Differential Regularization of a Non-relativistic Anyon Model

Differential regularization is applied to a field theory of a non-relativistic charged boson field $\phi$ with $\lambda (\phi {}^{*} \phi)^2$ self-interaction and coupling to a statistics-changing $U(1)$ Chern-Simons gauge field. Renormalized configuration-space amplitudes for all diagrams contributing to the $\phi {}^{*} \phi {}^{*} \phi \phi$ 4-point function, which is the only primitively divergent Green's function, are obtained up to 3-loop order. The renormalization group equations are explicitly checked, and the scheme dependence of the $\beta$-function is investigated. If the renormalization scheme is fixed to agree with a previous 1-loop calculation, the 2- and 3-loop contributions to $\beta(\lambda,e)$ vanish, and $\beta(\lambda,e)$ itself vanishes when the ``self-dual'' condition relating $\lambda$ to the gauge coupling $e$ is imposed.Comment: 22 pages in ReVTEX (with a plaintext PostScript figure appended at end), MIT CTP #221

Nonperturbative Formulas for Central Functions of Supersymmetric Gauge Theories

For quantum field theories that flow between ultraviolet and infrared fixed points, central functions, defined from two-point correlators of the stress tensor and conserved currents, interpolate between central charges of the UV and IR critical theories. We develop techniques that allow one to calculate the flows of the central charges and that of the Euler trace anomaly coefficient in a general N=1 supersymmetric gauge theory. Exact, explicit formulas for $SU(N_c)$ gauge theories in the conformal window are given and analysed. The Euler anomaly coefficient always satisfies the inequality $$. This is new evidence in strongly coupled theories that this quantity satisfies a four-dimensional analogue of the $c$-theorem, supporting the idea of irreversibility of the RG flow. Various other implications are discussed.Comment: latex, 27 page

Extremal Correlators in the AdS/CFT Correspondence

The non-renormalization of the 3-point functions $tr X^{k_1} tr X^{k_2} tr X^{k_3}$ of chiral primary operators in N=4 super-Yang-Mills theory is one of the most striking facts to emerge from the AdS/CFT correspondence. A two-fold puzzle appears in the extremal case, e.g. k_1 = k_2 + k_3. First, the supergravity calculation involves analytic continuation in the k_i variables to define the product of a vanishing bulk coupling and an infinite integral over AdS. Second, extremal correlators are uniquely sensitive to mixing of the single-trace operators $tr X^k$ with protected multi-trace operators in the same representation of SU(4). We show that the calculation of extremal correlators from supergravity is subject to the same subtlety of regularization known for the 2-point functions, and we present a careful method which justifies the analytic continuation and shows that supergravity fields couple to single traces without admixture. We also study extremal n-point functions of chiral primary operators, and argue that Type IIB supergravity requires that their space-time form is a product of n-1 two-point functions (as in the free field approximation) multiplied by a non-renormalized coefficient. This non-renormalization property of extremal n-point functions is a new prediction of the AdS/CFT correspondence. As a byproduct of this work we obtain the cubic couplings $t \phi \phi$ and $s \phi \phi$ of fields in the dilaton and 5-sphere graviton towers of Type IIB supergravity on $AdS_5 \times S^5$.Comment: 26 pages, LateX, no figure

Comments on 4-point functions in the CFT/AdS correspondence

We study the four--point function of chiral primaries corresponding to the dilaton--axion sector in supergravity in the $AdS_5$/CFT$_4$ correspondence. We find relations between some of the supergravity graphs and compute their leading singularities. We discuss the issue of logarithmic singularities and their significance for the OPE structure of the CFT.Comment: Latex, 12 pages, Some typos correcte