Delta-function interactions for the bosonic and spinning strings and the generation of Abelian gauge theory

Abstract

We construct contact interactions for bosonic and spinning strings. In the tensionless limit of the spinning string this reproduces the super-Wilson loop that couples spinor matter to Abelian gauge theory. Adding boundary terms that quantise the motion of charges results in a string model equivalent to spinor QED. The strings represent lines of electric flux connected to the charges. The purely bosonic model is spoilt by divergences that are excluded from the spinning model by world-sheet supersymmetry, indicating a preference for spinor matter.

QED as the tensionless limit of the spinning string with contact interaction

“…Since K is real the remaining contributions in the expansion (33) are suppressed by the common factor /y 2 1 α K 2 /4 arising from the second term in (34) which vanishes as the regulator is removed for all K 2 except those close to zero (in terms of ). Since K is ultimately to be integrated over we also need to consider the contribution of these small values, however for α large and small this factor behaves effectively as δ K 2 / 1 2 α ln y 1 2 and so is also suppressed in the tensionless limit -see [14] for further detail. We conclude that no divergent terms can be generated by insertions that approach each other far from the boundary.…”

“…Because of the Dirichlet boundary conditions however, points close to the boundary make a finite scale independent contribution. Set X = X c +X with X c a classical piece satisfying the boundary conditions (14) and Euler-Lagrange equations D DX c = 0, and X a quantum fluctuation. Integrating over X gives…”

“…Following Strassler we can include a background gauge field on the world-lines to source photon amplitudes, and, as explained in [14] the Green functions for the charged particles are obtained by including open world-lines with appropriate boundary conditions at their ends.…”

Worldline formalism for a confined scalar field

“…These consecutive rescalings of the x 2 -coordinate will be frequent in the subsequent calculations and can further be justified by first integrating over ω; in the limit T → 0, the result yields a representation of δ(x 2 ) plus higher-order corrections. To compute subleading contributions 5 Correlators involving δ-functions of this form have previously been treated in this way in the context of contact interactions between strings [17,18] and particles [19]. one should Taylor expand h(x) around x 2 = 0 and perform consistent calculations order by order in T .…”

Intersection of world-lines on curved surfaces and path-ordering of the Wilson loop

“…(1) is not the full field-strength resulting from the charge pair as it represents only a single line of force, but the full field-strength does emerge after summing over surfaces. In [5], [6] it was shown that the Wilson loop for Abelian gauge theory associated with a closed curve B in flat Euclidean space can be written as the partition function of a tensionless four-dimensional string whose world-sheet Σ spans B with an interaction that is the supersymmetric version of (1). To see how the contact interaction gives rise to the electromagnetic photon propagator first Fourier decompose the δ-function…”