The closed-string 3-loop amplitude and S-duality

The low-energy limit of the four-point 3-loop amplitude (including its overall coefficient) is computed in both type IIA and IIB superstring theories using the pure spinor formalism. The result is shown to agree with the prediction of the coefficient for the type IIB $D^6 R^4$ interaction made by Green and Vanhove based on S-duality considerations.Comment: 26 pages, harvmac. v3: factor of 3 in section 3.3 corrected, updated abstract and dropped Z_3-symmetry argumen

Generalized CoK\"ahler Geometry and an Application to Generalized K\"ahler Structures

In this paper we define the notion of a generalized coK\"ahler structure and prove that the product $M_{1}\times M_{2}$ of generalized contact metric manifolds $(M_i, \Phi_i,E_{\pm,i}, G_i)$, $i=1, 2$, where $M_{1}\times M_{2}$ is endowed with the product generalized complex structure induced from $\Phi_1$ and $\Phi_2$, is generalized K\"ahler if and only if $(M_i, \Phi_i, E_{\pm,i}, G_i) ,\ \ i=1,2$ are generalized coK\"ahler structures. We also prove that products of generalized coK\"ahler and generalized K\"ahler manifolds admit a generalized coK\"ahler structure. We use these product constructions to give nontrivial examples of generalized coK\"ahler structures. Finally, we show the analogs of these theorems hold in the setting of twisted generalized geometries. We use these theorems to construct new examples of twisted generalized K\"ahler structures on manifolds that do not admit a classical K\"ahler structure and we give examples of twisted generalized coK\"ahler structures on manifolds which do not admit a classical coK\"ahler structure.Comment: Final print version. To appear in Journal of Geometry and Physic

The two-loop superstring five-point amplitude and S-duality

The low-energy limit of the massless two-loop five-point amplitudes for both type IIA and type IIB superstrings is computed with the pure spinor formalism and its overall coefficient determined from first principles. For the type IIB theory, the five-graviton amplitude is found to be proportional to its tree-level counterpart at the corresponding order in $\alpha'$. Their ratio ties in with expectations based on S-duality since it matches the same modular function $E_{5/2}$ which relates the two-loop and tree-level four-graviton amplitudes. For R-symmetry violating states, the ratio between tree-level and two-loop amplitudes at the same $\alpha'$-order carries an additional factor of $-3/5$. Its S-duality origin can be traced back to a modular form derived from $E_{5/2}$.Comment: 42 pages, harvmac te