Exponential Networks, Quivers, and Conifold Mirrors

DOI

10.2991/978-94-6463-789-2_5How to use a DOI?

Keywords

BPS networks; representations of quivers; linear partitions; knot invariants

Abstract

We give an introduction and brief review of the “exponential networks approach” to counting BPS states in local Calabi-Yau threefolds. We begin by illustrating the basic ideas in the celebrated correspondence between the BPS spectrum of 4d N = 2 pure SU(2) gauge theory, finite geodesics on the Seiberg-Witten curve, and stable representations of the Kronecker quiver. We proceed by describing the reduction of special Lagrangian submanifolds to networks calibrated by the Liouville one-form on the mirror curve, and the resulting local junction rules and novel global features. We present an explicit correspondence between torus fixed points of the Hilbert scheme of points in the plane and anomaly free exponential networks attached to the quadratically framed pair of pants. We digress to explain how the BPS spectrum of the resolved conifold can be recovered from exponential networks on the mirror-like geometries attached to knots in S³. We end with an outlook on how to recover the knots-quivers correspondence from such constructions.

Copyright

© 2025 The Author(s)

Open Access

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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TY  - CONF
AU  - Raphael Senghaas
AU  - Johannes Walcher
PY  - 2025
DA  - 2025/07/17
TI  - Exponential Networks, Quivers, and Conifold Mirrors
BT  - Proceedings of the International Conference on Knots, Quivers and Beyond (ICKQB 2025)
PB  - Atlantis Press
SP  - 50
EP  - 64
SN  - 2352-541X
UR  - https://doi.org/10.2991/978-94-6463-789-2_5
DO  - 10.2991/978-94-6463-789-2_5
ID  - Senghaas2025
ER  -