[Paper Review] A tale of two cones: the Higgs Branch of Sp(n) theories with 2n flavours

Giulia Ferlito, Amihay Hanany|arXiv (Cornell University)|Sep 21, 2016

Algebraic structures and combinatorial modelsMathematics19 references19 citations

TL;DR

This paper demonstrates that the Higgs branch of Sp(n) gauge theories with 2n fundamental flavours is not a single hyperkähler cone but the union of two distinct hyperkähler cones intersecting along a shared subvariety. Using Hilbert series and highest weight generating functions, the authors identify the algebraic structure of the chiral ring and show that the Higgs branch decomposes into two cones isomorphic to C²/ℤ₂, with their intersection corresponding to a common hyperkähler subvariety, a phenomenon rare in N=2 4D theories.

ABSTRACT

The purpose of this short note is to highlight a particular phenomenon which concerns the Higgs branch of a certain family of 4d N = 2 theories with SO(2N) flavour symmetry. By studying the Higgs branch as an algebraic variety through Hilbert series techniques we find that it is not a single hyperkahler cone but rather the union of two cones with intersection a hyperkahler subvariety which we specify. This remarkable phenomenon is not only interesting per se but plays a crucial role in understanding the structure of all Higgs branches that are generated by mesons.

Motivation & Objective

• To investigate the algebraic structure of the Higgs branch in N=2 Sp(n) gauge theories with 2n fundamental flavours.
• To determine whether the Higgs branch is a single hyperkähler cone or a more complex union of cones.
• To identify the geometric and algebraic origin of this decomposition using Hilbert series and chiral ring techniques.
• To clarify the role of this phenomenon in the broader context of Higgs branches generated by mesonic operators.

Proposed method

• Employing Hilbert series techniques to compute the generating function of gauge-invariant chiral ring operators on the Higgs branch.
• Using highest weight generating functions to encode the representation-theoretic structure of the chiral ring.
• Applying the hyperkähler quotient construction to derive the chiral ring relations for classical gauge groups with bifundamental matter.
• Specializing to Sp(n) with 2n flavours and analyzing the Hilbert series to detect a decomposition into two cones.
• Comparing the Higgs branch structure to the Coulomb branch of the mirror dual quiver theory, using brane constructions and quiver gauge theories.
• Identifying the intersection of the two cones as a hyperkähler subvariety via plethystic logarithm analysis of the Hilbert series.

Experimental results

Research questions

• RQ1 Is the Higgs branch of Sp(n) with 2n flavours a single hyperkähler cone or a union of multiple cones?
• RQ2 What is the algebraic and geometric structure of the chiral ring in these theories, and how does it reflect the underlying gauge and global symmetries?
• RQ3 How does the decomposition into two cones arise from the Hilbert series, and what is the nature of their intersection?
• RQ4 Why is this phenomenon rare in N=2 4D theories, and what are its implications for Higgs branches generated by mesons?
• RQ5 How does the mirror symmetry of the theory manifest when one side is a single quiver and the other side involves two equivalent quivers?

Key findings

• The Higgs branch of Sp(n) with 2n flavours decomposes into two distinct hyperkähler cones, each isomorphic to C²/ℤ₂.
• The two cones intersect non-trivially along a common hyperkähler subvariety, which is explicitly identified as the origin in the Hilbert series decomposition.
• The Hilbert series of the Higgs branch is expressed as the sum of two C²/ℤ₂ Hilbert series minus one, confirming the union structure: HS = HS(C²/ℤ₂; t, x₁) + HS(C²/ℤ₂; t, x₂) - 1.
• This decomposition is confirmed via plethystic logarithm analysis, which reveals two sets of independent generators and their relations.
• The phenomenon is linked to the mirror dual theory, where the Coulomb branch of the mirror quiver theory corresponds to only one of the two cones, reflecting a duality ambiguity in the brane construction.
• The result generalizes earlier observations in SU(2) with 2 flavours and highlights a deeper algebraic structure in Higgs branches generated by mesonic operators.