Guillaume Bossard, Axel Kleinschmidt, Boris Pioline
SciPost Phys. 8, 054 (2020) ·
published 8 April 2020

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Unlike the $R^4$ and $\nabla^4 R^4$ couplings, whose coefficients are LanglandsEisenstein series of the Uduality group, the coefficient $\mathcal{E}^{(d)}_{(0,1)}$ of the $\nabla^6 R^4$ interaction in the lowenergy effective action of type II strings compactified on a torus $T^d$ belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplacetype equations. In earlier work, it was proposed that the exact coefficient is given by a twoloop integral in exceptional field theory, with the full spectrum of mutually 1/2BPS states running in the loops, up to the addition of a particular LanglandsEisenstein 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 nonperturbative corrections
which have the expected form for 1/8BPS instantons and bound states of 1/2BPS instantons and antiinstantons. The additional LanglandsEisenstein series arises from a subtle cancellation between the twoloop amplitude with 1/4BPS states running in the loops, and the threeloop amplitude with mutually 1/2BPS states in the loops. For $d=4$, the result is shown to coincide with an alternative proposal in terms of a covariantised genustwo string amplitude, due to interesting identities between the KawazumiZhang invariant of genustwo 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 CosnierHoreau, 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 sixderivative
couplings in the low energy effective action of threedimensional string vacua
with 16 supercharges. Based on perturbative computations up to twoloop,
supersymmetry and duality arguments, we conjecture that the exact coefficient
of the $\nabla^2(\nabla\phi)^4$ effective interaction is given by a genustwo
modular integral of a Siegel theta series for the nonperturbative Narain
lattice times a specific meromorphic Siegel modular form. The latter is
familiar from the DijkgraafVerlindeVerlinde (DVV) conjecture on exact
degeneracies of 1/4BPS dyons. We show that this Ansatz reproduces the known
perturbative corrections at weak heterotic coupling, including treelevel, one
and twoloop corrections, plus nonperturbative effects of order
$e^{1/g_3^2}$. 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
$\nabla^2 F^4$ effective interaction in fourdimensional 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 $e^{R^2}$ from TaubNUT instantons. We
show that instanton corrections from 1/4BPS black holes are precisely weighted
by the BPS index predicted from the DVV formula, including the detailed moduli
dependence. We also extract twoinstanton corrections from pairs of 1/2BPS
black holes, demonstrating consistency with supersymmetry and wallcrossing,
and estimate the size of instantonantiinstanton contributions.
Guillaume Bossard, Charles CosnierHoreau, Boris Pioline
SciPost Phys. 3, 008 (2017) ·
published 31 July 2017

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Threedimensional string models with halfmaximal supersymmetry are believed
to be invariant under a large Uduality group which unifies the S and T
dualities in four dimensions. We propose an exact, Uduality invariant formula
for fourderivative scalar couplings of the form $F(\Phi) (\nabla\Phi)^4$ in a
class of string vacua known as CHL $\mathbb{Z}_N$ heterotic orbifolds with $N$
prime, generalizing our previous work which dealt with the case of heterotic
string on $T^6$. We derive the Ward identities that $F(\Phi)$ must satisfy, and
check that our formula obeys them. We analyze the weak coupling expansion of
$F(\Phi)$, and show that it reproduces the correct treelevel and oneloop
contributions, plus an infinite series of nonperturbative contributions.
Similarly, the large radius expansion reproduces the exact $F^4$ coupling in
four dimensions, including both supersymmetric invariants, plus infinite series
of instanton corrections from halfBPS dyons winding around the large circle,
and from TaubNUT instantons. The summation measure for dyonic instantons
agrees with the helicity supertrace for halfBPS 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
1/2BPS dyonic states in all duality orbits.