Four lectures on closed string field theory

doi:10.1016/j.physrep.2020.01.003

Physics Reports

Volume 851, 3 April 2020, Pages 1-36

https://doi.org/10.1016/j.physrep.2020.01.003 Get rights and content

Abstract

The following notes derive from review lectures on closed string field theory given at the Galileo Galilei Institute for Theoretical Physics in March 2019.

Section snippets

Preface

These are notes for four lectures on the topic of closed string field theory (closed SFT). Videos of the lectures can be found by clicking here, though in four allotted slots I covered only the first two lectures. Even with all four lectures, the scope of what is covered is limited. The definitive reference is Zwiebach’s 1992 paper [1], which however only discusses the bosonic string. The superstring requires a few additional considerations which have only been taken up in recent years. A

Intro

Closed string field theory is a quantum field theory constructed in such a way that its Feynman diagram expansion computes string S-matrix elements. The utility of this formalism is that it seems necessary to give a fully complete and consistent definition of string perturbation theory. Applications have been especially highlighted in recent work by Sen and others [2], and include

•
Consistent treatment of divergences. Several kinds of divergences can appear in the standard worldsheet approach to

Lecture 1: Off-shell amplitudes

We begin by describing off-shell amplitudes in bosonic string theory. By this we have in mind, at the very least, some kind of continuation of physical amplitudes to generic momenta which are not constrained to lie on the mass shell. More precisely, we are looking for a multilinear map $〈 A_{g, n} | : {\hat{H}}^{\otimes n} \to {\hat{H}}^{\otimes 0}$ subject to the following conditions:

(1)
The map is defined on a vector space $\hat{H}$ satisfying $H_{Q} \subset \hat{H} \subseteq H,$ where $H$ is the full conformal field theory state space and $H_{Q}$ is the vector space of BRST invariant

Lecture 2: Feynman diagrams

The off-shell amplitudes of closed SFT are of a special kind, since they all derive from a common set of vertices connected by propagators to form Feynman diagrams. Usually the Feynman graph expansion is deduced from the action; however, we do not know the form of the closed SFT action (yet), and presently it is actually easier to go the other way: construct a Feynman graph expansion of off-shell amplitudes, and use this to deduce the necessary form of the action. To construct Feynman diagrams

Lecture 3: The action

We are now ready to formulate the action of closed SFT. We start with the classical action (no loop vertices) where the dynamical string field is a Grassmann even state $Φ \in \hat{H}$ of ghost number $2$ . The action can be expressed $S_{g = 0} = \frac{1}{2!} 〈 V_{0, 2} | Φ \otimes Φ + \frac{1}{3!} 〈 V_{0, 3} | Φ \otimes Φ \otimes Φ + \frac{1}{4!} 〈 V_{0, 4} | Φ \otimes Φ \otimes Φ \otimes Φ + \dots .$ The subscript $g = 0$ indicates that this is the classical action. In the last lecture we explained the procedure for constructing vertices $〈 V_{g, n} |$ by filling in “gaps” so that Feynman diagrams produce a continuous global section

Lecture 4: Closed superstring field theory

Closed super SFT is the field theory of fluctuations of a closed string background in superstring theory—the heterotic or one of the Type II string theories. Closed super SFT has been characterized in the RNS formalism, based on worldsheet theories with $N = (1, 0)$ (heterotic) or $N = (1, 1)$ (Type II) supersymmetry. The theories are structurally similar to closed bosonic SFT: The dynamical string field $Φ$ is Grassmann even and is subject to $b_{0}^{-}$ and level matching conditions; at the classical level it

Acknowledgments

I would like to thank the organizers of the workshop “String Theory from a worldsheet perspective” in spring 2019 for inviting me to give these lectures. I also thank the Galileo Galilei Institute for Theoretical Physics and INFN for hospitality and partial support during my stay at this workshop. I am grateful to R. Donagi for discussion and encouraging me to write up these notes, and S. Stieberger and O. Lechtenfeld for comments. This work was supported by ERDF and MŠMT, Czech Republic

References (39)

ZwiebachB.
Closed string field theory: Quantum action and the B-V master equation
Nuclear Phys. B
(1993)
SenA. et al.
A Proof of local background independence of classical closed string field theory
Nuclear Phys. B
(1994)
SonodaH. et al.
Covariant closed string theory cannot be cubic
Nuclear Phys. B
(1990)
FriedanD. et al.
Conformal invariance supersymmetry and string theory
Nuclear Phys. B
(1986)
KazamaY. et al.
Symmetry structures of superstring field theories
Nuclear Phys. B
(1986)
KazamaY. et al.
Space–time supersymmetry of the covariant superstring
Nucl. Phys. B
(1986)
TeraoH. et al.
Gauge invariant actions and gauge fixed actions of free superstring field theory
Phys. Lett. B
(1986)
YamronJ.P.
A gauge invariant action for the free Ramond string
Phys. Lett. B
(1986)
VerlindeE.P. et al.
Multiloop calculations in covariant superstring theory
Phys. Lett. B
(1987)
LechtenfeldO.
Superconformal ghost correlations on Riemann surfaces
Phys. Lett. B
(1989)
SenA.
BV master action for heterotic and type II string field theories
J. High Energy Phys.
(2016)
SenA.
Off-shell amplitudes in superstring theory
Fortschr. Phys.
(2015)
SenA. et al.
Filling the gaps with PCO’s
J. High Energy Phys.
(2015)

de LacroixC. et al.

Closed superstring field theory and its applications

Internat. J. Modern Phys. A

(2017)

J. Polchinski, String Theory, Vols. 1 & 2, Cambridge University...

SenA.

Supersymmetry restoration in superstring perturbation theory

J. High Energy Phys.

(2015)

SenA.

One loop mass renormalization of unstable particles in superstring theory

J. High Energy Phys.

(2016)

SenA.

Unitarity of superstring field theory

J. High Energy Phys.

(2016)

C. De Lacroix, H. Erbin, A. Sen, Analyticity and crossing symmetry of superstring loop amplitudes,...

SenA.

Background independence of closed superstring field theory

J. High Energy Phys.

(2018)

M. Cho, S. Collier, X. Yin, Strings in Ramond–Ramond backgrounds from the Neveu–Schwarz–Ramond formalism,...

MarklM.

Loop homotopy algebras in closed string field theory

Comm. Math. Phys.

(2001)

Cited by (59)

String vertices for the large N limit
2024, Nuclear Physics B
String vertices of open-closed string field theory on an arbitrary closed string background with N identical D-branes are investigated when N is large. We identify the relevant geometric master equation and solve it using open-closed hyperbolic genus zero string vertices with a milder systolic constraint. The limits corresponding to integrating out open or closed strings are investigated. We highlight the possible implications of our construction to AdS/CFT correspondence.
Tree-level amplitudes from the pure spinor superstring
2023, Physics Reports
On-shell action for type IIB supergravity and superstrings on AdS<inf>5</inf> × S<sup>5</sup>
2022, Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
AdS/CFT predicts that the value of the on-shell action for type IIB Supergravity (SUGRA) on $A d S_{5} \times S^{5}$ background must be a non-zero number completely determined from the boundary theory. We examine this statement within Sen's formalism for type IIB SUGRA and find that consistency with AdS/CFT requires us to add a specific boundary term to the action. We contrast our resolution with two other resolutions recently proposed in the literature in the context of different approaches to type IIB SUGRA. We explain how our resolution presents a strong benchmark for the possible boundary term of the complete spacetime action for type IIB superstring and how it may possibly lead to a piece of evidence for the strongest form of AdS/CFT conjecture in $A d S_{5} \times S^{5}$ . We also comment on the fate of the on-shell action for general self-dual p-form fields in Sen's formalism in any curved backgrounds.
The end of spacetime
2024, arXiv
Adding stubs to quantum string field theories
2024, arXiv
Strebel differentials and string field theory
2024, arXiv

View all citing articles on Scopus

View full text

Four lectures on closed string field theory

Abstract

Section snippets

Preface

Intro

Lecture 1: Off-shell amplitudes

Lecture 2: Feynman diagrams

Lecture 3: The action

Lecture 4: Closed superstring field theory

Acknowledgments

Nuclear Phys. B

Nuclear Phys. B

Nuclear Phys. B

Nuclear Phys. B

Nuclear Phys. B

Nucl. Phys. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

Phys. Lett. B

J. High Energy Phys.

Fortschr. Phys.

J. High Energy Phys.

Closed superstring field theory and its applications

Internat. J. Modern Phys. A

Supersymmetry restoration in superstring perturbation theory

J. High Energy Phys.

One loop mass renormalization of unstable particles in superstring theory

J. High Energy Phys.

Unitarity of superstring field theory

J. High Energy Phys.

Background independence of closed superstring field theory

J. High Energy Phys.

Loop homotopy algebras in closed string field theory

Comm. Math. Phys.