Abstract

Unlike the \mathcal{R}^4ℛ4 and \nabla^4\mathcal{R}^4∇4ℛ4 couplings, whose coefficients are Langlands–Eisenstein series of the U-duality group, the coefficient \mathcal{E}^{(d)}_{(0,1)}ℰ(0,1)(d) of the \nabla^6\mathcal{R}^4∇6ℛ4 interaction in the low-energy effective action of type II strings compactified on a torus T^dTd belongs to a more general class of automorphic functions, which satisfy Poisson rather than Laplace-type equations. In earlier work [1], 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 dd. 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=4d=4, the result is shown to coincide with an alternative proposal [2] 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.

Highlights

  • The result is complete for E5 = D5 = S pin(5, 5), we argue that it is complete for E6 and we compute the generic Fourier coefficients (3.89) for E7

  • The result is complete for E5 = S pin(5, 5) and E6, we argue that it is complete for E7 and we compute the generic abelian Fourier coefficients for E8 in Appendix E

  • The result of our analysis shows that the contributions from 1/2- and 1/4-BPS states up to three-loop reproduces the exact low energy effective action up to ∇6R4, leading to the conclusion that 1/8-BPS states do not contribute to this coupling

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Summary

Introduction and summary

One could in principle extract the constant terms with respect to the other maximal parabolic subgroups, e.g. the one relevant to the limit where the M-theory torus T d+1 decompactifies keeping its shape fixed, and characterise the behavior at all cusps Assuming that these constant terms agree with predictions from Mtheory, one may apply the the conjecture that the relevant U-duality groups do not admit cuspidal automorphic representations attached to suitably small nilpotent orbits [43, 44] to conclude that (1.13), suitably renormalised, is the full exact coupling in any dimension D ≥ 4. Following the same strategy for ∇6R4, we find that the two-loop amplitude with 1/4-BPS charges running in the loops is given by a similar integral as in (1.12), where φKtrZ(Ω2) is replaced by −E−S3LΛ(21)(τ)/V in the variables (1.14) Combining this with the exceptional field theory result, we get.

From particle to string multiplet
Laplace identities
Tensorial differential equations
Nilpotent orbits and BPS states
Integrating against cusp forms and against Eisenstein series
Weak coupling limit
Decompactification limit
Regularisation and divergences
F1 s 2
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