# Loop corrections to string scattering via gluing axiom

The usual method of calculating higher-loop corrections to string scattering is to consider the stringy CFT on Riemann surfaces with increasing genus, and take the integral over the Teichmuller space. However, the calculation becomes significantly more cumbersome for surfaces with higher genus. Afaik, it is unknown if the sum converges to a nonperturbative value or not.

Suppose we want to calculate the 1-loop correction to the scattering amplitude, which comes from the worldsheet with the topology of the torus. We could split the torus with a plane passing through its center. The intersection of the torus and the plane is a pair of circles – to which we intuitively associate intermediate string states which are automatically summed over for us when we are calculating the scattering amplitude for the torus worldsheet.

Can we sum over the intermediate states explicitly instead?

To clarify – instead of solving the (hard) problem of scattering on the torus worldsheet, we could solve the (easy) problem of scattering on the spherical worldsheet and then reconstruct the answer for the torus by composing the two spherical amplitudes and integrating over intermediate states. Those intermediate states correspond to two strings, so it should be just an integral over the basis in $\mathcal{H}^2$ where $\mathcal{H}$ is the Hilbert space of the string.

If now we assume $n$ intermediate strings instead of two, we will get the result for the Riemann surface with genus $n-1$.

This is in fact reminiscent of the gluing axiom of Atiyah-Singer. While CFT is not topologically invariant, string theory comes from the diffeo-invariant Polyakov action. The Hilbert space of the string is not just the Hilbert space of the worldsheet CFT – it is determined by (the cohomology of) the BRST operator, which encodes the hidden diffeomorphism symmetry of the gauge-fixed CFT action. Thus, naively I would expect string amplitudes to satisfy the gluing axiom. In fact, this is compatible (on the tree level) with the known property of the stringy amplitudes – crossing duality.

In this sense, string theory is presumably only different from TQFT in that its Hilbert space is infinite-dimensional, because it has local worldsheet degrees of freedom. It also follows that string dynamics is described by a certain infinite-dimensional Frobenius algebra.

Does it make sense? Has it been done before? Why didn't it work out? I would appreciate a link to the paper discussing this.

• It appears you think this calculation would be easy - have you done it and does it agree with the traditional amplitude for the torus? – ACuriousMind Apr 19 '18 at 17:40
• @ACuriousMind I was hoping it won’t come to this :) Its always nice if smith has already been done for you, and I am not a professional physicist, and don’t have a lot of time for this. But if this question remains unanswered, I guess the only way is to do the math myself. – Prof. Legolasov Apr 19 '18 at 17:46