We present an approach to the parametrization of (super) Schottky space obtained by sewing together three-punctured discs with strips. Different cubic ribbon graphs classify distinct sets of pinching parameters; we show how they are mapped onto each other. The parametrization is particularly well-suited to describing the region within (super) moduli space where open bosonic or Neveu-Schwarz string propagators become very long and thin, which dominates the IR behaviour of string theories. We show how worldsheet objects such as the Green’s function converge to graph theoretic objects such as the Symanzik polynomials in the α ′ → 0 limit, allowing us to see how string theory reproduces the sum over Feynman graphs. The (super) string measure takes on a simple and elegant form when expressed in terms of these parameters.

A parametrization of (super) moduli space near the corners corresponding to bosonic or Neveu–Schwarz open string degenerations is introduced for worldsheets of arbitrary topology. With this parametrization, Feynman graph polynomials arise as the α′→0 limit of objects on moduli space. Furthermore, the integration measures of string theory take on a very simple and elegant form.

Starting from the superstring amplitude describing interactions among D-branes with a constant world-volume field strength, we present a detailed analysis of how the open string degeneration limits reproduce the corresponding field theory Feynman diagrams. A key ingredient in the string construction is represented by the twisted (Prym) super differentials, as their periods encode the information about the background field. We provide an efficient method to calculate perturbatively the determinant of the twisted period matrix in terms of sets of super-moduli appropriate to the degeneration limits. Using this result we show that there is a precise one-to-one correspondence between the degeneration of different factors in the superstring amplitudes and one-particle irreducible Feynman diagrams capturing the gauge theory effective action at the two-loop level.

We study the field theory limit of multi-loop (super)string amplitudes, with the aim of clarifying their relationship to Feynman diagrams describing the dynamics of the massless states. We propose an explicit map between string moduli around degeneration points and Schwinger proper-times characterizing individual Feynman diagram topologies. This makes it possible to identify the contribution of each light string state within the full string amplitude and to extract the field theory Feynman rules selected by (covariantly quantized) string theory. The connection between string and field theory amplitudes also provides a concrete tool to clarify ambiguities related to total derivatives over moduli space: in the superstring case, consistency with the field theory results selects a specific prescription for integrating over supermoduli. In this paper, as an example, we focus on open strings supported by parallel D-branes, and we present two-loop examples drawn from bosonic and RNS string theories, highlighting the common features between the two setups.

We consider toroidal compactifications of bosonic string theory with particular regard to the phases (cocycles) necessary for a consistent definition of the vertex operators, the boundary states and the T-duality rules. We use these ingredients to compute the planar multi-loop partition function describing the interaction among magnetized or intersecting D-branes, also in presence of open string moduli. It turns out that unitarity in the open string channel crucially depends on the presence of the cocycles. We then focus on the 2-loop case and study the degeneration limit where this partition function is directly related to the tree-level 3-point correlators between twist fields. These correlators represent the main ingredient in the computation of Yukawa couplings and other terms in the effective action for D-brane phenomenological models. By factorizing the 2-loop partition function we are able to compute the 3-point couplings for abelian twist fields on generic non-factorized tori, thus generalizing previous expressions valid for the 2-torus.

We use the operator formalism to derive the bosonic contribution to the twisted open string partition function in toroidal compactifications. This amplitude describes, for instance, the planar interaction between g + 1 magnetized or intersecting D-branes. We write the result both in the closed and in the open string channel in terms of Prym differentials on the appropriate Riemann surface. Then we focus on the g = 2 case for a 2-torus. By factorizing the twisted partition function in the open string channel we obtain an explicit expression for the 3-twist field correlator, which is the main ingredient in the computation of Yukawa couplings in D-brane phenomenological models. This provides an alternative method for computing these couplings that does not rely on the stress-energy tensor technique.

We study systems of D3 and D(-1) branes in a NS-NS magnetic background and show that, when the brane configuration is stable, the physical degrees of freedom of the open strings with at least one end-point on the D-instantons describe the ADHM moduli of instantons for non-commutative gauge theories. We also prove that disk diagrams with mixed boundary conditions are the sources for the classical profile of the non-commutative gauge instantons in the singular gauge. We finally compare the string theory description in a large distance expansion with the non-commutative ADHM construction in the singular gauge and find complete agreement at perturbative level in the non-commutativity parameter.

We present the multiloop partition function of open bosonic string theory in the presence of a constant gauge field strength, and discuss its low-energy limit. The result is written in terms of twisted determinants and differentials on higher-genus Riemann surfaces, for which we provide an explicit representation in the Schottky parametrization. In the field theory limit, we recover from the string formula the two-loop Euler-Heisenberg effective action for adjoint scalars minimally coupled to the background gauge field.

This talk is based on work made with L. Magnea and R. Russo. We give an explicit expression of the multiloop partition function of open bosonic string theory in the presence of a constant gauge field strength. The Schottky parametrization allows to perform the field theory limit, which at two-loop level reproduces the Euler-Heisenberg effective action for adjoint scalars minimally coupled to the background gauge field.

The bosonization equivalence between the 2-dimensional Dirac and Laplacian operators can be used to derive new interesting identities involving Theta functions. We use these formulae to compute the multiloop partition function of the bosonic open string in presence of a constant electromagnetic field.

We study the Dirac and the Laplacian operators on orientable Riemann surfaces of arbitrary genus g. In particular we compute their determinants with twisted boundary conditions along the b-cycles. All the ingredients of the final results (including the normalizations) are explicitly written in terms of the Schottky parametrization of the Riemann surface. By using the bosonization equivalence, we derive a multi-loop generalization of the well-known g=1 product formulae for the Theta-functions. We finally comment on the applications of these results to the perturbative theory of open charged strings.

In string theory, there are various physical situations where the world-sheet fields have a shifted moding. For instance, this is the case for the twisted closed string in Z_N orbifold or for the charged open string in a constant electro-magnetic field. Because of this feature, it is quite challenging to give explicit formulae describing the string interaction, even for the bosonic case. In this note, we focus on the case of the charged open bosonic string and construct the 1-loop tadpole which is an object generating all 1-point functions from the annulus in the presence of an external field. In the operator formalism, this represents one of the basic building blocks for the construction of a general loop amplitude.

The multiloop amplitudes for the bosonic string in presence of a constant B -field are built by using the basic commutation relations for the open-string zero modes and oscillators. The open-string Green function on the annulus is obtained from the one-loop scattering amplitude among N tachyons. For higher loops, it is necessary to use the so-called three-reggeon vertex, which describes the emission from the open-string of another string and not simply of a tachyon. We find that the modifications to the three- (and multi-) reggeon vertex due to the B -field only affect the zero modes and can be written in a simple and elegant way. Therefore we can easily sew these vertices together and write the general expression for the multiloop N -reggeon vertex, which contains any loop string amplitude, in presence of the B -field. The field theory limit is also considered in some examples at two loops and reproduces exactly the results of a noncommutative scalar field theory.