nLab quiver gauge theory

Idea

General

A quiver gauge theory is a gauge theory – usually a super Yang-Mills theory – whose field content is determined by a quiver in that to each node $x_i$ of the quiver is assigned a gauge group $U(n_i)$ and to each edge a fermion field species which is charged as the fundamental representation under the gauge group of the target and the anti-fundamental representation of the domain of the quiver edge.

Such data naturally arises in geometric engineering of quantum field theory from D-brane models in string theory with D-branes located at certain singularities of a Calabi-Yau variety: the nodes $x_i$ of the quiver for a $(d+1)$-dimensional quiver gauge theory correspond to $n_i$ coincident $d$-D-branes and the edges of the quiver to open string states stretching between these branes. Under this interpretation, a representation of the quiver corresponds to giving all these fields a vacuum expectation value.

As fixed point locus

Consider a $U(N)$ Chan-Paton gauge field theory on D-branes at a $G$-orbifold singularity $\mathbb{C}^n\sslash G$ for $\Gamma$ a finite group.

Then the fermions transform in $\mathbf{c} \otimes (N \mathbf[r]) \otimes (N \mathbf{r}^\ast)$, where $\mathbf{r}$ denotes the regular representation of $\Gamma$ and $\mathbf{c}$ denotes the representation which defines the orbifold action.

Then the $\Gamma$-fixed point space inside this representation is

$$\big( \mathbf{c} \otimes (N \mathbf{r}) \otimes (N \mathbf{r}^\ast) \big)^\Gamma$$

is the corresponding weighted McKay quiver matrix.

This observation is due to Lawrence-Nekrasov-Vafa 98, Section 2.1

Properties

Relation between coherent sheaves and quiver representations

The representation theoretic aspects of the gauge theory thus obtained depend only on aspects which are already seen by the B-model topological string, which in this context plays the role of a simplified version of the physical superstring itself. By the discussion at TCFT this topological field theory is essentially determined by its derived category of B-branes, which is a derived category of coherent sheaves on the given target spacetime. Localized to suitable singularities the B-branes decompose into fractional branes which is mathematically reflected by the existence of exceptional collections of objects in the derived category. The endomorphism algebra of the direct sum of the objects in the exceptional collection turns out to be the path algebra of the corresponding quiver and under this identification the derived categories of coherent sheaves and of quiver representations are equivalent (Bondal 90). This equivalence to quiver representations gives one of the ways of getting a concrete combinatorial handle on derived categories of B-branes and hence on properties of quiver gauge theory.

Relation to Seiberg duality

Under geometric engineering of quantum field theory via D-branes situated at ADE-singularities in non-compact Calabi-Yau varieties as above, for instance Seiberg duality of the corresponding quiver gauge field theories may be understood in terms of equivalences of categories of derived quiver representations corresponding to mutations of exceptional collections etc. (Robles-Llana & Rocek 04).

Related concepts

• McKay correspondence

• knots-quivers correspondence

• AGT correspondence

• fractional D-brane

References

General

An original article is

• Michael Douglas, Gregory Moore, D-branes, Quivers, and ALE Instantons (arXiv:hep-th/9603167)

The understanding of the McKay quiver as the fixed point space in the fermion-representation under R-symmetry is due to

• A. Lawrence, Nikita Nekrasov, Cumrun Vafa, On Conformal Theories in Four Dimensions, Nucl.Phys. B533 (1998) 199-209 (arXiv:hep-th/9803015)

In

• Alexei Bondal, Helices, representations of quivers and Koszul algebras, in Helices and vector bundles, vol. 148 of London Math. Soc. Lecture Note Ser., pp. 75–95. Cambridge Univ. Press, Cambridge, 1990.

an exceptional collection in the derived category of coherent sheaves on a suitable Calabi-Yau variety cone is used to induce a quiver and make the derived category be equivalent to that of its quiver representations.

Review and discussion of further details:

• Yang-Hui He, Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities (arXiv:hep-th/0408142)

• Yang-Hui He, Quiver Gauge Theories: Finitude and Trichotomoty, Mathematics 2018, 6(12), 291 (doi:10.3390/math6120291)

• Aaron Bergman, Nicholas Proudfoot, Moduli Spaces for D-branes at the Tip of a Cone, JHEP0603:073, 2006 (arXiv:hep-th/0510158)

• Sheldon Katz, ADE Quiver representations and branes (pdf), lecture 2 of ADE Geometry and dualities (pdf)

• Taro Kimura, Quiver Gauge Theory, Chapter 2 in: Instanton Counting, Quantum Geometry and Algebra, Mathematical Physics Studies, Springer (2021) [doi:10.1007/978-3-030-76190-5_4]

In heterotic string theory in relation to Donaldson-Thomas theory:

• Yang-Hui He, Seung-Joo Lee, Quiver Structure of Heterotic Moduli, J. High Energ. Phys. (2012) 2012: 119 (arXiv:1208.3004)

With emphasis on string phenomenology:

• Angel Uranga, From quiver diagrams to particle physics (arXiv:hep-th/0007173)

Discussion from the point of view of worldsheet 2d CFT is in

• Wolfgang Lerche, A. Lutken, Christoph Schweigert, D-Branes on ALE Spaces and the ADE Classification of Conformal Field Theories, Nucl.Phys. B622 (2002) 269-278 (arXiv:hep-th/0006247)

Discussion for Sasakian manifolds includes

• Olaf Lechtenfeld, Alexander Popov, Richard Szabo, Sasakian quiver gauge theories and instantons on Calabi-Yau cones (arXiv:1412.4409)

Seiberg duality

• David Berenstein, Michael Douglas, Seiberg Duality for Quiver Gauge Theories (arXiv:hep-th/0207027)

• Subir Mukhopadhyay, Koushik Ray, Seiberg duality as derived equivalence for some quiver gauge theories (arXiv:hep-th/0309191)

• Daniel Robles-Llana, Martin Rocek, Quivers, Quotients, and Duality (arXiv:hep-th/0405230)

On an algorithm for constructing S-dual quiver gauge theories:

• Chiung Hwang, Sara Pasquetti, Matteo Sacchi, Rethinking mirror symmetry as a local duality on fields, Phys. Rev. D 106 (2022) 105014 [arXiv:2110.11362, doi:10.1103/PhysRevD.106.105014]

• Riccardo Comi, Chiung Hwang, Fabio Marino, Sara Pasquetti, Matteo Sacchi, The $SL(2, \mathbb{Z})$ dualization algorithm at work [arXiv:2212.10571]