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Register a new userThe a-theorem for the four-dimensional gauged vector model
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The discussion of renormalization group flows in four-dimensional conformal field theories has recently focused on the a-anomaly. It has recently been shown that there is a monotonic decreasing function which interpolates between the ultraviolet and infrared fixed points such that Delta a = a_UV - a_IR > 0. The analysis has been extended to weakly relevant and marginal deformations, though there are few explicit examples involving interacting theories. In this paper we examine the a-theorem in the context of the gauged vector model which couples the usual vector model to the Banks-Zaks model. We consider the model to leading order in the 1/N expansion, all orders in the coupling constant lambda, and to second order in g^2. The model has both an IR and UV fixed point, and satisfies Delta a > 0.
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A calculation of the renormalization group improved effective potential for the gauged U(N) vector model, coupled to $N_f$ fermions in the fundamental representation, computed to leading order in 1/N, all orders in the scalar self-coupling $lambda$, and lowest order in gauge coupling $g^2$, with $N_f$ of order $N$, is presented. It is shown that the theory has two phases, one of which is asymptotically free, and the other not, where the asymptotically free phase occurs if $0 < lambda /g^2 < {4/3} (frac{N_f}{N} - 1)$, and $frac{N_f}{N} < {11/2}$. In the asymptotically free phase, the effective potential behaves qualitatively like the tree-level potential. In the other phase, the theory exhibits all the difficulties of the ungauged $(g^2 = 0)$ vector model. Therefore the theory appears to be consistent (only) in the asymptotically free phase.
In this paper we introduce a new method for generating gauged sigma models from four-dimensional Chern-Simons theory and give a unified action for a class of these models. We begin with a review of recent work by several authors on the classical generation of integrable sigma models from four dimensional Chern-Simons theory. This approach involves introducing classes of two dimensional defects into the bulk on which the gauge field must satisfy certain boundary conditions. By solving the equations of motion of the gauge one finds an integrable sigma models by substituting the solution back into the action. This integrability is guaranteed because the gauge field is gauge equivalent to the Lax connection of the sigma model. By considering a theory with two four-dimensional Chern-Simons fields coupled together on two dimensional surfaces in the bulk we are able to introduce new classes of `gauged defects. By solving the bulk equations of motion we find a unified action for a set of genus zero integrable gauged sigma models. The integrability of these models is guaranteed as the new coupling does not break the gauge equivalence of the gauged fields to their Lax connections. Finally, we consider a couple of examples in which we derive the gauged Wess-Zumino-Witten and Nilpotent gauged Wess-Zumino-Witten models. This latter model is of note given one can find the conformal Toda models from it.
The formulation of the non-linear sigma model in terms of flat connection allows the construction of a perturbative solution of a local functional equation encoding the underlying gauge symmetry. In this paper we discuss some properties of the solution at the one-loop level in D=4. We prove the validity of a weak power-counting theorem in the following form: although the number of divergent amplitudes is infinite only a finite number of divergent amplitudes have to be renormalized. The proof uses the linearized functional equation of which we provide the general solution in terms of local functionals. The counteterms are given in terms of linear combinations of these invariants and the coefficients are fixed by a finite number of divergent amplitudes. The latter contain only insertions of the composite operators $phi_0$ (the constraint of the non-linear sigma model) and $F_mu$ (the flat connection). These amplitudes are at the top of a hierarchy implicit in the functional equation. As an example we derive the counterterms for the four-point amplitudes.
We compute the supersymmetric partition function of $mathcal{N}{=}1$ supersymmetric gauge theories with an $R$-symmetry on $mathcal{M}_4 cong mathcal{M}_{g,p}times S^1$, a principal elliptic fiber bundle of degree $p$ over a genus-$g$ Riemann surface, $Sigma_g$. Equivalently, we compute the generalized supersymmetric index $I_{mathcal{M}_{g,p}}$, with the supersymmetric three-manifold ${mathcal{M}_{g,p}}$ as the spatial slice. The ordinary $mathcal{N}{=}1$ supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological $A$-model for the abelianized gauge fields on the base $Sigma_g$. This $A$-model---or $A$-twisted two-dimensional $mathcal{N}{=}(2,2)$ gauge theory---encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on $T^2$ inside $Sigma_g times T^2$. Being defined by compactification on the torus, the $A$-model also enjoys natural modular properties, governed by the four-dimensional t Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.
In cosmological perturbation theory it is convenient to use the scalar, vector, tensor (SVT) basis as defined according to how these components transform under 3-dimensional rotations. In attempting to solve the fluctuation equations that are automatically written in terms of gauge-invariant combinations of these components, the equations are taken to break up into separate SVT sectors, the decomposition theorem. Here, without needing to specify a gauge, we solve the fluctuation equations exactly for some standard cosmologies, to show that in general the various gauge-invariant combinations only separate at a higher-derivative level. To achieve separation at the level of the fluctuation equations themselves one has to assume boundary conditions for the higher-derivative equations. While asymptotic conditions suffice for fluctuations around a dS background or a $k=0$ RW background, for fluctuations around a $k eq 0$ RW background one additionally has to require that the fluctuations be well-behaved at the origin. We show that in certain cases the gauge-invariant combinations themselves involve both scalars and vectors. For such cases there is no decomposition theorem for the individual SVT components themselves, but for the gauge-invariant combinations there still can be. Given the lack of manifest covariance in defining a basis with respect to 3-dimensional rotations, we introduce an alternate SVT basis whose components are defined according to how they transform under 4-dimensional general coordinate transformations. With this basis the fluctuation equations greatly simplify, and while one can again break them up into separate gauge-invariant sectors at the higher-derivative level, in general we find that even with boundary conditions we do not obtain a decomposition theorem in which the fluctuations separate at the level of the fluctuation equations themselves.
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