theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...

, a quiver diagram is a graph representing the

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic pa ...

content of a

gauge theory In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...

that describes

D-branes In string theory, D-branes, short for Dirichlet membrane, are a class of extended objects upon which open string (physics), strings can end with Dirichlet boundary conditions, after which they are named. D-branes are typically classified by their ...

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space that is locally a finite group quotient of a Euclidean space. D ...

s. Quiver diagrams may also be used to described

\mathcal = 2

supersymmetric gauge theories in four dimensions. Each node of the graph corresponds to a factor ''U''(''N'') of the

gauge group A gauge group is a group of gauge symmetries of the Yang–Mills gauge theory of principal connections on a principal bundle. Given a principal bundle P\to X with a structure Lie group G, a gauge group is defined to be a group of its vertical ...

, and each link represents a field in the bifundamental representation :

(M,\bar)

. The relevance of quiver diagrams for

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

was pointed out and studied by

Michael Douglas Michael Kirk Douglas (born September 25, 1944) is an American actor and film producer. He has received numerous accolades, including two Academy Awards, five Golden Globe Awards, a Primetime Emmy Award, the Cecil B. DeMille Award, and the ...

and Greg Moore. While string theorists use the words ''quiver diagram'', many of their colleagues in

particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...

call these diagrams ''mooses''.

Definition

For convenience, consider the supersymmetric

\mathcal  =1

gauge theory in four-dimensional spacetime. The quiver gauge theory is given by the following data: * Finite quiver

Q

* Each vertex

v\in \operatorname  (Q)

corresponds to a compact Lie group

G_

. This can be the unitary group

U(N)

, the special unitary group

SU(N)

, special orthogonal group

SO(N)

or symplectic group

USp(N)

. * The gauge group is the product

\textstyle \prod _G_

. * Each edge of Q

e\colon u\to v

corresponds to the defining representation

_\otimes N_

. There is a corresponding superfield

\Phi _

. This representation is called a bifundamental representation. For example, if

u

and

v

corresponds to

S U ( 2 )

and

S U ( 3 )

then the edge corresponds to a six-dimensional representation

_\otimes _

In this case, the quiver gauge theory is a four-dimensional

\mathcal  =1

supersymmetric gauge theory. The quiver gauge theory in higher dimensions can be defined similarly. The quiver is particularly convenient for representing conformal gauge theory. The structure of the quiver makes it easy to check whether the theory preserves conformal symmetry.

Definition

References

See also