Abstract
We analyze the pertubative contributions to the $D^4 R^4$ and $D^6 R^4$ couplings in the low-energy effective action of type II string theory compactified on a torus $T^d$, with particular emphasis on two-loop corrections. In general, it is necessary to introduce an infrared cut-off $\Lambda$ to separate local interactions from non-local effects due to the exchange of massless states. We identify the degenerations of the genus-two Riemann surface which are responsible for power-like dependence on $\Lambda$, and give an explicit prescription for extracting the $\Lambda$-independent effective couplings. These renormalized couplings are then shown to be eigenmodes of the Laplace operator with respect to the torus moduli, up to computable anomalous source terms arising in the presence of logarithmic divergences, in precise agreement with predictions from U-duality. Our results for the two-loop $D^6 R^4$ contribution also probe essential properties of the Kawazumi-Zhang invariant
Highlights
JHEP12(2015)102 consistent with the fact that supersymmetry requires these functions to be eigenmodes of the Laplacian on the moduli space Ed+1/Kd+1 with a specific eigenvalue [11, 18]
We analyze the pertubative contributions to the D4R4 and D6R4 couplings in the low-energy effective action of type II string theory compactified on a torus T d, with particular emphasis on two-loop corrections
They follow from similar anomalous terms appearing in the U-duality invariant Laplace-type equation for full D4R4 coupling E((0d,)0), which were determined in [23] using general consistency requirements and confirmed in [24]
Summary
The couplings E((md),n)D4m+6nR4 of interest in this work refer to local terms in the lowenergy expansion of the one-particule irreducible effective action of type II string theory compactified on a torus T d. In order to isolate the local part of the effective action, it is convenient to introduce an infrared cut-off Λ to separate the contribution of massless supergravity states from those of massive string states, and take the low-energy expansion of each parts separately [8, 34, 35]. The sum of the string theory and supergravity contributions to the coefficients of the local interation D4m+6nR4 has a finite limit as the cut-off Λ is removed, and defines the renormalized coupling E((md),n). The string theory contribution to the coefficient of the D4m+6nR4 term at h-loop is given by MΛh dμh F((md,,hn)) Γd,d,h(Ω; G, B). The supergravity contribution, corresponding to the integral over the complement of MΛh inside Mh, cancels these power-like terms, leaving a finite coefficient for the term D4m+6nR4 in the local effective action.
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