For semisimple Lie algebra targets
For discrete group targets
For discrete 2-group targets
For Lie 2-algebra targets
For targets extending the super Poincare Lie algebra
(such as the supergravity Lie 3-algebra, the supergravity Lie 6-algebra)
Chern-Simons-supergravity
for higher abelian targets
for symplectic Lie n-algebroid targets
for the $L_\infty$-structure on the BRST complex of the closed string:
higher dimensional Chern-Simons theory
topological AdS7/CFT6-sector
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 theory | of fundamental branes | and their second quantization | which in perturbation theory is given by |
---|---|---|---|
worldline formalism | particle | quantum field theory | scattering amplitudes |
2d CFT correlators | string | string field theory | string 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 .
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.
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.
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.
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?.
(…)
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 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
$(b_0 - \bar b_0) \Psi = 0$;
$(L_0 - \bar L_0) \Psi = 0$ (“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
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
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
Moreover, there is a bilinear inner product
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
in that for all $A \in \mathfrak{g}$ with $b_0^- A = 0$ we have
This is (Zwiebach, (2.61)).
The inner product satisfies for all $\Psi_1, \Psi_2$ of homogeneous degree the relation
Moreover, it is non-vanishing only on pairs of elements of total degree 5. (Zwiebach, (2.31)(2.44)).
From this one constructs the $(n+1)$-point functions
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
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
$(b_0 - \bar b_0) \vert\Psi\rangle = 0$;
$(L_0 - \bar L_0) \vert \Psi \rangle = 0$ (“level matching condition”);
$\vert \Psi \rangle^\dagger = - (\langle \Psi \vert)$ (“reality”);
$\vert \Psi\rangle$ is Grassmann even (…define…)
$ghostnumber \vert \Psi \rangle = 2$ (…define…)
This is (Zwiebach, (3.9))
The action functional of closed string field theory is
Since $[-]_1 = d_{BRST}$ is the BRST operator this starts out as
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.
The string BRST complex equipped with its $k$-ary interaction genus-0 interaction vertices
is an L-∞ algebra.
This is (Zwiebach, (4.12)). For more details on the $L_\infty$-structure see References – Relation to L-∞- and A-∞-algebra) .
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. 1.
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
The Weil algebra $W(\mathfrak{g})$ is similarly generated from $\{t^a, r^a\}$ with differential
and
Write
for the components of the bilinear pairing in this basis. By (3) it follows that we can indeed regard
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
for the components of the $(k+1)$-point function (4). Then compute
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
where in the last step we used the constraints on degrees given by def. 1.
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
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 $\Sigma$.
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)
(Kajiura-Stasheff 04, Markl 04).
Notice that this is half of the axioms of an $\infty$-Lie-Rinehart pair.
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)
A textbook-like account is in
Original articles are
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-Zwieback, 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
Haitang Yang, Barton Zwiebach, A Closed String Tachyon Vacuum ?, JHEP 0509:054,2005 (arXiv:hep-th/0506077)
Nicolas Moeller, Haitang Yang, The nonperturbative closed string tachyon vacuum to high level (arXiv:hep-th/0609208)
Nicolas Moeller, A tachyon lump in closed string field theory (arXiv:0804.0697)
and further detailed analysis is in
Nicolas Moeller, Closed Bosonic String Field Theory at Quintic Order: Five-Tachyon Contact Term and Dilaton Theorem, JHEP 0703:043,2007 (arXiv:hep-th/0609209)
Nicolas Moeller, Closed Bosonic String Field Theory at Quintic Order II: Marginal Deformations and Effective Potential, JHEP 0709:118,2007 (arXiv:0705.2102)
A ∞-Chern-Simons theory-type formulation of closed superstring field theory analogous to the bosonic version in (Zwiebach 93) is in
based on
See also
Theodore Erler, Sebastian Konopka, Ivo Sachs, Resolving Witten’s Superstring Field Theory, JHEP04(2014)150 (arXiv:1312.2948)
Theodore Erler, Sebastian Konopka, Ivo Sachs, NS-NS Sector of Closed Superstring Field Theory, JHEP08(2014)158 (arXiv:1403.0940)
The introduction of (Jurco-Muenster13) has a useful survey of the previous attempt, which we quote now:
The first attempt towards a field theory of superstrings was initiated by the work of Witten
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
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
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
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
A. 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.
See also
Reviews include
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
See also at L-infinity algebras in physics.
The A-infinity algebra structure of bosonic open string field theory in
For the topological string see
Discussion of the mathematical aspects is in
Jim Stasheff, Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space Talk given at the Conference on Topics in Geometry and Physics (1992) (arXiv:hep-th/9304061)
Martin Markl, Loop Homotopy Algebras in Closed String Field Theory, Comm. Math. Phys. 221(2):367–384, 2001. (arXiv:hep-th/9711045)
Alessandro Tomasiello, A-infinity structure and superpotentials (arXiv:hep-th/0107195)
Hiroshige Kajiura, Homotopy Algebra Morphism and Geometry of Classical String Field Theory (2001) (arXiv:hep-th/0112228)
Hiroshige Kajiura, Jim Stasheff, Homotopy algebras inspired by classical open-closed string field theory, Comm. Math. Phys. 263 (2006) 553–581 (2004) (arXiv:math/0410291)
Martin Markl, Operadic interpretation of $A_\infty$-algebras over $L_\infty$-algebras, appendix to (Kajiura-Stasheff 04)
Korbinian Münster, Ivo Sachs, Quantum Open-Closed Homotopy Algebra and String Field Theory (arXiv:1109.4101)
Korbinian Münster, Ivo Sachs, On Homotopy Algebras and Quantum String Field Theory (arXiv:1303.3444)
Surveys include
Hiroshige Kajiura, Jim Stasheff, Homotopy algebra of open–closed strings Geometry & Topology Monographs 13 (2008) 229–259 (pdf) (arXiv:hep-th/0606283)
Ivo Sachs, String theory and homotopy algebras, talk notes, Srni 2015 (pdf)
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
See also higher category theory and physics .
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
Homotopy Lie Superalgebra in Yang-Mills Theory (arXiv:0708.1773)
BV Yang-Mills as a Homotopy Chern-Simons via SFT (arXiv:0709.1411)
SFT-inspired Algebraic Structures in Gauge Theories (arXiv:0711.3843)
Conformal Field Theory and Algebraic Structure of Gauge Theory (arXiv:0812.1840)
References discussing independence of string field theories on the CFT (sigma-model background) in terms of which they are written down.
Ashoke Sen, Barton Zwiebach, Quantum Background Independence of Closed String Field Theory (arXiv:hep-th/9311009)
Ashoke Sen, Barton Zwiebach, Background Independent Algebraic Structures in Closed String Field Theory (arXiv:hep-th/9408053)
A review of the history of some related developments is given in
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)
Edward Witten, On background independent open string field theory, Phys.Rev. D46 (1992) 5467. (arXiv:hep-th/9208027)
Theodore Erler, Carlo Maccaferri, String Field Theory Solution for Any Open String Background (arXiv:1406.3021)
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
Martin Schnabl, Analytic solution for tachyon condensation in open string field theory (arXiv:hep-th/0511286)
Ian Ellwood, Martin Schnabl, Proof of vanishing cohomology at the tachyon vacuum, JHEP 0702:096,2007 (arXiv:hep-th/0606142)
Analogous discussion including also brane/anti-brane? pairs in superstring theory is in
Leonardo Rastelli, Ashoke Sen, Barton Zwiebach, Vacuum String Field Theory (arXiv:hep-th/0106010)
Ashoke Sen, Tachyon Dynamics in Open String Theory, Int.J.Mod.Phys.A20:5513-5656,2005 (arXiv:hep-th/0410103)
L. Bonora, N. Bouatta, C. Maccaferri, Towards open-closed string duality: Closed Strings as Open String Fields (arXiv:hep-th/0609182)
Theodore Erler, Tachyon Vacuum in Cubic Superstring Field Theory, JHEP 0801:013,2008 (arXiv:0707.4591)