Abstract
A ghor algebra is the path algebra of a dimer quiver on a surface, modulo relations that come from the perfect matchings of its quiver. Such algebras arise from abelian quiver gauge theories in physics. We show that a ghor algebra \(\Lambda \) on a torus is a dimer algebra (a quiver with potential) if and only if it is noetherian, and otherwise \(\Lambda \) is the quotient of a dimer algebra by homotopy relations. Furthermore, we classify the simple \(\Lambda \)-modules of maximal dimension and give an explicit description of the center of \(\Lambda \) using a special subset of perfect matchings. In our proofs we introduce formalized notions of Higgsing and the mesonic chiral ring from quiver gauge theory.
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Notes
Thanks to Francesco Benini, Mike Douglas, Peng Gao, Mauricio Romo, and James Sparks for discussions on the physics of non-cancellative dimers.
More correctly, weakly coupled superstring theory requires 10 dimensions.
Here we are considering theories with \(\mathcal {N} = 1\) supersymmetry.
More correctly, the F-term relations plus the D-term relations imply the equations of motion.
More precisely, there are domains of magnetization.
This is an example of ‘global’ symmetry breaking, meaning the symmetry is physically observable.
This is an example ‘gauge’ symmetry breaking, meaning the symmetry is not an actual observable symmetry of a physical system, but only an artifact of the math used to describe it (like a choice of basis for the matrix of a linear transformation).
This is another example of gauge symmetry breaking.
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The author thanks the referees for their helpful comments.
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Open access funding provided by Austrian Science Fund (FWF). The author was supported by the Austrian Science Fund (FWF) grant P 30549-N26. Part of this article is also based on work supported by the Simons Foundation and the Heilbronn Institute for Mathematical Research while the author held postdoctoral positions at the Simons Center for Geometry and Physics at Stony Brook University and the University of Bristol.
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Appendix A. A Brief Account of Higgsing with Quivers
Appendix A. A Brief Account of Higgsing with Quivers
Quiver gauge theories
According to string theory, our universe is 10 dimensional.Footnote 1,Footnote 2 In many string theories our universe has a product structure \(M \times Y\), where M is our usual 4-dimensional spacetime and Y is a 6-dimensional compact Calabi-Yau variety.
Let us consider a special class of gauge theories called ‘quiver gauge theories’, which can often be realized in string theory.Footnote 3 The input for such a theory is a quiver Q, a superpotential W, a dimension vector \(d \in \mathbb {N}^{Q_0}\), and a stability parameter \(\theta \in \mathbb {R}^{Q_0}\).
Let I be the ideal in \(\mathbb {C}Q\) generated by the partial derivatives of W with respect to the arrows in Q. These relations (called ‘F-term relations’) are classical equations of motion from a supersymmetric Lagrangian with superpotential W.Footnote 4 Denote by A the quiver algebra \(\mathbb {C}Q/I\).
According to these theories, the space X of \(\theta \)-stable representation isoclasses of dimension d is an affine chart on the compact Calabi-Yau variety Y. The ‘gauge group’ of the theory is the isomorphism group (i.e., change of basis) for representations of A.
Physicists view the elements of A as fields on X. More precisely, A may be viewed as a noncommutative ring of functions on X, where the evaluation of a function \(f \in A\) at a point \(p \in X\) (i.e., representation p) is the matrix \(f(p):= p(f)\) (up to isomorphism).
Vacuum expectation values
Given a path \(f \in A\) and a representation \(p \in X\), denote by \(f\left( \hspace{0.83328pt}\overline{\hspace{-0.83328pt}p\hspace{-0.83328pt}}\hspace{0.83328pt}\right) \) the matrix representing f in the vector space diagram on Q associated to p.
A field \(f \in A\) is ‘gauge-invariant’ if \(f(p) = f(p')\) whenever p and \(p'\) are isomorphic representations (i.e., they differ by a ‘gauge transformation’). If f is a path, then f will necessarily be a cycle in Q.
The ‘vacuum expectation value’ of a field is its expected (average) energy in the vacuum (similar to rest mass), and is abbreviated ‘vev’. In our case, the vev of a path \(f \in A\) at a point \(p \in X\) is the matrix \(f\left( \hspace{0.83328pt}\overline{\hspace{-0.83328pt}p\hspace{-0.83328pt}}\hspace{0.83328pt}\right) \), which is just the expected energy of f in \(M \times \{ p \}\).
Higgsing
Spontaneous symmetry breaking is a process where the symmetry of a physical system decreases, and a new property (typically mass) emerges.
For example, suppose a magnet is heated to a high temperature. Then all of its molecules, which are each themselves tiny magnets, jostle and wiggle about randomly. In this heated state the material has rotational symmetry and no net magnet field. However, as the material cools, one molecule happens to settle down first. As the neighboring molecules settle down, they align themselves with the first molecule, until all the molecules settle down in alignment with the first.Footnote 5 The orientation of the first settled molecule then determines the direction of magnetization for the whole material, and the material no longer has rotational symmetry. One says that the rotational symmetry of the heated magnet was spontaneously broken as it cooled, and a global magnetic field emerged.Footnote 6
Higgsing is a way of using spontaneous symmetry breaking to turn a quantum field theory with a massless field and more symmetry into a theory with a massive field and less symmetry. Here mass (vev’s) takes the place of magnetization, gauge symmetry (or the rank of the gauge group) takes the place of rotational symmetry, and energy scale (RG flow) takes the place of temperature.
The recent discovery of the Higgs boson at the Large Hadron Collider is another example of Higgsing.Footnote 7
Higgsing in quiver gauge theories
We now give our main example. Suppose an arrow a in a quiver gauge theory with dimension \(1^{Q_0}\) is contracted to a vertex e. We make two observations:
-
1.
the rank of the gauge group drops by one since the head and tail of a become identified as the single vertex e;
-
2.
a has zero vev at any representation where a is represented by zero, while e can never have zero vev since it is a vertex, and X only consists of representation isoclasses with dimension \(1^{Q_0}\).
We therefore see that contracting an arrow to a vertex is a form of Higgsing in quiver gauge theories with dimension \(1^{Q_0}\).Footnote 8
In the context of a 4-dimensional \(\mathcal {N} = 1\) quiver gauge theory with quiver Q, the Higgsing we consider in this article is related to RG flow. We start with a non-superconformal (strongly coupled) quiver theory Q which admits a low energy effective description, give nonzero vev’s to a set of bifundamental fields \(Q_1^*\), and obtain a new theory \(Q'\) that lies at a superconformal fixed point.
The mesonic chiral ring and the cycle algebra
The cycle algebra S, introduced in [4], is similar to the mesonic chiral ring in the corresponding quiver gauge theory. In such a theory, the mesonic operators, which are the gauge invariant operators, are generated by the cycles in the quiver. If the gauge group is abelian, then the dimension vector is \(1^{Q_0}\). In the case of a dimer theory with abelian gauge group, two disjoint cycles may share the same \(\bar{\tau }\psi \)-image, but take different values on a point of the vacuum moduli space. These cycles would then be distinct elements in the mesonic chiral ring, although they would be identified in the cycle algebra S; see [4, Remark 3.17].
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Beil, C. Dimer Algebras, Ghor Algebras, and Cyclic Contractions. Algebr Represent Theor 27, 547–582 (2024). https://doi.org/10.1007/s10468-023-10224-y
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DOI: https://doi.org/10.1007/s10468-023-10224-y