[Paper Review] Background Independent String Field Theory

Itzhak Bars, Dmitry Rychkov|arXiv (Cornell University)|Jul 17, 2014

Computational Physics and Python Applications3 citations

TL;DR

This paper introduces a background-independent Moyal star formalism in bosonic open string field theory using a novel $σ$-basis that parameterizes open string coordinates by the worldsheet parameter $σ$. The formalism employs a half-phase-space formulation with $x_+^M(\sigma)$ and $p_-^M(\sigma)$, rendering interactions and the BRST operator independent of spacetime backgrounds. A key result is that the BRST operator emerges as a solution to a purely cubic, background-independent equation $Q \star Q = 0$, enabling non-perturbative computations and revealing spontaneous symmetry breaking of an OSp(d|2) superalgebra.

ABSTRACT

We develop a new background independent Moyal star formalism in bosonic open string field theory. The new star product is formulated in a half-phase-space, and because phase space is independent of any background fields, the interactions are background independent. In this basis there is a large amount of symmetry, including a supersymmetry OSp(d|2) that acts on matter and ghost degrees of freedom, and simplifies computations. The BRST operator that defines the quadratic kinetic term of string field theory may be regarded as the solution of the equation of motion A*A=0 of a purely cubic background independent string field theory. We find an infinite number of non-perturbative solutions to this equation, and are able to associate them to the BRST operator of conformal field theories on the worldsheet. Thus, the background emerges from a spontaneous-type breaking of a purely cubic highly symmetric theory. The form of the BRST field breaks the symmetry in a tractable way such that the symmetry continues to be useful in practical perturbative computations as an expansion around some background. The new Moyal basis is called the $σ$-basis, where $σ$ is the worldsheet parameter of an open string. A vital part of the new star product is a natural and crucially needed mid-point regulator in this continuous basis, so that all computations are finite. The regulator is removed after renormalization and then the theory is finite only in the critical dimension. Boundary conditions for D-branes at the endpoints of the string are naturally introduced and made part of the theory as simple rules in algebraic computations. A byproduct of our approach is an astonishing suggestion of the formalism: the roots of ordinary quantum mechanics may originate in the rules of non-commutative interactions in string theory.

Motivation & Objective

• To develop a background-independent formulation of open string field theory that eliminates dependence on specific spacetime backgrounds.
• To reformulate string-string interactions using a new Moyal star product in a half-phase-space basis ($x_+, p_-$) derived from the worldsheet parameter $\sigma$.
• To show that the BRST operator arises as a solution to a purely cubic equation $Q \star Q = 0$, enabling non-perturbative analysis.
• To incorporate D-brane boundary conditions naturally through algebraic rules in the new formalism.
• To demonstrate that the theory retains OSp(d|2) supersymmetry acting on matter and ghost fields, even after background breaking.

Proposed method

• Formulate string field theory in the $\sigma$-basis, where string fields are functionals $A(x_+(\sigma), p_-(\sigma))$ of symmetric and antisymmetric parts of string coordinates and momenta.
• Define a new Moyal star product $\star$ in this half-phase-space basis, which is independent of any background fields due to the phase space's intrinsic invariance.
• Represent the BRST operator $\hat{Q}$ as an anticommutator $\hat{Q}A = \{Q(x,p), A(x,p)\}_\star$, with $Q$ satisfying $Q \star Q = 0$.
• Introduce a midpoint regulator in the continuous $\sigma$-basis to ensure finiteness of all computations, which is removed after renormalization.
• Implement D-brane boundary conditions as algebraic constraints in the $\sigma$-basis, without modifying the star product or action.
• Use a supertrace formalism with Grassmann integration to compute gauge invariance and derive the BRST transformation $\delta_\Lambda A = [Q, \Lambda]_\star + g_0 \{\partial_{\bar{x}^b}A, \partial_{\bar{x}^b}\Lambda\}_\star$.

Experimental results

Research questions

• RQ1 Can string field theory be formulated in a background-independent way using a phase-space basis derived from the worldsheet parameter $\sigma$?
• RQ2 How can the BRST operator be represented algebraically as a solution to a purely cubic equation in a star-product formalism?
• RQ3 What is the role of the OSp(d|2) superalgebra in a background-independent string field theory, and how is it spontaneously broken?
• RQ4 How can D-brane boundary conditions be incorporated into the algebraic structure of the theory without reference to conformal field theory?
• RQ5 Can finite, non-perturbative computations be performed in this formalism, and how does it compare to conventional approaches?

Key findings

• The new Moyal star product in the $\sigma$-basis is fully background independent because it is defined on half-phase space, independent of any spacetime or worldsheet background fields.
• The BRST operator $\hat{Q}$ is realized as $\{Q, A\}_\star$, where $Q$ satisfies $Q \star Q = 0$, making the kinetic term of the action purely cubic and background independent.
• An infinite number of non-perturbative solutions to $Q \star Q = 0$ are found, which can be associated with BRST charges of conformal field theories on the worldsheet.
• The theory exhibits OSp(d|2) supersymmetry acting on both matter and ghost degrees of freedom, which is spontaneously broken by the choice of $Q$ field, but remains useful for perturbative computations.
• D-brane boundary conditions are naturally encoded as algebraic constraints in the $\sigma$-basis, simplifying their inclusion in computations.
• The formalism passes perturbative checks, reproducing known results such as the Siegel gauge and BRST invariance, and enables new non-perturbative computations via the cubic action.