Guillaume Bossard, Axel Kleinschmidt, Boris Pioline
SciPost Phys. 8, 054 (2020) ·
published 8 April 2020
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Unlike the R4 and ∇4R4 couplings, whose coefficients are Langlands--Eisenstein series of the U-duality group, the coefficient E(d)(0,1) of the ∇6R4 interaction in the low-energy effective action of type II strings compactified on a torus Td belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work, it was proposed that the exact coefficient is given by a two-loop integral in exceptional field theory, with the full spectrum of mutually 1/2-BPS states running in the loops, up to the addition of a particular Langlands--Eisenstein series. Here we compute the weak coupling and large radius expansions of these automorphic functions for any d. We find perfect agreement with perturbative string theory up to genus three, along with non-perturbative corrections which have the expected form for 1/8-BPS instantons and bound states of 1/2-BPS instantons and anti-instantons. The additional Langlands--Eisenstein series arises from a subtle cancellation between the two-loop amplitude with 1/4-BPS states running in the loops, and the three-loop amplitude with mutually 1/2-BPS states in the loops. For d=4, the result is shown to coincide with an alternative proposal in terms of a covariantised genus-two string amplitude, due to interesting identities between the Kawazumi--Zhang invariant of genus-two curves and its tropical limit, and between double lattice sums for the particle and string multiplets, which may be of independent mathematical interest.
Guillaume Bossard, Charles Cosnier-Horeau, Boris Pioline
SciPost Phys. 7, 028 (2019) ·
published 5 September 2019
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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 ∇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/g23. We also examine the weak coupling expansions in type I and type II string duals and find agreement with known perturbative results, as well as new predictions for higher genus perturbative contributions. In the limit where a circle in the internal torus decompactifies, our Ansatz predicts the exact ∇2F4 effective interaction in four-dimensional CHL string vacua, along with infinite series of exponentially suppressed corrections of order e−R from Euclideanized BPS black holes winding around the circle, and further suppressed corrections of order 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.
Guillaume Bossard, Charles Cosnier-Horeau, Boris Pioline
SciPost Phys. 3, 008 (2017) ·
published 31 July 2017
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Three-dimensional string models with half-maximal supersymmetry are believed to be invariant under a large U-duality group which unifies the S and T dualities in four dimensions. We propose an exact, U-duality invariant formula for four-derivative scalar couplings of the form F(Φ)(∇Φ)4 in a class of string vacua known as CHL ZN heterotic orbifolds with N prime, generalizing our previous work which dealt with the case of heterotic string on T6. We derive the Ward identities that F(Φ) must satisfy, and check that our formula obeys them. We analyze the weak coupling expansion of F(Φ), and show that it reproduces the correct tree-level and one-loop contributions, plus an infinite series of non-perturbative contributions. Similarly, the large radius expansion reproduces the exact F4 coupling in four dimensions, including both supersymmetric invariants, plus infinite series of instanton corrections from half-BPS dyons winding around the large circle, and from Taub-NUT instantons. The summation measure for dyonic instantons agrees with the helicity supertrace for half-BPS dyons in 4 dimensions in all charge sectors. In the process we clarify several subtleties about CHL models in D=4 and D=3, in particular we obtain the exact helicity supertraces for 1/2-BPS dyonic states in all duality orbits.