Analytical Solutions of Open String Field Theory

Abstract

In this work we review Schnabl's construction of the tachyon vacuum solution to bosonic covariant open string field theory and the results that followed.We survey the state of the art of string field theory research preceding this construction focusing on Sen's conjectures and the results obtained using level truncation methods.The tachyon vacuum solution can be described in various ways. We describe its geometric representation using wedge states, its formal algebraic representation as a pure-gauge solution and its oscillator representation. We also describe the analytical proofs of some of Sen's conjectures for this solution.The tools used in the context of the vacuum solution can be adapted to the construction of other solutions, namely various marginal deformations. We present some of the approaches used in the construction of these solutions.The generalization of these ideas to open superstring field theory is explained in detail. We start from the exposition of the problems one faces in the construction of superstring field theory. We then present the cubic and the non-polynomial versions of superstring field theory and discuss a proposal suggesting their classical equivalence. Finally, the bosonic solutions are generalized to this case. In particular, we focus on the (somewhat surprising) generalization of the tachyon solution to the case of a theory with no tachyons.

Kinks of open superstring field theory

“…According to Sen's first and second 'conjectures' [49,50,51,52,53,54], 7 the tachyon vacuum corresponds to an unstable (non-BPS) Dp-brane (whose volume V p+1 we normalize to 1), which decays into a stable configuration. Therefore, at its local maximum the effective tachyon potential equals the tension of the non-BPS Dp-brane, which is [49] T…”

“…Otherwise, the quartic term is rescaled as 1/2 = σ → 7 As a point of fact these are no longer conjectures, since they have been proven or strongly supported numerically in the level truncation scheme [55,56,57,58,59] and analytically [60,61,62,63,64,65]. See [7] for an updated discussion on the subject and references.…”

“…The same problem is neither trivial nor of mere academic interest as far as non-local theories are concerned. In fact, the simplest example of a non-local scalar with a static (i.e., zero-momentum) double-well potential is provided by the tachyon of open string field theory (OSFT, reviewed in [4,5,6,7]). In the case of nonchiral bilocal supersymmetric OSFT [8,9,10], the Euclidean (1/2, 1)-level effective equation for a homogeneous tachyon is (up to a field redefinition)…”

Analytic solutions for Dp branes in SFT

“…There is a general expectation that an OSFT defined on a particular boundary conformal field theory (BCFT) has classical solutions describing other boundary conformal field theories [7,8]. Previously analytic solutions were constructed describing the tachyon vacuum [5,6,9,10,11,12,13,14,15,16,17,18,19,20,21] and of those describing a general marginal boundary deformations of the initial BCFT [22,23,24,25,26,27,28,29,30,31], see also the reviews [32,33]. In this panorama an element was missing: the solutions describing inhomogeneous and relevant boundary deformations of the initial BCFT were not known, though their existence was predicted [7,8,34].…”

Generalized states in SFT

“…The search for exact (analytic) solutions in the SFT à la Witten, [1], has been characterized not only by considerable successes (see [2,3,7,4,5,6] and [8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30], as well as [33,34,38,39,35,36,37] and the reviews [31,32]), but also by remarkable improvements in the mathematical language of the theory. An example of the latter is the astonishingly simple form taken by such analytic solutions thanks especially to the so called K, B, c algebra.…”