Classification of compactified su(N c ) gauge theories with fermions in all representations
Received: May
ed su(Nc) gauge theories
Mohamed M. Anber 1
Loc Vincent-Genod 0
Portland
U.S.A.
Lausanne
Switzerland
0 Institute de Theorie des Phenomenes Physiques, Ecole Polytechnique Federale de Lausanne
1 Department of Physics, Lewis & Clark College
We classify su(Nc) gauge theories on R representations obeying periodic boundary conditions along S1. In particular, we single out the class of theories that is asymptotically free and weakly coupled in the infrared, and therefore, is amenable to semi-classical treatment. Our study is conducted by carefully identifying the vacua inside the a ne Weyl chamber using Verma bases and Frobenius formula techniques. Theories with fermions in pure representations are generally strongly coupled. The only exceptions are the four-index symmetric representation of su(2) and adjoint representation of su(Nc). However, we nd a plethora of admissible theories with fermions in mixed representations. A sub-class of these theories have degenerate perturbative vacua separated by domain walls. In particular, su(Nc) theories with fermions in the mixed representations adjoint fundamental and adjoint two-index symmetric admit degenerate vacua that spontaneously break the parity P, charge conjugation C, and time reversal T symmetries. These are the rst examples of strictly weakly coupled gauge theoS1 with spontaneously broken C, P, and T symmetries. We also compute the fermion zero modes in the background of monopole-instantons. The monopoles and their composites (topological molecules) proliferate in the vacuum leading to the con nement of electric charges. Interestingly enough, some theories have also accidental degenerate vacua, which are not related by any symmetry. These vacua admit di erent numbers of fermionic zero modes, and hence, di erent kinds of topological molecules. The lack of symmetry, however, indicates that such degeneracy might be lifted by higher order corrections. Finally, we study the general phase structure of adjoint fundamental theories in the small circle and decompacti cation limits. ArXiv ePrint: 1704.08277
Con nement; Field Theories in Lower Dimensions; Solitons Monopoles and
-
3
S1 with massless fermions in higher
ries on R
Instantons
1 Introduction
2 Theory and formulation
3 Asymptotically and anomaly free theories
3.1
Asymptotically free theories
3.2 Anomalies
5 Computation of traces
5.1
Constructing the weights using Verma bases
5.2 The Frobenius formula
6 The global minima of the e ective potential
6.1
6.2
The a ne Weyl chamber
Analytical solutions
6.3 Numerical investigation
4 Integrating out the Kaluza Klein tower: the e ective potential
7 The ow of the 3-D coupling constant
8 The admissible class of theories
8.1
Perturbative vacua and the role of discrete symmetries
8.1.1
8.1.2
Theories with a unique vacuum
Theories with multiple vacua
9
Monopole-instantons and fermion zero modes on R
3
S
1
10 su(Nc) theories with fermions in G
F : a detailed study
11 Summary and future directions
11.1 Future directions
A Lie algebra and conventions
B The Casimir and trace operators, and the dimension of representation
C Cubic Dynkin index
D Constructing the weights using Verma bases
E Frobenius formula and traces of the asymptotically free theories
F Computing the index using Frobenius formula
Introduction
Con ning gauge theories that are analytically calculable in four dimensions are scarce.
In fact, Seiberg-Witten theory on R4 [1, 2] and certain QCD-like theories on R
3
S
1
(where S1 is a spatial rather than thermal circle) [3] are the only two known examples.
Indeed, it has been known for a long time [4{6] that compactifying a gauge theory on a
circle provides a mechanism for the gauge group to spontaneously break to its maximum
abelian subgroup, and hence, for the theory to admit monopole-instantons. The
monopoleinstantons or their composites proliferate in the vacuum causing electric probe charges to
con ne [3]. Surprisingly, the class of con ning gauge theories on a circle has not yet been
su(Nc) ! u(1)Nc 1, and hence, to have an analytical control over the theory. The second
homotopy group 2 SU(Nc)=U(1)Nc 1 = Z (the set of integers) is not trivial, and
therefore, one expects to have stable nonperturbative solutions of the eld equations. These
are the monopoles (or dyons) in 3 + 1-D and monopole-instantons in 3-D. The monopoles
carry magnetic charge, and hence, their proliferation in the vacuum causes electric charges
to con ne. This is an example of the celebrated dual superconductivity [9{11]. The
original mechanism of the monopole condensation in nonabelian theories was introduced by
Polyakov in the context of the 3-D Georgi-Glashow model, while lifting the mechanism to
four dimensions was realized in the seminal work of Seiberg and Witten on N = 2 super
Yang-Mills. A crucial ingredient in this theory is the scalars in the supermultiplet that
cause the gauge group to abelianize, and hence, the monopoles to form. Then, one ca (...truncated)