The twisted open string partition function and Yukawa couplings

TL;DR

The paper develops an operator-formalism derivation of the bosonic twisted open string partition function on toroidal backgrounds, focusing on open strings between magnetized or intersecting D-branes. It expresses the twisted partition function in both closed and open string channels using Schottky group data and Prym differentials, and analyzes the $g=2$ case on $T^2$ to extract explicit $3$-twist Yukawa couplings by degeneration. The method yields both quantum (non-zero mode) and classical (zero-mode) contributions and agrees with previous results obtained by stress-tensor techniques and by earlier CFT approaches, providing a cross-check and a string-theoretic derivation of Yukawa couplings that naturally accommodates D-brane fluxes and T-duality. The work shows that the partition function factorizes into diagrams labeled by a degeneracy index $\mathcal{I}$, reproducing the observed dependence on $U$ and flux data and connecting to world-sheet instanton contributions in intersecting brane setups.

Abstract

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.

Figures (5)

• Figure 1: \ref{['fact']}a represents the twisted 2-loop partition function in the open string channel in a generic point of the world-sheet moduli space: on the three borders there are different magnetic fields $F_i$; \ref{['fact']}b is the degeneration limit we are interested in, which is obtained by focusing on the corner of the world-sheet moduli space defined in (\ref{['ftl']}); \ref{['fact']}c is the factorization of \ref{['fact']}b into two twisted $3$-string vertices and three propagators: this is obtained by focusing only on the leading term in the expansion of the previous point .
• Figure 2: \ref{['clos-par']}a represents the partition function under study from a space-time point of view; \ref{['clos-par']}b is a representation from the world-sheet point of view: the world-sheet is the upper half part of the complex plane that is outside all disks.
• Figure 3: \ref{['open-par']}a is a space-time representation of the same partition function of Fig. \ref{['clos-par']} in the open string channel; \ref{['open-par']}b is the corresponding world-sheet surface in the Schottky parametrization.
• Figure 4: This figure represents the special case $g=2$ of the partition function depicted in Fig. \ref{['open-par']}. Notice that for this case we choose a slightly different the world-sheet parametrization (\ref{['open-par2']}b) with the respect to the general case: we choose to have $\xi_1=\infty$ and $\eta_1=0$ and to swap the order of $\xi_2$ and $\eta_2$.
• Figure 5: This represents the path of integration ${\cal C}$ resulting from the combination of the integrals in the square parenthesis of (\ref{['2lexp']}). Starting from the dot and following the arrows one can check that this is a closed circuit on the branched Riemann surface. The $\epsilon_1$-cut ($\epsilon_2$-cut) are along the border $B_1$ ($B_2$).