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
for higher abelian targets
for symplectic Lie n-algebroid targets
|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 -graded vector space with respect to the ghost number grading, on those elements that satisfy
(“level matching condition”),
We shall write for this graded vector space. See (Markl, section 1)
This is equipped for each with a -ary operation
These operations are graded-symmetric: for all of homogeneous degree and for all we have
Moreover, there is a bilinear inner product
in that for all with we have
This is (Zwiebach, (2.61)).
The inner product satisfies for all 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 -point functions
These are still graded-symmetric in all arguments: for all of homogeneous degree and all we have
(“level matching condition”);
is Grassmann even (…define…)
This is (Zwiebach, (3.9))
The action functional of closed string field theory is
Since 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 -ary interaction genus-0 interaction vertices
is an L-∞ algebra.
For simplicity of notation we discuss this as if were finite-dimensional. The argument for the infinite-dimensional case follows analogously.
The Weil algebra is similarly generated from with differential
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 ).
Therefore to see that this is an invariant polynomial it remains to check that it is -closed. To see this, first introduce the notation
for the components of the -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 satisfies the defining equation of an invariant polynomial on the proper configuration space. The non-degeneracy is due to (2).
for a -valued differential form on some . So the closed string field theory action looks like that of -Chern-Simons theory over an odd-graded .
When considering open and closed strings jointly, then in addition to the closed string sector being encoded by an L-∞ algebra as above, the open string sector is encoded in an A-∞ algebra and the former acts on the latter by homotopy derivations (see also at derivations on algebras over a dg-operad)
Notice that this is half of the axioms of an -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
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
and further detailed analysis is in
The introduction of (Jurco-Muenster13) has a useful survey of the previous attempt, which we quote now:
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)
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 .
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)
A review of the history of some related developments is given 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
L. Bonora, N. Bouatta, C. Maccaferri, Towards open-closed string duality: Closed Strings as Open String Fields (arXiv:hep-th/0609182)