Damiano Anselmi

Higher-spin current multiplets in operator-product expansions

Irreducible currents with arbitrary spin are constructed in a general spacetime dimension, in the free-field limit and, at the bare level, in the presence of interactions. For the n -dimensional generalization of the (conformal) vector field, the (n /2-1)-form is used. Two-point functions and higher-spin central charges are evaluated at one-loop. As an application, the higher-spin hierarchies generated by the stress tensor operator-product expansion are computed in supersymmetric theories. The results exhibit an interesting universality.

Anomalies, Unitarity, and Quantum Irreversibility

Abstract The trace anomaly in external gravity is the sum of three terms at criticality: the square of the Weyl tensor, the Euler density and □R, with coefficients, properly normalized, called c, a, and a′, the latter being ambiguously defined by an additive constant. Considerations about unitarity and positivity properties of the induced actions allow us to show that the total RG flows of a and a′ are equal and therefore the a′-ambiguity can be consistently removed through the identification a′=a. The picture that emerges clarifies several long-standing issues. The interplay between unitarity and renormalization implies that the flux of the renormalization group is irreversible. A monotonically decreasing a-function interpolating between the appropriate values is naturally provided by a′. The total a-flow is expressed non-perturbatively as the invariant (i.e., scheme-independent) area of the graph of the beta function between the fixed points. We test this prediction to the fourth loop order in perturbation theory, in QCD with Nf≲11/2Nc and in supersymmetric QCD. There is agreement also in the absence of an interacting fixed point (QED and ϕ4-theory). Arguments for the positivity of a are also discussed.

A note on the holographic beta and C functions

Abstract The holographic RG flow in AdS/CFT correspondence naturally defines a holographic scheme in which the central charge c and the beta function are related by the formula c =−2cβ a β b G ab , where G ab is the metric of the kinetic term of the supergravity scalars. In particular, the metric in the space of couplings is f ab =2 cG ab . We perform some checks of that result and we compare it with the quantum field theory expectations. We discuss alternative definitions of the c -function. In particular, we compare, for a particular supersymmetric flow, the holographic c -function with the central charge computed directly from the two-point function of the stress-energy tensor.

Subleading corrections and central charges in the AdS/CFT correspondence

Abstract We explore subleading contributions to the two basic central charges c and a of four-dimensional conformal field theories in the AdS/CFT scheme. In particular we probe subleading corrections to the difference c − a from the string-theory side. In the N =4 CFT, c − a vanish identically consistently with the string-theory expectations. However, for N =1 and N =2 CFTs, the U R (1) anomaly, which is proportional to c − a , is subleading in the large N limit for theories in the AdS/CFT context and one expects string one-loop R 2 and B ∧ R ∧ R terms in the low energy effective action. We identify these terms as coming from the R 4 terms. Similar considerations apply to the U R (1) 3 anomaly which is, however, subleading only for N =2 theories. As a result, a string one-loop term B ∧ F ∧ F should exist in the low energy effective action of the N =4 five-dimensional supergravity. The U R (1) 3 term is leading for the N =1 CFT and it is indeed present in the N =2 five-dimensional supergravity.

Quantum irreversibility in arbitrary dimension

Abstract Some recent ideas are generalized from four dimensions to the general dimension n. In quantum field theory, two terms of the trace anomaly in external gravity, the Euler density Gn and □n/2−1R, are relevant to the problem of quantum irreversibility. By adding the divergence of a gauge-invariant current, Gn can be extended to a new notion of Euler density G n , linear in the conformal factor. We call it pondered Euler density. This notion relates the trace-anomaly coefficients a and a′ of Gn and □n/2−1R in a universal way (a=a′) and gives a formula expressing the total RG flow of a as the invariant area of the graph of the beta function between the fixed points. I illustrate these facts in detail for n=6 and check the prediction to the fourth-loop order in the ϕ3-theory. The formula of quantum irreversibility for general n even can be extended to n odd by dimensional continuation. Although the trace anomaly in external gravity is zero in odd dimensions, I show that the odd-dimensional formula has a predictive content.

Quantum conformal algebras and closed conformal field theory

Abstract We investigate the quantum conformal algebras of N = 2 and N = 1 supersymmetric gauge theories. Phenomena occurring at strong coupling are analysed using the Nachtmann theorem and very general, model-independent, arguments. The results lead us to introduce a novel class of conformal field theories, identified by a closed quantum conformal algebra. We argue that they are the exact solution to the strongly coupled large- N c limit of the open conformal field theories. We study the basic properties of closed conformal field theory and work out the operator product expansion of the conserved current multiplet ⊤. The OPE structure is uniquely determined by two central charges, c and a , where c has a direct interpretation as the central extension of the algebra, while the combination 1 − a c is a structure constant. When the ratio c a is different from 1, the multiplet ⊤ contains the stress-tensor, R -currents and finite mass operators. On the other hand, an open algebra contains an infinite tower of non-conserved currents, organized in pairs and singlets with respect to renormalization mixing. ⊤ mixes with a second multiplet ⊤ ∗ and the main consequence is that c and a have different subleading corrections. The closed algebra simplifies considerably at c = a , where it coincides with the N = 4 one and ⊤ contains just the stress-tensor.

Towards the classification of conformal field theories in arbitrary even dimension

Abstract I identify the class of even-dimensional conformal field theories that is most similar to two-dimensional conformal field theory. In this class the formula, elaborated recently, for the irreversibility of the renormalization-group flow applies also to massive flows. This implies a prediction for the ratio between the coefficient of the Euler density in the trace anomaly (charge a ) and the stress-tensor two-point function (charge c ). More precisely, the trace anomaly in external gravity is quadratic in the Ricci tensor and the Ricci scalar and contains a unique central charge. I check the prediction in detail in four, six and eight dimensions, and then in arbitrary even dimension.

Large-N expansion, conformal field theory and renormalization-group flows in three dimensions

I study a class of interacting conformal field theories and conformal windows in three dimensions, formulated using the Parisi large-N approach and a modified dimensional-regularization technique. Bosons are associated with composite operators and their propagators are dynamically generated by fermion bubbles. Renormalization-group flows between pairs of interacting fixed points satisfy a set of non-perturbative g1/g dualities. There is an exact relation between the beta function and the anomalous dimension of the composite boson. Non-abelian gauge fields have a non-renormalized and quantized gauge coupling, although no Chern-Simons term is present. A problem of the naive dimensional-regularization technique for these theories is uncovered and removed with a non-local, evanescent, non-renormalized kinetic term. The models are expected to be a fruitful arena for the study of odd-dimensional conformal field theory.

Irreversibility and higher-spin conformal field theory

I discuss the properties of the central charges c and a for higher-derivative and higher-spin theories (spin 2 included). Ordinary gravity does not admit a straightforward identification of c and a in the trace anomaly, because it is not conformal. On the other hand, higher-derivative theories can be conformal, but have negative c and a. A third possibility is to consider higher-spin conformal field theories. They are not unitary, but have a variety of interesting properties. Bosonic conformal tensors have a positive-definite action, equal to the square of a field strength, and a higher-derivative gauge invariance. There exists a conserved spin-2 current (not the canonical stress tensor) defining positive central charges c and a. I calculate the values of c and a and study the operator-product structure. Higher-spin conformal spinors have no gauge invariance, admit a standard definition of c and a and can be coupled to Abelian and non-Abelian gauge fields in a renormalizable way. At the quantum level, they contribute to the one-loop beta function with the same sign as ordinary matter, admit a conformal window and non-trivial interacting fixed points. There are composite operators of high spin and low dimension, which violate the Ferrara-Gatto-Grillo theorem. Finally, other theories, such as conformal antisymmetric tensors, exhibit more severe internal problems. This research is motivated by the idea that fundamental quantum field theories should be renormalization-group (RG) interpolations between ultraviolet and infrared conformal fixed points, and quantum irreversibility should be a general principle of nature.

A New Perspective on the Philosophical Implications of Quantum Field Theory

I discuss issues concerning the philosophical foundations andimplications of quantum field theory, renormalization inparticular. A new understanding of the correspondence principle,an unexpected role of perturbation theory and, most of all, acriterion to reduce the set of consistent theories frominfinitely many to finitely many, are the key concepts of atheoretical set-up that appears to overcome in a natural wayvarious consistency problems of quantum mechanics and offerseveral hints for further developments.