Abstract
Motivated by precision counting of BPS black holes, we analyze six-derivative couplings in the low energy effective action of three-dimensional string vacua with 16 supercharges. Based on perturbative computations up to two-loop, supersymmetry and duality arguments, we conjecture that the exact coefficient of the \nabla^2(\nabla\phi)^4∇2(∇ϕ)4 effective interaction is given by a genus-two modular integral of a Siegel theta series for the non-perturbative Narain lattice times a specific meromorphic Siegel modular form. The latter is familiar from the Dijkgraaf-Verlinde-Verlinde (DVV) conjecture on exact degeneracies of 1/4-BPS dyons. We show that this Ansatz reproduces the known perturbative corrections at weak heterotic coupling, including tree-level, one- and two-loop corrections, plus non-perturbative effects of order e^{-1/g_3^2}e−1/g32. We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results, . In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact \nabla^2 F^4∇2F4 effective interaction in four-dimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order e^{-R}e−R from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order e^{-R^2}e−R2 from Taub-NUT instantons. We show that instanton corrections from 1/4-BPS black holes are precisely weighted by the BPS index predicted from the DVV formula, including the detailed moduli dependence. We also extract two-instanton corrections from pairs of 1/2-BPS black holes, demonstrating consistency with supersymmetry and wall-crossing, and estimate the size of instanton-anti-instanton contributions.
Highlights
Background and executive summarywe recall relevant facts about the moduli space, duality group and BPS spectrum of heterotic CHL models in D = 4 and D = 3, and summarize the main features of our Ansatz for the exact ∇2(∇φ)4 and ∇2F 4 couplings in these models
We demonstrated that the Ansätze (1.4) and (1.5) satisfy the relevant supersymmetric Ward identities, and that their asymptotic expansion at weak heterotic string coupling g3 → 0 reproduces the known perturbative contributions, up to one-loop and two-loop, respectively, plus an infinite series of O(e−1/g32) corrections ascribed to NS5-instantons, Kaluza–Klein (6,1)-branes and
After conjecturing the precise form of the U-duality group G3(Z) ⊂ O(2k, 8, Z) in D = 3, we proposed an exact formula for Fabcd analogous to (1.4), F (2k,8)
Summary
Providing a statistical origin of the thermodynamic entropy of black holes is a key goal for any theory of quantum gravity. We demonstrated that the Ansätze (1.4) and (1.5) satisfy the relevant supersymmetric Ward identities, and that their asymptotic expansion at weak heterotic string coupling g3 → 0 reproduces the known perturbative contributions, up to one-loop and two-loop, respectively, plus an infinite series of O(e−1/g32) corrections ascribed to NS5-instantons, Kaluza–Klein (6,1)-branes and. We studied the weak coupling and large radius limits of the Ansatz (1.6), and found that it reproduces correctly the known tree-level and one-loop contributions in the limit g3 → 0, powerlike corrections in the limit R → ∞, as well as infinite series of instanton corrections consistent with the known helicity supertrace of 1/2-BPS states in D = 4, for all orbits of the U-duality group G4(Z). The effective coupling provides a duality-invariant generating function for the indices (or helicity supertraces) counting 1/4-BPS black hole micro-states in 4 dimensions, which arise as coefficients of exponentially suppressed contributions in the large radius limit;. The reader is invited to refer to [22] for more details on points discussed cursorily
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