String Field Theory

Main ideas

String field theory is a field theoretical framework for string theory (String field theory is a formulation of string theory as a field theory in space-time with an infinite number of massive fields.). As such, it can potentially be used for studying non-perturbative effects of the theory, for evaluating scattering amplitudes using Feynman diagram techniques and for exploring various string vacua by studying classical solutions.

Main results

Simple analytic solution to cubic Neveu-Schwarz String Field Theory including the $GSO(-)$ sector is presented. This solution is an analog of the Erler-Schnabl solution for bosonic case and one of the authors solution for the pure $GSO(+)$ case. Gauge transformations of the new solution to others known solutions for the $NS$ string tachyon condensation are constructed explicitly. This gauge equivalence manifestly supports the early observed fact that these solutions have the same value of the action density.

We construct a new solution of the superstring equation of motion and show that this solution satisfies two of Sen’s conjectures and does not require “phantom terms.”

A class of exact analytic solutions in the modified cubic fermionic string field theory with the GSO(-) sector is presented. This class contains the GSO(-) tachyon field and reproduces the correct value for the nonBPS D-brane tension.

Neveu-Schwarz ghost slivers in pictures zero and minus one are constructed. In particular, using algebraic methods $\beta$, $\gamma$ ghost sliver in the -1 picture is obtained. The algebraic method consists in solving a projector equation in an algebra, where the multiplication is defined by a pure 3-string vertex without any insertions at the string midpoint. We show that this projector is a sliver in a twisted version of $\beta$, $\gamma$ conformal theory. We also show that the product of the twisted $b$, $c$ and $\beta$, $\gamma$ ghost slivers solves an equation that appears after a special rescaling of super VSFT.

In the framework of the Sen conjectures a construction of vacuum superstring field theory on a non-BPS brane is discussed. A distinguished feature of this theory is a presence of a ghost kinetic operator mixing GSO+/- sectors. A candidate for such kinetic operator with zero cohomology is discussed.

Using algebraic methods the Neveu-Schwarz fermionic matter sliver is constructed. Inspirited by the wedge algebra two equations for the sliver, linear and quadratic, are considered. It is shown that both equations give the same nontrivial answer. The sliver is considered also using CFT methods where it is defined as the limit of the wedge states in the NS sector of the superstring.

In this lecture notes we explain and discuss some ideas concerning noncommutative geometry in general, as well as noncommutative field theories and string field theories. We consider noncommutative quantum field theories emphasizing an issue of their renormalizability and the UV/IR mixing. Sen's conjectures on open string tachyon condensation and their application to the D-brane physics have led to wide investigations of the covariant string field theory proposed by Witten about 15 years ago. We review main ingredients of cubic (super)string field theories using various formulations: functional, operator, conformal and the half string formalisms. The main technical tools that are used to study conjectured D-brane decay into closed string vacuum through the tachyon condensation are presented. We describe also methods which are used to study the cubic open string field theory around the tachyon vacuum: construction of the sliver state, comma'' and matrix representations of vertices.

The gauge invariance of cubic open superstring field theory is considered in a framework of level truncation, and applications to the tachyon condensation problem are discussed. As it is known, in the bosonic case the Feynman-Siegel gauge is not universal within the level truncation method. We explore another gauge that is more suitable for calculation of the tachyon potential for fermionic string at level (2,6). We show that this new gauge has no restrictions on the region of its validity at least at this level.

It has been conjectured that at the stationary point of the tachyon potential for the non-BPS D-brane or brane-anti-D-brane pair, the negative energy density cancels the brane tension. We study this conjecture using a cubic superstring field theory with insertion of a double-step inverse picture changing operator. We compute the tachyon potential at levels (1/2,1) and (2,6). In the first case we obtain that the value of the potential at the minimum is 97.5% of the non BPS D-brane tension. Using a special gauge in the second case we get 105.8% of the tension.

We modify Witten's action for the NSR superstring. Our proposal is based on the use of new “double-step” picture-changing operators. The modified action, in contrast to Witten's, generates N-point tree-level amplitudes which agree with the Koba-Nielsen amplitudes. The action is gauge invariant without any reference to a regularization and manifestly space-time supersymmetric. The algebra of the gauge transformations is closed and the classical equations of motion do not contain any picture-changing insertions.

In the framework of the background formalism we analyse possible versions of the Witten-type NSR superstring field theory. We find the picture for string fields to be uniquely fixed by the requirement that the perturbative classical solutions are well-defined. This uniquely defined picture and the corresponding action are different from the ones in Witten's theory and coincide with the ones proposed from different reasons in our previous paper. Following the same background method we calculate the tree-level scattering amplitudes for the new action and argue that in contrast to the ones in Witten's original theory, the amplitudes are singularity-free and hence there is no need to add any tree-level counterterms. We also prove the amplitudes to reproduce correctly the first quantized results.

In the framework of the background formalism we analyse possible versions of the Witten-type NSR superstring field theory. We find the picture for string fields to be uniquely fixed by the requirement that the perturbative classical solutions are well-defined. This uniquely defined picture and the corresponding action are different from the ones in Witten's theory and coincide with the ones proposed from different reasons in our previous paper. Following the same background method we calculate the tree-level scattering amplitudes for the new action and argue that in contrast to the ones in Witten's original theory, the amplitudes are singularity-free and hence there is no need to add any tree-level counterterms. We also prove the amplitudes to reproduce correctly the first quantized results.

We show that Witten's action for the NSR superstring on the space of non-smooth vectors generated by SUSY transformations depends on regularization that demonstrates the presence of anomalies in the theory. The necessity of choosing the regularization is connected with the appearance of the product of picture-changing operators at coincident points. Using one of the possible types of regularization, we get the minimal formulation involving truncated fields, and adopting another one, we get the scheme implied in Witten's paper.

A non-associative nilpotent algebra called string field algebra, proposed recently to describe the gauge-invariant interaction of the bosonic string, is considered. It is shown that the properties of the string field algebra can be derived from the requirement of closure of the algebra of gauge transformations. The gauge invariant theories of both the open and closed strings are different realizations of the same abstract string field algebra. The gauge invariant interacting theory of the matter string field is presented.

A gauge-covariant theory for the interacting bosonic open string is presented. The interaction of strings is interpreted as defining a commutative, nonassociative algebra with a Z2-grading and the classical string field theory is formulated in the language of this algebra. The even elements of the algebra are nilpotent elements and this imposes strong restrictions on the possible forms of the string interaction and moreover defines it in a unique way.

Main publications

• I.Ya. Aref'eva, R.V. Gorbachev, On Gauge Equivalence of Tachyon Solutions in Cubic Neveu-Schwarz String Field Theory; arXiv:1004.5064
• R.V. Gorbachev, New solution of the superstring equation of motion, Theor.Math.Phys. 162 (2010) 90-94
• I. Ya. Aref'eva, R. V. Gorbachev, P. B. Medvedev, Tachyon Solution in Cubic Neveu-Schwarz String Field Theory, Theor.Math.Phys. 158(2009) 320-332, arXiv:0804.2017
• I. Ya. Aref'eva, A. A. Giryavets, A. S. Koshelev, NS Ghost Slivers, Phys.Lett. B 536 (2002) 138-146; arXiv:hep-th/0203227
• I. Ya. Aref'eva, D. M. Belov, A. A. Giryavets, Construction of the Vacuum String Field Theory on a non-BPS Brane, JHEP 0209 (2002) 050; arXiv:hep-th/0201197
• I. Ya. Aref'eva, A. A. Giryavets, P. B. Medvedev, NS Matter Sliver, Phys.Lett. B 532 (2002) 291-296; arXiv:hep-th/0112214
• I. Ya. Aref'eva, D. M.Belov, A. A.Giryavets, A. S.Koshelev and P. B.Medvedev, Noncommutative Field Theories and (Super)String Field Theories; arXiv:hep-th/0111208
• I. Ya. Aref'eva, D. M. Belov, A. S. Koshelev, P. B. Medvedev, Gauge Invariance and Tachyon Condensation in Cubic Superstring Field Theory, Nucl.Phys. B638 (2002) 21-40; arXiv:hep-th/0107197
• I. Ya. Aref'eva, D. M. Belov, A. S. Koshelev and P. B. Medvedev, Tachyon condensation in cubic superstring field theory, Nucl.Phys.B 638 , (2002) 3-20; arXiv:hep-th/0011117
• I. Ya. Arefeva, P. B. Medvedev, A. P. Zubarev, New Representation For String Field Solves The Consistency Problem For Open Superstring Field Theory, Nucl.Phys.B 341 (1990) 464-498
• I. Ya. Arefeva, P. B. Medvedev, A. P. Zubarev, , Background Formalism For Superstring Field Theory, Phys.Lett.B 240 (1990) 356-362
• I. Ya. Arefeva , P. B. Medvedev,, Anomalies In Witten's Field Theory Of The Nsr String, Phys.Lett.B 212 (1988) 299
• I. Ya. Aref'eva and I. V. Volovich, String Field Algebra., Phys.Lett.B 182 (1986) 312
• I. Ya. Aref'eva and I. V. Volovich, Gauge Invariant String Interaction And Nonassociative Algebra, Phys.Lett.B 182 (1986) 159