Abstract
We deform a defect conformal field theory by an exactly marginal bulk operator and we consider the dependence on the marginal coupling of flat and spherical defect expectation values. For even dimensional spherical defects we find a logarithmic divergence which can be related to a $a$-type defect anomaly coefficient. This coefficient, for defect theories, is not invariant on the conformal manifold and its dependence on the coupling is controlled to all orders by the one-point function of the associated exactly marginal operator. For odd-dimensional defects, the flat and spherical case exhibit different qualitative behaviors, generalizing to arbitrary dimensions the line-circle anomaly of superconformal Wilson loops. Our results also imply a non-trivial coupling dependence for the recently proposed defect $C$-function. We finally apply our general result to a few specific examples, including superconformal Wilson loops and R\'enyi entropy.
Highlights
AND RESULTSExtended probes play a distinguished role in a wide range of physical phenomena
The correct way to interpret this result is to affirm that there is no universal part of the flat defect expectation value which depends on the marginal coupling
Even if some symmetry protects the flat expectation value from the aforementioned divergences or if one could identify a universal part after renormalization, the result would not depend on λ
Summary
Extended probes play a distinguished role in a wide range of physical phenomena. Wilson and ’t Hooft lines, boundaries, interfaces and twist operators provide physically interesting examples of a broad class of observables denoted as defects. Similar phenomena have been observed in all those cases where the expectation value of the circular Wilson loop could be computed exactly, such as N 1⁄4 2 theories in four dimensions [7,8] or N ≥ 2 theories in three dimensions [9,10,11,12] It is a natural question whether such an anomaly is a more general feature of conformal defects, i.e., whether it is always true that the flat defect expectation value is different from the spherical one. This is in agreement with general expectations, since we know that for even p the universal part of the defect expectation value is given by the coefficient of the logarithmic divergence This coefficient can be expressed as a linear combination of Weyl invariants and the expectation value of the spherical defect is related to the a-type anomaly. This is no contradiction with usual results since under a defect RG-flow a candidate defect C-function should not depend on defect marginal couplings, but could depend in general on bulk marginal couplings
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