Open string fields and D‐branes

Abstract

These lecture notes review some modern developments in open string field theory. We begin with a discussion of the physical motivations and general framework of open SFT. We then briefly introduce and contrast the two basic formalisms, Boundary SFT and cubic SFT, and summarize the main results about tachyon condensation obtained with both approaches. Next, we review the Vacuum SFT conjecture, according to which D‐brane solutions correspond to string fields that are projectors of Witten's *‐algebra. We describe the Boundary CFT technique to compute *‐products and construct projector solutions. We argue that VSFT accounts precisely for the expected moduli space of D‐branes.

Sobolev Spaces on Locally Compact Abelian Groups: Compact Embeddings and Local Spaces

“…Classically, Equation ( 6) is known as an equation with an infinite number of derivatives (see [4] and references therein). Such equations have been considered in the mathematical literature since the 1930's, but only recently physicists have found reasons to study nonlinear equations such as (6), see for instance [4,5,21,28,30]. The existence of very serious proposals claiming that p-adic and non-commutative mathematics are relevant to physics (see for instance [28,30]), makes it natural, even necessary, to consider equations of physical relevance in contexts other than Euclidean space or (pseudo-)Riemannian manifolds, as stated in Section 1.…”

“…and this Lagrangian is but an approximation to the highly sophisticated bosonic string action considered in [30] which contains an infinite number of fields and yields -via a formal application of the variational principle-an infinite number of equations for infinitely many variables, see for instance [21]. Topological groups appear therefore as a natural testing ground for gathering a better understanding of ( 1) and (2).…”

Analytical Solutions of Open String Field Theory

“…Another generalization of Schnabl's solution was considered in [183], where a B-field was included 45 . It was shown that Schnabl's solution (in its CFT form) is not modified.…”

“…• Shorter modern general introduction to string field theory can be found in [43,44]. • The issue of tachyon condensation in string field theory and vacuum string field theory is addressed in [45]. • General and extensive reviews of tachyon condensation covering also other frameworks for addressing this subject are [46,47,48,49,50].…”

On linear differential equations with infinitely many derivatives

“…Differential equations with infinitely many derivatives appear frequently in modern physics, and special cases of such "nonlocal" equations have been extensively studied in string theory, quantum gravitation theory and cosmology, see for instance [1,5,6,4,7,8,10,11,16,18,28,38,39] and also [30,36,40]. The mathematical study of such equations began over a century ago (see e.g.…”