Abstract
We consider the algebraic couplings in the tree level effective action of the heterotic string. We show how these couplings can be computed from closed string field theory. When the light fields we are interested in are charged under an underlying mathcal{N} = 2 R-charge in the left-moving sector, their quartic effective potential localizes at the boundary of the worldsheet moduli space, in complete analogy to the previously studied open string case. In particular we are able to compute the quartic closed string field theory potential without resorting to any explicit expression for the 3- and the 4-strings vertices but only using the L∞ relations between them. As a non trivial example we show how the heterotic Yang-Mills quartic potential arises in this way.
Highlights
A quantum consistent theory necessarily includes closed strings as dynamical degrees of freedom and from this perspective, the simplest model to consider is a theory with only closed strings since, at least perturbatively, open strings cannot be created by closed strings interacting between themselves
Closed string field theory is not as explicit as open string field theory because the fundamental vertices defining its interactions necessarily include integrations over implicitly-defined internal regions of the moduli space of punctured Riemann surfaces, together with local coordinates around punctures, for which we generally do not have closed form expressions. Since these data are necessary for constructing off-shell amplitudes, it seems that a direct approach towards analytic computations in closed string field theory is still not really available, progress in this direction is happening [45,46,47,48,49,50]
To do so we start from the WZW-like formulation of heterotic string field theory [54, 55] in the NS sector and we classically integrate out the massive fields by solving their equation of motion in terms of the massless fields, in analogy to what has been done in the early days by Berkovits and Schnabl [57] for the open superstring
Summary
The action for heterotic SFT in the large Hilbert space [54, 55] expanded up to quartic order in the NS sector is. The string field Φ is a combination of states consisting in the left moving sector of an. N = 1 matter SCFT with central charge c = 15 together with the (b, c) and (β, γ) systems while, in the right-moving sector, of a c = 26 matter CFT together with the (b, c) system. The string field Φ carries ghost number +1 and picture number 0 and satisfies the level matching conditions b−0 Φ = L−0 Φ = 0 ,. Following Zwiebach’s standard notation [56], the inner product is defined as. In order to guarantee that the action is invariant under an appropriate non-linear gauge transformation [54, 55], the multi-string products satisfy the relations of an L∞ algebra which, up to cubic order are explicitly.
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