Non-renormalisation conditions in type II string theory and maximal supergravity

Abstract

This paper considers general features of the derivative expansion of Feynman diagram contributions to the four-graviton scattering amplitude in eleven-dimensional supergravity compactified on a two-torus. These are translated into statements about interactions of the form D 2k R 4 in type II superstring theories, assuming the standard M-theory/string theory duality relationships, which provide powerful constraints on the effective interactions. In the ten-dimensional IIA limit we find that there can be no perturbative contributions beyond k string loops (for k > 0). Furthermore, the genus h = k contributions are determined exactly by the one-loop eleven-dimensional supergravity amplitude for all values of k. A plausible interpretation of these observations is that the sum of h-loop Feynman diagrams of maximally extended supergravity is less divergent than might be expected and could be ultraviolet finite in dimensions d < 4 + 6/h -the same bound as for N = 4 Yang-Mills.

The ultra-violet question in maximally supersymmetric field theories

“…in D = 11, although it is difficult to see how such a conclusion could be justified using known field-theoretic methods. The stronger suggestion [21], that supergravity might be finite to all orders, could also be argued if the divergence structure of maximal Yang-Mills and supergravity theories were the same in all dimensions, a possibility that would be consistent with currently known explicit calculations (but so are the divergence expectations shown in Table 1). Similar suggestions have also been obtained from the finite structure of one-loop amplitudes in maximal supergrav-ity [15].…”

“…Such a scenario might pose puzzling questions for the superstring programme, where it has been assumed that ordinary supergravity theories need string ultraviolet completions in order to form consistent quantum theories. On the other hand, there are hints from superstring theory [19] that, if extrapolated to the field theory limit [20], would suggest that these cancellations might continue up to nine loops in N = 8 supergravity, and it has also been suggested, again from a string theory perspective [21], that N = 8 supergravity might be finite to all orders, although it is not clear exactly what one can learn from superstring theory about purely perturbative field-theory divergences [22].…”

Nondecoupling of Maximal Supergravity from the Superstring

“…Consider a four-dimensional graviton scattering amplitude where the graviton momenta are below the four-dimensional Planck scale. According to (1) and (2), the ten-dimensional Planck length ' 10 is given by…”

“…In particular, we conjecture that pure N 8 supergravity in four dimensions is in the Swampland. There has recently has been some speculation that fourdimensional N 8 supergravity might be ultraviolet finite to all orders in perturbation theory [1][2][3]. If true, this would raise the question of whether N 8 supergravity might be a consistent theory that is decoupled from its string theory extension.…”

One loop superstring effective actions andN=8supergravity

“…Besides, the study of higher order corrections in N = 8 supergravity is particularly relevant considering the recent claims that this theory may actually be eight-loop finite [16] or even ultraviolet finite [17].…”