Insight: Some one loop gravitational interactions in string theory

Anirban Basu

Anirban Basu is a researcher at Harish-Chandra Research Institute, Allahabad

String theory yields ultraviolet finite scattering amplitudes in theories of gravity coupled to matter. While the matter content of the theory is dependent on the compactification, the presence of gravity in the spectrum is universal. Hence, this is drastically different from the high energy behavior of conventional quantum field theories of point like excitations because such amplitudes are generically ultraviolet divergent. While in quantum field theory the ultraviolet divergences arise from short distance effects which manifest themselves as divergences arising from high momentum modes in loop integrals in various Feynman diagrams, these divergences are absent in string theory where analogous loop integrals involve an integration over the fundamental domain of the moduli space of two dimensional Riemann surfaces which is the Euclidean worldsheet of the string propagating in the background spacetime. The fundamental domain precisely excludes the regions of moduli space which yield the ultraviolet divergences in quantum field theory. The ultraviolet finiteness of string theory makes it, among other reasons, particularly attractive in the quest for a theory of quantum gravity. On the other hand, there are infrared divergences that arise from the boundaries of moduli space in calculating string amplitudes which reproduce expectations from quantum field theory, which must be the case as string theory must reproduce field theory at large distances. Hence, their cancellation proceeds as in field theory.

However, calculating these loop amplitudes in perturbative string theory is not an entirely trivial exercise. In the absence of Ramond–Ramond backgrounds, tree level amplitudes have been calculated in superstring theory. The one loop amplitudes, which are more complicated, have also been calculated in several cases. Going beyond one loop is technically challenging and much remains to be done.

Now these perturbative amplitudes along with non–perturbative corrections yield the complete interactions in string theory. These non–perturbative contributions involve scattering of the external states in the background of various solitons or instantons which are very heavy at weak coupling. Some of these calculations can be performed exactly in certain string compactifications. An example is the toroidal compactification of type II string theory which preserves 32 supersymmetries. A class of BPS (Bogomolnyi–Prasad–Sommerfield) interactions in such compactifications, which preserve a fraction of the supersymmetry, can be exactly calculated. This is a consequence of the enormous supersymmetry the theory possesses. These BPS interactions in the perturbative regime involve, among certain others, some leading terms in the low momentum expansion of the four graviton amplitude.

What about the non–BPS interactions that are not protected by supersymmetry? These are not expected to yield simple expressions like their BPS counterparts, and hence are useful to analyze. In this article in Classical and Quantum Gravity, I have studied the one loop perturbative contributions to such local interactions that arise from the low momentum expansion of the four graviton amplitude. At the technical level, the analysis is considerably more involved than that for the BPS amplitudes. In the uncompactified theory in 10 dimensional flat spacetime, this analysis has been performed by Eric D’Hoker, Michael Green and Pierre Vanhove. My focus has been to perform the calculation when the type II theory has been toroidally compactified. Needless to say, the 10 dimensional result is obtained when the various radii of the torus become infinitely large. I have obtained a second order differential equation of the Poisson type which these amplitudes satisfy, with a very specific structure of source terms. These equations along with appropriate boundary conditions can be solved for compactifications on torii in arbitrary dimensions. I have solved them for the simplest case of compactification on a circle, which yields a non–trivial dependence of the amplitudes on the radius of the circle. The various numerical factors involve Riemann zeta functions that arise from infinite sums involving momentum and winding modes along the circle. The appearance of Riemann zeta functions and their generalizations — the multiple zeta functions — are natural in perturbative computations in string theory.

Thus, while BPS amplitudes are the simplest objects to analyze in supersymmetric theories as they are protected by supersymmetry, the non–BPS amplitudes do not enjoy such properties and their analysis yields invaluable information about the theory. Such amplitudes have not been studied much in string compactifications, and this work yields a simple example where I have calculated such amplitudes at one loop.

Read the full article for free* in Classical and Quantum Gravity:
Non-BPS interactions from the type II one loop four graviton amplitude
Anirban Basu 2016 Class. Quantum Grav. 33 125028

*until 23 June 2016

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