nLab string field theory

Contents

Phenomenology

$\infty$-Chern-Simons theory

∞-Chern-Weil theory

∞-Chern-Simons theory

∞-Wess-Zumino-Witten theory

Contents

Idea

String field theory is supposed to be something like a quantum field theory which is the second quantization of the string in string theory, following this analogy:

worldvolume field theoryof fundamental branesand their second quantizationwhich in perturbation theory is given by
worldline formalismparticlequantum field theoryscattering amplitudes
2d CFT correlatorsstringstring field theorystring scattering amplitudes

Recall that perturbative string theory is a higher dimensional version of the Feynman perturbation series in quantum field theory. This Feynman perturbation series may be understood as computing the path integral over the Lagrangian of the given quantum field theory. String field theory is the attempt to identify this Lagrangian description corresponding to the string perturbation series.

So string field theory is the attempt to complete the following analogy:

Feynman perturbation series : QFT Lagrangian :: String perturbation theory : String field theory .

Motivation

The original hope was that string field theory would be a way to embed the string perturbation series prescription into a more coherent non-perturbative framework.

Achievements

The most detailed insight that has come out of the study of string field theory is the full understanding of the role of the “tachyon” field in bosonic perturbative string theory. In the bosonic version of the theory one of the excitations of the string is a quantum that appears to have imaginary mass. Such “tachyonic” quanta appear in ordinary field theory when the perturbation series is developed around an extremum of the QFT action functional that is not a local minimum, but a local maximum: it indicates that the classical configuration around which the perturbation series computes the quantum corrections is dynamically unstable and time evolution will tend to evolve it to the next local minimum. In the perturbative quantum description the movement to the next local minimum manifests itself in the condensation (as in Bose-Einstein condensation?) of the tachyon field. This is called tachyon condensation.

Shortly after its conception it was suspected that the tachyon that appears in the perturbation theory of the bosonic string is similarly an indication that the bosonic string’s perturbation series has to be understood as being a perturbation about a local maximum of some action functional. String field theory aimed to provide that notion of action functional. And indeed, in bosonic string field theory one has a kind of higher action functional and may compute the “tachyon potential” that it implies. It indeed has a local maximum at the point about which the ordinary bosonic string perturbation series is a perturbative expansion, while a local minimum is foun nearby.

Ashoke Sen conjectured the statement – now known as Sen's conjecture – that the depth of this tachyon potential, i.e. the energy density difference between this local maximum and this local minimum corresponds precisely to the energy density of the space-filling D25-brane that is seen in perturbative string theory. This would mean that the condensation of the bosonic string’s tachyon corresponds to the decay of the unstable space-filling D25 brane.

The detailed quantitative confirmation of Sen’s conjecture has been one of the main successes of string field theory. In the course of this a detailed algebraic description of the “true closed bosonic string vacuum”, i.e. of the theory at that local tachyon potential minimum has been found. However, the algebraic expressions involved tend to be hard to handle in their complexity.

There are numerical indications that indeed as the D25-brane decays, the remaining vacuum contains (only) closed strings. See the references below.

Shortcomings

The shortcoming of the current development of string field theory can probably be summarized as follows:

• it has been studied as a theory of a classical action functional. Little is known about the true quantum effects of the string field theory action functional.

• the best understanding exists for bosonic open string field theory, while closed and supersymmetric string field theory has remained much less accessible.

In terms of higher category theory

Closed string field theory is governed by an L-infinity algebra of interactions, open string field theory by an A-infinity algebra and open-closed string field theory by a mixture of both: an open-closed homotopy algebra?.

(…)

Bosonic closed string field theory

So far string field theory is defined in terms of an action functional. So, strictly speaking, it is defined as a classical field theory. The corresponding quantum master action is known, but apart from that not much detail about the quantization of this action has been considered in the literature.

The interaction terms

The unrestricted configuration space of string field theory is the subcomplex of the BRST complex of the closed (super-)string, regarded as a $\mathbb{N}$-graded vector space with respect to the ghost number grading, on those elements $\Psi$ that satisfy

1. $(b_0 - \bar b_0) \Psi = 0$;

2. $(L_0 - \bar L_0) \Psi = 0$ (“level matching condition”),

We shall write $\mathfrak{g}$ for this graded vector space. See (Markl, section 1)

This is equipped for each $k \in \mathbb{N}$ with a $k$-ary operation

$[-,\cdots,-]_k : \mathfrak{g}^{\otimes k} \to \mathfrak{g}$

given by the (k+1)-point function of the string (the amplitude for $k$ closed strings coming in and merging into a single outgoing string). For $k = 1$ this is the BRST operator

$[-]_1 = d_{BRST} \,.$

These operations are graded-symmetric: for all $\{\Psi_j\}$ of homogeneous degree $deg \Psi_j$ and for all $0 \leq i \lt k$ we have

(1)$[\Psi_1 , \cdots, \Psi_i, \Psi_{i+1}, \cdots, \Psi_k]_k = (-1)^{(deg \Psi_i)(deg \Psi_{i+1})} [\Psi_1 , \cdots, \Psi_{i+1}, \Psi_{i}, \cdots, \Psi_k]_k .$

Moreover, there is a bilinear inner product

$\langle -,- \rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{C}$

coming from the Hilbert space inner product of string states (Zwiebach (2.60)). This is non-degenerate on elements $\Psi$ which are annihilated by the ghost operator

$b_0^- := b_0 - \bar b_0$

in that for all $A \in \mathfrak{g}$ with $b_0^- A = 0$ we have

(2)$(\forall B \in \mathfrak{g} \,:\, \langle A,B\rangle = 0) \Rightarrow A = 0 \,.$

This is (Zwiebach 93, (2.61)).

The inner product satisfies for all $\Psi_1, \Psi_2$ of homogeneous degree the relation

(3)$\langle \Psi_1 , \psi_2 \rangle = (-1)^{(deg \Psi_1 + 1) (deg \Psi_2 + 1)} \langle \Psi_2, \Psi_1 \rangle$

Moreover, it is non-vanishing only on pairs of elements of total degree 5. (Zwiebach 93, (2.31)(2.44)).

From this one constructs the $(n+1)$-point functions

(4)$\{ \Psi_0, \Psi_1, \cdots, \Psi_k \} := \langle \Psi_0, [\Psi_1, \cdots, \Psi_k]_k \rangle \,.$

These are still graded-symmetric in all arguments: for all $\{\Psi_j\}$ of homogeneous degree $deg \Psi_j$ and all $0 \leq i \lt k$ we have

(5)$\{\Psi_0, \cdots, \Psi_i, \Psi_{i+1}, \cdots, \Psi_k\} = (-1)^{(deg \Psi_i)(deg \Psi_{i+1})} \{\Psi_0, \cdots, \Psi_{i+1}, \Psi_{i}, \cdots, \Psi_k\} \,.$

The action functional

Definition

The proper configuration space of string field theory is the sub-complex of the BRST complex of the closed (super-)string on those elements $\Psi$ for which

1. $(b_0 - \bar b_0) \vert\Psi\rangle = 0$;

2. $(L_0 - \bar L_0) \vert \Psi \rangle = 0$ (“level matching condition”);

3. $\vert \Psi \rangle^\dagger = - (\langle \Psi \vert)$ (“reality”);

4. $\vert \Psi\rangle$ is Grassmann even (…define…)

5. $ghostnumber \vert \Psi \rangle = 2$ (…define…)

This is (Zwiebach 93, (3.9))

The action functional of closed string field theory is

$S : \Psi \mapsto \sum_{k = 1}^\infty \frac{1}{(k+1)!} \langle \Psi, [\Psi, \cdots, \Psi]_k\rangle \,.$

Since $[-]_1 = d_{BRST}$ is the BRST operator this starts out as

$S : \Psi = \frac{1}{2}\langle \Psi , d_{BRST} \Psi \rangle + \frac{1}{3} \langle \Psi, [\Psi, \Psi]_2\rangle + \cdots \,.$

This immediately gives the equations of motion by variation with respect to $\Psi$ (Zwiebach 93, (4.46)). The analogous statement for the superstring in in (Sen 15 (2.22)).

As an $\infty$-Chern-Simons theory

The above action functional for closed string field theory turns out to have a general abstract meaning in higher category theory/homotopy theory. We spell out here how the action functional for closed string field theory is an example of an ∞-Chern-Simons theory in that it arises precisely as the Chern-Simons element of the binary pairing regarded as a binary invariant polynomial on the L-∞ algebra of string fields.

Proposition

The string BRST complex equipped with its $k$-ary interaction genus-0 interaction vertices

$(\mathfrak{g}, \{[-,\cdots,-]_k\})$

is an L-∞ algebra.

This is (Zwiebach 93, (4.12)). For more details on the $L_\infty$-structure see References – Relation to L-∞- and A-∞-algebra) .

Proposition

The inner product $\langle -,-\rangle$ satisfies the definition of a non-degenerate invariant polynomial on this $L_\infty$-algebra when restricted to fields of even degree as in def. .

Proof

For simplicity of notation we discuss this as if $\mathfrak{g}$ were finite-dimensional. The argument for the infinite-dimensional case follows analogously.

Let $\{t_a\}$ be a basis of $\mathfrak{g}$ with dual basis $\{t^a\}$. Then the Chevalley-Eilenberg algebra of $\mathfrak{g}$ is generated from the $\{t^a\}$ with differential given by

$d_{CE(\mathfrak{g})} : t^a \mapsto - \sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, \cdots, t_{a_k}]_k \,\, t^{a_1} \wedge \cdots \wedge t^{a_k} \,.$

The Weil algebra $W(\mathfrak{g})$ is similarly generated from $\{t^a, r^a\}$ with differential

$d_{W(\mathfrak{g})} : t^a \mapsto - \sum_{k = 1}^\infty \frac{1}{k!} [t_{a_1}, \cdots, t_{a_k}] \,\, t^{a_1} \wedge \cdots \wedge t^{a_k} + r^a$

and

$d_{W(\mathfrak{g})} : r^a \mapsto \sum_{k = 1}^\infty \frac{1}{k!} [t_{a_0}, t_{a_1}, \cdots, t_{a_k}]_{k+1} \, r^{a_0} \wedge t^{a_1} \wedge \cdots \wedge t^{a_k} \,.$

Write

$P_{a b} := \langle t_a, t_b\rangle$

for the components of the bilinear pairing in this basis. By (3) it follows that we can indeed regard

$P_{a b} r^a \wedge r^b \in W(\mathfrak{g})$

as an element in the Weil algebra (since $deg r^a = deg t^a + 1$).

Therefore to see that this is an invariant polynomial it remains to check that it is $d_W$-closed. To see this, first introduce the notation

$C_{a_0, \cdots, a_k} := \{t_{a_0}, \cdots, t_{a_k}\}$

for the components of the $(k+1)$-point function (4). Then compute

\begin{aligned} d_{W(\mathfrak{g})} P_{a b } r^a \wedge r^b & = 2 P_{a b} r^a \wedge \left( \sum_{k = 1}^\infty [t_{a_1}, \cdots, t_{a_k}]_k \, r^{a_1} \wedge t^{a_2}\wedge \cdots \wedge t^{a_k} \right) \\ & = 2 \sum_{k = 1}^\infty C_{a_0, a_1, \cdots, a_k} \, r^{a_0} \wedge r^{a_1} \wedge t^{a_2} \wedge \cdots \wedge t^{a_k} \end{aligned} \,.

This expression vanishes term-by-term by the symmetry properties (5) when restricted to fields of even degree: by first switching the factors in the wedge product and then relabelling the indices we obtain

\begin{aligned} C_{a_0, a_1, \cdots, a_k} r^{a_0} \wedge r^{a_1} &= (-1)^{(deg t_{a_0} + 1)(deg t_{a_1} + 1) + (deg t_{a_0})(deg t_{a_1})} C_{a_0, a_1, \cdots, a_k} r^{a_0} \wedge r^{a_1} \\ & = (-1)^{deg t_{a_0} + deg t_{a_1} + 1} C_{a_0, a_1, \cdots, a_k} r^{a_0} \wedge r^{a_1} \\ & = (-1) C_{a_0, a_1, \cdots, a_k} r^{a_0} \wedge r^{a_1} \end{aligned} \,,

where in the last step we used the constraints on degrees given by def. .

This shows that $\langle-,-\rangle$ satisfies the defining equation of an invariant polynomial on the proper configuration space. The non-degeneracy is due to (2).

From the discussion at Chern-Simons element in the section Canonical Chern-Simons element we have that the Lagrangian of the infinity-Chern-Simons theory defined by the data $(\mathfrak{g}, \langle -,-\rangle)$ is

$L : A \mapsto \langle A, d A\rangle + \sum_{k = 1}^\infty \frac{2}{(k+1)!} \langle A , [A, \cdots , A]_k\rangle$

for $A$ a $\mathfrak{g}$-valued differential form on some $\Sigma$. So the closed string field theory action looks like that of $\infty$-Chern-Simons theory over an odd-graded $\Sigma$.

Open-Closed string field theory

When considering open and closed strings jointly, then in addition to the closed string sector being encoded by an L-∞ algebra $\mathfrak{g}_{closed}$ as above, the open string sector is encoded in an A-∞ algebra $A_{open}$ and the former acts on the latter by homotopy derivations (see also at derivations on algebras over a dg-operad)

$\mathfrak{g}_{closed} \longrightarrow Der (A_{open})$

Notice that this is half of the axioms of an $\infty$-Lie-Rinehart pair.

Superstring field theory

The maybe most wide-spread attempt to generalize the above to superstring field theory replaces the Chern-Simons-type action with a Wess-Zumino-Witten theory-type action, see at WZW-type superstring field theory .

A formulation of superstring field theory more on line with the Chern-Simons type bosonic theory is proposed in (Jurco-Muenster 13). See also the introduction there for a survey of the literature

Closed string field theory has been argued to arise from the dynamics of Wilson loops in the IKKT matrix model in (Fukuma-Kawai-Kitazawa-Tsuchiya 97)

References

General

A textbook-like account is in

Bosonic string field theory

Open SFT

Original articles are

Closed SFT

The fundamental work of Zwiebach on closed SFT is summed up in

Brief reviews include

The explicit identification of the Einstein-Hilbert action for gravity coupled to the action for the B-field and the dilaton in the lowest orders of the CSFT action is discussed for instance in Yang-Zwiebach, section 3.1 and in

• Bang-Gui Liu, General coordinate transformation and gravitational action from closed bosonic string field theory, Class. Quantum Grav. 6 (1989)

• Masako Asano, Mitsuhiro Kato, Closed string field theory in a-gauge (arXiv:1206.3901)

Discussion of the expected closed string tachyon vacuum is in

and further detailed analysis is in

Superstring field theory

The generalization to field theory for the superstring, where picture number compliates the RR-field-sector, has (only) more recently seen considerable development,

• Hiroshi Kunitomo, Yuji Okawa, Complete action for open superstring field theory, Prog. Theor. Exp. Phys. (2016) 023B01 (arXiv:1508.00366)

A survey is in

• Yuji Okawa, Recent developments in the construction of superstring field theory I, 2016 (pdf)

based on

with further developments in

Interpretation of picture number as a grading on differential forms on supermanifolds induced from a choice ofintegral top form is due to

and has been further amplified in

• Ashoke Sen, Background Independence of Closed Superstring Field Theory, JHEP02(2018)155 (arXiv:1711.08468)

• Corinne de Lacroix, Harold Erbin, Sitender Pratap Kashyap, Ashoke Sen, Mritunjay Verma, Closed Superstring Field Theory and its Applications, International Journal of Modern Physics AVol. 32, No. 28n29, 1730021 (2017) (arXiv:1703.06410)

For previous constructions, the introduction of

based on

• Chungsheng James Yeh, Topics in superstring theory, PhD thesis, Berkeley 1993 (SPIRE)

has a useful survey, which we quote now:

Quote from (Jurco-Muenster13):

The first attempt towards a field theory of superstrings was initiated by the work of Witten

• Edward Witten, Interacting field theory of open superstrings, Nuclear Physics B, Volume 276, Issue 2 (1986)

by seeking a Chern-Simons like action for open superstrings similar to the one of open bosonic string field theory (Witten 86). The major obstacle compared to the bosonic string is the necessity of picture changing operators. Indeed, the cubic superstring theory of (Witten 86a) turns out to be inconsistent due to singularities arising form the collision of picture changing operators

• C. Wendt, Scattering amplitudes and contact interactions in Witten’s superstring field theory, Nuclear Physics B, Volume 314, Issue 1.

In order to circumvent this problem, another approach was pursued which sets the string field into a different picture

• C.R. Preitschopf, C.B. Thorn, S. Yost, Superstring field theory Nuclear Physics B, Volume 337, Issue 2.

• I.Ya. Aref’eva, P.B. Medvedev, A.P. Zubarev, New representation for string field solves the consistency problem for open superstring field theory, Nuclear Physics B, Volume 341, Issue 2.

but upon including the Ramond sector, the modified superstring field theory suffers from similar inconsistencies

• M. Kroyter, Superstring field theory equivalence: Ramond sector, Journal of High Energy Physics, Volume 2009, Issue 10.

These two approaches are based on the small Hilbert space, the state space including the reparametrization ghosts and superghosts as they arise from gaugefixing. Upon bosonization of the superghosts, an additional zero mode arises which allows the formulation of a WZW like action for the NS sector of open superstring field theory

In contrast to bosonic string field theory, BV quantization of this theory is more intricate than simply relaxing the ghost number constraint for the fields of the classical action

• Nathan Berkovits, Constrained BV description of string field theory, Journal of High Energy Physics, Volume 2012, Issue 3.

• M. Kroyter, Y. Okawa, M. Schnabl, S. Torii, Barton Zwiebach, Open superstring eld theory I: gauge xing, ghost structure, and propagator, Journal of High Energy Physics, Volume 2012, Issue 3.

Finally, there is a formulation of open superstring field theory that differs from all other approaches in not fixing the picture of classical fields

• M. Kroyter, Superstring field theory in the democratic picture, Advances in Theoretical and Mathematical Physics, Volume 15, Number 3.

On the other hand, the construction of bosonic closed string field theory (Zwiebach 92) takes its origin in the moduli space of closed Riemann surfaces. Vertices represent a subspace of the moduli space, such that the moduli space decomposes uniquely into vertices and graphs,and do not apriori require a background. Graphs are constructed from the vertices by sewing together punctures along prescribed local coordinates around the punctures. But an assignment of local coordinates around the punctures, globally on the moduli space, is possible only up to rotations. This fact implies the level matching condition and via gauge invariance also the $b_0^- = 0$ constraint.

In an almost unnoticed work (Yeh), the geometric approach developed in bosonic closed string field theory, as described in the previous paragraph, has been generalized to the context of superstring field theory. Neveu-Schwarz punctures behave quite similar to punctures in the bosonic case, but a Ramond puncture describes a divisor on a super Riemann surface rather than a point. As a consequence, local coordinates around Ramond punctures, globally defined over super moduli space, can be fixed only up to rotations and translation in the Ramond divisor.

A given background provides forms on super moduli space

• Alexander Belopolsky, New Geometrical Approach to Superstrings (arXiv:hep-th/9703183)

• L. Alvarez-Gaume, P. Nelson, C. Gomez, G. Sierra, C. Vafa, Fermionic strings in the operator formalism, Nuclear Physics B, Volume 311, Issue 2.

in the sense of geometric integration theory on supermanifolds, and in particular the geometric meaning of picture changing operators has been clarified

Integrating along an odd direction in moduli space inevitably generates a picture changing operator. Thus, the ambiguity of defining local coordinates around Ramond punctures produces a picture changing operator associated with the vector field generating translations in the Ramond divisor. The bpz inner product plus the additional insertions originating from the sewing define the symplectic form relevant for BV quantization. As in the bosonic case, we require that the symplectic form has to be non-degenerate, but the fact that the picture changing operator present in the Ramond sector has a non-trivial kernel, forces to impose additional restrictions besides the level matching and $b_0^- = 0$ constraint on the state space. The purpose of (Jurco-Muenster 13) is to describe the construction of type II superstring field theory in the geometric approach.

Older reviews include

More recently there is

• Tomoyuki Takezaki, Open superstring field theory including the Ramond sector based on the supermoduli space (arXiv:1901.02176)

Effective field theory

Discussion of Wilsonian effective field theory of string field theory includes

• R. Brustein, S.P.De Alwis, Renormalization group equation and non-perturbative effects in string-field theory, Nuclear Physics B Volume 352, Issue 2, 25 March 1991, Pages 451-468 (doi:10.1016/0550-3213(91)90451-3)

• Brustein and K. Roland, ?Space-time versus world sheet renormalization group equation in string theory,? Nucl. Phys. B372, 201 (1992) (doi:10.1016/0550-3213(92)90317-5)

• Ashoke Sen, Wilsonian Effective Action of Superstring Theory, J. High Energ. Phys. (2017) 2017: 108 (arXiv:1609.00459)

Relation to $A_\infty$- and $L_\infty$-algebras

The L-infinity algebra structure on Zwiebach’s bosonic closed string fields is apparently due to a comment by Jim Stasheff to the conference contribution

It appears in print in

• Barton Zwiebach, Closed string field theory: Quantum action and the B-V master equation , Nucl.Phys. B390 (1993) 33

The A-infinity algebra structure of bosonic open string field theory in

For the topological string see

Discussion for heterotic string field theory is in

• Hiroshi Kunitomo, Tatsuya Sugimoto, Heterotic string field theory with cyclic L-infinity structure (arXiv:1902.02991)

Discussion of the mathematical aspects is in

Surveys include

Discussion of the CSFT-action as of the form of ∞-Chern-Simons theory is in section 4.4 of

From all this one might expect analogously a super L-∞ algebra underlying closed superstring field theory. This does not seem to materialzed yet in the literature, though. The closest is maybe the structure described in

For Yang-Mills theory

The idea that open strings on $N$ coincident D-branes exhibit gauge enhancement to $U(N)$-gauge field theory is due to

There, this is called an “obvious guess” (first line on p. 8). Subsequently, most authors cite this obvious guess as a fact; for instance the review

By actual computation in open string field theory “convincing evidence” (see bottom of p. 22) was found, numerically, in

Similar numerical derivation, as well as exact derivation at zero momentum, is in

The first full derivation seems to be due to

which is surveyed in

That on D0-branes this reproduces the BFSS matrix model and on D(-1)-branes the IKKT matrix model is shown in

• Taejin Lee, Covariant Open String Field Theory on Multiple D$p$-Branes (arXiv:1703.06402)

Discussion of the L-infinity algebra higher Chern-Simons theory of the Yang-Mills theory that appears to lowest order as the effective QFT in open string field theory is for instance in

Background independence

References discussing independence of string field theories on the CFT (sigma-model background) in terms of which they are written down.

For closed string field theory

A review of the history of some related developments is given in

• Sabbir Rahman, Manifest background independent formulation of string field theory (newsgroup comment)

Closed string field theory has also been argued to arise from the dynamics of Wilson loops in the IKKT matrix model in (Fukuma-Kawai-Kitazawa-Tsuchiya 97)

• M. Fukuma, H. Kawai, Y. Kitazawa, A. Tsuchiya, String Field Theory from IIB Matrix Model, Nucl.Phys.B510:158-174,1998 (arXiv:hep-th/9705128)

Tachyon dynamics, decaying D-branes and Sen’s conjecture

Sen's conjecture about the open bosonic string tachyon and the decay of the D25-brane originates in

Hints for the decay of the space-filling D25-brane in open bosonic string field theory and the resulting closed string vacuum were discussed in articles like

• Ian Ellwood, Washington Taylor, Open string field theory without open strings, Phys.Lett. B512 (2001) 181-188 (arXiv:hep-th/0103085)

• Bo Feng, Yang-Hui He, Nicolas Moeller, Testing the Uniqueness of the Open Bosonic String Field Theory Vacuum (arXiv:hep-th/0103103)

A breakthrough were then the analytic solutions describing the bosonic string tachyon vacuum in

Analogous discussion including also brane/anti-brane pairs in superstring theory is in

Relation to higher spin gauge theory

The idea that higher spin gauge theory appears as the limiting case of string field theory where the string tension vanishes goes back to

and is further developed for instance in

And conversely:

Last revised on April 29, 2019 at 11:35:51. See the history of this page for a list of all contributions to it.