string field theory

{:quote: .un_remark style="border:solid #0000cc;background: #add8e6;border-width:2px 1px;padding:0 1em;margin:0 1em;"}

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### $\infty$-Chern-Simons theory
+--{: .hide}
[[!include infinity-Chern-Simons theory - contents]]
=--
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#Contents#
* table of contents
{:toc}

## 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:

[[!include second quantization -- table]]

Recall that [[string theory|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
 {#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 condensate|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](#BosonicStringVacuumAndSenConjecture).

### 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 open string field theory

(...)

## 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 [[BV-BRST formalism|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 ([[superstring|super]]-)[[string]], regarded as a $\mathbb{N}$-[[graded vector space]] with respect to the ghost number grading, on those elements $\Psi$ that satisfy

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

1. $(L_0 - \bar L_0) \Psi = 0$ ("level matching condition"),

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

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 [[correlator|(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 complex|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

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

([Zwiebach, (4.4)](#Zwiebach93)).

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)](#Zwiebach93)). This is non-degenerate on elements $\Psi$ which are annihilated by the [[BRST complex|ghost]] operator

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

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

\[
  \label{NonDegeneracyOfInnerProduct}
  (\forall B \in \mathfrak{g} \,:\,
  \langle A,B\rangle = 0)
  \Rightarrow
  A = 0
  \,.
\]

This is ([Zwiebach 93, (2.61)](#Zwiebach93)).

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

\[
  \label{GradedSymmetryBilinearPairing}
  \langle \Psi_1 , \psi_2 \rangle
  =
  (-1)^{(deg \Psi_1 + 1) (deg \Psi_2 + 1)}
  \langle \Psi_2, \Psi_1 \rangle
\]

([Zwiebach 93, (2.50)](#Zwiebach93)).

Moreover, it is non-vanishing only on pairs of elements of total degree 5. ([Zwiebach 93, (2.31)(2.44)](#Zwiebach93)).

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

\[
  \label{CorrelationFunctions}
  \{ \Psi_0, \Psi_1, \cdots, \Psi_k \}
  :=
  \langle \Psi_0, [\Psi_1, \cdots, \Psi_k]_k \rangle
  \,.
\]

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

\[
  \label{GradedSymmetrykPointFunction}
  \{\Psi_0, \cdots, \Psi_i, \Psi_{i+1}, \cdots, \Psi_k\}
  =
  (-1)^{(deg \Psi_i)(deg \Psi_{i+1})}
  \{\Psi_0, \cdots, \Psi_{i+1}, \Psi_{i}, \cdots, \Psi_k\}
  \,.
\]

([Zwiebach 93, (4.36)](#Zwiebach93)).

### The action functional

+-- {: .num_defn #LabelConfigurationSpace}
###### Definition

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

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

1. $(L_0 - \bar L_0) \vert \Psi \rangle = 0$ ("level matching condition");

1. $\vert \Psi \rangle^\dagger = - (\langle \Psi \vert)$ ("reality");

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

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

=--

This is ([Zwiebach 93, (3.9)](#Zwiebach93))

The _[[action functional]]_ of closed string field theory is

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

([Zwiebach 93, (4.41)](#Zwiebach93))

Since $[-]_1 = d_{BRST}$ is the [[BRST complex|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)](#Zwiebach93)). The analogous statement for the [[superstring]] in in ([Sen 15 (2.22)](#Sen15)).

(See also at _[string theory FAQ -- What are the equations of string theory?](https://ncatlab.org/nlab/show/string+theory+FAQ#WhatAreTheEquations)_).

### As an $\infty$-Chern-Simons theory
 {#AsAnInfinityCSTheory}

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 [[schreiber:∞-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.

+-- {: .num_prop}
###### 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)](#Zwiebach93)). For more details on the $L_\infty$-structure see _[References -- Relation to L-∞- and A-∞-algebra](ReferencesHomotopyAlgebra))_ .

+-- {: .num_prop}
###### 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. \ref{LabelConfigurationSpace}.

=--

+-- {: .proof}
###### 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 (eq:GradedSymmetryBilinearPairing) 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 (eq:CorrelationFunctions). 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 (eq:GradedSymmetrykPointFunction) 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. \ref{LabelConfigurationSpace}.

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

=--

From the discussion at [[Chern-Simons element]] in the section [Canonical Chern-Simons element](http://ncatlab.org/nlab/show/Chern-Simons+element#CanonicalChernSimonsElement) we have that the [[Lagrangian]] of the [[schreiber: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 $\mathfrak{g}$-[[infinity-Lie algebroid valued differential form|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
 {#OpenClosedStringFieldTheory}

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](derivation#OfAlgebrasOverADGOperad)_)

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

([Kajiura-Stasheff 04](#KajiuraStasheff04), [Markl 04](#Markl04)).

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

## Superstring field theory
 {#SuperstringFieldTheory}

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](#JurcoMuenster13)). See also the introduction there for a survey of the literature

## Related concepts

* [[schreiber:∞-Chern-Simons theory]]

* [[higher dimensional Chern-Simons theory]]

  * [[1d Chern-Simons theory]]

  * [[2d Chern-Simons theory]]

  * [[3d Chern-Simons theory]]

  * [[4d Chern-Simons theory]]

  * [[5d Chern-Simons theory]]

  * [[6d Chern-Simons theory]]

  * [[7d Chern-Simons theory]]

  * [[AKSZ sigma-models]]

  * [[infinite-dimensional Chern-Simons theory]]

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](#FukumaKawaiKitazawaTsuchiya97))

## References

### General

Introduction and survey:

* [[Leonardo Rastelli]], _String Field Theory_ in _Encyclopedia of Mathematical Physics_ ([arXiv:hep-th/0509129](http://arxiv.org/abs/hep-th/0509129))

* W. Siegel, _Introduction to string field theory_ ([arXiv:hep-th/0107094](http://arxiv.org/abs/hep-th/0107094))

* [[Ashoke Sen]], _String Field Theory as World-sheet UV Regulator_ ([arXiv:1902.00263](https://arxiv.org/abs/1902.00263))

* {#Erbin20a} [[Harold Erbin]], *String Field Theory – A Modern Introduction*, Lecture Notes in Physics **980**, Springer (2021) [[arXiv:2301.01686](https://arxiv.org/abs/2301.01686), [doi:10.1007/978-3-030-65321-7](https://doi.org/10.1007/978-3-030-65321-7)]

* {#Erbin20b} [[Harold Erbin]], _String theory: a field theory perspective_, 2020 ([pdf](http://www.lpthe.jussieu.fr/~erbin/files/reviews/string_theory.pdf))

* [[Harold Erbin]], _[String Field Theory -- Website](http://string-field-theory.org/)_

* [[Theodore Erler]], *Four Lectures on Closed String Field Theory*, Physics Reports **851** (2020) 1-36 [[arXiv:1905.06785](https://arxiv.org/abs/1905.06785), [doi:10.1016/j.physrep.2020.01.003](https://doi.org/10.1016/j.physrep.2020.01.003)&rbrack

* Carlo Maccaferri, *String Field Theory*, in: *Oxford Research Encyclopedia of Physics* [[arXiv:2308.00875](https://arxiv.org/abs/2308.00875)]

### Bosonic string field theory
 {#ReferencesBosonicSFT}

#### Open SFT
 {#ReferencesBosonocOSFT}

Original articles:

* {#Witten86} [[Edward Witten]], *Noncommutative Geometry and String Field Theory*, Nucl. Phys B **268** (1986) 253-294 [doi:10.1016/0550-3213(86)90155-0]

  > (cf. also [here](fusion+bundle#RelationToStringFieldTheory) at *[[fusion bundle]]*)

* [[Edward Witten]], _On background independent open string field theory_,  Phys.Rev. D46 (1992) 5467. ([arXiv:hep-th/9208027](http://arxiv.org/abs/hep-th/9208027))

See also

* Matej Kudrna, [[Martin Schnabl]], _Universal Solutions in Open String Field Theory_ ([arXiv:1812.03221](https://arxiv.org/abs/1812.03221))

On a proposal for a background-independent formalism for bosonic open string field theory, exhibiting in particular the [[orthosymplectic super Lie algebra|orthosymplectic group]] $OSp(d|2)$ as a symmetry:

* [[Itzhak Bars]], Dmitry Rychkov. *Background Independent String Field Theory* [[arXiv:1407.4699](https://arxiv.org/abs/1407.4699)]

#### Closed SFT
 {#ReferencesBosonicCSFT}

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

* {#Zwiebach93} [[Barton Zwiebach]], *Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation*, Nucl. Phys. B **390** 1 (1993) 33-152 [[arXiv:hep-th/9206084](http://arxiv.org/abs/hep-th/9206084), doi:10.1016/0550-3213(93)90388-6]

Brief reviews include

* [[Barton Zwiebach]], _Closed String Field Theory: An Introduction_ ([arXiv:hep-th/9305026](http://arxiv.org/abs/hep-th/9305026))

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](#YangZwiebach) 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](http://arxiv.org/abs/1206.3901))

Discussion of the expected closed string tachyon [[vacuum]] is in

* {#YangZwiebach} Haitang Yang, [[Barton Zwiebach]], _A Closed String Tachyon Vacuum ?_, JHEP 0509:054,2005 ([arXiv:hep-th/0506077](http://arxiv.org/abs/hep-th/0506077))

* [[Nicolas Moeller]], Haitang Yang, _The nonperturbative closed string tachyon vacuum to high level_ ([arXiv:hep-th/0609208](http://arxiv.org/abs/hep-th/0609208))

* [[Nicolas Moeller]], _A tachyon lump in closed string field theory_ ([arXiv:0804.0697](http://arxiv.org/abs/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](http://arxiv.org/abs/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](http://arxiv.org/abs/0705.2102))

### Superstring field theory
 {#ReferencesSuperstringFieldTheory}

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](https://arxiv.org/abs/1508.00366))

A survey is in

* {#Okawa16} Yuji Okawa, _Recent developments in the construction of superstring field theory I_, 2016 ([pdf](http://www-het.ph.tsukuba.ac.jp/~ishibash/SFT16/okawa.pdf))

based on

* {#ErlerKonopkaSach13} [[Theodore Erler]], Sebastian Konopka, [[Ivo Sachs]], _Resolving Witten's Superstring Field Theory_, JHEP04(2014)150 ([arXiv:1312.2948](http://arxiv.org/abs/1312.2948))

* {#ErlerKonopkaSach14} [[Theodore Erler]], Sebastian Konopka, [[Ivo Sachs]], _NS-NS Sector of Closed Superstring Field Theory_, JHEP08(2014)158 ([arXiv:1403.0940](http://arxiv.org/abs/1403.0940))

* [[Ashoke Sen]], _Gauge Invariant 1PI Effective Superstring Field Theory: Inclusion of the Ramond Sector_ ([arXiv:1501.00988](https://arxiv.org/abs/1501.00988))

* [[Theodore Erler]], Sebastian Konopka, [[Ivo Sachs]], _Ramond Equations of Motion in Superstring Field Theory_ ([arXiv:1506.05774](https://arxiv.org/abs/1506.05774))

* {#Sen15} [[Ashoke Sen]], _BV Master Action for Heterotic and Type II String Field Theories_, JHEP02(2016)087 ([arXiv:1508.05387](http://arxiv.org/abs/1508.05387))

with further developments in

* Yuji Okawa, [[Barton Zwiebach]], _Heterotic String Field Theory_ ([arXiv:hep-th/0406212](http://arxiv.org/abs/hep-th/0406212))

* Roji Pius, [[Ashoke Sen]], _Cutkosky Rules for Superstring Field Theory_ ([arXiv:1604.01783](http://arxiv.org/abs/1604.01783))

* [[Ashoke Sen]], _Reality of Superstring Field Theory Action_ ([arXiv:1606.03455](http://arxiv.org/abs/1606.03455))

* Hiroshi Kunitomo, Tatsuya Sugimoto, _Type II superstring field theory with cyclic L-infinity structure_ ([arxiv:1911.04103](https://arxiv.org/abs/1911.04103))

Interpretation of [[picture number]] as a grading on [[differential forms on supermanifolds]] induced from a choice of[[integration over supermanifolds|integral top form]] is due to

* [[Alexander Belopolsky]], _Picture changing operators in supergeometry and superstring theory_ ([arXiv:hep-th/9706033](http://arxiv.org/abs/hep-th/9706033)).

and has been further amplified in

* {#Witten12} [[Edward Witten]], appendix D of _Notes On Super Riemann Surfaces And Their Moduli_ ([arXiv:1209.2459](http://arxiv.org/abs/1209.2459))

* R. Catenacci, P.A. Grassi, S. Noja, _Superstring Field Theory, Superforms and Supergeometry_ ([arXiv:1807.09563](https://arxiv.org/abs/1807.09563))

See also

* [[Ashoke Sen]], _Background Independence of Closed Superstring Field Theory_, JHEP02(2018)155 ([arXiv:1711.08468](https://arxiv.org/abs/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](https://arxiv.org/abs/1703.06410))

For previous constructions, the introduction of

* {#JurcoMuenster13} [[Branislav Jurco]], Korbinian Muenster, _Type II Superstring Field Theory: Geometric Approach and Operadic Description_, JHEP 1304:126 (2013) [arXiv/1303.2323](http://arxiv.org/abs/1303.2323) doi

based on

* {#Yeh93} Chungsheng James Yeh, _Topics in superstring theory_, PhD thesis, Berkeley 1993 ([SPIRE](http://inspirehep.net/record/366138))

has a useful survey, which we quote now:

+-- {: quote}
###### Quote from ([Jurco-Muenster13](#JurcoMuenster13)):

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

* {#Witten86a} [[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](#Witten86)). The major obstacle compared to the bosonic string is the necessity of [[picture changing operators]]. Indeed, the cubic superstring theory of ([Witten 86a](#Witten86a)) 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 [[RR-field|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

* [[Nathan Berkovits]], _Super-Poincare Invariant Superstring Field Theory_, ([hep-th/9503099](http://arxiv.org/abs/hep-th/9503099))

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](#Zwiebach)) 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](#Yeh93)), 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](http://arxiv.org/abs/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 [[integration over supermanifolds|geometric integration theory on supermanifolds]], and in particular the geometric meaning
of [[picture changing operators]] has been clarified

* [[Alexander Belopolsky]], _Picture changing operators in supergeometry and superstring theory_ ([arXiv:hep-th/9706033](http://arxiv.org/abs/hep-th/9706033)).

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](#JurcoMuenster13)) is to describe the construction of type II superstring field theory in the geometric approach.

=--

Older reviews include

* [[Nathan Berkovits]], _Review of open superstring field theory_ , ([arXiv:hep-th/0105230](http://arxiv.org/abs/hep-th/0105230))

More recently there is

* Tomoyuki Takezaki, _Open superstring field theory including the Ramond sector based on the supermoduli space_ ([arXiv:1901.02176](https://arxiv.org/abs/1901.02176))

* Hiroshi Kunitomo, *Type II superstring field theory revisited* ([arXiv:2106.07917](https://arxiv.org/abs/2106.07917))

### 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)

* {#Sen16} [[Ashoke Sen]], _Wilsonian Effective Action of Superstring Theory_, J. High Energ. Phys. (2017) 2017: 108 ([arXiv:1609.00459](https://arxiv.org/abs/1609.00459))

### Relation to $A_\infty$- and $L_\infty$-algebras
 {#ReferencesHomotopyAlgebra}

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

* {#Zwiebach89} [[Barton Zwiebach]], *Issues In Covariant Closed String Theory*, pp 192-200 in: *Proceedings of [10th and Final Workshop on Grand Unification](http://inspirehep.net/record/966930)*, 20-22 Apr 1989. Chapel Hill, North Carolina [[spire:282685](http://inspirehep.net/record/282685)]

It appears in print in

* [[Barton Zwiebach]], *Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation*, Nucl. Phys. B **390** 1 (1993) 33-152 [[arXiv:hep-th/9206084](http://arxiv.org/abs/hep-th/9206084), doi:10.1016/0550-3213(93)90388-6]

See also at _[[L-infinity algebras in physics]]_.

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

* [[Matthias Gaberdiel]], [[Barton Zwiebach]], _Tensor Constructions of Open String Theories I: Foundations_ ([arXiv:hep-th/9705038](http://arxiv.org/abs/hep-th/9705038))

For the topological string see

* [[Manfred Herbst]], _Quantum A-infinity Structures for Open-Closed Topological Strings_ ([arXiv:hep-th/0602018](http://arxiv.org/abs/hep-th/0602018))

Discussion for [[heterotic string theory|heterotic]] string field theory is in

* {#KunitomoSugimoto19} Hiroshi Kunitomo, Tatsuya Sugimoto, _Heterotic string field theory with cyclic L-infinity structure_ ([arXiv:1902.02991](https://arxiv.org/abs/1902.02991))

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](http://arxiv.org/abs/hep-th/9304061))

* {#Markl} [[Martin Markl]], _Loop Homotopy Algebras in Closed String Field Theory_, Comm. Math. Phys. 221(2):367--384, 2001. ([arXiv:hep-th/9711045](http://arxiv.org/abs/hep-th/9711045))

* [[Alessandro Tomasiello]], _A-infinity structure and superpotentials_ ([arXiv:hep-th/0107195](http://arxiv.org/abs/hep-th/0107195))

* [[Hiroshige Kajiura]], _Homotopy Algebra Morphism and Geometry of Classical String Field Theory_ (2001) ([arXiv:hep-th/0112228](http://arxiv.org/abs/hep-th/0112228))

* {#KajiuraStasheff04} [[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](http://arxiv.org/abs/math/0410291))

* {#Markl04} [[Martin Markl]], _Operadic interpretation of $A_\infty$-algebras over $L_\infty$-algebras_, appendix to ([Kajiura-Stasheff 04](#KajiuraStasheff04))

* Korbinian Münster, [[Ivo Sachs]], _Quantum Open-Closed Homotopy Algebra and String Field Theory_ ([arXiv:1109.4101](http://arxiv.org/abs/1109.4101))

* Korbinian Münster, [[Ivo Sachs]], _On Homotopy Algebras and Quantum String Field Theory_ ([arXiv:1303.3444](http://arxiv.org/abs/1303.3444))

* {#DoubekJurcoMuenster13} [[Martin Doubek]], [[Branislav Jurco]], Korbinian Muenster, _Modular operads and the quantum open-closed homotopy algebra_, J. High Energ. Phys. 2015, 1–55 (2015) ([arXiv:1308.3223](https://arxiv.org/abs/1308.3223), doi:10.1007/JHEP12(2015)158)

Surveys include

* [[Jim Stasheff]], _[[A Survey of Cohomological Physics]]_

* [[Hiroshige Kajiura]], [[Jim Stasheff]], _Homotopy algebra of open–closed strings_ Geometry & Topology Monographs 13 (2008) 229–259 ([pdf](http://msp.warwick.ac.uk/gtm/2008/13/gtm-2008-13-010s.pdf)) ([arXiv:hep-th/0606283](http://arxiv.org/abs/hep-th/0606283))

* {#Sachs15} [[Ivo Sachs]], _String theory and homotopy algebras_, talk notes, Srni 2015 ([pdf](http://www.math.muni.cz/~srni/Prednasky/Sachs.pdf))

* {#Sachs19} [[Ivo Sachs]], _Homotopy Algebras in String Field Theory_, in: Proceedings of _[[Higher Structures in M-Theory 2018]]_ Fortschritte der Physik, Special Issue Volume 67, Issue 8-9 ([arXiv:1903.02870](https://arxiv.org/abs/1903.02870), [doi:10.1002/prop.201910013](https://doi.org/10.1002/prop.201910013))

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

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

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

* Yuji Okawa, [[Barton Zwiebach]], _Heterotic string field theory_ ([arXiv:hep-th/0406212](http://arxiv.org/abs/hep-th/0406212))

See also _[[higher category theory and physics]]_ .

#### For Yang-Mills theory
 {#ReferencesOnYMTheoryAspects}

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

* {#Witten95} [[Edward Witten]], section 3 of _Bound States Of Strings And $p$-Branes_, Nucl.Phys.B460:335-350, 1996 ([arXiv:hep-th/9510135](https://arxiv.org/abs/hep-th/9510135))

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

* {#Myers03} [[Robert Myers]], section 3 of _Nonabelian Phenomena on D-branes_, Class.Quant.Grav. 20 (2003) ([arXiv:hep-th/0303072](https://arxiv.org/abs/hep-th/0303072))

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

* {#ColettiSigalovTaylor03} Erasmo Coletti, Ilya Sigalov, [[Washington Taylor]], _Abelian and nonabelian vector field effective actions from string field theory_, JHEP 0309 (2003) 050 ([arXiv:hep-th/0306041](https://arxiv.org/abs/hep-th/0306041))

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

* [[Nathan Berkovits]], [[Martin Schnabl]], _Yang-Mills Action from Open Superstring Field Theory_, JHEP 0309 (2003) 022 ([arXiv:hep-th/0307019](https://arxiv.org/abs/hep-th/0307019))

The first full derivation seems to be due to

* {#Lee16} [[Taejin Lee]], _Covariant open bosonic string field theory on multiple D-branes in the proper-time gauge_ ([arXiv:1609.01473](https://arxiv.org/abs/1609.01473))

which is surveyed in

* {#Lee17} [[Taejin Lee]], _Deformation of the Cubic Open String Field Theory_, Phys. Lett. B 768 (2017) 248 ([arXiv:1701.06154](https://arxiv.org/abs/1701.06154))

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](https://arxiv.org/abs/1703.06402))

Discussion of the [[L-infinity algebra]] [[schreiber:infinity-Chern-Simons theory|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

* [[Anton Zeitlin]],

  _Homotopy Lie Superalgebra in Yang-Mills Theory_ ([arXiv:0708.1773](http://arxiv.org/abs/0708.1773))

  _BV Yang-Mills as a Homotopy Chern-Simons via SFT_ ([arXiv:0709.1411](http://arxiv.org/abs/0709.1411))

  _SFT-inspired Algebraic Structures in Gauge Theories_ ([arXiv:0711.3843](http://arxiv.org/abs/0711.3843))

  _Conformal Field Theory and Algebraic Structure of Gauge Theory_ ([arXiv:0812.1840](http://arxiv.org/abs/0812.1840))

### Background independence
 {#ReferencesBackgroundIndependence}

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
 {#ReferencesBackgroundIndependenceForClosed}

* [[Ashoke Sen]], [[Barton Zwiebach]], _Quantum Background Independence of Closed String Field Theory_ ([arXiv:hep-th/9311009](http://arxiv.org/abs/hep-th/9311009))

* [[Ashoke Sen]], [[Barton Zwiebach]], _Background Independent Algebraic Structures in Closed String Field Theory_ ([arXiv:hep-th/9408053](http://arxiv.org/abs/hep-th/9408053))

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

* Sabbir Rahman, _Manifest background independent formulation of string field theory_ ([newsgroup comment](http://www.natscience.com/Uwe/Forum.aspx/physics-research/503/Manifest-background-independent-formulation-of-string-field))

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](#FukumaKawaiKitazawaTsuchiya97))

* {#FukumaKawaiKitazawaTsuchiya97} 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](http://arxiv.org/abs/hep-th/9705128))

#### For open string field theory

* [[Edward Witten]], _On background independent open string field theory_,  Phys.Rev. D46 (1992) 5467. ([arXiv:hep-th/9208027](http://arxiv.org/abs/hep-th/9208027))

* [[Theodore Erler]], Carlo Maccaferri, _String Field Theory Solution for Any Open String Background_ ([arXiv:1406.3021](http://arxiv.org/abs/1406.3021))

### Tachyon dynamics, decaying D-branes and Sen's conjecture
 {#BosonicStringVacuumAndSenConjecture}

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

* [[Ashoke Sen]], _Universality of the Tachyon Potential_,  	JHEP 9912:027,1999 ([arXiv:hep-th/9911116](http://arxiv.org/abs/hep-th/9911116))

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](http://arxiv.org/abs/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](http://arxiv.org/abs/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](http://arxiv.org/abs/hep-th/0511286))

* Ian Ellwood, [[Martin Schnabl]], _Proof of vanishing cohomology at the tachyon vacuum_, JHEP 0702:096,2007 ([arXiv:hep-th/0606142](http://arxiv.org/abs/hep-th/0606142))

Analogous discussion [[Sen's conjecture]] including also brane/[[anti-brane]] pairs in [[superstring]] theory:

* [[Leonardo Rastelli]], [[Ashoke Sen]], [[Barton Zwiebach]], _Vacuum String Field Theory_ ([arXiv:hep-th/0106010](http://arxiv.org/abs/hep-th/0106010))

* [[Ashoke Sen]], _Tachyon Dynamics in Open String Theory_,  	Int.J.Mod.Phys.A20:5513-5656,2005 ([arXiv:hep-th/0410103](http://arxiv.org/abs/hep-th/0410103))

* L. Bonora, N. Bouatta, C. Maccaferri, _Towards open-closed string duality: Closed Strings as Open String Fields_ ([arXiv:hep-th/0609182](http://arxiv.org/abs/hep-th/0609182))

* [[Theodore Erler]], _Tachyon Vacuum in Cubic Superstring Field Theory_, JHEP 0801:013,2008 ([arXiv:0707.4591](http://arxiv.org/abs/0707.4591))

* [[Theodore Erler]], _Analytic Solution for Tachyon Condensation in Berkovits' Open Superstring Field Theory_, JHEP 1311 (2013) 007 ([arXiv:1308.4400](https://arxiv.org/abs/1308.4400))

### 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

* {#HenneauxTeitelboim87} [[Marc Henneaux]], [[Claudio Teitelboim]], section 2 of _First And Second Quantized Point Particles Of Any Spin_, in [[Claudio Teitelboim]], [[Jorge Zanelli]] (eds.) _Santiago 1987, Proceedings, Quantum mechanics of fundamental systems 2_, pp. 113-152. Plenum Press.

* {#Gross88} [[David Gross]], _High-Energy Symmetries Of String Theory_, Phys. Rev. Lett. 60 (1988) 1229.

and is further developed for instance in

* {#SagnottiTsulaia03} [[Auguste Sagnotti]], M. Tsulaia, _On higher spins and the tensionless limit of String Theory_, Nucl.Phys.B682:83-116,2004 ([arXiv:hep-th/0311257](http://arxiv.org/abs/hep-th/0311257))

* {#Bonelli03} G. Bonelli, _On the Tensionless Limit of Bosonic Strings, Infinite Symmetries and Higher Spins_, Nucl.Phys. B669 (2003) 159-172 ([arXiv:hep-th/0305155](http://arxiv.org/abs/hep-th/0305155))

* [[Auguste Sagnotti]], M. Taronna, _String Lessons for Higher-Spin Interactions_, Nucl.Phys.B842:299-361,2011 ([arXiv:1006.5242](http://arxiv.org/abs/1006.5242))

* {#Sagnotti11} [[Auguste Sagnotti]], _Notes on Strings and Higher Spins_ ([arXiv:1112.4285](http://arxiv.org/abs/1112.4285))

And conversely:

* Rakibur Rahman, Massimo Taronna, _From Higher Spins to Strings: A Primer_ in [[Stefan Fredenhagen]] (ed.) _Introduction to Higher Spin Theory_ ([arXiv:1512.07932](https://arxiv.org/abs/1512.07932))

* [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], _String Theory as a Higher Spin Theory_, J. High Energ. Phys. (2016) 2016: 85 ([arXiv:1512.07237](https://arxiv.org/abs/1512.07237), doi:10.1007/JHEP09(2016)085)

[[!redirects String field theory]]

[[!redirects open string field theory]]
[[!redirects closed string field theory]]

[[!redirects bosonic string field theory]]

[[!redirects closed bosonic string field theory]]
[[!redirects open bosonic string field theory]]

[[!redirects superstring field theory]]


String field theory > history

< [[String field theory]]
[[!redirects String field theory -- history]]


string geometry

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### Context
#### Higher spin geometry
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=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The analog of [[spin geometry]] as one passes from [[spinning particles]] to [[spinning strings]]. To some extent the step from the [[spin group]] to the [[string 2-group]].

## Related concepts

* [[spin geometry]], [[fivebrane geometry]]

* [[Dirac-Ramond operator]], [[Dirac operator on smooth loop space]]

* [[Witten genus]] [[Stolz conjecture]]

## References

* _The String Geometry Research Network_ ([website](http://www-app.uni-regensburg.de/Fakultaeten/MAT/waldorf/Stringgeometry/index.php?show=home)).

detailed discussion of the relation to [[spin geometry]] in [[smooth loop spaces]] is in

* [[Konrad Waldorf]], _String geometry vs. spin geometry on loop spaces_ ([arXiv:1403.5656](http://arxiv.org/abs/1403.5656))

related remarks are briefly in

* Chris Kottke with [[Richard Melrose]], _A very smooth Dirac operator on loop space_ ([pdf](http://www.math.uiuc.edu/~palbin/CRM2012/KottkeAbstract.pdf))


string group

> under construction

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###Context###
#### Cohomology
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[[!include cohomology - contents]]
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#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
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=--

#Contents#
* autoamtic table of contents goes here
{:toc}

## Definition

The **string group** $String(n)$  is defined to be, as a [[topological group]], the [[Whitehead tower|3-connected cover]] of the [[Spin group]] $Spin(n)$, for any $n \in \mathbb{N}$.

Notice that for $n\ge3$, $Spin(n)$ itself is the [[Whitehead tower|simply connected cover]] of the [[special orthogonal group]] $SO(n)$, which in turn is the connected component (of the identity)  of the [[orthogonal group]] $O(n)$. Hence $String(n)$ is one element in the [[Whitehead tower]] of $\mathrm{O}(n)$:

$$
  \cdots \to Fivebrane(n) \to String(n) \to Spin(n) \to SO(n) \to \mathrm{O}(n)
  \,.
$$

The next higher connected group is called the [[Fivebrane group]].

The [[homotopy group]]s of $O(n)$ are for $k \in \mathbb{N}$ and for sufficiently large $n$

$$
  \array{
     \pi_{8k+0}(O) & = \mathbb{Z}_2
     \\
     \pi_{8k+1}(O) & = \mathbb{Z}_2
     \\
     \pi_{8k+2}(O) & = 0
     \\
     \pi_{8k+3}(O) & = \mathbb{Z}
     \\
     \pi_{8k+4}(O) & = 0
     \\
     \pi_{8k+5}(O) & = 0
     \\
     \pi_{8k+6}(O) & = 0
     \\
     \pi_{8k+7}(O) & = \mathbb{Z}
  }
  \,.
$$

By [[Whitehead tower|co-killing]] these groups step by step one gets

$$
  \array{
     cokill~this &&&& to~get
     \\
     \\
     \pi_{0}(O) & = \mathbb{Z}_2 &&& SO
     \\
     \pi_{1}(O) & = \mathbb{Z}_2 &&& Spin
     \\
     \pi_{2}(O) & = 0
     \\
     \pi_{3}(O) & = \mathbb{Z} &&& String
     \\
     \pi_{4}(O) & = 0
     \\
     \pi_{5}(O) & = 0
     \\
     \pi_{6}(O) & = 0
     \\
     \pi_{7}(O) & = \mathbb{Z} &&& Fivebrane
  }
  \,.
$$

## Definition by co-killing of $\pi_3$

More in detail this means the following.

First notice that since by construction $\pi_i(\mathcal{B}Spin(n)) = 0 $ for $0 \leq i \leq 3$, by the [[Hurewicz theorem]] we have for the degree 4 [[integral cohomology]] group of the [[classifying space]] $B Spin(n)$ that

$$
  H^4(B Spin(n)) \simeq \pi_4(B Spin(n))
  \simeq
  \mathbb{Z}
  \,.
$$

The generator of this group is called the **fractional first Pontryagin class** and denoted

$$
  \frac{1}{2}p_1 : B Spin(n) \to B^4 \mathbb{Z} \simeq K(\mathbb{Z},4)
$$

because the ordinary first [[Pontryagin class]] $p_1 : \mathcal{B} SO(n) \to K(\mathbb{Z},4)$ fits into a diagram

$$
  \array{
    B Spin(n) &\stackrel{\frac{1}{2} p_1}{\to}&
    B^4 \mathbb{Z}
    \\
    \downarrow && \downarrow^{\cdot 2}
    \\
    B SO(n) &\stackrel{p_1}{\to}&
    B^4 \mathbb{Z}
  }
  \,,
$$

where the right vertical morphism comes from multiplication by 2 in $\mathbb{Z}$.

This says that _after being pulled back_ to $\mathcal{B} Spin(n)$ the first [[Pontryagin class]] is 2 times the generator of the degree 4 [[integral cohomology]] group of $\mathcal{B}Spin(n)$ and hence that generator is called one half of $p_1$, denoted $\frac{1}{2}p_1$ (by slight abuse of notation).

The [[delooping]] of the String-group as a [[topological group]] is the [[homotopy fiber]] of this fractional Pontyagin class, i.e. the [[homotopy pullback]]

$$
  \array{
    B String(n) &\to& {*}
    \\
    \downarrow && \downarrow
    \\
    B Spin(n) &\to& \mathcal{B}^4 \mathbb{Z}
  }
$$

in [[Top]].

In other words: $B String(n)$ is the $U(1)$- 2-[[gerbe]] or $B^2 U(1)$ [[principal ∞-bundle]] on $B Spin(n)$ whose class is $\frac{1}{2}p_1 \in H^4(B Spin(n), 4)$.

## Models

### As a topological group

There is a model due to Stolz and Teichner in 'What is an elliptic object?'...

### As a smooth 2-group

While $Spin(n)$ is not just a [[topological group]] but a (finite dimensional) [[Lie group]], $String(n)$ cannot have the structure of a finite dimensional [[Lie group]], due to the fact that the third [[homotopy group]] is nontrivial for every (finite dimensional) Lie group, while for $\pi_3(String(n)) = 0$ by the very definition of $String(n)$.

However, one can define an _infinite-dimensional_ Lie group with the correct properties to be a model of $String(n)$ ([Nikolaus-Sachse-Wockel 2013](#NikolausSachseWockel)).

There are also smooth models of $String(n)$ in the form of [[2-group]]s. See [[string 2-group]].

## Role in string theory

The reason for the name is that in [[string theory]], for (blah) to be well-defined, it is necessary for the
structure group of (blah) to lift to (blah).

See [[String structure]].

If one considers passing to the (free) [[loop space]] of spacetime and then doing [[quantum mechanics]], the requirement of the previous paragraph is that the structure group lifts to ... (cite Killingback, Mickelsson, Schreiber, Witten,...)

## Generalization to other groups

One may consider the universal 3-connected cover of any general [[compact space|compact]], [[simple Lie group|simple]] and [[simply connected space|simply connected]] [[Lie group]] $G$, in complete analogy to the case $G = Spin(n)$. Accordingly one speaks of string-groups $String_G$.

Of these the case $G = $ [[E8]] is the other one relevant in [[string theory]]: see [[Green-Schwarz mechanism]].

## Related concepts

$\cdots \to$ [[Fivebrane group]] $\to$ **string group** $\to$ [[spin group]] $\to$ [[special orthogonal group]] $\to$ [[orthogonal group]].

[[!include image of J -- table]]

* [[string orientation of tmf]]

[[!include table of orthogonal groups and related]]

## References

Originally the String-group was just known by its generic name: with $\mathcal{B} O \langle 8 \rangle$ being the topologist's notation for the [[n-connected|7-connected]] cover of the [[delooping]]/[[classifying space]] $\mathcal{B}O$ of the group $O$.

When it was realized that lifts of the structure maps $X \to \mathcal{B}O$ of the [[tangent bundle]] of a [[manifold]] $X$ through the projection $\mathcal{B}O\langle 8 \rangle \to \mathcal{B}O$ -- now called a [[String structure]] -- play the same role in [[string theory]] as a [[Spin structure]] does in ordinary [[quantum mechanics]], the term _String group_ for $\Omega (\mathcal{B}O\langle 8 \rangle)$ was suggested.

Following some inquiries by [[Jim Stasheff]] and confirmed in private email by [[Haynes Miller]] it seems that the first one to propose the term _$String$ group_ for the group known to topologists as $\Omega (\mathcal{B}O\langle 8\rangle)$ was [[Haynes Miller]].

A model of the string group by [[local nets]] of fermions is discussed in

* [[Stefan Stolz]], [[Peter Teichner]], _The spinor bundle on loop space_ (2005) ([pdf](http://web.me.com/teichner/Math/Surveys_files/MPI.pdf))

Many more models exist by now in terms of [[geometric realization]] of a model for the [[string 2-group]]. See there for more references.

A good review is in the introduction of

* [[Chris Schommer-Pries]], _Central Extensions of Smooth 2-Groups and a Finite-Dimensional String 2-Group_ ([arXiv:0911.2483](http://arxiv.org/abs/0911.2483))
{#Schommer-Pries}

In

* {#NikolausSachseWockel} [[Thomas Nikolaus]], [[Christoph Sachse]], [[Christoph Wockel]], _A Smooth Model for the String Group_, Int. Math. Res. Not. IMRN 16 (2013) 3678-3721, doi:[10.1093/imrn/rns154](https://dx.doi.org/10.1093/imrn/rns154), ([arXiv:1104.4288](http://arxiv.org/abs/1104.4288))

it is shown that the topological string group does admit a [[Frechet manifold]] [[Lie group]] structure.

For relation to [[conformal nets]] see

* Bas Janssens, _Notes on Defects & String Group_ ([pdf](http://www.bjadres.nl/WorkInProgress/Defect&String.pdf))

[[!redirects String group]]


string length scale

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+-- {: .hide}
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* table of contents
{:toc}

## Idea

In [[perturbative string theory]], the _string length scale_ $\ell_s$ is a [[unit]] of [[length]] which sets the [[scale]] for the extension of [[strings]] described by [[sigma-model]] [[worldsheet]] [[field theories]]. Specifically, the [[Nambu-Goto action]] for the string is nothing but the relativistic [[volume]]-[[functional]], and the string length determines in which [[units]] the volume is measured.

The [[square]] of the string length is known as the _Regge slope_ (as in: [[slope]] of [[Regge trajectories]]) and traditionally denoted by $\alpha^\prime$:

$$
  \alpha^\prime \;=\; \ell_s^2
  \,.
$$

The inverse of the string length squared/Regge slope is called the _string tension_, traditionally denoted

$$
  T_s \;=\; \frac{1}{2\pi \alpha^\prime} = \frac{1}{2 \pi \ell_s^2}
  \,.
$$

This way the [[Nambu-Goto action]] for the string with proper [[units]] attached is

$$
  L_{NG} \;=\; T vol_{\Sigma}
  \,,
$$

where $vol_{\Sigma}$ is the (induced) [[volume form]] on the [[worldsheet]] $\Sigma$.

## Properties

### Relation to Planck length and string coupling

For discussion of relation to [[Planck length]] and [[string coupling constant]] see at [[non-perturbative effect]] the section _[Worldsheet and brane instantons](non-perturbative+effect#WorldsheetAndBraneInstantons)_

### Vanishing tension limit

In the [[limit of a sequence|limit]] $T_s \to 0$, $\ell_s \to \infty$ of vanishing string tension, [[string field theory]] is supposed to become Vasiliev's [[higher spin gauge theory]]. See there for more.

## Related concepts

* [[Regge slope]]

[[!include fundamental scales -- contents]]

## References

For instance:

* {#Tong09} [[David Tong]], around p. 16 in _Lectures on String Theory_ ([arXiv:0908.0333](https://arxiv.org/abs/0908.0333))

[[!redirects string length scales]]

[[!redirects alpha']]

[[!redirects string scale]]
[[!redirects string scales]]

[[!redirects string length]]
[[!redirects string lengths]]

[[!redirects Regge slope]]
[[!redirects Regge slopes]]

[[!redirects string tension]]
[[!redirects string tensions]]

[[!redirects string's tension]]
[[!redirects string's tensions]]


String Lie 2-algebra

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## Idea

The _string Lie 2-algebra_ is the [[infinitesimal object|infinitesimal approximation]] to the [[Lie 2-group]] that is called the [[string 2-group]].

It is a shifted [[L-infinity-algebra|∞-Lie algebra]] central extension

$$
  0 \to \mathbf{b} \mathfrak{u}(1) \to \mathfrak{string}(n)
  \to \mathfrak{so}(n) \to 0
$$

of the [[Lie algebra]] $\mathfrak{so}(n)$ by the [[Lie 2-algebra]] $\mathbf{b} \mathfrak{u}(1)$ which is classified by the canonical (up to normalization) [[Lie algebra cohomology|Lie algebra 3-cocycle]] $\mu$ on $\mathfrak{so}(n)$, which may itself be understood as a morphism

$$
  \mu : \mathfrak{so}(n) \to b^2 \mathfrak{u}(1)
  \,.
$$

When $\mu$ is normalized such that it represents the image in [[deRham cohomology]] of the generator of the [[integral cohomology]] $H^3(X,Spin(n))$ of the [[Spin group]], then the [[Lie integration]] of the  String Lie 2-algebra is the [[String Lie 2-group]].

## Definition

We spell out first an explicit algebraic realization of the string Lie 2-algebra and then give its abstract definition as a [[homotopy fiber]] or [[principal ∞-bundle]].

### In components {#InComponents}

As with any [[L-∞ algebra]], we may define the String Lie 2-algebra $\mathfrak{string}(n)$ equivalently in terms of its [[Chevalley-Eilenberg algebra]].

There are various equivalent models we discuss a small one with a trinary bracket, and an infinite dimensional model which is however strict in that it comes from a [[differential crossed module]].

#### Skeletal model

Write $\mathfrak{g} := \mathfrak{so}(n)$ in the following. The [[Chevalley-Eilenberg algebra]] $CE(\mathfrak{g})$ of $\mathfrak{g}$ has a degree 3 element

$$
  \mu = \langle -, [-,-]\rangle
  \,,
$$

well defined up to normalization ($\langle - ,- \rangle$ is the canonical bilinear symmetric [[invariant polynomial]] on $\mathfrak{g}$ and $[-,-]$ the Lie bracket), which is closed in $CE(\mathfrak{g})$

$$
  d_{\mathfrak{g}} \mu = 0
  \,.
$$

Hence this is the canonical (up to normalization) 3-cocycle in the [[Lie algebra cohomology]] of $\mathfrak{g}$.

The Chevalley--Eilenberg algebra of $\mathfrak{string}(n)$ is

$$
  CE(\mathfrak{string}(n)) = (\wedge^\bullet (\mathfrak{g}^* \oplus \langle b\rangle), d_{\mathfrak{string}})
  \,,
$$

where

* $b$ is a single new generator in degree 2;

* the differental $d_{\mathfrak{string}}$ coincides with $d_{\mathfrak{g}}$ on $\mathfrak{g}^*$:

  $$
    d_{\mathfrak{string}} |_{\mathfrak{g}^*} = d_{\mathfrak{g}}
  $$

* on the new generator it is defined by

  $$
    d_{\mathfrak{string}} : b \mapsto \mu
    \,.
  $$

That the differential defined this way is indeed of degree +1 and squares to 0 is precisely the fact that $\mu$ is a degree 3-cocycle of $\mathfrak{g}$.

One can equivalently describe the $L_\infty$-algebra structure of $\mathfrak{string}(n)$ in terms of lots of brackets
$$
[-,-,\dots,-]_k:\wedge^k \mathfrak{string}(n)\to \mathfrak{string}(n),
$$
of degree $2-k$.
In addition to the Lie bracket of $\mathfrak{g}$, there is only a further nontrivial bracket: it is the 3-bracket
$$
[-,-,-]_3:\wedge^3 \mathfrak{g}\to \langle b\rangle^*
$$
given by
$$
[x,y,z]_3=\mu(x,y,z)\cdot \beta,
$$
where $\beta:\langle b\rangle\to\mathbb{R}$ is the dual of $b$.

#### Strict Lie 2-algebra model

**Proposition** The string Lie 2-algebra given above is equivalent to the infinite-dimensional Lie 2-algebra coming from the [[differential crossed module]]

$$
  \hat \Omega \mathfrak{g} \to P \mathfrak{g}
$$

of the [[universal central extension]] of the [[loop Lie algebra]] mapping into the path Lie algebra, which acts on the former in the evident way.

This is proven in [BCSS](#BCSS).

### As a homotopy fiber
 {#AsHomotopyFiber}

Up to equivalence, the string Lie 2-algebra is the [[homotopy fiber]] of the cocycle $\mu : \mathfrak{so}(n) \to \mathbf{b}^2 \mathfrak{u}(1)$, hence is the canonical $\mathbf{b} \mathfrak{u}(1)$-[[principal ∞-bundle]] over $\mathfrak{so}(n)$.

Here we take by definition the [[(∞,1)-category]] of [[∞-Lie algebroid]]s to be that [[presentable (∞,1)-category|presented]] by the opposite (after passing to [[Chevalley-Eilenberg algebras]]) of the [[model structure on dg-algebras]]. For a detailed discussion of the recognition of this [[homotopy fiber]] see section 3.1 and specifically example 3.5.4 of ([Fiorenza-Rogers-Schreiber 13](#FiorenzaRogersSchreiber13)).

In terms of dg-algebras, the cocycle is dually a morphism

$$
  CE(\mathfrak{so}(n)) \leftarrow CE(\mathbf{b}^2 \mathfrak{u}(1)) : \mu
$$

and the [[homotopy fiber]] in question is dually modeled by the [[homotopy pushout]]

$$
  \array{
    && 0
    \\
    && \uparrow
    \\
    CE(\mathfrak{so}(n)) &\stackrel{\mu}{\leftarrow}&
    CE(\mathbf{b}^2 \mathfrak{u}(1))
  }
  \,.
$$

By the general rules for computing [[homotopy pushout]]s, this may be computed by an ordinary [[pushout]] if we choose a  [[resolution]] of $CE(\mathbf{b}^2 \mathfrak{u}(1)) \to 0$ by a cofibration and ensure that all three objects in the pushout diagram are cofibrations.

For the resolution we take the standard one by the CE-algebra of the $\mathbf{b}^2 \mathfrak{u}(1)$-[[universal principal ∞-bundle]] $\mathbf{e b} \mathfrak{u}(1)$, which is the dg-algebra

$$
  CE(\mathbf{e b} \mathfrak{u}(1)) = (\wedge^\bullet( \langle b\rangle \oplus \langle c\rangle ), d)
$$

where $b$ is a generator of degree 2, $c$ one of degree 3 and the differential is given by

$$
  d b = c
$$

and

$$
  d c = 0
  \,.
$$

The morphism $CE(\mathbf{b}^2 \mathfrak{u}(1)) \to CE(e b \mathfrak{u}(1))$ is the one that identifies the two degree-3 generators.

Now $CE(\mathbf{b}^2 \mathfrak{u}(1))$ and $CE(\mathbf{e b} \mathfrak{u}(1))$ are [[Sullivan algebra]]s, hence are cofibrant objects in the [[model structure on dg-algebra]]s. The dg-algebra $CE(\mathfrak{g})$ is not quite a Sullivan algebra, but almost: it is a [[semifree dga]] and only fails to have the filtering property on the differential.  This is sufficient for computing the desired [[homotopy fiber]], as discussed at [∞-Lie algebra cohomology -- Extensions](infinity-Lie+algebra+cohomology#Extensions).

One observes now that

$$
  \array{
    CE(\mathfrak{string}) &\leftarrow& CE(\mathbf{e b} \mathfrak{u}(1))
    \\
    \uparrow && \uparrow
    \\
    CE(\mathfrak{so}(n)) &\leftarrow& CE(\mathbf{b}^2 \mathfrak{u}(1))
  }
$$

is a [[pushout]] diagram.  Dually, this exhibits $\mathfrak{string}$ as the [[(∞,1)-pullback]]

$$
  \array{
    \mathfrak{string}(n) &\to& *
    \\
    \downarrow && \downarrow
    \\
    \mathfrak{so}(n) &\stackrel{\mu}{\to}& \mathbf{b}^2 \mathfrak{u}(1)
  }
  \,.
$$

And this may be taken to be the abstract definition of the string Lie 2-algebra.

By the general logic of [[fiber sequence|fiber sequences]] this implies that also

$$
  \mathbf{b} \mathfrak{u}(1) \to \mathfrak{string} \to \mathfrak{so}(n)
$$

is a fiber sequence. By analogous reasoning as before, we see that this is modeled by the ordinary pushout

$$
  \array{
    CE(\mathbf{b} \mathfrak{u}(1)) &\leftarrow& *
    \\
    \uparrow && \uparrow
    \\
    CE(\mathfrak{string}) &\leftarrow& CE(\mathfrak{g})
  }
  \,.
$$

This is indeed a homotopy pushout even without resolving the point, because $CE(\mathfrak{string}) \leftarrow CE(\mathfrak{g})$ is already a cofibration, being the pushout of a cofibration by the above.

### As the prequantum line 2-bundle of a Courant algebroid

The [[delooping]] of a [[semisimple Lie algebra]] $\mathfrak{g}$ to a 1-object [[L-infinity algebroid]] $b \mathfrak{g}$ carries the [[Killing form]] as a quadratic bilinear [[invariant polynomial]] and is as such a [[symplectic Lie n-algebroid]] over the point for $n = 2$, hence a [[Courant Lie 2-algebroid]] over the point.

As described at [[symplectic infinity-groupoid]] one can consider the higher analog of [[geometric quantization]] of these objects. This is again the homotopy fiber as [above](AsHomotopyFiber).

### As a Heisenberg Lie 2-algebra

The String Lie 2-algebra identifies also with the [[Heisenberg Lie 2-algebra]]
of the [[string]] [[sigma-model]] for the specialization to the [[WZW model]] ([Baez-Rogers 10](#BaezRogers10)). See at _[[2-plectic geometry]]_ for more.

## Generalizations

More generally, for $\mathfrak{g}$ an [[L-infinity-algebra|∞-Lie algebra]] and $\mu \in CE(\mathfrak{g})$ an $\infty$-Lie algebra cocycle (a closed element in the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$) of degree $k$, there is a corresponding shifted central extension

$$
  0 \to \mathbf{b}^{k-2} \mathfrak{u}(1) \to \mathfrak{g}_\mu  \to \mathfrak{g} \to 0
  \,.
$$

For instance the [[supergravity Lie 3-algebra]] is such an extension of the [[super Poincare Lie algebra]] by a [[super Lie algebra]] 4-cocycle.

## Related concepts

* [[special orthogonal Lie algebra]]

* [[fivebrane Lie 6-algebra]]

* [[type II supergravity Lie 2-algebra]]

* [[supergravity Lie 3-algebra]], [[supergravity Lie 6-algebra]]

* [[string Lie algebroid]]

## References

In one incarnation or other the String Lie 2-algebra has been considered in literature of [[dg-algebra]]s, but its [[Lie theory|Lie theoretic]] interpretation as a Lie 2-algebra has been made fully explicit only in

* {#BaezCrans03} [[John Baez]], [[Alissa Crans]], Higher-dimensional Algebra V: Lie 2-algebras, _Theory and Applications of Categories_ **12** (2004), 492-528.  ([web](http://www.tac.mta.ca/tac/volumes/12/15/12-15abs.html))
([arXiv:math.QA/0307263](http://arxiv.org/abs/math.QA/0307263))

In

* [[Andre Henriques]], _Integrating $L_\infty$-algebras_ ([arXiv:0603563](http://arxiv.org/abs/math/0603563))

the string Lie 2-algebra is integrated to the [[string 2-group]] using the general abstract method described at [[Lie integration]].

In

* {#BCSS} [[John Baez]], [[Alissa Crans]], [[Urs Schreiber]] and [[Danny Stevenson]], From loop groups to 2-groups, _Homotopy, Homology and Applications_ **9** (2007), 101-135.  [[arXiv:math.QA/0504123](http://arxiv.org/abs/math.QA/0504123), [hha:1201127333](https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-9/issue-2/From-loop-groups-to-2-groups/hha/1201127333.full)]

the equivalent strict model given by a differential crossed module is found, which is then integrated termwise as ordinary Lie algebras to a [[crossed module]] of Frechet-Lie groups, hence to a Lie [[strict 2-group]] model of the String Lie 2-group.

The string Lie 2-algebra as the [[Heisenberg Lie 2-algebra]] on the group $G$ is discussed in

* {#BaezRogers10} [[John Baez]], [[Chris Rogers]], _Categorified Symplectic Geometry and the String Lie 2-Algebra_, Homology Homotopy Appl. Volume 12, Number 1 (2010), 221-236. ([arXiv:0901.4721](http://arxiv.org/abs/0901.4721),[euclid](https://projecteuclid.org/euclid.hha/1296223828)).

The string Lie 2-algebra is also considered in a certain context in

* Sati, Schreiber, Stasheff, _$L_\infty$-algebra connections_

where also the relation to the [[supergravity Lie 3-algebra]] and other structures is discussed.

The super-$L_\infty$-version of the string $L_\infty$-algebra was considered in

* [[John Baez]], [[John Huerta]], _Division Algebras and Supersymmetry II_ ([arXiv:1003.3436](http://arxiv.org/abs/1003.3436)).

See also [[division algebra and supersymmetry]].

Discussion of the string Lie 2-algebra as the homotopy fiber of the underlying 3-cocycle is around prop. 3.3.96 in

* _[[schreiber:differential cohomology in a cohesive topos]]_

and example 3.5.4 in

* {#FiorenzaRogersSchreiber13} [[Domenico Fiorenza]], [[Chris Rogers]], [[Urs Schreiber]], _[[schreiber:L-∞ algebras of local observables from higher prequantum bundles]]_ ([arXiv:1304.6292](http://arxiv.org/abs/1304.6292))

More on this in

* {#Schmidt19} [[Lennart Schmidt]], _Twisted Weil Algebras for the String Lie 2-Algebra_, in [[Christian Saemann]], [[Urs Schreiber]], [[Martin Wolf]] (eds.) _[Higher Structures in M-Theory](http://www.maths.dur.ac.uk/lms/109/index.html)_ [Durham Symposium](http://www.maths.dur.ac.uk/lms/) 2018, Fortschritte der Physik 2019 ([arXiv:1903.02873](https://arxiv.org/abs/1903.02873))

[[!redirects string Lie 2-algebra]]


String Math 2017

#Contents#
* table of contents
{:toc}

## Overview

[String Math 2017](https://stringmath2017.desy.de/) is a conference about [[string theory]] and [[quantum field theory]] and its intersection with other fields of [[mathematics]].

## Talks

### July 24

* [[Shing-Tung Yau]], *Aspects of Calabi-Yau variety and SCFTs in various dimensions* ([slides](https://stringmath2017.desy.de/sites/sites_conferences/site_stringmath2017/content/e45470/e56510/e56535/SM-Yau.pdf))

### July 25

(...)

### July 26

(...)

### July 27

(...)

### July 28

(...)

## See also

* [[String Math 2022]]


string network

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{:toc}

## Idea

There are $1/4$-[[BPS state]] solutions of [[type IIB supergravity]] which locally exhibit [[black branes]] in the shape of three $(p_i,q_i)$-[[strings]] jointly [[brane intersection|intersecting]]. Globally many of such 3-string junctions may combine to form a more complicated _network_ of [[black brane|black]] string configurations, hence called a _string network_ ([Sen 97](#Sen97)).

## Properties

### M-Theory lift to membrane triple junctions

The lift to [[11d supergravity]]/[[M-theory]] of 3-string junctions is supposed to be given by [[membrane triple junctions]].

## Related concepts

[[!include brane bound states -- table]]

## References

The concept goes back to

* {#Sen97} [[Ashoke Sen]], _String Network_, JHEP 9803:005,1998 ([arXiv:hep-th/9711130](https://arxiv.org/abs/hep-th/9711130))

* Soo-Jong Rey, Jung-Tay Yee, _BPS Dynamics of Triple $(p,q)$ String Junction_, Nucl.Phys. B526 (1998) 229-240 ([arXiv:hep-th/9711202](https://arxiv.org/abs/hep-th/9711202))

* [[Keshav Dasgupta]], [[Sunil Mukhi]], _BPS Nature of 3-String Junctions_, Phys.Lett. B423 (1998) 261-264 ([arXiv:hep-th/9711094](https://arxiv.org/abs/hep-th/9711094))

Via [[tropical geometry]]:

* Koushik Ray, _String networks as tropical curves_, JHEP 0809:098, 2008 ([arXiv:0804.1870](https://arxiv.org/abs/0804.1870))

Relation to [[Lie algebras]] and [[Lie algebra representations]]:

* Oliver DeWolfe, [[Barton Zwiebach]], _String Junctions for Arbitrary Lie Algebra Representations_,  Nucl. Phys. B541 (1999) 509-565 ([arXiv:hep-th/9804210](https://arxiv.org/abs/hep-th/9804210))

* Oliver DeWolfe, Tamas Hauer, [[Amer Iqbal]], [[Barton Zwiebach]], _Uncovering the Symmetries on $[p,q]$ 7-branes: Beyond the Kodaira Classification_, Adv. Theor. Math. Phys. 3 (1999) 1785-1833 ([arXiv:hep-th/9812028](https://arxiv.org/abs/hep-th/9812028))

Relation to the [[D=6 N=(2,0) SCFT]] on [[M5-branes]]:

* Stefano Bolognesi, Kimyeong Lee, _$1/4$ BPS String Junctions and N3 Problem in 6-dim $(2,0)$ Superconformal Theories_, Phys. Rev. D.84. 126018 ([arXiv:1105.5073](https://arxiv.org/abs/1105.5073))

See also

* Oleg Lunin, _Brane webs and 1/4-BPS geometries_,  Journal of High Energy Physics, Volume 2008, JHEP09(2008) ([arXiv:0802.0735](https://arxiv.org/abs/0802.0735))

The lift to [[11d supergravity]]/[[M-theory]] via [[membrane triple junctions]] is discussed in

* {#KroghLee97} Morten Krogh, Sangmin Lee, _String Network from M-theory_, Nucl. Phys. B516:241-254, 1998 ([arXiv:hep-th/9712050](https://arxiv.org/abs/hep-th/9712050))

* {#Ramadevi99} P. Ramadevi, _Supergravity Solution for Three-String Junction in M-Theory_, JHEP 0006 (2000) 005 ([arXiv:hep-th/9906247](https://arxiv.org/abs/hep-th/9906247))

[[!redirects string networks]]

[[!redirects string junction]]
[[!redirects string junctions]]

[[!redirects string triple junction]]
[[!redirects string triple junctions]]

[[!redirects triple string junction]]
[[!redirects triple string junctions]]

[[!redirects string web]]
[[!redirects string webs]]


string orientation of tmf

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=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Higher spin geometry
+-- {: .hide}
[[!include higher spin geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _string orientation of tmf_ is the universal [[orientation in generalized cohomology]] for [[tmf]]-cohomology ("universal [[elliptic cohomology]]), given by a [[homomorphism]]

$$
  \sigma \;\colon\; M String \longrightarrow tmf
$$

of [[E-∞ rings]], from the [[String structure]]-[[Thom spectrum]] to the [[tmf]]-[[spectrum]].  This is refinement of the [[Witten genus]] (see there for more)

$$
  w \;\colon\; \Omega^{String,rat}_\bullet \longrightarrow MF_\bullet
$$

(with values in the ring of [[modular forms]]) which it reproduces on [[homotopy groups]]

$$
  w \simeq \pi_\bullet(\sigma)
  \,.
$$

For this reason the string orientation of [[tmf]] is also referred to as the "topological Witten genus".

All this is due to ([Ando-Hopkins-Strickland 01](#AndoHopkinsStrickland01), [Ando-Hopkins-Rezk 10](#AndoHopkinsRezk10)).

See the Idea-section at _[[tmf]]_ and at _[[Witten genus]]_ for more background.

## Construction via Cubical structure

The construction proceeds via the relation between
orientations in complex orientable cohomology theory and [[cubical structures on line bundles]], see there for more.

## Properties

### Relation to twists of $tmf$

(...) relation to the [[twisted cohomology|twists]] of [[tmf]]-[[cohomology theory]] (...)

$$
  \array{
      && B String
      \\
      & \swarrow && \searrow
      \\
     \ast && && B Spin
     \\
      & \searrow && \swarrow
      \\
      && B^3 U(1)
      \\
      && \downarrow
      \\
      && B GL_1(tmf)
  }
$$

([ABG 10, (8.1)](#ABG10))

## Related concepts

* [[spin orientation of Tate K-theory]]

* [[sigma-orientation]]

* [[logarithmic cohomology operation]]

[[!include genera and partition functions - table]]

## References

The construction/identification of the string orientation and its relation to the Witten genus is due to

* {#AndoHopkinsStrickland01} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _Elliptic spectra, the Witten genus and the theorem of the cube_, Invent. Math. 146 (2001) 595–687 MR1869850 ([doi:10.1007/s002220100175](https://doi.org/10.1007/s002220100175), [pdf](http://www.math.rochester.edu/people/faculty/doug/otherpapers/musix.pdf))

* {#AndoHopkinsStrickland02} [[Matthew Ando]], [[Michael Hopkins]], [[Neil Strickland]], _The sigma orientation is an H-infinity map_, American Journal of Mathematics Vol. 126, No. 2 (Apr., 2004), pp. 247-334 ([arXiv:math/0204053](http://arxiv.org/abs/math/0204053), [doi:10.1353/ajm.2004.0008](https://doi.org/10.1353/ajm.2004.0008))

* {#AndoHopkinsRezk10} [[Matthew Ando]], [[Mike Hopkins]], [[Charles Rezk]], _Multiplicative orientations of $KO$-theory and the spectrum of topological modular forms_, 2010 ([pdf](http://www.math.uiuc.edu/~mando/papers/koandtmf.pdf))

following announcements of results in

* {#Hopkins02} [[Michael Hopkins]], _Algebraic topology and modular forms_ in Proceedings of the International Congress of Mathematicians,
Vol. I (Beijing, 2002), pages 291–317, Beijing, 2002. Higher Ed. Press ([arXiv:math/0212397](http://arxiv.org/abs/math/0212397))

which in turn follows the general program outlined in

* [[Michael Hopkins]], _Topological modular forms, the Witten genus, and the theorem of the cube_, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Z¨urich, 1994) (Basel), Birkhäuser, 1995, 554–565. MR 97i:11043 ([pdf](http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0554.0565.ocr.pdf))

An alternative construction using the [[derived algebraic geometry]] of the [[moduli stack of elliptic curves]] is sketched in

* [[Jacob Lurie]], section 5.3 of _[[A Survey of Elliptic Cohomology]]_

In fact the construction there is a refinement of the orientation of just $tmf$ to one of all $E_\infty$-rings $A$ carrying a [[derived elliptic curve]] $E \to Spec(A)$.

Discussion in relation to the [[twisted cohomology|twists]] of [[tmf]]-cohomology is in

* {#ABG10} [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))

with some related chat in _[[schreiber:Quantization via Linear homotopy types]]_.

Possible hints for further relation between [[2d SCFT]] and [[tmf]] are presented in

* [[Davide Gaiotto]], [[Theo Johnson-Freyd]], _Holomorphic SCFTs with small index_ ([arXiv:1811.00589](https://arxiv.org/abs/1811.00589))

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[[!redirects String orientation of tmf]]

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[[!redirects String-orientation of tmf]]