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Register a new userThe mixed black hole partition function for the STU model
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We evaluate the mixed partition function for dyonic BPS black holes using the recently proposed degeneracy formula for the STU model. The result factorizes into the OSV mixed partition function times a proportionality factor. The latter is in agreement with the measure factor that was recently conjectured for a class of N=2 black holes that contains the STU model.
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The type IIA superstring partition function Z_IIA on the euclidean attractor geometry AdS_2 x S^2 x CY_3, computes the modified elliptic genus Z_BH of the associated black hole. The hybrid formalism of superstrings defined as a conformally invariant sigma model on the coset supermanifold PSU(1,1|2)/U(1)xU(1), together with Calabi-Yau and chiral boson CFTs, is used to calculate Z_IIA. The sigma model action on AdS_2 x S^2 is explicitly written in U(1)xU(1) invariant variables. The N=2 generators of AdS_2 x S^2 x CY_3 are enlarged and embedded in an N=4 topological algebra. The world sheet superconformal invariance is then used to construct a nilpotent BRST operator, in contrast to the kappa symmetry analysis used by Beasely et. al. in hep-th/0608021. The sigma model action is explicitly shown to be closed under this BRST operator. Localization arguments are then used to deform the world sheet path integral with the addition of a BRST exact term, where, contributions arise only from the center of AdS_2 and, the north and south poles of S^2. This leads to the OSV result Z_BH = Z_IIA = |Z_top|^2, where |Z_top|^2 is the square of the topological string partition function.
We use the techniques of supersymmetric localization to compute the BPS black hole entropy in N=2 supergravity. We focus on the n_v+1 vector multiplets on the black hole near horizon background which is AdS_2 x S^2 space. We find the localizing saddle point of the vector multiplets by solving the localization equations, and compute the exact one loop partition function on the saddle point. Furthermore, we propose the appropriate functional integration measure. Through this measure, the one loop determinant is written in terms of the radius of the physical metric, which depends on the localizing saddle point value of the vector multiplets. The result for the one loop determinant is consistent with the logarithmic corrections to the BPS black hole entropy from vector multiplets.
The general solutions of the radial attractor flow equations for extremal black holes, both for non-BPS with non-vanishing central charge Z and for Z=0, are obtained for the so-called stu model, the minimal rank-3 N=2 symmetric supergravity in d=4 space-time dimensions. Comparisons with previous results, as well as the fake supergravity (first order) formalism and an analysis of the BPS bound all along the non-BPS attractor flows and of the marginal stability of corresponding D-brane configurations, are given.
The macroscopic entropy and the attractor equations for BPS black holes in four-dimensional N=2 supergravity theories follow from a variational principle for a certain `entropy function. We present this function in the presence of R^2-interactions and non-holomorphic corrections. The variational principle identifies the entropy as a Legendre transform and this motivates the definition of various partition functions corresponding to different ensembles and a hierarchy of corresponding duality invariant inverse Laplace integral representations for the microscopic degeneracies. Whenever the microscopic degeneracies are known the partition functions can be evaluated directly. This is the case for N=4 heterotic CHL black holes, where we demonstrate that the partition functions are consistent with the results obtained on the macroscopic side for black holes that have a non-vanishing classical area. In this way we confirm the presence of a measure in the duality invariant inverse Laplace integrals. Most, but not all, of these results are obtained in the context of semiclassical approximations. For black holes whose area vanishes classically, there remain discrepancies at the semiclassical level and beyond, the nature of which is not fully understood at present.
The duality symmetries of the STU-model of Sen and Vafa are very restrictive. This is utilized to determine the holomorphic function that encodes its two-derivative Wilsonian effective action and its couplings to the square of the Weyl tensor to fifth order in perturbation theory. At fifth order some ambiguities remain which are expected to resolve themselves when proceeding to the next order. Subsequently, a corresponding topological string partition function is studied in an expansion in terms of independent invariants of $S$, $T$ and $U$, with coefficient functions that depend on an effective duality invariant coupling constant $u$, which is defined on a Riemann surface $mathbb{C}$. The coefficient function of the invariant that is independent of $S$, $T$ and $U$ is determined to all orders by resummation. The other functions can be solved as well, either algebraically or by solving differential equations whose solutions have ambiguities associated with integration constants. This determination of the topological string partition function, while interesting in its own right, reveals new qualitative features in the result for the Wilsonian action, which would be difficult to appreciate otherwise. It is demonstrated how eventually the various ambiguities are eliminated by comparing the results for the effective action and the topological string. While we only demonstrate this for the leading terms, we conjecture that this will hold in general for this model.
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