Non-abelian finite gauge theories (original) (raw)

On conformal field theories in four dimensions

Nuclear Physics B, 1998

Extending recent work of Kachru and Silverstein, we consider "orbifolds" of 4-dimensional N = 4 SU (n) super-Yang-Mills theories with respect to discrete subgroups of the SU (4) R-symmetry which act nontrivially on the gauge group. We show that for every discrete subgroup of SU (4) there is a canonical choice of imbedding of the discrete group in the gauge group which leads to theories with a vanishing one-loop beta-function. We conjecture that these give rise to (generically non-supersymmetric) conformal theories. The gauge group is ⊗ i SU (Nn i ) where n i denote the dimension of the irreducible representations of the corresponding discrete group; there is also bifundamental matter, dictated by associated quiver diagrams. The interactions can also be read off from the quiver diagram. For SU (3) and SU (2) subgroups this leads to superconformal theories with N = 1 and N = 2 respectively.

Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories

2012

Seiberg-Witten geometry of mass deformed N=2 superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space M of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space of holomorphic G-bundles on a (possibly degenerate) elliptic curve defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group G. The integrable systems underlying, or, rather, overlooking the special geometry of M are identified. The moduli spaces of framed G-instantons on R^2xT^2, of G-monopoles with singularities on R^2xS^1, the Hitchin systems on curves with punctures, as well as various spin chains play an important role in our story. We also comment on the higher dimensional theories. In the companion paper the quantum integrable systems and their connections to the representation theory of quantum affine algebras will be discussed

Quantum space-times and finite effects in 4D super Yang–Mills theories

Nuclear Physics B, 2000

The truncation in the number of single-trace chiral primary operators of N = 4 SYM and its conjectured connection with gravity on quantum spacetimes are elaborated. The model of quantum spacetime we use is AdS 5 q ×S 5 q for q a root of unity. The quantum sphere is defined as a homogeneous space with manifest SU q (3) symmetry, but as anticipated from the field theory correspondence, we show that there is a hidden SO q (6) symmetry in the constrution. We also study some properties of quantum space quotients as candidate models for the quantum spacetime relevant for some Z n quiver quotients of the N = 4 theory which break SUSY to N = 2. We find various qualitative agreements between the proposed models and the properties of the corresponding finite N gauge theories.

Classification of 4d $ \mathcal{N} $ =2 gauge theories

Journal of High Energy Physics, 2013

We classify all possible four-dimensional N =2 supersymmetric UV-complete gauge theories composed of semi-simple gauge groups and hypermultiplets. We also give appropriate references for all theories with known Seiberg-Witten solutions. Contents 1 Introduction 1 2 Defining data of N =2 gauge theories 4 3 The classification, part I 5 3.1 Listing available nodes 6 3.2 Listing available n-gons 6 3.3 Reduction to the classification of conformal theories 7 3.4 Classification of theories with 3-gons 10 3.5 Classification of theories with simple gauge groups 10 4 The classification, part II 14 4.1 Three rare cases 15 4.2 Overall structure of the remaining theories 15 4.3 Branches 17 4.4 Main classification 24 4.5 Status of Seiberg-Witten solutions 39

Pure N = 2 mathcalN=2\mathcal{N}=2mathcalN=2 super Yang-Mills and exact WKB

Journal of High Energy Physics, 2015

We apply exact WKB methods to the study of the partition function of pure N = 2 i-deformed gauge theory in four dimensions in the context of the 2d/4d correspondence. We study the partition function at leading order in 2 / 1 (i.e. at large central charge) and in an expansion in 1. We find corrections of the form ∼ exp[− SW periods 1 ] to this expansion. We attribute these to the exchange of the order of summation over gauge instanton number and over powers of 1 when passing from the Nekrasov form of the partition function to the topological string theory inspired form. We conjecture that such corrections should be computable from a worldsheet perspective on the partition function. Our results follow upon the determination of the Stokes graphs associated to the Mathieu equation with complex parameters and the application of exact WKB techniques to compute the Mathieu characteristic exponent.

Non-perturbative studies of N=2 conformal quiver gauge theories

Fortschritte der Physik, 2015

We study N = 2 super-conformal field theories in four dimensions that correspond to mass-deformed linear quivers with n gauge groups and (bi-)fundamental matter. We describe them using Seiberg-Witten curves obtained from an M-theory construction and via the AGT correspondence. We take particular care in obtaining the detailed relation between the parameters appearing in these descriptions and the physical quantities of the quiver gauge theories. This precise map allows us to efficiently reconstruct the non-perturbative prepotential that encodes the effective IR properties of these theories. We give explicit expressions in the cases n = 1, 2, also in the presence of an Ω-background in the Nekrasov-Shatashvili limit. All our results are successfully checked against those of the direct microscopic evaluation of the prepotentialà la Nekrasov using localization methods.

Instanton expansions for mass deformed N = 4 super Yang-Mills theories

Nuclear Physics B, 1998

We derive modular anomaly equations from the Seiberg-Witten-Donagi curves for softly broken N = 4 SU (n) gauge theories. From these equations we can derive recursion relations for the pre-potential in powers of m 2 , where m is the mass of the adjoint hypermultiplet. Given the perturbative contribution of the pre-potential and the presence of "gaps" we can easily generate the m 2 expansion in terms of polynomials of Eisenstein series, at least for relatively low rank groups. This enables us to determine efficiently the instanton expansion up to fairly high order for these gauge groups, e. g. eighth order for SU (3). We find that after taking a derivative, the instanton expansion of the pre-potential has integer coefficients. We also postulate the form of the modular anomaly equations, the recursion relations and the form of the instanton expansions for the SO(2n) and E n gauge groups, even though the corresponding Seiberg-Witten-Donagi curves are unknown at this time.

N = 2 super Yang-Mills and subgroups of

Nuclear Physics B, 1996

We discuss SL(2, Z) subgroups appropriate for the study of N = 2 Super Yang-Mills with N f = 2n flavors. Hyperelliptic curves describing such theories should have coefficients that are modular forms of these subgroups. In particular, uniqueness arguments are sufficient to construct the SU (3) curve, up to two numerical constants, which can be fixed by making some assumptions about strong coupling behavior. We also discuss the situation for higher groups. We also include a derivation of the closed form β-function for the SU (2) and SU theories without matter, and the massless theories with N f = n.

$${\mathcal{N} = 2}$$ N = 2 Quiver Gauge Theories on A-type ALE Spaces

Letters in Mathematical Physics, 2014

We survey and compare recent approaches to the computation of the partition functions and correlators of chiral BPS observables in N = 2 gauge theories on ALE spaces based on quiver varieties and the minimal resolution X k of the A k−1 toric singularity C 2 /Z k , in light of their recently conjectured duality with two-dimensional coset conformal field theories. We review and elucidate the rigorous constructions of gauge theories for a particular family of ALE spaces, using their relation to the cohomology of moduli spaces of framed torsion free sheaves on a suitable orbifold compactification of X k . We extend these computations to generic N = 2 superconformal quiver gauge theories, obtaining in these instances new constraints on fractional instanton charges, a rigorous proof of the Nekrasov master formula, and new quantizations of Hitchin systems based on the underlying Seiberg-Witten geometry. N = 2 QUIVER GAUGE THEORIES ON ALE SPACES provide a rigorous proof of the Nekrasov master formula for X k . We further explore the Seiberg-Witten geometry and elaborate on the interpretations of these gauge theories as quantizations of certain Hitchin systems. Two appendices at the end of the paper summarise some technical details which are used in the main text: in Appendix A we discuss some aspects of equivariant cohomology which are used to compute instanton partition functions, while in Appendix B we list some of the factors which enter the explicit expressions for the partition functions.