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Toric Duality, Seiberg Duality and Picard-Lefschetz Transformations

Sebastian Franco, Amihay Hanany

TL;DR

This work addresses the non-uniqueness of 4d ${\cal N}=1$ gauge theories on D3-branes probing toric singularities, showing that toric duality can be understood through Seiberg duality and Picard-Lefschetz monodromies. It develops an integrated framework using $(p,q)$ webs and local mirror symmetry to compute quivers, establish toric duals, and generalize dualities beyond node-based Seiberg duality via fractional Seiberg duals. A key contribution is the identification of a PL monodromy group larger than Seiberg duals and the corresponding Diophantine invariants that classify all dual theories. The results provide a systematic method to enumerate IR-equivalent gauge theories associated with toric singularities and deepen the connection between toric geometry and four-dimensional gauge dynamics.$d=4$ ${\cal N}=1$.

Abstract

Toric Duality arises as an ambiguity in computing the quiver gauge theory living on a D3-brane which probes a toric singularity. It is reviewed how, in simple cases Toric Duality is Seiberg Duality. The set of all Seiberg Dualities on a single node in the quiver forms a group which is contained in a larger group given by a set of Picard-Lefschetz transformations. This leads to elements in the group (sometimes called fractional Seiberg Duals) which are not Seiberg Duality on a single node, thus providing a new set of gauge theories which flow to the same universality class in the Infra Red.

Toric Duality, Seiberg Duality and Picard-Lefschetz Transformations

TL;DR

This work addresses the non-uniqueness of 4d gauge theories on D3-branes probing toric singularities, showing that toric duality can be understood through Seiberg duality and Picard-Lefschetz monodromies. It develops an integrated framework using webs and local mirror symmetry to compute quivers, establish toric duals, and generalize dualities beyond node-based Seiberg duality via fractional Seiberg duals. A key contribution is the identification of a PL monodromy group larger than Seiberg duals and the corresponding Diophantine invariants that classify all dual theories. The results provide a systematic method to enumerate IR-equivalent gauge theories associated with toric singularities and deepen the connection between toric geometry and four-dimensional gauge dynamics. .

Abstract

Toric Duality arises as an ambiguity in computing the quiver gauge theory living on a D3-brane which probes a toric singularity. It is reviewed how, in simple cases Toric Duality is Seiberg Duality. The set of all Seiberg Dualities on a single node in the quiver forms a group which is contained in a larger group given by a set of Picard-Lefschetz transformations. This leads to elements in the group (sometimes called fractional Seiberg Duals) which are not Seiberg Duality on a single node, thus providing a new set of gauge theories which flow to the same universality class in the Infra Red.

Paper Structure

This paper contains 6 sections, 16 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Toric duality
  3. $(p,q)$ webs
  4. 4d theories via local mirror symmetry
  5. Picard-Lefschetz transformations
  6. Diophantine equations from invariant traces

Figures (5)

  • Figure 1: Quivers for the two phases of $F_0$. Nodes represent $U(N)$ gauge theories, with $N$ the number of D3 branes. Each arrow represents a bi-fundamental field transforming under the two gauge groups associated to the nodes it connects.
  • Figure 2: (a) $(p.q)$ web and (b) quiver diagram for $dP_0$.
  • Figure 3: Possible blowups of $dP_1$. They correspond to two inequivalent phases of $dP_2$.
  • Figure 4: Picard-Lefschetz monodromy as reordering of a configuration of 7-branes. We have indicated in the figure the expression of (\ref{['monodromy']}) as a transformation of $[p,q]$ charges.
  • Figure 5: A sequence of Picard-Lefschetz transformations for $F_0$. In this case, model 2 cannot be obtained by any combination of Seiberg dualities.The ranks of the gauge groups (up to an overall rescaling) are denoted in red.