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.
Prof. Pioline: "We appreciate the careful read..."
in Report on Exact effective interactions and 1/4BPS dyons in heterotic CHL orbifolds