Eisenstein series and string thresholds.
Let \(G(\mathbb Z)\) be the mapping class group \(S\ell(d,\mathbb Z)\), the \(T\)-duality group \(SO(d,d,\mathbb Z)\), or the non-perturbative \(U\)-duality group \(E_{d+ 1(d+1)} (\mathbb Z)\). The authors investigate the relevance of Eisenstein series for representing certain \(G(\mathbb Z)\)-invariant string theory amplitudes which receive corrections from BPS states only. Using \(G(\mathbb Z)\)-invariant mass formulae, they construct invariant modular functions on the symmetric space \(K \backslash G(\mathbb R)\) associated to a maximal compact subgroup \(K\) of \(G(\mathbb R)\), which generalize the standard nonholomorphic Eisenstein series arising in harmonic analysis on the fundamental domain of the Poincaré upper half plane. By comparing the asymptotics and eigenvalues of the Eisenstein series under second order differential operators with quantities arising in one-loop and \(g\)-loop string amplitudes, the authors also obtain a \(T\)-duality invariant representation of the latter, conjecture their non-perturbative \(U\)-duality invariant extension, and analyze the resulting non-perturbative effects.
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