String field theory

String theory
Fundamental objects
String Cosmic string Brane D-brane
Perturbative theory
Bosonic Superstring (Type I, Type II, Heterotic)
Non-perturbative results
S-duality T-duality U-duality M-theory F-theory AdS/CFT correspondence
Phenomenology
Phenomenology Cosmology Landscape
Mathematics
Geometric Langlands correspondence Mirror symmetry Monstrous moonshine Vertex algebra
Related concepts Theory of everything Conformal field theory Quantum gravity Supersymmetry Supergravity Twistor string theory N = 4 supersymmetric Yang–Mills theory Kaluza–Klein theory Multiverse Holographic principle
Theorists Aganagić Arkani-Hamed Atiyah Banks Berenstein Bousso Cleaver Curtright Dijkgraaf Distler Douglas Duff Dvali Ferrara Fischler Friedan Gates Gliozzi Gopakumar Green Greene Gross Gubser Gukov Guth Hanson Harvey Hořava Horowitz Gibbons Kachru Kaku Kallosh Kaluza Kapustin Klebanov Knizhnik Kontsevich Klein Linde Maldacena Mandelstam Marolf Martinec Minwalla Moore Motl Mukhi Myers Nanopoulos Năstase Nekrasov Neveu Nielsen van Nieuwenhuizen Novikov Olive Ooguri Ovrut Polchinski Polyakov Rajaraman Ramond Randall Randjbar-Daemi Roček Rohm Sagnotti Scherk Schwarz Seiberg Sen Shenker Siegel Silverstein Sơn Staudacher Steinhardt Strominger Sundrum Susskind 't Hooft Townsend Trivedi Turok Vafa Veneziano Verlinde Verlinde Wess Witten Yau Yoneya Zamolodchikov Zamolodchikov Zaslow Zumino Zwiebach
History Glossary
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String field theory (SFT) is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. This is accomplished at the level of perturbation theory by finding a collection of vertices for joining and splitting strings, as well as string propagators, that give a Feynman diagram-like expansion for string scattering amplitudes. In most string field theories, this expansion is encoded by a classical action found by second-quantizing the free string and adding interaction terms. As is usually the case in second quantization, a classical field configuration of the second-quantized theory is given by a wave function in the original theory. In the case of string field theory, this implies that a classical configuration, usually called the string field, is given by an element of the free string Fock space.

Contents

• Light-cone string field theory
• Free covariant string field theory
• Witten's cubic open string field theory
• Gauge invariance
• Equations of motion
• Quantization
• Supersymmetric covariant open string field theories
• Modified cubic superstring field theory
• Berkovits superstring field theory
• Other formulations of covariant open superstring field theory
• Covariant closed string field theory
• Covariant heterotic string field theory
• See also
• References

The principal advantages of the formalism are that it allows the computation of off-shell amplitudes and, when a classical action is available, gives non-perturbative information that cannot be seen directly from the standard genus expansion of string scattering. In particular, following the work of Ashoke Sen, ^[1] it has been useful in the study of tachyon condensation on unstable D-branes. It has also had applications to topological string theory, ^[2] non-commutative geometry, ^[3] and strings in low dimensions. ^[4]

String field theories come in a number of varieties depending on which type of string is second quantized: Open string field theories describe the scattering of open strings, closed string field theories describe closed strings, while open-closed string field theories include both open and closed strings.

In addition, depending on the method used to fix the worldsheet diffeomorphisms and conformal transformations in the original free string theory, the resulting string field theories can be very different. Using light cone gauge, yields light-cone string field theories whereas using BRST quantization, one finds covariant string field theories. There are also hybrid string field theories, known as covariantized light-cone string field theories which use elements of both light-cone and BRST gauge-fixed string field theories. ^[5]

A final form of string field theory, known as background independent open string field theory, takes a very different form; instead of second quantizing the worldsheet string theory, it second quantizes the space of two-dimensional quantum field theories. ^[6]

Light-cone string field theory

Light-cone string field theories were introduced by Stanley Mandelstam ^{[7] [8]} and developed by Mandelstam, Michael Green, John Schwarz and Lars Brink. ^{[9] [10] [11] [12] [13]} An explicit description of the second-quantization of the light-cone string was given by Michio Kaku and Keiji Kikkawa. ^{[14] [15]}

Light-cone string field theories were the first string field theories to be constructed and are based on the simplicity of string scattering in light-cone gauge. For example, in the bosonic closed string case, the worldsheet scattering diagrams naturally take a Feynman diagram-like form, being built from two ingredients, a propagator,

and two vertices for splitting and joining strings, which can be used to glue three propagators together,

These vertices and propagators produce a single cover of the moduli space of ${\displaystyle n}$-point closed string scattering amplitudes so no higher order vertices are required. ^[16] Similar vertices exist for the open string.

When one considers light-cone quantized superstrings, the discussion is more subtle as divergences can arise when the light-cone vertices collide. ^[17] To produce a consistent theory, it is necessary to introduce higher order vertices, called contact terms, to cancel the divergences.

Light-cone string field theories have the disadvantage that they break manifest Lorentz invariance. However, in backgrounds with light-like Killing vectors, they can considerably simplify the quantization of the string action. Moreover, until the advent of the Berkovits string ^[18] it was the only known method for quantizing strings in the presence of Ramond–Ramond fields. In recent research, light-cone string field theory played an important role in understanding strings in pp-wave backgrounds. ^[19]

Free covariant string field theory

An important step in the construction of covariant string field theories (preserving manifest Lorentz invariance) was the construction of a covariant kinetic term. This kinetic term can be considered a string field theory in its own right: the string field theory of free strings. Since the work of Warren Siegel, ^[20] it has been standard to first BRST-quantize the free string theory and then second quantize so that the classical fields of the string field theory include ghosts as well as matter fields. For example, in the case of the bosonic open string theory in 26-dimensional flat spacetime, a general element of the Fock-space of the BRST quantized string takes the form (in radial quantization in the upper half plane),

${\displaystyle |\Psi \rangle =\int d^{26}p\left(T(p)c_{1}e^{ip\cdot X}|0\rangle +A_{\mu }(p)\partial X^{\mu }c_{1}e^{ip\cdot X}|0\rangle +\chi (p)c_{0}e^{ip\cdot X}|0\rangle +\ldots \right),}$

where ${\displaystyle |0\rangle }$ is the free string vacuum and the dots represent more massive fields. In the language of worldsheet string theory, ${\displaystyle T(p)}$, ${\displaystyle A_{\mu }(p)}$, and ${\displaystyle \chi (p)}$ represent the amplitudes for the string to be found in the various basis states. After second quantization, they are interpreted instead as classical fields representing the tachyon ${\displaystyle T}$, gauge field ${\displaystyle A_{\mu }}$ and a ghost field ${\displaystyle \chi }$.

In the worldsheet string theory, the unphysical elements of the Fock space are removed by imposing the condition ${\displaystyle Q_{B}|\Psi \rangle =0}$ as well as the equivalence relation ${\displaystyle |\Psi \rangle \sim |\Psi \rangle +Q_{B}|\Lambda \rangle }$. After second quantization, the equivalence relation is interpreted as a gauge invariance, whereas the condition that ${\displaystyle |\Psi \rangle }$ is physical is interpreted as an equation of motion. Because the physical fields live at ghostnumber one, it is also assumed that the string field ${\displaystyle |\Psi \rangle }$ is a ghostnumber one element of the Fock space.

In the case of the open bosonic string a gauge-unfixed action with the appropriate symmetries and equations of motion was originally obtained by André Neveu, Hermann Nicolai and Peter C. West. ^[21] It is given by

${\displaystyle S_{\text{free open}}(\Psi )={\tfrac {1}{2}}\langle \Psi |Q_{B}|\Psi \rangle \ ,}$

where ${\displaystyle \langle \Psi |}$ is the BPZ-dual of ${\displaystyle |\Psi \rangle }$. ^[22]

For the bosonic closed string, construction of a BRST-invariant kinetic term requires additionally that one impose ${\displaystyle (L_{0}-{\tilde {L}}_{0})|\Psi \rangle =0}$ and ${\displaystyle (b_{0}-{\tilde {b}}_{0})|\Psi \rangle =0}$. The kinetic term is then

${\displaystyle S_{\text{free closed}}={\tfrac {1}{2}}\langle \Psi |(c_{0}-{\tilde {c}}_{0})Q_{B}|\Psi \rangle \ .}$

Additional considerations are required for the superstrings to deal with the superghost zero-modes.

Witten's cubic open string field theory

The best studied and simplest of covariant interacting string field theories was constructed by Edward Witten. ^[23] It describes the dynamics of bosonic open strings and is given by adding to the free open string action a cubic vertex:

${\displaystyle S(\Psi )={\tfrac {1}{2}}\langle \Psi |Q_{B}|\Psi \rangle +{\tfrac {1}{3}}\langle \Psi ,\Psi ,\Psi \rangle }$,

where, as in the free case, ${\displaystyle \Psi }$ is a ghostnumber one element of the BRST-quantized free bosonic open-string Fock-space.

The cubic vertex,

${\displaystyle \langle \Psi _{1},\Psi _{2},\Psi _{3}\rangle }$

is a trilinear map which takes three string fields of total ghostnumber three and yields a number. Following Witten, who was motivated by ideas from noncommutative geometry, it is conventional to introduce the ${\displaystyle *}$-product defined implicitly through

${\displaystyle \langle \Sigma |\Psi _{1}*\Psi _{2}\rangle =\langle \Sigma ,\Psi _{1},\Psi _{2}\rangle \ .}$

The ${\displaystyle *}$-product and cubic vertex satisfy a number of important properties (allowing the ${\displaystyle \Psi _{i}}$ to be general ghost number fields):

In these equations, ${\displaystyle gn(\Psi )}$ denotes the ghost number of ${\displaystyle \Psi }$.

Supersymmetric covariant open string field theories

There are two main constructions of supersymmetric extensions of Witten's cubic open string field theory. The first is very similar in form to its bosonic cousin and is known as modified cubic superstring field theory. The second, due to Nathan Berkovits is very different and is based on a WZW-type action.

Covariant closed string field theory

Covariant closed string field theories are considerably more complicated than their open string cousins. Even if one wants to construct a string field theory which only reproduces tree-level interactions between closed strings, the classical action must contain an infinite number of vertices ^[45] consisting of string polyhedra. ^{[46] [47]}

If one demands that on-shell scattering diagrams be reproduced to all orders in the string coupling, one must also include additional vertices arising from higher genus (and hence higher order in ${\displaystyle \hbar }$) as well. In general, a manifestly BV invariant, quantizable action takes the form ^[48]

${\displaystyle S(\Psi )=\hbar \sum _{g\geq 0}(\hbar g_{c})^{g-1}\sum _{n\geq 0}{\frac {1}{n!}}\{\Psi ^{n}\}_{g}}$

where ${\displaystyle \{\Psi ^{n}\}_{g}}$ denotes an ${\displaystyle n}$th order vertex arising from a genus ${\displaystyle g}$ surface and ${\displaystyle g_{c}}$ is the closed string coupling. The structure of the vertices is in principle determined by a minimal area prescription, ^[49] although, even for the polyhedral vertices, explicit computations have only been performed to quintic order. ^{[50] [51]}

Covariant heterotic string field theory

A formulation of the NS sector of the heterotic string was given by Berkovits, Okawa and Zwiebach. ^[52] The formulation amalgamates bosonic closed string field theory with Berkovits' superstring field theory.

See also

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References