this paper for publication.
However if the authors choose to do so, I would suggest substantial modifications in the text structure:
1) Review sections 2 and 3 seem too be rather excessive. This is a bulky textbook material the reader might easily acquire from other sources. And it is hard to grasp it from a concise summary of this paper. For example, parallel transport variables introduced in (3.12) seem to be never used outside this section.
2) What review material seems to be missing is from references [6] and [8] where the cut-off proposal to mimic D4-brane was born. The very proposal why a suitable cut-off would mimic quiver framing is not apparent:
a) As a naive answer to the question why the BPS spectrum is equivalent to EQUIVARIANT fixed points I would suggest that the D4-divisor was chosen in an equivariant way -- as a (x,y)-plane in C3. However there are two more apparent choices (x,z) and (y,z) that would be distinguished by the last term in superpotential (1.3): -IJB_2 or -IJB_1. Should those be other punctures in Figure 9?
b) Surely, it is natural to ask what to do if the D4-divisor (or even D2-divisor) is chosen in a more generic way, say two intersecting planes in C3?
c) It seems to be not very clear if eq.(3.33) has a priori origin. Do we first calculate the BPS spectrum then compare it with fixed pints on Hilb(C2) and afterwards identify dual objects in two spectra, or is there a mechanism to dualize D4-brane to a sLag on the mirror curve directly?
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