A simple analytic solution for Tachyon condensation

Abstract

In this paper we present a new and simple analytic solution for tachyon condensation in open bosonic string field theory. Unlike the B 0 gauge solution, which requires a carefully regulated discrete sum of wedge states subtracted against a mysterious "phantom" counter term, this new solution involves a continuous integral of wedge states, and no regularization or phantom term is necessary. Moreover, we can evaluate the action and prove Sen's conjecture in a mere few lines of calculation. the ψ N term does not contribute to the energy in the ordinary level expansion [1,4], since as a state in the Fock space it vanishes identically.By now the regularization and phantom piece are better understood [2,5,6,7,8,9,10], and there is little doubt that the B 0 gauge solution is for practical purposes nonsingular. Yet, no one has found an adequate definition of the solution-or gauge equivalent alternative-which does not require the regulated sum and phantom piece.In this note, we present an alternative solution for the tachyon vacuum which avoids the above complications. Instead of a discrete sum, the solution involves a continuous integral over wedge states, and no regularization or mysterious phantom term is necessary.Moreover, evaluation of the action and the proof of Sen's conjectures is, in contrast to the B 0 gauge, very straightforward.Broad classes of generalizations of the B 0 gauge solution have been constructed in [11,12,13,14,7]. Note in particular that our new solution is a special case of the solutions considered in [7], though our analysis will be quite different. This paper is organized as follows. After some algebraic and notational preliminaries, in Section 2 we present the new solution for the tachyon vacuum, comment on its structure, and prove the equations of motion. In Sec.2.1 we prove Sen's conjectures, specifically proving the absence of open string states and giving a very simple calculation of the brane tension. In Sec.2.2 we comment on the relation between pure gauge solutions and the phantom piece, and in Sec.2.3 we compute the closed string tadpole and demonstrate that it vanishes. In Section 3 we investigate the energy of the new vacuum in level truncation. As a warmup exercise, in Sec.3.1 we consider the L 0 level expansion. Due to the remarkable simplicity of our solution, we can solve the L 0 expansion exactly; we resum the expansion to confirm Sen's conjecture up to better than one part in 10 million. In Sec.3.2 we consider the "true" level expansion in terms of eigenstates of L 0 . Surprisinglyunlike the Siegel gauge or B 0 gauge tachyon condensates-we find that the expansion for the energy does not converge. In order to understand this phenomenon, in section Sec.3.3we consider a toy model of our solution where the L 0 level expansion, though divergent, can be solved exactly. In the end, we are able to resum the L 0 expansion of our solution and confirm Sen's conjecture to better than 99%. We end with some discussion.

A simple analytic solution for tachyon condensation

“…There are many questions related to the tachyon vacuum that have yet to be understood: Giving an analytic construction of the tachyon potential, understanding vacuum string field theory and multiple D-branes [24,41,42], recovering closed string physics around the tachyon vacuum, and finding an analytic tachyon vacuum in superstring field theory [43,44,45,46,47,48]. Perhaps this solution could inspire new approaches to marginal deformations [49,50,51,52,53,54,55,56], or help in the construction of lump solutions [57].…”

Простое аналитическое решение для конденсации тахиoнов

“…По поводу более явных определений K, B, c см. работу [4]. Итак, запишем новое решение для тахионного вакуума:…”

“…Никакой регуляризации или "фантомной части" решительно не требуется. На рисунке изображено наложение решения уравнения (5) в виде корреляционной функции конформной теории поля на цилиндре [4]. Важное свойство этого решения состоит в том, что оно включает интегрирование по wedge-состояниям, сколь угодно близким к единичному.…”

“…В некотором смысле в этом и состоит "секрет" того, почему это решение можно выразить без исполь-зования регуляризационной или "фантомной" частей. Однако единичное струнное поле представляет собой "непослушный" объект, и (как обсуждается в работе [4]) энергия этого решениия расходится при разложении по уровням. Эта расходимость возникает из суммирования ряда за пределами его радиуса сходимости, что очень похоже на энергию решения Шнабля в разложении по L 0 -уровням [1].…”

“…Однако интересно было бы найти другое решение, в котором подобные тонкости не возникают. В настоящей статье представлено такое решение, полученное недавно совместно в работе [4]. …”