On RR couplings on D-branes at order $O(α'^2)$

TL;DR

This work analyzes RR couplings on D-branes at order ${\cal O}(\alpha'^2)$, showing that nonlinear T-duality requires replacing ${F}^{(p)}$ with ${\cal F}^{(p)}=d{\cal C}^{(p-1)}$ where ${\cal C}=e^{B}C$, and that only B-field gauge-invariant terms with worldvolume indices survive (e.g., $B\wedge dC^{(p-3)}$) once one also uses the replacement $B\to B+2\pi\alpha'f$ to restore gauge invariance. The authors test these couplings by comparing the low-energy field-theory amplitudes to disk-level string theory amplitudes for one RR and two NSNS states, demonstrating consistency with nonlinear T-duality. They analyze two RR-polarization cases: for $\varepsilon_1^{ij}$ the string amplitude has no massless open-string pole and aligns with the field-theory predictions via RR/$H$-tensor structures, while for $\varepsilon_1^{i}$ a massless open-string pole appears and the amplitude reproduces the field-theory results at ${\cal O}(\alpha'^2)$. Collectively, the results confirm the proposed nonlinear RR couplings on D-branes and provide a robust method to enforce nonlinear T-duality invariance in higher-derivative brane actions, with implications for precise D-brane effective actions in string theory.

Abstract

Recently, it has been found that there are couplings of the RR field strength $F^{(p)}$ and the B-field strength $H$ on the world volume of D$_p$-branes at order ${\cal O}(α'^2)$. These couplings which have both world-volume and transverse indices, are invariant under the linear T-duality transformations. Consistency with the nonlinear T-duality indicates that the RR field strength $F^{(p)}$ in these couplings should be replaced by ${\cal F}^{(p)}=d{\cal C}^{(p-1)}$ where ${\cal C}=e^{B}C$. This replacement, however, reproduces some non-gauge invariant terms. On the other hand, the nonlinear terms are invariant under the linear T-duality transformations at the level of two B-fields. This allows one to remove some of the nonlinear terms in ${\cal F}^{(p)}$. We fix this by comparing the nonlinear couplings with the S-matrix element of one RR and two NSNS vertex operators. Our results indicate that in the expansion of ${\cal F}^{(p)}$ one should keep only the B-field gauge invariant terms, e.g. $B\wedge dC^{(p-3)}$ where both indices of B-field lie along the brane. Moreover, in this case one should replace $B$ with $B+2πα'f$ to have the $B$-field gauge invariance.