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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{string field theory} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{string_theory}{}\paragraph*{{String theory}}\label{string_theory} [[!include string theory - contents]] \hypertarget{chernsimons_theory}{}\paragraph*{{$\infty$-Chern-Simons theory}}\label{chernsimons_theory} [[!include infinity-Chern-Simons theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{motivation}{Motivation}\dotfill \pageref*{motivation} \linebreak \noindent\hyperlink{Achievements}{Achievements}\dotfill \pageref*{Achievements} \linebreak \noindent\hyperlink{shortcomings}{Shortcomings}\dotfill \pageref*{shortcomings} \linebreak \noindent\hyperlink{in_terms_of_higher_category_theory}{In terms of higher category theory}\dotfill \pageref*{in_terms_of_higher_category_theory} \linebreak \noindent\hyperlink{bosonic_open_string_field_theory}{Bosonic open string field theory}\dotfill \pageref*{bosonic_open_string_field_theory} \linebreak \noindent\hyperlink{bosonic_closed_string_field_theory}{Bosonic closed string field theory}\dotfill \pageref*{bosonic_closed_string_field_theory} \linebreak \noindent\hyperlink{the_interaction_terms}{The interaction terms}\dotfill \pageref*{the_interaction_terms} \linebreak \noindent\hyperlink{the_action_functional}{The action functional}\dotfill \pageref*{the_action_functional} \linebreak \noindent\hyperlink{AsAnInfinityCSTheory}{As an $\infty$-Chern-Simons theory}\dotfill \pageref*{AsAnInfinityCSTheory} \linebreak \noindent\hyperlink{OpenClosedStringFieldTheory}{Open-Closed string field theory}\dotfill \pageref*{OpenClosedStringFieldTheory} \linebreak \noindent\hyperlink{SuperstringFieldTheory}{Superstring field theory}\dotfill \pageref*{SuperstringFieldTheory} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{ReferencesBosonicSFT}{Bosonic string field theory}\dotfill \pageref*{ReferencesBosonicSFT} \linebreak \noindent\hyperlink{ReferencesBosonocOSFT}{Open SFT}\dotfill \pageref*{ReferencesBosonocOSFT} \linebreak \noindent\hyperlink{ReferencesBosonicCSFT}{Closed SFT}\dotfill \pageref*{ReferencesBosonicCSFT} \linebreak \noindent\hyperlink{ReferencesSuperstringFieldTheory}{Superstring field theory}\dotfill \pageref*{ReferencesSuperstringFieldTheory} \linebreak \noindent\hyperlink{effective_field_theory}{Effective field theory}\dotfill \pageref*{effective_field_theory} \linebreak \noindent\hyperlink{ReferencesHomotopyAlgebra}{Relation to $A_\infty$- and $L_\infty$-algebras}\dotfill \pageref*{ReferencesHomotopyAlgebra} \linebreak \noindent\hyperlink{ReferencesOnYMTheoryAspects}{For Yang-Mills theory}\dotfill \pageref*{ReferencesOnYMTheoryAspects} \linebreak \noindent\hyperlink{ReferencesBackgroundIndependence}{Background independence}\dotfill \pageref*{ReferencesBackgroundIndependence} \linebreak \noindent\hyperlink{ReferencesBackgroundIndependenceForClosed}{For closed string field theory}\dotfill \pageref*{ReferencesBackgroundIndependenceForClosed} \linebreak \noindent\hyperlink{for_open_string_field_theory}{For open string field theory}\dotfill \pageref*{for_open_string_field_theory} \linebreak \noindent\hyperlink{BosonicStringVacuumAndSenConjecture}{Tachyon dynamics, decaying D-branes and Sen's conjecture}\dotfill \pageref*{BosonicStringVacuumAndSenConjecture} \linebreak \noindent\hyperlink{relation_to_higher_spin_gauge_theory}{Relation to higher spin gauge theory}\dotfill \pageref*{relation_to_higher_spin_gauge_theory} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{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. \emph{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 . \hypertarget{motivation}{}\subsubsection*{{Motivation}}\label{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. \hypertarget{Achievements}{}\subsubsection*{{Achievements}}\label{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 \emph{condensation} (as in [[Bose-Einstein condensation]]) of the tachyon field. This is called \emph{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 \emph{[[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 \hyperlink{BosonicStringVacuumAndSenConjecture}{references below}. \hypertarget{shortcomings}{}\subsubsection*{{Shortcomings}}\label{shortcomings} The shortcoming of the current development of string field theory can probably be summarized as follows: \begin{itemize}% \item it has been studied as a theory of a \emph{classical} action functional. Little is known about the true quantum effects of the string field theory action functional. \item the best understanding exists for bosonic open string field theory, while closed and supersymmetric string field theory has remained much less accessible. \end{itemize} \hypertarget{in_terms_of_higher_category_theory}{}\subsubsection*{{In terms of higher category theory}}\label{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]]. \hypertarget{bosonic_open_string_field_theory}{}\subsection*{{Bosonic open string field theory}}\label{bosonic_open_string_field_theory} (\ldots{}) \hypertarget{bosonic_closed_string_field_theory}{}\subsection*{{Bosonic closed string field theory}}\label{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. \hypertarget{the_interaction_terms}{}\subsubsection*{{The interaction terms}}\label{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 \begin{enumerate}% \item $(b_0 - \bar b_0) \Psi = 0$; \item $(L_0 - \bar L_0) \Psi = 0$ (``level matching condition''), \end{enumerate} We shall write $\mathfrak{g}$ for this graded vector space. See (\hyperlink{Markl}{Markl, section 1}) This is equipped for each $k \in \mathbb{N}$ with a $k$-ary operation \begin{displaymath} [-,\cdots,-]_k : \mathfrak{g}^{\otimes k} \to \mathfrak{g} \end{displaymath} 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]] \begin{displaymath} [-]_1 = d_{BRST} \,. \end{displaymath} 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 \begin{equation} [\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 . \label{GradedSymmetrykBracket}\end{equation} (\hyperlink{Zwiebach93}{Zwiebach, (4.4)}). Moreover, there is a bilinear [[inner product]] \begin{displaymath} \langle -,- \rangle : \mathfrak{g} \otimes \mathfrak{g} \to \mathbb{C} \end{displaymath} coming from the [[Hilbert space]] inner product of string states (\hyperlink{Zwiebach93}{Zwiebach (2.60)}). This is non-degenerate on elements $\Psi$ which are annihilated by the [[BRST complex|ghost]] operator \begin{displaymath} b_0^- := b_0 - \bar b_0 \end{displaymath} in that for all $A \in \mathfrak{g}$ with $b_0^- A = 0$ we have \begin{equation} (\forall B \in \mathfrak{g} \,:\, \langle A,B\rangle = 0) \Rightarrow A = 0 \,. \label{NonDegeneracyOfInnerProduct}\end{equation} This is (\hyperlink{Zwiebach93}{Zwiebach 93, (2.61)}). The inner product satisfies for all $\Psi_1, \Psi_2$ of homogeneous degree the relation \begin{equation} \langle \Psi_1 , \psi_2 \rangle = (-1)^{(deg \Psi_1 + 1) (deg \Psi_2 + 1)} \langle \Psi_2, \Psi_1 \rangle \label{GradedSymmetryBilinearPairing}\end{equation} (\hyperlink{Zwiebach93}{Zwiebach 93, (2.50)}). Moreover, it is non-vanishing only on pairs of elements of total degree 5. (\hyperlink{Zwiebach93}{Zwiebach 93, (2.31)(2.44)}). From this one constructs the $(n+1)$-point functions \begin{equation} \{ \Psi_0, \Psi_1, \cdots, \Psi_k \} := \langle \Psi_0, [\Psi_1, \cdots, \Psi_k]_k \rangle \,. \label{CorrelationFunctions}\end{equation} 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 \begin{equation} \{\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\} \,. \label{GradedSymmetrykPointFunction}\end{equation} (\hyperlink{Zwiebach93}{Zwiebach 93, (4.36)}). \hypertarget{the_action_functional}{}\subsubsection*{{The action functional}}\label{the_action_functional} \begin{defn} \label{LabelConfigurationSpace}\hypertarget{LabelConfigurationSpace}{} 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 \begin{enumerate}% \item $(b_0 - \bar b_0) \vert\Psi\rangle = 0$; \item $(L_0 - \bar L_0) \vert \Psi \rangle = 0$ (``level matching condition''); \item $\vert \Psi \rangle^\dagger = - (\langle \Psi \vert)$ (``reality''); \item $\vert \Psi\rangle$ is Grassmann even (\ldots{}define\ldots{}) \item $ghostnumber \vert \Psi \rangle = 2$ (\ldots{}define\ldots{}) \end{enumerate} \end{defn} This is (\hyperlink{Zwiebach93}{Zwiebach 93, (3.9)}) The \emph{[[action functional]]} of closed string field theory is \begin{displaymath} S : \Psi \mapsto \sum_{k = 1}^\infty \frac{1}{(k+1)!} \langle \Psi, [\Psi, \cdots, \Psi]_k\rangle \,. \end{displaymath} (\hyperlink{Zwiebach93}{Zwiebach 93, (4.41)}) Since $[-]_1 = d_{BRST}$ is the [[BRST complex|BRST operator]] this starts out as \begin{displaymath} S : \Psi = \frac{1}{2}\langle \Psi , d_{BRST} \Psi \rangle + \frac{1}{3} \langle \Psi, [\Psi, \Psi]_2\rangle + \cdots \,. \end{displaymath} This immediately gives the [[equations of motion]] by variation with respect to $\Psi$ (\hyperlink{Zwiebach93}{Zwiebach 93, (4.46)}). The analogous statement for the [[superstring]] in in (\hyperlink{Sen15}{Sen 15 (2.22)}). (See also at \emph{\href{https://ncatlab.org/nlab/show/string+theory+FAQ#WhatAreTheEquations}{string theory FAQ -- What are the equations of string theory?}}). \hypertarget{AsAnInfinityCSTheory}{}\subsubsection*{{As an $\infty$-Chern-Simons theory}}\label{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. \begin{prop} \label{}\hypertarget{}{} The string [[BRST complex]] equipped with its $k$-ary interaction genus-0 interaction vertices \begin{displaymath} (\mathfrak{g}, \{[-,\cdots,-]_k\}) \end{displaymath} is an [[L-∞ algebra]]. \end{prop} This is (\hyperlink{Zwiebach93}{Zwiebach 93, (4.12)}). For more details on the $L_\infty$-structure see \emph{\href{ReferencesHomotopyAlgebra}{References -- Relation to L-∞- and A-∞-algebra})} . \begin{prop} \label{}\hypertarget{}{} 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}. \end{prop} \begin{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 \begin{displaymath} 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} \,. \end{displaymath} The [[Weil algebra]] $W(\mathfrak{g})$ is similarly generated from $\{t^a, r^a\}$ with differential \begin{displaymath} 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 \end{displaymath} and \begin{displaymath} 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} \,. \end{displaymath} Write \begin{displaymath} P_{a b} := \langle t_a, t_b\rangle \end{displaymath} for the components of the bilinear pairing in this basis. By \eqref{GradedSymmetryBilinearPairing} it follows that we can indeed regard \begin{displaymath} P_{a b} r^a \wedge r^b \in W(\mathfrak{g}) \end{displaymath} 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 \begin{displaymath} C_{a_0, \cdots, a_k} := \{t_{a_0}, \cdots, t_{a_k}\} \end{displaymath} for the components of the $(k+1)$-point function \eqref{CorrelationFunctions}. Then compute \begin{displaymath} \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} \,. \end{displaymath} This expression vanishes term-by-term by the symmetry properties \eqref{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{displaymath} \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} \,, \end{displaymath} 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 \eqref{NonDegeneracyOfInnerProduct}. \end{proof} From the discussion at [[Chern-Simons element]] in the section \href{http://ncatlab.org/nlab/show/Chern-Simons+element#CanonicalChernSimonsElement}{Canonical Chern-Simons element} we have that the [[Lagrangian]] of the [[schreiber:infinity-Chern-Simons theory]] defined by the data $(\mathfrak{g}, \langle -,-\rangle)$ is \begin{displaymath} L : A \mapsto \langle A, d A\rangle + \sum_{k = 1}^\infty \frac{2}{(k+1)!} \langle A , [A, \cdots , A]_k\rangle \end{displaymath} 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 $\Sigma$. \hypertarget{OpenClosedStringFieldTheory}{}\subsection*{{Open-Closed string field theory}}\label{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 \emph{\href{derivation#OfAlgebrasOverADGOperad}{derivations on algebras over a dg-operad}}) \begin{displaymath} \mathfrak{g}_{closed} \longrightarrow Der (A_{open}) \end{displaymath} (\hyperlink{KajiuraStasheff04}{Kajiura-Stasheff 04}, \hyperlink{Markl04}{Markl 04}). Notice that this is half of the axioms of an $\infty$-[[Lie-Rinehart pair]]. \hypertarget{SuperstringFieldTheory}{}\subsection*{{Superstring field theory}}\label{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 \emph{[[WZW-type superstring field theory]]} . A formulation of superstring field theory more on line with the Chern-Simons type bosonic theory is proposed in (\hyperlink{JurcoMuenster13}{Jurco-Muenster 13}). See also the introduction there for a survey of the literature \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[schreiber:∞-Chern-Simons theory]] \item [[higher dimensional Chern-Simons theory]] \begin{itemize}% \item [[1d Chern-Simons theory]] \item [[2d Chern-Simons theory]] \item [[3d Chern-Simons theory]] \item [[4d Chern-Simons theory]] \item [[5d Chern-Simons theory]] \item [[6d Chern-Simons theory]] \item [[7d Chern-Simons theory]] \item [[AKSZ sigma-models]] \item [[infinite-dimensional Chern-Simons theory]] \end{itemize} \end{itemize} Closed string field theory has been argued to arise from the [[dynamics]] of [[Wilson loops]] in the [[IKKT matrix model]] in (\hyperlink{FukumaKawaiKitazawaTsuchiya97}{Fukuma-Kawai-Kitazawa-Tsuchiya 97}) \hypertarget{references}{}\subsection*{{References}}\label{references} \hypertarget{general}{}\subsubsection*{{General}}\label{general} \begin{itemize}% \item [[Leonardo Rastelli]], \emph{String Field Theory} in \emph{Encyclopedia of Mathematical Physics} (\href{http://arxiv.org/abs/hep-th/0509129}{arXiv:hep-th/0509129}) \end{itemize} A textbook-like account is in \begin{itemize}% \item W. Siegel, \emph{Introduction to string field theory} (\href{http://arxiv.org/abs/hep-th/0107094}{arXiv:hep-th/0107094}) \item [[Ashoke Sen]], \emph{String Field Theory as World-sheet UV Regulator} (\href{https://arxiv.org/abs/1902.00263}{arXiv:1902.00263}) \end{itemize} \hypertarget{ReferencesBosonicSFT}{}\subsubsection*{{Bosonic string field theory}}\label{ReferencesBosonicSFT} \hypertarget{ReferencesBosonocOSFT}{}\paragraph*{{Open SFT}}\label{ReferencesBosonocOSFT} Original articles are \begin{itemize}% \item [[Edward Witten]], \emph{Noncommutative Geometry and String Field Theory} , Nucl. Phys B268 , 253, (1986) () \item [[Edward Witten]], \emph{On background independent open string field theory}, Phys.Rev. D46 (1992) 5467. (\href{http://arxiv.org/abs/hep-th/9208027}{arXiv:hep-th/9208027}) \end{itemize} See also \begin{itemize}% \item Matej Kudrna, [[Martin Schnabl]], \emph{Universal Solutions in Open String Field Theory} (\href{https://arxiv.org/abs/1812.03221}{arXiv:1812.03221}) \end{itemize} \hypertarget{ReferencesBosonicCSFT}{}\paragraph*{{Closed SFT}}\label{ReferencesBosonicCSFT} The fundamental work of Zwiebach on closed SFT is summed up in \begin{itemize}% \item [[Barton Zwiebach]], \emph{Closed string field theory: Quantum action and the B-V master equation} , Nucl.Phys. B390 (1993) 33 (\href{http://arxiv.org/abs/hep-th/9206084}{arXiv:hep-th/9206084}) \end{itemize} Brief reviews include \begin{itemize}% \item [[Barton Zwiebach]], \emph{Closed String Field Theory: An Introduction} (\href{http://arxiv.org/abs/hep-th/9305026}{arXiv:hep-th/9305026}) \end{itemize} 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 \hyperlink{YangZwiebach}{Yang-Zwiebach, section 3.1} and in \begin{itemize}% \item Bang-Gui Liu, \emph{General coordinate transformation and gravitational action from closed bosonic string field theory}, Class. Quantum Grav. 6 (1989) \item Masako Asano, Mitsuhiro Kato, \emph{Closed string field theory in a-gauge} (\href{http://arxiv.org/abs/1206.3901}{arXiv:1206.3901}) \end{itemize} Discussion of the expected closed string tachyon [[vacuum]] is in \begin{itemize}% \item Haitang Yang, [[Barton Zwiebach]], \emph{A Closed String Tachyon Vacuum ?}, JHEP 0509:054,2005 (\href{http://arxiv.org/abs/hep-th/0506077}{arXiv:hep-th/0506077}) \item [[Nicolas Moeller]], Haitang Yang, \emph{The nonperturbative closed string tachyon vacuum to high level} (\href{http://arxiv.org/abs/hep-th/0609208}{arXiv:hep-th/0609208}) \item [[Nicolas Moeller]], \emph{A tachyon lump in closed string field theory} (\href{http://arxiv.org/abs/0804.0697}{arXiv:0804.0697}) \end{itemize} and further detailed analysis is in \begin{itemize}% \item [[Nicolas Moeller]], \emph{Closed Bosonic String Field Theory at Quintic Order: Five-Tachyon Contact Term and Dilaton Theorem}, JHEP 0703:043,2007 (\href{http://arxiv.org/abs/hep-th/0609209}{arXiv:hep-th/0609209}) \item [[Nicolas Moeller]], \emph{Closed Bosonic String Field Theory at Quintic Order II: Marginal Deformations and Effective Potential}, JHEP 0709:118,2007 (\href{http://arxiv.org/abs/0705.2102}{arXiv:0705.2102}) \end{itemize} \hypertarget{ReferencesSuperstringFieldTheory}{}\subsubsection*{{Superstring field theory}}\label{ReferencesSuperstringFieldTheory} The generalization to field theory for the [[superstring]], where [[picture number]] compliates the [[RR-field]]-sector, has (only) more recently seen considerable development, \begin{itemize}% \item Hiroshi Kunitomo, Yuji Okawa, \emph{Complete action for open superstring field theory}, Prog. Theor. Exp. Phys. (2016) 023B01 (\href{https://arxiv.org/abs/1508.00366}{arXiv:1508.00366}) \end{itemize} A survey is in \begin{itemize}% \item Yuji Okawa, \emph{Recent developments in the construction of superstring field theory I}, 2016 (\href{http://www-het.ph.tsukuba.ac.jp/~ishibash/SFT16/okawa.pdf}{pdf}) \end{itemize} based on \begin{itemize}% \item [[Theodore Erler]], Sebastian Konopka, [[Ivo Sachs]], \emph{Resolving Witten's Superstring Field Theory}, JHEP04(2014)150 (\href{http://arxiv.org/abs/1312.2948}{arXiv:1312.2948}) \item [[Theodore Erler]], Sebastian Konopka, [[Ivo Sachs]], \emph{NS-NS Sector of Closed Superstring Field Theory}, JHEP08(2014)158 (\href{http://arxiv.org/abs/1403.0940}{arXiv:1403.0940}) \item [[Ashoke Sen]], \emph{Gauge Invariant 1PI Effective Superstring Field Theory: Inclusion of the Ramond Sector} (\href{https://arxiv.org/abs/1501.00988}{arXiv:1501.00988}) \item [[Theodore Erler]], Sebastian Konopka, [[Ivo Sachs]], \emph{Ramond Equations of Motion in Superstring Field Theory} (\href{https://arxiv.org/abs/1506.05774}{arXiv:1506.05774}) \item [[Ashoke Sen]], \emph{BV Master Action for Heterotic and Type II String Field Theories}, JHEP02(2016)087 (\href{http://arxiv.org/abs/1508.05387}{arXiv:1508.05387}) \end{itemize} with further developments in \begin{itemize}% \item Yuji Okawa, [[Barton Zwiebach]], \emph{Heterotic String Field Theory} (\href{http://arxiv.org/abs/hep-th/0406212}{arXiv:hep-th/0406212}) \item Roji Pius, [[Ashoke Sen]], \emph{Cutkosky Rules for Superstring Field Theory} (\href{http://arxiv.org/abs/1604.01783}{arXiv:1604.01783}) \item [[Ashoke Sen]], \emph{Reality of Superstring Field Theory Action} (\href{http://arxiv.org/abs/1606.03455}{arXiv:1606.03455}) \end{itemize} 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 \begin{itemize}% \item [[Alexander Belopolsky]], \emph{Picture changing operators in supergeometry and superstring theory} (\href{http://arxiv.org/abs/hep-th/9706033}{arXiv:hep-th/9706033}). \end{itemize} and has been further amplified in \begin{itemize}% \item [[Edward Witten]], appendix D of \emph{Notes On Super Riemann Surfaces And Their Moduli} (\href{http://arxiv.org/abs/1209.2459}{arXiv:1209.2459}) \item R. Catenacci, P.A. Grassi, S. Noja, \emph{Superstring Field Theory, Superforms and Supergeometry} (\href{https://arxiv.org/abs/1807.09563}{arXiv:1807.09563}) \end{itemize} See also \begin{itemize}% \item [[Ashoke Sen]], \emph{Background Independence of Closed Superstring Field Theory}, JHEP02(2018)155 (\href{https://arxiv.org/abs/1711.08468}{arXiv:1711.08468}) \item Corinne de Lacroix, Harold Erbin, Sitender Pratap Kashyap, [[Ashoke Sen]], Mritunjay Verma, \emph{Closed Superstring Field Theory and its Applications}, International Journal of Modern Physics AVol. 32, No. 28n29, 1730021 (2017) (\href{https://arxiv.org/abs/1703.06410}{arXiv:1703.06410}) \end{itemize} For previous constructions, the introduction of \begin{itemize}% \item [[Branislav Jurco]], Korbinian Muenster, \emph{Type II Superstring Field Theory: Geometric Approach and Operadic Description}, JHEP 1304:126 (2013) \href{http://arxiv.org/abs/1303.2323}{arXiv/1303.2323} \end{itemize} based on \begin{itemize}% \item Chungsheng James Yeh, \emph{Topics in superstring theory}, PhD thesis, Berkeley 1993 (\href{http://inspirehep.net/record/366138}{SPIRE}) \end{itemize} has a useful survey, which we quote now: \begin{uremark} The first attempt towards a field theory of superstrings was initiated by the work of Witten \begin{itemize}% \item [[Edward Witten]], \emph{Interacting field theory of open superstrings}, Nuclear Physics B, Volume 276, Issue 2 (1986) \end{itemize} by seeking a Chern-Simons like action for open superstrings similar to the one of open bosonic string field theory (\hyperlink{Witten86}{Witten 86}). The major obstacle compared to the bosonic string is the necessity of [[picture changing operators]]. Indeed, the cubic superstring theory of (\hyperlink{Witten86a}{Witten 86a}) turns out to be inconsistent due to singularities arising form the collision of picture changing operators \begin{itemize}% \item C. Wendt, \emph{Scattering amplitudes and contact interactions in Witten's superstring field theory}, Nuclear Physics B, Volume 314, Issue 1. \end{itemize} In order to circumvent this problem, another approach was pursued which sets the string field into a different picture \begin{itemize}% \item C.R. Preitschopf, C.B. Thorn, S. Yost, \emph{Superstring field theory} Nuclear Physics B, Volume 337, Issue 2. \item I.Ya. Aref'eva, P.B. Medvedev, A.P. Zubarev, \emph{New representation for string field solves the consistency problem for open superstring field theory}, Nuclear Physics B, Volume 341, Issue 2. \end{itemize} but upon including the [[RR-field|Ramond sector]], the modified superstring field theory suffers from similar inconsistencies \begin{itemize}% \item M. Kroyter, \emph{Superstring field theory equivalence: Ramond sector}, Journal of High Energy Physics, Volume 2009, Issue 10. \end{itemize} 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 \begin{itemize}% \item [[Nathan Berkovits]], \emph{Super-Poincare Invariant Superstring Field Theory}, (\href{http://arxiv.org/abs/hep-th/9503099}{hep-th/9503099}) \end{itemize} 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 \begin{itemize}% \item [[Nathan Berkovits]], \emph{Constrained BV description of string field theory}, Journal of High Energy Physics, Volume 2012, Issue 3. \item M. Kroyter, Y. Okawa, M. Schnabl, S. Torii, [[Barton Zwiebach]], \emph{Open superstring eld theory I: gauge xing, ghost structure, and propagator, Journal of High Energy Physics, Volume 2012, Issue 3.} \end{itemize} Finally, there is a formulation of open superstring field theory that differs from all other approaches in not fixing the picture of classical fields \begin{itemize}% \item M. Kroyter, \emph{Superstring field theory in the democratic picture}, Advances in Theoretical and Mathematical Physics, Volume 15, Number 3. \end{itemize} On the other hand, the construction of bosonic closed string field theory (\hyperlink{Zwiebach}{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 (\hyperlink{Yeh93}{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 \begin{itemize}% \item [[Alexander Belopolsky]], \emph{New Geometrical Approach to Superstrings} (\href{http://arxiv.org/abs/hep-th/9703183}{arXiv:hep-th/9703183}) \item L. Alvarez-Gaume, P. Nelson, C. Gomez, G. Sierra, C. Vafa, \emph{Fermionic strings in the operator formalism}, Nuclear Physics B, Volume 311, Issue 2. \end{itemize} 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 \begin{itemize}% \item [[Alexander Belopolsky]], \emph{Picture changing operators in supergeometry and superstring theory} (\href{http://arxiv.org/abs/hep-th/9706033}{arXiv:hep-th/9706033}). \end{itemize} 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 (\hyperlink{JurcoMuenster13}{Jurco-Muenster 13}) is to describe the construction of type II superstring field theory in the geometric approach. \end{uremark} Older reviews include \begin{itemize}% \item [[Nathan Berkovits]], \emph{Review of open superstring field theory} , (\href{http://arxiv.org/abs/hep-th/0105230}{arXiv:hep-th/0105230}) \end{itemize} More recently there is \begin{itemize}% \item Tomoyuki Takezaki, \emph{Open superstring field theory including the Ramond sector based on the supermoduli space} (\href{https://arxiv.org/abs/1901.02176}{arXiv:1901.02176}) \end{itemize} \hypertarget{effective_field_theory}{}\subsubsection*{{Effective field theory}}\label{effective_field_theory} Discussion of Wilsonian [[effective field theory]] of string field theory includes \begin{itemize}% \item R. Brustein, S.P.De Alwis, \emph{Renormalization group equation and non-perturbative effects in string-field theory}, Nuclear Physics B Volume 352, Issue 2, 25 March 1991, Pages 451-468 () \item Brustein and K. Roland, ?Space-time versus world sheet renormalization group equation in string theory,? Nucl. Phys. B372, 201 (1992) () \item [[Ashoke Sen]], \emph{Wilsonian Effective Action of Superstring Theory}, J. High Energ. Phys. (2017) 2017: 108 (\href{https://arxiv.org/abs/1609.00459}{arXiv:1609.00459}) \end{itemize} \hypertarget{ReferencesHomotopyAlgebra}{}\subsubsection*{{Relation to $A_\infty$- and $L_\infty$-algebras}}\label{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 \begin{itemize}% \item [[Barton Zwiebach]], \emph{Issues In Covariant Closed String Theory}, pages 192-200 in Proceedings of \emph{\href{http://inspirehep.net/record/966930}{10th and Final Workshop on Grand Unification}}, 20-22 Apr 1989. Chapel Hill, North Carolina (\href{http://inspirehep.net/record/282685?ln=de}{spire}) \end{itemize} It appears in print in \begin{itemize}% \item [[Barton Zwiebach]], \emph{Closed string field theory: Quantum action and the B-V master equation} , Nucl.Phys. B390 (1993) 33 (\href{http://arxiv.org/abs/hep-th/9206084}{arXiv:hep-th/9206084}) \end{itemize} See also at \emph{[[L-infinity algebras in physics]]}. The [[A-infinity algebra]] structure of bosonic open string field theory in \begin{itemize}% \item [[Matthias Gaberdiel]], [[Barton Zwiebach]], \emph{Tensor Constructions of Open String Theories I: Foundations} (\href{http://arxiv.org/abs/hep-th/9705038}{arXiv:hep-th/9705038}) \end{itemize} For the topological string see \begin{itemize}% \item [[Manfred Herbst]], \emph{Quantum A-infinity Structures for Open-Closed Topological Strings} (\href{http://arxiv.org/abs/hep-th/0602018}{arXiv:hep-th/0602018}) \end{itemize} Discussion for [[heterotic string theory|heterotic]] string field theory is in \begin{itemize}% \item Hiroshi Kunitomo, Tatsuya Sugimoto, \emph{Heterotic string field theory with cyclic L-infinity structure} (\href{https://arxiv.org/abs/1902.02991}{arXiv:1902.02991}) \end{itemize} Discussion of the mathematical aspects is in \begin{itemize}% \item [[Jim Stasheff]], \emph{Closed string field theory, strong homotopy Lie algebras and the operad actions of moduli space} Talk given at the \emph{Conference on Topics in Geometry and Physics} (1992) (\href{http://arxiv.org/abs/hep-th/9304061}{arXiv:hep-th/9304061}) \item [[Martin Markl]], \emph{Loop Homotopy Algebras in Closed String Field Theory}, Comm. Math. Phys. 221(2):367--384, 2001. (\href{http://arxiv.org/abs/hep-th/9711045}{arXiv:hep-th/9711045}) \item [[Alessandro Tomasiello]], \emph{A-infinity structure and superpotentials} (\href{http://arxiv.org/abs/hep-th/0107195}{arXiv:hep-th/0107195}) \item [[Hiroshige Kajiura]], \emph{Homotopy Algebra Morphism and Geometry of Classical String Field Theory} (2001) (\href{http://arxiv.org/abs/hep-th/0112228}{arXiv:hep-th/0112228}) \item [[Hiroshige Kajiura]], [[Jim Stasheff]], \emph{Homotopy algebras inspired by classical open-closed string field theory}, Comm. Math. Phys. 263 (2006) 553--581 (2004) (\href{http://arxiv.org/abs/math/0410291}{arXiv:math/0410291}) \item [[Martin Markl]], \emph{Operadic interpretation of $A_\infty$-algebras over $L_\infty$-algebras}, appendix to (\hyperlink{KajiuraStasheff04}{Kajiura-Stasheff 04}) \item Korbinian M\"u{}nster, [[Ivo Sachs]], \emph{Quantum Open-Closed Homotopy Algebra and String Field Theory} (\href{http://arxiv.org/abs/1109.4101}{arXiv:1109.4101}) \item Korbinian M\"u{}nster, [[Ivo Sachs]], \emph{On Homotopy Algebras and Quantum String Field Theory} (\href{http://arxiv.org/abs/1303.3444}{arXiv:1303.3444}) \end{itemize} Surveys include \begin{itemize}% \item [[Jim Stasheff]], \emph{[[A Survey of Cohomological Physics]]} \item [[Hiroshige Kajiura]], [[Jim Stasheff]], \emph{Homotopy algebra of open--closed strings} Geometry \& Topology Monographs 13 (2008) 229--259 (\href{http://msp.warwick.ac.uk/gtm/2008/13/gtm-2008-13-010s.pdf}{pdf}) (\href{http://arxiv.org/abs/hep-th/0606283}{arXiv:hep-th/0606283}) \item [[Ivo Sachs]], \emph{String theory and homotopy algebras}, talk notes, Srni 2015 (\href{http://www.math.muni.cz/~srni/Prednasky/Sachs.pdf}{pdf}) \item [[Ivo Sachs]], \emph{Homotopy Algebras in String Field Theory}, Proceedings of LMS/EPSRC Symposium \emph{\href{http://www.maths.dur.ac.uk/lms/109/index.html}{Higher Structures in M-Theory}}, Fortschritte der Physik 2019 (\href{https://arxiv.org/abs/1903.02870}{arXiv:1903.02870}) \end{itemize} Discussion of the CSFT-action as of the form of [[schreiber:∞-Chern-Simons theory]] is in section 4.4 of \begin{itemize}% \item [[Urs Schreiber]], \emph{[[schreiber:differential cohomology in a cohesive topos]]} \end{itemize} 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 \begin{itemize}% \item Yuji Okawa, [[Barton Zwiebach]], \emph{Heterotic string field theory} (\href{http://arxiv.org/abs/hep-th/0406212}{arXiv:hep-th/0406212}) \end{itemize} See also \emph{[[higher category theory and physics]]} . \hypertarget{ReferencesOnYMTheoryAspects}{}\paragraph*{{For Yang-Mills theory}}\label{ReferencesOnYMTheoryAspects} The idea that open strings on $N$ coincident [[D-branes]] exhibit [[gauge enhancement]] to $U(N)$-gauge field theory is due to \begin{itemize}% \item [[Edward Witten]], section 3 of \emph{Bound States Of Strings And $p$-Branes}, Nucl.Phys.B460:335-350, 1996 (\href{https://arxiv.org/abs/hep-th/9510135}{arXiv:hep-th/9510135}) \end{itemize} 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 \begin{itemize}% \item [[Robert Myers]], section 3 of \emph{Nonabelian Phenomena on D-branes}, Class.Quant.Grav. 20 (2003) (\href{https://arxiv.org/abs/hep-th/0303072}{arXiv:hep-th/0303072}) \end{itemize} By actual computation in [[open string field theory]] ``convincing evidence'' (see bottom of p. 22) was found, numerically, in \begin{itemize}% \item Erasmo Coletti, Ilya Sigalov, [[Washington Taylor]], \emph{Abelian and nonabelian vector field effective actions from string field theory}, JHEP 0309 (2003) 050 (\href{https://arxiv.org/abs/hep-th/0306041}{arXiv:hep-th/0306041}) \end{itemize} Similar numerical derivation, as well as exact derivation at zero momentum, is in \begin{itemize}% \item [[Nathan Berkovits]], [[Martin Schnabl]], \emph{Yang-Mills Action from Open Superstring Field Theory}, JHEP 0309 (2003) 022 (\href{https://arxiv.org/abs/hep-th/0307019}{arXiv:hep-th/0307019}) \end{itemize} The first full derivation seems to be due to \begin{itemize}% \item [[Taejin Lee]], \emph{Covariant open bosonic string field theory on multiple D-branes in the proper-time gauge} (\href{https://arxiv.org/abs/1609.01473}{arXiv:1609.01473}) \end{itemize} which is surveyed in \begin{itemize}% \item [[Taejin Lee]], \emph{Deformation of the Cubic Open String Field Theory}, Phys. Lett. B 768 (2017) 248 (\href{https://arxiv.org/abs/1701.06154}{arXiv:1701.06154}) \end{itemize} That on [[D0-branes]] this reproduces the [[BFSS matrix model]] and on [[D(-1)-branes]] the [[IKKT matrix model]] is shown in \begin{itemize}% \item [[Taejin Lee]], \emph{Covariant Open String Field Theory on Multiple D$p$-Branes} (\href{https://arxiv.org/abs/1703.06402}{arXiv:1703.06402}) \end{itemize} 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 \begin{itemize}% \item [[Anton Zeitlin]], \emph{Homotopy Lie Superalgebra in Yang-Mills Theory} (\href{http://arxiv.org/abs/0708.1773}{arXiv:0708.1773}) \emph{BV Yang-Mills as a Homotopy Chern-Simons via SFT} (\href{http://arxiv.org/abs/0709.1411}{arXiv:0709.1411}) \emph{SFT-inspired Algebraic Structures in Gauge Theories} (\href{http://arxiv.org/abs/0711.3843}{arXiv:0711.3843}) \emph{Conformal Field Theory and Algebraic Structure of Gauge Theory} (\href{http://arxiv.org/abs/0812.1840}{arXiv:0812.1840}) \end{itemize} \hypertarget{ReferencesBackgroundIndependence}{}\subsubsection*{{Background independence}}\label{ReferencesBackgroundIndependence} References discussing independence of string field theories on the [[CFT]] ([[sigma-model]] background) in terms of which they are written down. \hypertarget{ReferencesBackgroundIndependenceForClosed}{}\paragraph*{{For closed string field theory}}\label{ReferencesBackgroundIndependenceForClosed} \begin{itemize}% \item [[Ashoke Sen]], [[Barton Zwiebach]], \emph{Quantum Background Independence of Closed String Field Theory} (\href{http://arxiv.org/abs/hep-th/9311009}{arXiv:hep-th/9311009}) \item [[Ashoke Sen]], [[Barton Zwiebach]], \emph{Background Independent Algebraic Structures in Closed String Field Theory} (\href{http://arxiv.org/abs/hep-th/9408053}{arXiv:hep-th/9408053}) \end{itemize} A review of the history of some related developments is given in \begin{itemize}% \item Sabbir Rahman, \emph{Manifest background independent formulation of string field theory} (\href{http://www.natscience.com/Uwe/Forum.aspx/physics-research/503/Manifest-background-independent-formulation-of-string-field}{newsgroup comment}) \end{itemize} Closed string field theory has also been argued to arise from the [[dynamics]] of [[Wilson loops]] in the [[IKKT matrix model]] in (\hyperlink{FukumaKawaiKitazawaTsuchiya97}{Fukuma-Kawai-Kitazawa-Tsuchiya 97}) \begin{itemize}% \item M. Fukuma, H. Kawai, Y. Kitazawa, A. Tsuchiya, \emph{String Field Theory from IIB Matrix Model}, Nucl.Phys.B510:158-174,1998 (\href{http://arxiv.org/abs/hep-th/9705128}{arXiv:hep-th/9705128}) \end{itemize} \hypertarget{for_open_string_field_theory}{}\paragraph*{{For open string field theory}}\label{for_open_string_field_theory} \begin{itemize}% \item [[Edward Witten]], \emph{On background independent open string field theory}, Phys.Rev. D46 (1992) 5467. (\href{http://arxiv.org/abs/hep-th/9208027}{arXiv:hep-th/9208027}) \item [[Theodore Erler]], Carlo Maccaferri, \emph{String Field Theory Solution for Any Open String Background} (\href{http://arxiv.org/abs/1406.3021}{arXiv:1406.3021}) \end{itemize} \hypertarget{BosonicStringVacuumAndSenConjecture}{}\subsubsection*{{Tachyon dynamics, decaying D-branes and Sen's conjecture}}\label{BosonicStringVacuumAndSenConjecture} [[Sen's conjecture]] about the open [[bosonic string]] [[tachyon]] and the decay of the [[D25-brane]] originates in \begin{itemize}% \item [[Ashoke Sen]], \emph{Universality of the Tachyon Potential}, JHEP 9912:027,1999 (\href{http://arxiv.org/abs/hep-th/9911116}{arXiv:hep-th/9911116}) \end{itemize} 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 \begin{itemize}% \item Ian Ellwood, Washington Taylor, \emph{Open string field theory without open strings}, Phys.Lett. B512 (2001) 181-188 (\href{http://arxiv.org/abs/hep-th/0103085}{arXiv:hep-th/0103085}) \item Bo Feng, Yang-Hui He, Nicolas Moeller, \emph{Testing the Uniqueness of the Open Bosonic String Field Theory Vacuum} (\href{http://arxiv.org/abs/hep-th/0103103}{arXiv:hep-th/0103103}) \end{itemize} A breakthrough were then the analytic solutions describing the bosonic string tachyon vacuum in \begin{itemize}% \item [[Martin Schnabl]], \emph{Analytic solution for tachyon condensation in open string field theory} (\href{http://arxiv.org/abs/hep-th/0511286}{arXiv:hep-th/0511286}) \item Ian Ellwood, [[Martin Schnabl]], \emph{Proof of vanishing cohomology at the tachyon vacuum}, JHEP 0702:096,2007 (\href{http://arxiv.org/abs/hep-th/0606142}{arXiv:hep-th/0606142}) \end{itemize} Analogous discussion including also brane/[[anti-brane]] pairs in [[superstring]] theory is in \begin{itemize}% \item [[Leonardo Rastelli]], [[Ashoke Sen]], [[Barton Zwiebach]], \emph{Vacuum String Field Theory} (\href{http://arxiv.org/abs/hep-th/0106010}{arXiv:hep-th/0106010}) \item [[Ashoke Sen]], \emph{Tachyon Dynamics in Open String Theory}, Int.J.Mod.Phys.A20:5513-5656,2005 (\href{http://arxiv.org/abs/hep-th/0410103}{arXiv:hep-th/0410103}) \item L. Bonora, N. Bouatta, C. Maccaferri, \emph{Towards open-closed string duality: Closed Strings as Open String Fields} (\href{http://arxiv.org/abs/hep-th/0609182}{arXiv:hep-th/0609182}) \item [[Theodore Erler]], \emph{Tachyon Vacuum in Cubic Superstring Field Theory}, JHEP 0801:013,2008 (\href{http://arxiv.org/abs/0707.4591}{arXiv:0707.4591}) \end{itemize} \hypertarget{relation_to_higher_spin_gauge_theory}{}\subsubsection*{{Relation to higher spin gauge theory}}\label{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 \begin{itemize}% \item [[Marc Henneaux]], [[Claudio Teitelboim]], section 2 of \emph{First And Second Quantized Point Particles Of Any Spin}, in [[Claudio Teitelboim]], [[Jorge Zanelli]] (eds.) \emph{Santiago 1987, Proceedings, Quantum mechanics of fundamental systems 2}, pp. 113-152. Plenum Press. \item [[David Gross]], \emph{High-Energy Symmetries Of String Theory}, Phys. Rev. Lett. 60 (1988) 1229. \end{itemize} and is further developed for instance in \begin{itemize}% \item [[Auguste Sagnotti]], M. Tsulaia, \emph{On higher spins and the tensionless limit of String Theory}, Nucl.Phys.B682:83-116,2004 (\href{http://arxiv.org/abs/hep-th/0311257}{arXiv:hep-th/0311257}) \item G. Bonelli, \emph{On the Tensionless Limit of Bosonic Strings, Infinite Symmetries and Higher Spins}, Nucl.Phys. B669 (2003) 159-172 (\href{http://arxiv.org/abs/hep-th/0305155}{arXiv:hep-th/0305155}) \item [[Auguste Sagnotti]], M. Taronna, \emph{String Lessons for Higher-Spin Interactions}, Nucl.Phys.B842:299-361,2011 (\href{http://arxiv.org/abs/1006.5242}{arXiv:1006.5242}) \item [[Auguste Sagnotti]], \emph{Notes on Strings and Higher Spins} (\href{http://arxiv.org/abs/1112.4285}{arXiv:1112.4285}) \end{itemize} And conversely: \begin{itemize}% \item Rakibur Rahman, Massimo Taronna, \emph{From Higher Spins to Strings: A Primer} in [[Stefan Fredenhagen]] (ed.) \emph{Introduction to Higher Spin Theory} (\href{https://arxiv.org/abs/1512.07932}{arXiv:1512.07932}) \item [[Matthias Gaberdiel]], [[Rajesh Gopakumar]], \emph{String Theory as a Higher Spin Theory}, J. High Energ. Phys. (2016) 2016: 85 (\href{https://arxiv.org/abs/1512.07237}{arXiv:1512.07237}, ) \end{itemize} [[!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]] \end{document}