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The two-point function of exactly marginal operators leads to a universal contribution to the trace anomaly in even dimensions. We study aspects of this trace anomaly, emphasizing its interpretation as a sigma model, whose target space M is the space of conformal field theories (a.k.a. the conformal manifold). When the underlying quantum field theory is supersymmetric, this sigma model has to be appropriately supersymmetrized. As examples, we consider in some detail N=(2,2) and N=(0,2) supersymmetric theories in d=2 and N=2 supersymmetric theories in d=4. This reasoning leads to new information about the conformal manifolds of these theories, for example, we show that the manifold is Kahler-Hodge and we further argue that it has vanishing Kahler class. For N=(2,2) theories in d=2 and N=2 theories in d=4 we also show that the relation between the sphere partition function and the Kahler potential of M follows immediately from the appropriate sigma models that we construct. Along the way we find several examples of potential trace anomalies that obey the Wess-Zumino consistency conditions, but can be ruled out by a more detailed analysis.
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Hairy black holes in the gravitational decoupling setup are studied from the perspective of conformal anomalies. Fluctuations of decoupled sources can be computed by measuring the way the trace anomaly-to-holographic Weyl anomaly ratio differs from unit. Therefore the gravitational decoupling parameter governing three hairy black hole metrics is then bounded to a range wherein one can reliably emulate AdS/CFT with gravitational decoupled solutions, in the tensor vacuum regime.
We calculate conformal anomalies in noncommutative gauge theories by using the path integral method (Fujikawas method). Along with the axial anomalies and chiral gauge anomalies, conformal anomalies take the form of the straightforward Moyal deformation in the corresponding conformal anomalies in ordinary gauge theories. However, the Moyal star product leads to the difference in the coefficient of the conformal anomalies between noncommutative gauge theories and ordinary gauge theories. The $beta$ (Callan-Symanzik) functions which are evaluated from the coefficient of the conformal anomalies coincide with the result of perturbative analysis.
I present two calculations of the holographic Weyl anomalies induced by Chern-Simons gravity theories alternative to the ones presented in the literature. The calculations presented here rest on the extension from Chern-Simons to Transgression forms as lagrangians, motivated by gauge invariance, which automatically yields the boundary terms suitable to regularize the theory. The procedure followed here sheds light in the structure of Chern-Simons gravities and their regularization.
We compute conformal anomalies for conformal field theories with free conformal scalars and massless spin $1/2$ fields in hyperbolic space $mathbb{H}^d$ and in the ball $mathbb{B}^d$, for $2leq dleq 7$. These spaces are related by a conformal transformation. In even dimensional spaces, the conformal anomalies on $mathbb{H}^{2n}$ and $mathbb{B}^{2n}$ are shown to be identical. In odd dimensional spaces, the conformal anomaly on $mathbb{B}^{2n+1}$ comes from a boundary contribution, which exactly coincides with that of $mathbb{H}^{2n+1}$ provided one identifies the UV short-distance cutoff on $mathbb{B}^{2n+1}$ with the inverse large distance IR cutoff on $mathbb{H}^{2n+1}$, just as prescribed by the conformal map. As an application, we determine, for the first time, the conformal anomaly coefficients multiplying the Euler characteristic of the boundary for scalars and half-spin fields with various boundary conditions in $d=5$ and $d=7$.
This paper presents two methods to compute scale anomaly coefficients in conformal field theories (CFTs), such as the c anomaly in four dimensions, in terms of the CFT data. We first use Euclidean position space to show that the anomaly coefficient of a four-point function can be computed in the form of an operator product expansion (OPE), namely a weighted sum of OPE coefficients squared. We compute the weights for scale anomalies associated with scalar operators and show that they are not positive. We then derive a different sum rule of the same form in Minkowski momentum space where the weights are positive. The positivity arises because the scale anomaly is the coefficient of a logarithm in the momentum space four-point function. This logarithm also determines the dispersive part, which is a positive sum over states by the optical theorem. The momentum space sum rule may be invalidated by UV and/or IR divergences, and we discuss the conditions under which these singularities are absent. We present a detailed discussion of the formalism required to compute the weights directly in Minkowski momentum space. A number of explicit checks are performed, including a complete example in an 8-dimensional free field theory.
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