Kenneth A. Intriligator

Lectures on Supersymmetric Gauge Theories and Electric-Magnetic Duality

We review some of the recent work on the dynamics of four dimensional, supersymmetric gauge theories. The kinematics are largely determined by holomorphy and the dynamics are governed by duality. The results shed light on the phases of gauge theories. Some results and interpretations which have not been published before are also included.

Exact superpotentials, quantum vacua and duality in supersymmetric SP(Nc) gauge theories

Abstract We study N = 1 supersymmetric SP ( N c ) gauge theories with N f flavors of quarks in the fundamental representation. Depending on N f and N c , we find exact, dynamically generated superpotentials, smooth quantum moduli spaces of vacua, quantum moduli spaces of vacua with additional massless composites at strong coupling, confinement without chiral symmetry breaking, non-trivial fixed points of the renormalization group, and massless magnetic quarks and gluons.

Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(nc) gauge theories

Abstract We study supersymmetric SO(Nc) gauge theories with Nf flavors of quarks in the vector representation. Among the phenomena we find are dynamically generated superpotentials with physically inequivalent branches, smooth moduli spaces of vacua, confinement and oblique confinement, confinement without chiral symmetry breaking, massless composites (glueballs, exotics, monopoles and dyons), non-trivial fixed points of the renormalization group and massless magnetic quarks and gluons. Our analysis sheds new light on a recently found duality in N = 1 supersymmetric theories. The dual forms of some of the theories exhibit “quantum symmetries” which involve non-local transformations on the fields. We find that in some cases the duality has both S and T transformations generating SL( 2, Z ) (only an S3 quotient of which is realized nontrivially). They map the original non-Abelian electric theory to magnetic and dyonic non-Abelian theories. The magnetic theory gives a weak coupling description of confinement while the dyonic theory gives a weak coupling description of oblique confinement. Our analysis also shows that ] the duality in N = 1 is a generalization of the Montonen-Olive duality of N = 4 theories.

Phases of N = 1 supersymmetric gauge theories in four dimensions

We exhibit N = 1 supersymmetric field theories in confining, Coulomb and Higgs phases. The superpotential and the gauge kinetic terms are holomorphic and can be determined exactly in the variuos phases. The Coulomb phase generically has points with massless monopoles. When they condense, the theory undergoes a phase transition to a confining phase. When there are points in the Coulomb phase with massless electric charges, their condensation leads to a transition to a Higgs phase. When the Higgs and confinement phases are distinct, we expect to find massless interacting gluons at the transition point between them.

New examples of duality in chiral and non-chiral supersymmetric gauge theories

Abstract We present evidence for renormalization group fixed points with dual magnetic descriptions in fourteen new classes of four-dimensional N = 1 supersymmetric models. Nine of these classes are chiral and many involve two or three gauge groups. These theories are generalizations of models presented earlier by Seiberg, by Kutasov and Schwimmer, and by the present authors. The different classes are interrelated; one can flow from one class to another using confinement or symmetry breaking.

Proposal for a simple model of dynamical SUSY breaking

Abstract We discuss supersymmetric SU (2) gauge theory with a single matter field in the I = 3 2 representation. This theory has a moduli space of exactly degenerate vacua. Classically it is the complex plane with an orbifold singularity at the origin. There seem to be two possible candidates for the quantum theory at the origin. In both the global chiral symmetry is unbroken. The first is interacting quarks and gluons at a non-trivial infrared fixed point — a non-Abelian Coulomb phase. The second, which we consider more likely, is a confining phase where the singularity is simply smoothed out. If this second, more likely, possibility is realized, supersymmetry will dynamically break when a tree level superpotential is added. This would be the simplest known gauge theory which dynamically breaks supersymmetry.

“Integrating in” and exact superpotentials in 4d

Abstract We discuss integrating out matter fields and integrating in matter fields in four dimensional supersymmetric gauge theories. Highly nontrivial exact superpotentials can be easily obtained by starting from a known theory and integrating in matter.

New RG fixed points and duality in supersymmetric SP(Nc) and SO(Nc) gauge theories

Abstract We present evidence for new, non-trivial RG fixed points with dual magnetic descriptions in N = 1 supersymmetric SP ( N c ) and SO ( N c ) gauge theories. The SP ( N c ) case involves matter X in the antisymmetric tensor representation and N f flavors of quarks Q in the fundamental representation. The SO ( N c ) case involves matter X in the symmetric tensor representation and N f flavors of quarks Q in the vector representation of SO ( N c ). Perturbing these theories by superpotentials W ( X ), we find a variety of interesting RG fixed points with dual descriptions. The duality in these theories is similar to that found by Kutasov and by Kutasov and Schwimmer in SU ( N c ) with adjoint X and N f quarks in the fundamental.

Anomaly matching and a Hopf–Wess–Zumino term in six-dimensional, N=(2,0) field theories

Abstract We point out that the low energy theory of 6d N =(2,0) field theories, when away from the origin of the moduli space of vacua, necessarily includes a new kind of Wess–Zumino term. The form of this term is related to the Hopf invariant associated with π 7 ( S 4 ). The coefficient of the Wess–Zumino term is fixed by an anomaly matching relation for a global flavor symmetry. For example, in the context of a single M5 brane probe in the background of N distant M5 branes, the probe must have the Hopf–WZ term with coefficient proportional to N ( N +1). Various related checks and observations are made. We also point out that there are skyrmionic strings, and propose that they are the W -boson strings.

Exact N=2 Landau-Ginzburg flows

Abstract We find exactly solvable N = 2-supersymmetric flows whose infrared fixed point sare the N = 2 minimal models. The exact S -matrices and the Casimir energy (a c -function) are determined along the entire renormalization-group trajectory. The c -functions run from c = 3 (asymptotically) to the N = 2 minimal-model values, leading us to interpret these theories as the Landau-Ginzburg models with superpotential X k +2 . The calculation of the elliptic genus is consistent with this interpretation. We also find an integrable model in this hierarchy with spontaneously broken supersymmetry and superpotential X , and a series of integrable models with (0,2) supersymmetry. The flows exhibit interesting behavior in the UV, including a relation to the N = 2 super sine-Gordon model. We speculate about the relation between the kinetic term and the cigar target-space metric.