Abstract
We find general relations between the on-shell gravitational trace anomaly AN, and the logarithmic correction ΔSN to the entropy of “large” BPS extremal black holes in N⩾2 supergravity theories in D=4 space–time dimensions (recently computed by Sen, 2011 [1]). For (generalized) self-mirror theories (all having AN=0), we obtain the result ΔSN=−ΔS8−N=2−N/2, whereas for generic theories the trace anomaly A˜N of the fully dualized theory turns out to coincide with 2ΔSN, up to a model-independent shift: A˜N=2ΔSN−1. We also speculate on N=1 theories displaying “large” extremal black hole solutions.
Highlights
A generalized notion of mirror symmetry was suggested [2], under which AN = − 1 24 ρ, (1.1)occurring in the on-shell1 gravitational trace anomaly [3, 4, 5] gμν T μν R∗μνρσ Rμ∗νρσ, (1.2)
As we will see below, results crucially depend on the dualization of 3- and 2- form fields, which naturally arise from M -theory compactifications; it is remarkable that the trace anomaly coefficient AN of the undualized theory does vanish for N = 8, 6 and 5 “pure” supergravities, if the corresponding graviton multiplet is properly defined as containing form fields of degree higher than one [6]
On the other hand, another class of N = 1, D = 4 theories, complementary to the one discussed above, can be considered. Such a class, which we will dub “minimally coupled”, cannot be obtained as consistent truncation of N = 2 theories, and its kinetic vector matrix is constant: fIJ ∼ δIJ (I, J = 1, ..., nV 1). This implies that the complex scalar fields from the nc chiral multiplets are not involved in an attractor dynamics in the near horizon geometry of the “large” extremal black holes (BHs) under consideration5, which at bosonic level can be regarded as a ReissnerNordstrom (RN) extremal BH coupled to a set of spectator scalar fields and uncharged vectors
Summary
A generalized notion of mirror symmetry was suggested [2], under which. changes sign. As we will see below, results crucially depend on the dualization of 3- and 2- form fields, which naturally arise from M -theory compactifications; it is remarkable that the trace anomaly coefficient AN of the undualized theory does vanish for N = 8, 6 and 5 “pure” supergravities, if the corresponding graviton multiplet is properly defined as containing form fields of degree higher than one [6] (see [2]).
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