My research focuses on the relationship between string theory and quantum field theory. In particular, I am looking at the relationship between Feynman diagrams in quantum field theory and diagrams in string theory, working with Lorenzo Magnea, Rodolfo Russo (my supervisor) and Stefano Sciuto.

One of the often-repeated benefits of string theory is that (at least for closed strings) there is only one integral at each order in perturbation theory¹. But string scattering amplitudes have the property that they reduce to quantum field theory (QFT) amplitudes in the limit of infinite string tension, α′→ 0: open string scattering amplitudes reduce to Yang-Mills gauge theory amplitudes [2] and closed string amplitudes reduce to graviton scattering amplitudes [3].

We know that Yang-Mills amplitudes can be written as an asymptotic series in terms of Feynman diagrams, so can we find a correspondence between string theory loop integrals and Feynman diagrams? Although there is topologically only one orientable surface with boundaries and no handles at each loop level, integrals in string theory should be carried out over the compactification of moduli space; on the compactification divisor which lives at the boundary of moduli space, the surfaces can degenerate (i.e. develop ‘pinches’) in topologically distinct ways. The divergences in string amplitudes all come from these boundaries of moduli space. This region also dominates the Yang-Mills amplitudes, and we find a correspondence between topologically distinct degenerations and topologically distinct Feynman graphs.

There are many different ways to parametrize moduli space (see [4] for a summary). The one which is most useful for exploring the connection between string amplitude degenerations and QFT Feynman graphs is parametrization by Schottky groups. A Schottky group is a particular crystallographic group of Möbius transformations associated to a Riemann surface Σ with a choice of canonical homology basis (modulo fixing the PGL(2,C) invariance). Geometrically, it is isomorphic to the subgroup of the fundamental group of Σ generated by the a-cycles. To obtain Σ, we subtract all the limit points of the Schottky group from C and then quotient the remainder by the Schottky group.

The relevance of the Schottky group to string theory has been known since the early days of dual models, when it arose naturally from sewing tree-level “reggeon” (open string) vertices [5, 6, 7]. It reappeared in the more rigorous context of sewing BRST-invariant vertices and propagators in the late 1980’s [8]. There was also a ‘super’ version for superstrings sewn at Neveu-Schwarz (NS) punctures in which the Schottky group is a crystallographic subgroup of the group of super-projective transformations (isomorphic to OSp(2|1)) [9].

The benefit of the Schottky group is that there is a simple algebraic correspondence between the multipliers² k_i of some of the Schottky group elements, and the Schwinger parameters of associated QFT Feynman diagrams [10]. Heuristically, it goes like k_i = exp(-t_i∕α′). It is typically quite straightforward to expand a string amplitude integrand as an infinite power series in the multipliers. When we change integration variables to the dimensionful Schwinger parameters and take the α′→ 0 limit, only finitely many terms remain—and these are the Feynman diagrams.

Not only can string theory amplitudes be partitioned into a sum over Feynman diagrams topologically, but also field-by-field, including the Faddeev-Popov ghosts for the spacetime gauge theory. Ghost edges in Feynman diagrams come from the Schottky-group expansion of the Faddeev-Popov determinant, while gluon edges in Feynman diagrams come from the Schottky group expansion of the determinant of the Laplacian on the worldsheet.

There is no gauge-invariant notion of an individual Feynman diagram, and indeed the correspondence only holds at the level of Feynman diagrams if the Yang-Mills theory is written in a certain non-linear gauge originally used by Gervais and Neveu [11]; the gauge condition is roughly ∂_μA^μ + igA_μA^μ = 0. Although this gauge looks complicated, it actually simplifies many calculations in QFT [12]; the only gauge choice made in string theory is the standard (super-)conformal gauge.

To test the correspondence, we have calculated the NS sector of the two loop vacuum amplitude in a model with N parallel, separated D₃-branes, each with a constant background U(1) gauge field in the x¹–x² direction. The corresponding QFT is a U(N) Yang-Mills theory in four dimensions coupled to six adjoint scalars, with a background U(N) gauge field switched on. The background fields are implemented on the worldsheets by going to the double surface and giving the CFTs monodromies around the b-cycles; see e.g. [13]. The background affects fields with different spin differently; this allows us to quickly see which terms in the string amplitude correspond to which Feynman diagrams, which would be more opaque otherwise. This model can also be used to give a string theory computation of the Callan-Symanzik β function for Yang-Mills theory.

There are a number of obvious directions our work can be extended in. First of all, we can see if the correspondence between Feynman diagram topologies and topologically distinct degenerations of the worldsheet still holds at higher loop level. It would also be interesting to investigate how the formalism holds for amplitudes with external bosons. It would be most interesting to see if the correspondence could be generalized to include Feynman diagrams with fermion edges; the obstruction is that quotienting by a super-projective transformation is equivalent to sewing a pair of NS punctures, and the NS sector of superstrings corresponds to spacetime bosons. The superconformal structure of a worldsheet is completely different near Ramond punctures [14], which are needed for spacetime fermions.

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