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String field theory (SFT) is a formalism in string theory in which the dynamics of relativistic strings is reformulated in the language of quantum field theory. This is accomplished at the level of perturbation theory by finding a collection of vertices for joining and splitting strings, as well as string propagators, that give a Feynman diagram-like expansion for string scattering amplitudes. In most string field theories, this expansion is encoded by a classical action found by second-quantizing the free string and adding interaction terms. As is usually the case in second quantization, a classical field configuration of the second-quantized theory is given by a wave function in the original theory. In the case of string field theory, this implies that a classical configuration, usually called the string field, is given by an element of the free string Fock space.

The principal advantages of the formalism are that it allows the computation of off-shell amplitudes and, when a classical action is available, gives non-perturbative information that cannot be seen directly from the standard genus expansion of string scattering. In particular, following the work of Ashoke Sen,[1] it has been useful in the study of tachyon condensation on unstable D-branes. It has also had applications to topological string theory,[2] non-commutative geometry,[3] and strings in low dimensions.[4]

String field theories come in a number of varieties depending on which type of string is second quantized: Open string field theories describe the scattering of open strings, closed string field theories describe closed strings, while open-closed string field theories include both open and closed strings.

In addition, depending on the method used to fix the worldsheet diffeomorphisms and conformal transformations in the original free string theory, the resulting string field theories can be very different. Using light cone gauge, yields light-cone string field theories whereas using BRST quantization, one finds covariant string field theories. There are also hybrid string field theories, known as covariantized light-cone string field theories which use elements of both light-cone and BRST gauge-fixed string field theories.[5]

A final form of string field theory, known as background independent open string field theory, takes a very different form; instead of second quantizing the worldsheet string theory, it second quantizes the space of two-dimensional quantum field theories.[6]

Light-cone string field theory

[edit]

Light-cone string field theories were introduced by Stanley Mandelstam[7][8] and developed by Mandelstam, Michael Green, John Schwarz and Lars Brink.[9][10][11][12][13] An explicit description of the second-quantization of the light-cone string was given by Michio Kaku and Keiji Kikkawa.[14][15]

Light-cone string field theories were the first string field theories to be constructed and are based on the simplicity of string scattering in light-cone gauge. For example, in the bosonic closed string case, the worldsheet scattering diagrams naturally take a Feynman diagram-like form, being built from two ingredients, a propagator,

and two vertices for splitting and joining strings, which can be used to glue three propagators together,

These vertices and propagators produce a single cover of the moduli space of -point closed string scattering amplitudes so no higher order vertices are required.[16] Similar vertices exist for the open string.

When one considers light-cone quantized superstrings, the discussion is more subtle as divergences can arise when the light-cone vertices collide.[17] To produce a consistent theory, it is necessary to introduce higher order vertices, called contact terms, to cancel the divergences.

Light-cone string field theories have the disadvantage that they break manifest Lorentz invariance. However, in backgrounds with light-like Killing vectors, they can considerably simplify the quantization of the string action. Moreover, until the advent of the Berkovits string[18] it was the only known method for quantizing strings in the presence of Ramond–Ramond fields. In recent research, light-cone string field theory played an important role in understanding strings in pp-wave backgrounds.[19]

Free covariant string field theory

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An important step in the construction of covariant string field theories (preserving manifest Lorentz invariance) was the construction of a covariant kinetic term. This kinetic term can be considered a string field theory in its own right: the string field theory of free strings. Since the work of Warren Siegel,[20] it has been standard to first BRST-quantize the free string theory and then second quantize so that the classical fields of the string field theory include ghosts as well as matter fields. For example, in the case of the bosonic open string theory in 26-dimensional flat spacetime, a general element of the Fock-space of the BRST quantized string takes the form (in radial quantization in the upper half plane),

where is the free string vacuum and the dots represent more massive fields. In the language of worldsheet string theory, , , and represent the amplitudes for the string to be found in the various basis states. After second quantization, they are interpreted instead as classical fields representing the tachyon , gauge field and a ghost field .

In the worldsheet string theory, the unphysical elements of the Fock space are removed by imposing the condition as well as the equivalence relation . After second quantization, the equivalence relation is interpreted as a gauge invariance, whereas the condition that is physical is interpreted as an equation of motion. Because the physical fields live at ghostnumber one, it is also assumed that the string field is a ghostnumber one element of the Fock space.

In the case of the open bosonic string a gauge-unfixed action with the appropriate symmetries and equations of motion was originally obtained by André Neveu, Hermann Nicolai and Peter C. West.[21] It is given by

where is the BPZ-dual of .[22]

For the bosonic closed string, construction of a BRST-invariant kinetic term requires additionally that one impose and . The kinetic term is then

Additional considerations are required for the superstrings to deal with the superghost zero-modes.

Witten's cubic open string field theory

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The best studied and simplest of covariant interacting string field theories was constructed by Edward Witten.[23] It describes the dynamics of bosonic open strings and is given by adding to the free open string action a cubic vertex:

,

where, as in the free case, is a ghostnumber one element of the BRST-quantized free bosonic open-string Fock-space.

The cubic vertex,

is a trilinear map which takes three string fields of total ghostnumber three and yields a number. Following Witten, who was motivated by ideas from noncommutative geometry, it is conventional to introduce the -product defined implicitly through

The -product and cubic vertex satisfy a number of important properties (allowing the to be general ghost number fields):

  1. Cyclicity :
  2. BRST invariance :

    For the -product, this implies that acts as a graded derivation

  3. Associativity

    In terms of the cubic vertex,

In these equations, denotes the ghost number of .

Gauge invariance

[edit]

These properties of the cubic vertex are sufficient to show that is invariant under the Yang–Mills-like gauge transformation,

where is an infinitesimal gauge parameter. Finite gauge transformations take the form

where the exponential is defined by,

Equations of motion

[edit]

The equations of motion are given by the following equation:

Because the string field is an infinite collection of ordinary classical fields, these equations represent an infinite collection of non-linear coupled differential equations. There have been two approaches to finding solutions: First, numerically, one can truncate the string field to include only fields with mass less than a fixed bound, a procedure known as "level truncation".[24] This reduces the equations of motion to a finite number of coupled differential equations and has led to the discovery of many solutions.[25][26] Second, following the work of Martin Schnabl [27] one can seek analytic solutions by carefully picking an ansatz which has simple behavior under star multiplication and action by the BRST operator. This has led to solutions representing marginal deformations, the tachyon vacuum solution[28] and time-independent D-brane systems.[29]

Quantization

[edit]

To consistently quantize one has to fix a gauge. The traditional choice has been Feynman–Siegel gauge,

Because the gauge transformations are themselves redundant (there are gauge transformations of the gauge transformations), the gauge fixing procedure requires introducing an infinite number of ghosts via the BV formalism.[30] The complete gauge fixed action is given by

where the field is now allowed to be of arbitrary ghostnumber. In this gauge, the Feynman diagrams are constructed from a single propagator and vertex. The propagator takes the form of a strip of worldsheet of width and length

There is also an insertion of an integral of the -ghost along the red line. The modulus, is integrated from 0 to .

The three vertex can be described as a way of gluing three propagators together, as shown in the following picture:

In order to represent the vertex embedded in three dimensions, the propagators have been folded in half along their midpoints. The resulting geometry is completely flat except for a single curvature singularity where the midpoints of the three propagators meet.

These Feynman diagrams generate a complete cover of the moduli space of open string scattering diagrams. It follows that, for on-shell amplitudes, the n-point open string amplitudes computed using Witten's open string field theory are identical to those computed using standard worldsheet methods.[31][32]

Supersymmetric covariant open string field theories

[edit]

There are two main constructions of supersymmetric extensions of Witten's cubic open string field theory. The first is very similar in form to its bosonic cousin and is known as modified cubic superstring field theory. The second, due to Nathan Berkovits is very different and is based on a WZW-type action.

Modified cubic superstring field theory

[edit]

The first consistent extension of Witten's bosonic open string field theory to the RNS string was constructed by Christian Preitschopf, Charles Thorn and Scott Yost and independently by Irina Aref'eva, P. B. Medvedev and A. P. Zubarev.[33][34] The NS string field is taken to be a ghostnumber one picture zero string field in the small Hilbert space (i.e. ). The action takes a very similar form to bosonic action,

where,

is the inverse picture changing operator. The suggested picture number extension of this theory to the Ramond sector might be problematic.

This action has been shown to reproduce tree-level amplitudes and has a tachyon vacuum solution with the correct energy.[35] The one subtlety in the action is the insertion of picture changing operators at the midpoint, which imply that the linearized equations of motion take the form

Because has a non-trivial kernel, there are potentially extra solutions that are not in the cohomology of .[36] However, such solutions would have operator insertions near the midpoint and would be potentially singular, and importance of this problem remains unclear.

Berkovits superstring field theory

[edit]

A very different supersymmetric action for the open string was constructed by Nathan Berkovits. It takes the form[37]

where all of the products are performed using the -product including the anticommutator , and is any string field such that and . The string field is taken to be in the NS sector of the large Hilbert space, i.e. including the zero mode of . It is not known how to incorporate the R sector, although some preliminary ideas exist.[38]

The equations of motion take the form

The action is invariant under the gauge transformation

The principal advantage of this action is that it free from any insertions of picture-changing operators. It has been shown to reproduce correctly tree level amplitudes[39] and has been found, numerically, to have a tachyon vacuum with appropriate energy.[40][41] The known analytic solutions to the classical equations of motion include the tachyon vacuum[42] and marginal deformations.

Other formulations of covariant open superstring field theory

[edit]

A formulation of superstring field theory using the non-minimal pure-spinor variables was introduced by Berkovits.[43] The action is cubic and includes a midpoint insertion whose kernel is trivial. As always within the pure-spinor formulation, the Ramond sector can be easily treated. However, it is not known how to incorporate the GSO- sectors into the formalism.

In an attempt to resolve the allegedly problematic midpoint insertion of the modified cubic theory, Berkovits and Siegel proposed a superstring field theory based on a non-minimal extension of the RNS string,[44] which uses a midpoint insertion with no kernel. It is not clear if such insertions are in any way better than midpoint insertions with non-trivial kernels.

Covariant closed string field theory

[edit]

Covariant closed string field theories are considerably more complicated than their open string cousins. Even if one wants to construct a string field theory which only reproduces tree-level interactions between closed strings, the classical action must contain an infinite number of vertices [45] consisting of string polyhedra.[46][47]

If one demands that on-shell scattering diagrams be reproduced to all orders in the string coupling, one must also include additional vertices arising from higher genus (and hence higher order in ) as well. In general, a manifestly BV invariant, quantizable action takes the form[48]

where denotes an th order vertex arising from a genus surface and is the closed string coupling. The structure of the vertices is in principle determined by a minimal area prescription,[49] although, even for the polyhedral vertices, explicit computations have only been performed to quintic order.[50][51]

Covariant heterotic string field theory

[edit]

A formulation of the NS sector of the heterotic string was given by Berkovits, Okawa and Zwiebach.[52] The formulation amalgamates bosonic closed string field theory with Berkovits' superstring field theory.

See also

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References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

String field theory

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String field theory is a second-quantized, gauge-invariant of that treats strings as fields in an infinite-dimensional on the target , with string fields defined as elements of the state space of a two-dimensional describing the string . This framework extends the principles of point-particle to strings, providing an off-shell, background-independent description that encompasses all consistent string theories, including bosonic open and closed strings, superstrings, and heterotic strings. The development of string field theory began in the 1970s with early covariant approaches to string quantization, such as the work of Kaku and Kikkawa in 1974, which laid groundwork for field-theoretic treatments of strings. A pivotal advancement came in 1986 with Edward Witten's introduction of cubic open bosonic string field theory, which proposed an action functional S=12⟨Ψ,QΨ⟩+13⟨Ψ,Ψ∗Ψ⟩S = \frac{1}{2} \langle \Psi, Q \Psi \rangle + \frac{1}{3} \langle \Psi, \Psi * \Psi \rangle, where Ψ\Psi is the string field, QQ is the BRST operator, and ∗* denotes the star product derived from string overlaps. This formulation resolved ambiguities in perturbation theory by expressing amplitudes as sums over Feynman diagrams in the string field theory, with vertices corresponding to Riemann surfaces. Key formulations include open string field theory, which describes strings ending on D-branes and incorporates both Neveu-Schwarz (NS) and Ramond () sectors for superstrings, using picture-changing operators to handle fermionic degrees of freedom. Closed string field theory, developed by Zwiebach in –, employs L∞_\infty algebraic structures and multi-string interaction vertices on higher-genus surfaces to capture closed string dynamics in a fully quantum-consistent manner. Bosonic versions focus on tachyonic instabilities and use minimal-area metrics for regularization, while superstring extensions, completed in the , provide actions for type II and heterotic theories that respect . One of the primary advantages of string field theory is its potential for definitions of string interactions, as demonstrated by exact solutions like Schnabl's 2005 analytic , which resolves the of the bosonic string and describes decay via condensation. It also proves unitarity and ultraviolet finiteness beyond , relates open and closed string couplings through relations like gc=go2g_c = g_o^2, and facilitates background-independent deformations, such as marginal operators that shift the metric or other fields. Applications extend to effective field theories via homotopy transfer methods, matching low-energy actions, and explorations of , including contributions to black hole in the AdS/CFT correspondence.

Fundamentals

Definition and Motivation

String field theory (SFT) is a second-quantized formulation of that treats the wavefunctional of a string as a field propagating in an infinite-dimensional . This framework extends principles to one-dimensional extended objects, where strings serve as the fundamental entities rather than point particles, allowing for a systematic description of their . Central to SFT is the string field Ψ\Psi, which resides in a comprising functionals over the space of string configurations, often linked to the loop space or representations of the . The development of SFT is motivated by the shortcomings of first-quantized , which relies on path integrals over individual worldsheets and encounters difficulties in managing interactions and multi-string configurations. In first-quantized approaches, defining off-shell amplitudes, resolving divergences, and capturing effects like condensation prove challenging, limiting the theory's ability to handle complex string dynamics comprehensively. SFT overcomes these issues by second-quantizing strings, analogous to how second-quantizes point particles to describe many-body systems and interactions via Feynman diagrams. Furthermore, SFT generalizes the worldsheet path integral of first-quantized into a field-theoretic action over the entire space of string configurations, yielding a local and Lorentz-invariant principle for string interactions. This enables a background-independent treatment of string propagation and scattering, with manifest gauge invariance that accommodates the infinite of strings while ensuring consistency across perturbative and regimes. By providing such a unified structure, SFT facilitates deeper insights into the quantum properties of strings, including their role in unifying and .

Historical Development

The origins of string field theory trace back to the , emerging from the dual resonance models of and early efforts in light-cone quantization. In 1974, Takehiko Kaku and Keiji Kikkawa introduced a field-theoretic framework for relativistic strings, building on Stanley Mandelstam's light-cone description of interacting strings without fermions, which focused on tree-level interactions in the light-cone gauge. These works provided the initial conceptual foundation for treating strings as extended fields rather than point particles, addressing the dynamics of open strings in a perturbative expansion. The 1980s marked significant breakthroughs in both light-cone and covariant formulations. In 1984, Warren Siegel advanced covariant string field theory, developing a gauge-fixed approach using BRST invariance for bosonic open strings, though full interacting quantum consistency issues persisted. Concurrently, Alexander Polyakov's 1981 path-integral formulation of bosonic strings enabled covariant quantization techniques, paving the way for BRST-based methods. The decade culminated in Edward Witten's 1986 proposal of a cubic open string field theory, which reformulated the dynamics using an associative star product and the BRST operator, achieving gauge invariance for interacting bosonic open strings. In the 1990s, extensions to supersymmetric theories addressed anomalies in bosonic models, with modified cubic formulations appearing for superstrings. Nathan Berkovits introduced a non-polynomial superstring field theory in 1995, using a Wess-Zumino-Witten-like action in the large to ensure super-Poincaré invariance and resolve picture-changing issues. Challenges persisted for closed string field theory, where quantum consistency required the Batalin-Vilkovisky formalism, as developed by in 1993, but full gauge-invariant interactions remained elusive for supersymmetric cases. Key contributors like began exploring condensation within these frameworks, linking string field theory to phenomena. From the 2000s onward, focus shifted toward applications, with limited new foundational formulations but notable progress in numerical solutions for condensation. Researchers like Seyed Faroogh Moosavian and collaborators in the employed level truncation and methods to numerically solve equations in bosonic and heterotic string field theories, providing insights into decay and vacuum structure without relying on analytic closed forms. In the , further progress included formulations of open-closed superstring field theory, integrating type II dynamics more comprehensively. This era emphasized computational verification of conjectures, such as Sen's potential, while highlighting ongoing difficulties in closed supersymmetric extensions.

Light-Cone Formulations

Open Strings

In the light-cone gauge formulation of open field theory, coordinates are decomposed into light-cone components defined as x±=x0±x1x^\pm = x^0 \pm x^1, where x0x^0 and x1x^1 are the time and longitudinal spatial directions, respectively, with the transverse coordinates xix^i (for i=2,…,25i = 2, \dots, 25) spanning the remaining dimensions in 26-dimensional . The gauge is fixed by imposing X+(τ,σ)=p+τX^+(\tau, \sigma) = p^+ \tau, where p+p^+ is the conserved light-cone serving as the , and P+(σ)=p+P^+(\sigma) = p^+ constant along the , which eliminates the longitudinal X−X^- and the reparametrization ghosts, leaving only physical transverse modes. This choice ensures that the theory is non-covariant under full Lorentz transformations but preserves manifest SO(24) Lorentz invariance in the transverse directions.90560-6) The open string field Ψ\Psi is introduced as a functional depending on the transverse coordinates Xi(σ)X^i(\sigma) and the light-cone momentum p+p^+, expanded in the Fock space of transverse creation and annihilation operators αni\alpha_n^i and αni†\alpha_n^{i\dagger} satisfying [αni,αmj†]=δn,mδij[\alpha_n^i, \alpha_m^{j\dagger}] = \delta_{n,m} \delta^{ij}. Specifically, Ψ=Ψ[Xi(σ),p+,τ]\Psi = \Psi[X^i(\sigma), p^+, \tau], where τ\tau parametrizes the worldsheet time evolution, and the field creates or annihilates open strings with endpoints at fixed boundary conditions σ=0,πp+\sigma = 0, \pi p^+. The mass-shell condition arises from the Virasoro constraint L0Ψ=0L_0 \Psi = 0, with L0=12p⊥2+∑n=1∞nNn−1=α′m2/2L_0 = \frac{1}{2} p_\perp^2 + \sum_{n=1}^\infty n N_n - 1 = \alpha' m^2 / 2, ensuring physical states satisfy the open string spectrum m2=(N−1)/α′m^2 = (N - 1)/\alpha' for integer mode number NN. This representation facilitates a second-quantized description where multi-string states are built from single-string Fock space tensors.90560-6) The free action for the open string field in light-cone gauge takes the Hamiltonian form S0=−∫DXi dp+ dτ Tr[Ψ†p+(i∂τ+H)Ψ]S_0 = -\int \mathcal{D}X^i \, dp^+ \, d\tau \, \mathrm{Tr} \left[ \Psi^\dagger p^+ (i \partial_\tau + H) \Psi \right], where H = \int_0^{\pi p^+} \frac{d\sigma}{2\pi} \left( \frac{1}{2} P_i^2 + \frac{1}{2} (X^i')^2 \right) is the transverse Hamiltonian, equivalent to H=L0H = L_0, and the trace is over the . An alternative worldsheet-integrated expression is S=∫dτ dσ ⟨Ψ∣(∂τ−L0)∣Ψ⟩S = \int d\tau \, d\sigma \, \langle \Psi | (\partial_\tau - L_0) | \Psi \rangle, capturing the free evolution without interactions. This action yields (i∂τ+L0)Ψ=0(i \partial_\tau + L_0) \Psi = 0, directly enforcing the mass-shell condition and enabling straightforward quantization via of oscillators.90560-6) Interactions are incorporated through vertex operators in the light-cone frame, with the leading cubic term given by the three-string vertex Sint=g∫dτ D3Z d3p+ δ(∑p+)δ[Z˙1(σ)+Z˙2(πp2+−σ)−Z˙3(σ)]Ψ1Ψ2Ψ3S_\mathrm{int} = g \int d\tau \, \mathcal{D}^3 Z \, d^3 p^+ \, \delta\left( \sum p^+ \right) \delta\left[ \dot{Z}_1(\sigma) + \dot{Z}_2(\pi p_2^+ - \sigma) - \dot{Z}_3(\sigma) \right] \Psi_1 \Psi_2 \Psi_3, where ZZ denotes collective transverse coordinates and the delta functions enforce momentum conservation and string joining/splitting at interaction points. Higher-point amplitudes factorize into products of these three-point couplings, represented diagrammatically by Mandelstam diagrams, which depict worldsheet histories of string interactions as polygonal surfaces with parameters τi\tau_i integrated over, yielding scattering amplitudes like A=gN−2∫∏dτi exp⁡(−∑pr−τr)A = g^{N-2} \int \prod d\tau_i \, \exp\left( -\sum p_r^- \tau_r \right) for NN-string processes. These diagrams triangulate the of open string worldsheets, ensuring duality and Regge behavior in the bosonic theory.90560-6) The light-cone formulation offers key advantages for open string field theory, including manifest unitarity due to the positive-definite transverse norm and the absence of ghosts, as well as computational simplicity in perturbative calculations of scattering amplitudes via explicit oscillator matrix elements. It facilitates theorems and explicit evaluations of loop diagrams, making it particularly useful for verifying consistency conditions like the no-ghost in 26 dimensions, though full Lorentz invariance requires additional constraints. This approach has been instrumental in early developments of dual resonance models and remains valuable for numerical studies of string interactions.90560-6)

Closed Strings

Closed string field theory in the light-cone gauge builds upon the open string formulation by accounting for the closed topology, which introduces independent left- and right-moving modes along the string. The string field is represented as a state |Ψ(p^+)⟩ in the Fock space of transverse oscillators, depending on the light-cone momentum p^+ and incorporating both left (α_n^i for n > 0) and right (\bar{α}_n^i for n > 0) movers, with i labeling transverse directions (1 to D-2). This separation arises from the periodic boundary conditions of the closed string, allowing left and right sectors to evolve independently except through level-matching constraints. A key feature is the level-matching condition L_0 = \bar{L}0, where L_0 = (\alpha_0')^2 / 2 + \sum{n=1}^\infty n N_n - a and \bar{L}0 = (\alpha_0')^2 / 2 + \sum{n=1}^\infty n \tilde{N}_n - a are the Virasoro generators for the left and right sectors, respectively, with N_n (\tilde{N}_n) the number operators and a the normal-ordering constant (a=1 for bosonic strings). This condition, equivalent to \sum n (N_n - \tilde{N}_n) = 0 in oscillator basis, ensures the total in the σ direction vanishes, consistent with the closed string's translational invariance. In the light-cone gauge, reparametrization invariance is fixed by setting X^+ = τ (worldsheet time) and eliminating longitudinal modes X^-, while ghosts associated with reparametrization are absent due to the gauge choice; only transverse oscillators for both sectors remain dynamical. The free theory is governed by a quadratic Hamiltonian H_0 = \int dp^+ \sum_{\tilde{N}} E_{\tilde{N}} A^\dagger_{\tilde{N}}(p^+) A_{\tilde{N}}(p^+), where the string field operators A_{\tilde{N}} create states at level \tilde{N} = (N_L, N_R) satisfying level-matching, and the energy E_{\tilde{N}} = p^- = (1/p^+) (M^2 + 2(N_L + N_R - 2)), with M^2 the mass-squared from zero modes. The vacuum is defined by the absence of excitations in longitudinal directions, annihilated by transverse lowering operators. Interactions are introduced via cubic (and higher) vertices that sew left movers among incoming/outgoing strings separately from right movers, as in the three-string vertex |V(1,2,3)⟩ \propto \delta(p^+_1 + p^+_2 + p^+_3) \exp\left[ \sum V(\alpha^\dagger_1, \alpha^\dagger_2, \alpha^\dagger_3) + \tilde{V}(\bar{\alpha}^\dagger_1, \bar{\alpha}^\dagger_2, \bar{\alpha}^\dagger_3) \right], where V and \tilde{V} are bilinear forms ensuring momentum conservation in transverse directions. This framework faces challenges in perturbative calculations, particularly with modular invariance, as the light-cone gauge does not manifestly preserve the full SL(2,Z) symmetry of the ; off-shell continuations and careful integration over moduli are required to recover consistent higher-genus amplitudes. Historically, the light-cone closed string field theory was developed after the open string version, with foundational contributions from Mandelstam's interacting string picture in 1973 and Cremmer-Gervais's extension to amplitudes in 1974, enabling applications to the expansion where tree-level exchanges correspond to genus-zero surfaces and loops to higher genera.

Covariant Bosonic Open String Field Theories

Free Formulation

In the covariant quantization of the free bosonic open string field theory, the string field Ψ\Psi is defined as a Grassmann-even functional taking values in the full Hilbert space of the matter and ghost sectors, constructed from the Fock space generated by the transverse oscillators αnμ\alpha^\mu_n for μ=0,…,25\mu = 0, \dots, 25 and the ghost oscillators bm,cnb_m, c_n. This formulation incorporates the BRST symmetry directly into the field content, avoiding the need for subsidiary conditions on physical states. The BRST charge operator QBQ_B, which generates the symmetry, is expressed in terms of the Virasoro generators and ghost modes as QB=∑n∈Zc−nLn(m)−12∑m,n∈Z(m−n):c−mc−nbm+n:,Q_B = \sum_{n \in \mathbb{Z}} c_{-n} L_n^{(m)} - \frac{1}{2} \sum_{m,n \in \mathbb{Z}} (m-n) : c_{-m} c_{-n} b_{m+n} : , where Ln(m)L_n^{(m)} are the matter Virasoro generators, the colons denote normal ordering, and the sums run over integer modes for the open string. The nilpotency QB2=0Q_B^2 = 0 holds in the critical dimension D=26D=26, ensuring consistency of the theory without central charge anomalies. The free action for the string field is given by the quadratic functional S=12⟨Ψ∣QBΨ⟩,S = \frac{1}{2} \langle \Psi | Q_B \Psi \rangle , where the inner product ⟨⋅∣⋅⟩\langle \cdot | \cdot \rangle is the bilinear pairing defined via the BPZ (Belavin-Polyakov-Zamolodchikov) inner product on the space of states, integrated over the with appropriate half-density weight to account for the reparametrization invariance. This action is gauge invariant under the residual BRST transformations δΨ=QBΛ\delta \Psi = Q_B \Lambda, where Λ\Lambda is an arbitrary string field of ghost number 1, reflecting the redundancy in the description of physical configurations. The derived from varying are simply QBΨ=0Q_B \Psi = 0, whose solutions span the BRST cohomology groups representing the physical of the theory in D=26D=26 dimensions. To construct explicit solutions, the string field Ψ\Psi is expanded in a basis of states, comprising primary fields of ghost number 1 (such as the vertex operator c1eik⋅Xc_1 e^{i k \cdot X}) and their obtained by acting with and oscillators, ensuring the expansion respects the level-matching and ghost number constraints. This free formulation provides the foundation for extending to interacting theories by adding higher-order terms while preserving the BRST structure.

Witten's Cubic Theory

Witten's cubic theory provides a covariant formulation of interacting open bosonic string field theory, extending the free BRST action by incorporating a nonlinear cubic interaction term that encodes string interactions via a noncommutative star algebra. The full classical action is given by S=−12⟨Ψ∣Q∣Ψ⟩−13⟨Ψ∣Ψ∗Ψ⟩,S = -\frac{1}{2} \langle \Psi | Q | \Psi \rangle - \frac{1}{3} \langle \Psi | \Psi * \Psi \rangle, where Ψ\Psi is the string field, QQ is the BRST charge, ⟨⋅∣⋅⟩\langle \cdot | \cdot \rangle denotes the bilinear BPZ inner product on the space of states, and ∗* represents the star product. This action is gauge invariant under infinitesimal transformations of the form δΨ=QΛ+Ψ∗Λ−Λ∗Ψ\delta \Psi = Q \Lambda + \Psi * \Lambda - \Lambda * \Psi, where Λ\Lambda is a ghost number zero string field. The star product ∗* is a noncommutative, associative multiplication on the string field space, defined geometrically through the overlap of string worldsheets and analytically via (CFT) correlators on the upper half-plane. Specifically, for string fields AA, BB, and CC, the product satisfies ⟨A∣B∗C⟩=∫dμ ⟨A,B,C⟩\langle A | B * C \rangle = \int d\mu \, \langle A, B, C \rangle, where ⟨A,B,C⟩\langle A, B, C \rangle is the three-point CFT correlation function of the fields inserted at specific points on a disk with boundary conditions corresponding to open string endpoints, and dμd\mu is a measure ensuring associativity derived from the (OPE) in the underlying CFT. This construction interprets string interactions as algebraic operations in a , with the midpoint gluing of half-strings providing the intuitive picture of the three-string vertex. The string field Ψ\Psi resides in the large of the boundary (BCFT) describing the open string endpoints, specifically the subspace of states with ghost number one and vanishing anti-ghost zero mode, ensuring the correct grading for the BPZ inner product and BRST . This space includes off-shell extensions of physical open string states, allowing for a field-theoretic treatment of interactions while preserving the structure over the matter and ghost sectors. In , the action yields Feynman rules analogous to those in : the kinetic term provides the ∼1/(L0−a)\sim 1/(L_0 - a), where L0L_0 is the Virasoro generator and aa is the intercept, while the cubic term defines the three-string vertex via the star product, with higher-point interactions generated through associative multiplication. These rules reproduce the first-quantized open string elements at tree level through minimal-area metrics on the , summing over gluings that correspond to the standard Polyakov path integral. Despite its successes, Witten's theory exhibits dependence on the specific underlying CFT for the definition of the star product and correlators, limiting its direct applicability to backgrounds beyond the perturbative without redefinition. Additionally, at higher , quantum corrections introduce anomalies, such as ghost number nonconservation from gravitational contributions on the , complicating the consistency of loop amplitudes and requiring careful regularization.

Gauge Invariance and Fixing

In Witten's cubic open string field theory for the bosonic string, the theory possesses a gauge symmetry that generalizes the BRST invariance of the underlying to the second-quantized description. The infinitesimal gauge transformations are given by δΨ=QΛ+Ψ⋆Λ−Λ⋆Ψ,\delta \Psi = Q \Lambda + \Psi \star \Lambda - \Lambda \star \Psi, where Ψ\Psi is the string field of ghost number 1 in the full tensor product of matter and ghost Hilbert spaces, QQ is the BRST charge operator satisfying Q2=0Q^2 = 0, Λ\Lambda is a gauge parameter string field of ghost number 0, and ⋆\star denotes the non-commutative star product defined on the space of string fields. These transformations close under the BRST algebra, ensuring that the variation of a gauge-transformed field under another transformation remains a valid symmetry operation, consistent with the nilpotency of QQ and the associativity of the star product. The cubic action S=−12⟨Ψ∣QΨ⟩−13⟨Ψ∣Ψ⋆Ψ⟩S = -\frac{1}{2} \langle \Psi | Q \Psi \rangle - \frac{1}{3} \langle \Psi | \Psi \star \Psi \rangle, where ⟨⋅∣⋅⟩\langle \cdot | \cdot \rangle is the bilinear BPZ inner product, is invariant under these gauge transformations. The variation δS\delta S vanishes due to the cyclicity of the star product in the inner product, ⟨A∣B⋆C⟩=⟨B⋆C∣A⟩=⟨C∣A⋆B⟩\langle A | B \star C \rangle = \langle B \star C | A \rangle = \langle C | A \star B \rangle (up to total derivatives that integrate to zero), combined with the properties Q2=0Q^2 = 0 and the odd grading of QQ under the star algebra. This invariance ensures that the theory describes only physical , free from unphysical artifacts of the gauge choice. To quantize the theory and define a perturbative expansion, a gauge must be fixed to eliminate the redundancies introduced by the infinite-dimensional gauge symmetry. The Siegel gauge, defined by the condition b0∣Ψ⟩=0b_0 |\Psi \rangle = 0 where b0b_0 is the zero-mode of the antighost field b(z)b(z), is commonly employed as it simplifies the kinetic operator while preserving key symmetries. In this gauge, the takes the form of the inverse BRST operator, effectively 1/Q1/Q in momentum space (modulo ghost insertions c0b0c_0 b_0), facilitating computations analogous to those in ordinary field theories. Recent work has explored the light-cone gauge within the covariant formulation, providing additional insights into procedures. The gauge transformations generate orbits in the space of string fields, rendering configurations differing by a gauge transformation physically equivalent. The physical subspace corresponds to the BRST cohomology at ghost number 1, consisting of equivalence classes [Ψ][\Psi] where Ψ\Psi satisfies QΨ=0Q \Psi = 0 modulo exact terms Ψ=QΛ\Psi = Q \Lambda for some Λ\Lambda. This cohomology precisely captures the gauge-invariant observables, such as on-shell massless vector fields and higher-spin states, aligning with the spectrum of open string theory. Unlike the abelian case, the non-abelian structure of the gauge symmetry in Witten's theory—arising from the non-commutative star algebra—complicates , as the infinite tower of gauge-for-gauge redundancies requires careful treatment of ghost fields and can lead to issues like Gribov copies or obstructions not present in finite-dimensional Yang-Mills theories.

Equations of Motion

The in Witten's cubic open string field theory are derived by varying with respect to the string field Ψ\Psi, which resides in the space of off-shell states in the full of the open bosonic . The action takes the form S=−12⟨Ψ∣QΨ⟩−13⟨Ψ∣Ψ⋆Ψ⟩,S = -\frac{1}{2} \langle \Psi | Q \Psi \rangle - \frac{1}{3} \langle \Psi | \Psi \star \Psi \rangle, where QQ is the BRST charge and ⋆\star denotes the star product defined via the gluing of string configurations along the midpoints. Varying the action yields ⟨δΨ∣QΨ+Ψ⋆Ψ⟩=0\langle \delta \Psi | Q \Psi + \Psi \star \Psi \rangle = 0, implying the dynamical equation QΨ+Ψ⋆Ψ=0.\labeleq:eom(1)Q \Psi + \Psi \star \Psi = 0. \tag{1}\label{eq:eom} This equation, first introduced by Witten, governs the dynamics of the open bosonic . The structure of \eqref{eq:eom} naturally factorizes into a free kinetic term [Q](/page/Q)Ψ[Q](/page/Q) \Psi and a nonlinear interaction term Ψ⋆Ψ\Psi \star \Psi, mirroring the separation in ordinary field theories but with infinite . Perturbative solutions can be expanded as a in the string coupling, recovering the familiar tree-level string amplitudes through the factorization of the star product. However, the equation also admits exact non-perturbative solutions, such as the tachyon vacuum representing the condensation of the open string and the disappearance of the unstable . These solutions describe a stable vacuum with no open string excitations, consistent with expectations from boundary (CFT). In level-truncated approximations, where the string field is restricted to states below a fixed level, numerical solutions reveal effective tachyon potentials approaching the expected exponential form V(T)∝−τe−T2V(T) \propto -\tau e^{-T^2}, with intermediate approximations exhibiting butterfly and heart-like shapes that refine toward the true vacuum as truncation levels increase. The equations of motion ensure consistency with fundamental principles in the covariant formulation, implying unitarity through the nilpotency of QQ (Q2=0Q^2 = 0) and the hermiticity of the BRST operator, which projects physical states onto the cohomology. Causality is preserved, as demonstrated by the vanishing of retarded commutators outside the light cone in the free theory, extending to interactions via the associative star algebra. Gauge invariance, which leaves the action unchanged, further supports this consistency by allowing the elimination of unphysical degrees of freedom without altering the dynamics. Moreover, solutions to \eqref{eq:eom} correspond to marginal deformations in the space of two-dimensional CFTs on the worldsheet, where the string field acts as a generator of conformal symmetry breaking and restoration, linking the field-theoretic description to the underlying conformal structure.

Quantization Procedures

Quantization in Witten's cubic open bosonic string field theory can be approached through both path integral and operator formalisms, each addressing the challenges posed by the infinite-dimensional nature of the string field, gauge symmetries, and the underlying structure. The defines the partition function as Z=∫DΨ exp⁡(iS[Ψ])Z = \int \mathcal{D}\Psi \, \exp(i S[\Psi]), where S[Ψ]S[\Psi] is the cubic action and the functional measure DΨ\mathcal{D}\Psi is constructed to respect the Weyl invariance of the matter-ghost at central charge c=26c = 26. This ensures the absence of conformal anomalies in on-shell amplitudes, with the measure derived from the Zamolodchikov metric on the space of primary fields. Gauge invariance, stemming from the BRST symmetry δΨ=QΛ\delta \Psi = Q \Lambda, is handled using the Faddeev-Popov procedure within the Batalin-Vilkovisky (BV) formalism, which introduces antifields and satisfies the (S,S)=0(S, S) = 0 to account for the open algebra of gauge transformations. In the operator formalism, the field ∣Ψ⟩|\Psi\rangle is represented as a state in the full of the and b,cb,c sectors with total ghost number 1 and Grassmann-even, ensuring it lies in the large . Physical states correspond to elements of the H∗(Q)H^* (Q), where QQ is the BRST charge Q=∮dz2πi(cTm+cTgh)Q = \oint \frac{dz}{2\pi i} (c T_m + c T_{gh}), with TmT_m and TghT_{gh} the and stress tensors; cohomology classes at ghost number 1 describe massless gauge fields and higher excitations consistent with the open . The non-commutative star product ⋆\star and BPZ inner product are defined via three-point conformal correlators on the upper half-plane, regularized through of oscillators to subtract divergences from the continuous of the Hamiltonian L0L_0, with the midpoint insertion ensuring associativity up to gauge transformations. , typically the gauge b0∣Ψ⟩=0b_0 |\Psi\rangle = 0, propagates the Faddeev-Popov ghosts into the action, yielding a quadratic kinetic term ⟨Ψ∣c0L0∣Ψ⟩\langle \Psi | c_0 L_0 | \Psi \rangle and Feynman rules for . At loop level, computing amplitudes requires careful treatment of the sector and conformal in the star algebra, where multiloop diagrams arise from iterated star products and propagators. Conformal anomalies are avoided on-shell by the BRST condition, but off-shell extensions encounter issues with the non-local star product and number conservation, leading to potential divergences in higher-genus surfaces unless regulated by the full CFT correlators. In bosonic theory, the absence of superghosts simplifies this compared to superstring cases, but anomalies in the can appear if the central charge deviates from 26, emphasizing the role of the . An practical approximation for quantization is the level truncation method, which projects the infinite-dimensional string field onto a finite-dimensional subspace spanned by basis states up to a maximum oscillator level LL for fields and interactions up to level I>LI > L, solving the truncated numerically. This scheme converges rapidly for low-energy physics, as demonstrated in studies of condensation where the vacuum energy approaches the D25- tension TD25=12π2g2T_{D25} = \frac{1}{2\pi^2 g^2} (in units α′=1\alpha' = 1, with gg the open string coupling) with errors below 0.1% at level (10,20), providing evidence for the stability of the vacuum and the vanishing of the tension post-condensation. Unitarity of the theory follows from the operator formalism and BRST structure, with elements matching those of the first-quantized open via the KLT relations linking open and closed amplitudes, ensuring positive probabilities and no ghosts in physical subspaces; direct proofs confirm that the inner product preserves unitarity order by order in .

Supersymmetric Open String Field Theories

Modified Cubic Superstring Field Theory

The modified cubic superstring field theory extends Witten's cubic open string field theory to the supersymmetric case by incorporating fermionic and addressing the challenges posed by the picture number in the superghost sector. The superstring field Ψ\Psi resides in the Neveu-Schwarz (NS) sector of the open superstring , carrying ghost number 1 and picture number -1, which ensures compatibility with the BRST operator QQ at picture -2 for the kinetic term. The Ramond (R) sector is included indirectly through the action of picture-changing operators, avoiding the need for separate fields at picture -1/2 while maintaining GSO projection via appropriate Chan-Paton factors and parity assignments. This formulation adjusts for superconformal by defining interactions on super-Riemann surfaces, where the supercoodinates account for the fermionic . The action takes the form S=−12⟨Ψ∣QΨ⟩−13⟨Ψ∣γ(Ψ⋆Ψ)⟩,S = -\frac{1}{2} \langle \Psi | Q \Psi \rangle - \frac{1}{3} \langle \Psi | \gamma (\Psi \star \Psi) \rangle, where ⋆\star denotes the star product of fields, and γ\gamma is the picture-changing operator {Q,ξ}\{Q, \xi\} (with ξ\xi the inverse picture-changing field) inserted at the of the interaction to raise the total picture number by +1 and preserve conservation in the cubic vertex. This modification handles the fermionic statistics by anticommuting the fields in the interaction term, distinguishing it from the purely bosonic case. The cubic vertex is constructed using conformal mapping to the upper half-plane with supercoodinates, ensuring superconformal invariance and reproducing tree-level amplitudes in the NS sector upon . Gauge invariance is achieved under transformations δΨ=QΛ+γ(Ψ⋆Λ±Λ⋆Ψ),\delta \Psi = Q \Lambda + \gamma (\Psi \star \Lambda \pm \Lambda \star \Psi), where Λ\Lambda is a ghost number 0, picture -1 field, and the ±\pm sign reflects the Grassmann parity for NS fields. This structure enforces the correct GSO projection by projecting onto even or odd combinations as needed for the . However, subtleties arise with picture number conservation due to the non-invertibility of γ\gamma at certain points, leading to potential singularities in higher-order interactions and ill-defined finite gauge transformations in the R sector. Despite these issues, the theory achieves partial success in perturbative calculations, correctly deriving the low-energy (e.g., the Maxwell term for gauge fields) and approximating the potential up to level truncation, though full non-perturbative solutions remain challenging.

Berkovits Non-Polynomial Superstring Field Theory

The Berkovits non-polynomial superstring field theory provides a covariant formulation for open superstring field theory that incorporates both the Neveu-Schwarz (NS) and Ramond (R) sectors using the pure spinor formalism, thereby avoiding the picture-changing issues inherent in earlier cubic formulations. The string field Ψ is defined as a state in the large of the pure spinor variables, with ghost number 0 and picture number 0 for the leading component Φ_{(0,0)}, and it takes the form of superfields that encode the full supersymmetric spectrum without separate treatment of the sectors. The pure spinor variables consist of bosonic ghosts λ^α satisfying the pure spinor constraint λ γ^m λ = 0, along with their conjugate momenta, allowing for a manifestly spacetime supersymmetric description. The action S[Ψ] is non-polynomial in the string field and is constructed to be gauge invariant under the pure spinor BRST symmetry. In the NS sector, it takes the form S=12∫[G−1(QG)G−1(η0G)−∫01dt G^−1∂tG^ {G^−1(QG^),G^−1(η0G^)}],S = \frac{1}{2} \int \left[ G^{-1} (Q G) G^{-1} (\eta_0 G) - \int_0^1 dt \, \hat{G}^{-1} \partial_t \hat{G} \, \{ \hat{G}^{-1} (Q \hat{G}), \hat{G}^{-1} (\eta_0 \hat{G}) \} \right], where G = e^{\Phi_{(0,0)}}, \hat{G} = e^{t \Phi_{(0,0)}}, Q is the BRST operator, and η_0 is the zero mode of the bosonic ghost. This expression arises from exponentiating the string field to generate all orders in interactions via the wedge product, ensuring consistency in the interacting theory. The BRST operator in the pure spinor formalism is Q = \oint \frac{dz}{2\pi i} \lambda^\alpha d_\alpha, where d_α incorporates the fermionic and ghost contributions, and it satisfies Q^2 = 0 due to the pure spinor constraint. The wedge product * between string fields is defined through Witten's star product adapted to the pure spinor space, facilitating the multi-string interactions in the exponential structure. This formulation offers several key advantages: it realizes manifest spacetime supersymmetry at the level of the action, eliminates the need for picture-changing operators by working in the large , and provides an exact description without contact-term divergences. Gauge invariance is maintained under transformations δ e^V = Q(Λ) e^V + e^V \tilde{Q}(\tilde{Λ}), where V is related to the string field, and the pure spinor BRST symmetry enables systematic via the Batalin-Vilkovisky formalism. Exact solutions, such as those for condensation on non-BPS D-branes and marginal deformations at higher mass levels, have been constructed, illustrating the theory's utility for studies.

Other Covariant Formulations

One prominent alternative covariant formulation of open superstring field theory is the WZW-like approach, which constructs the action in a manner analogous to the Wess-Zumino-Witten model on a supergroup, leveraging the structure of affine Lie algebras for current interactions. In this framework, the string field resides in the , and the action incorporates both Neveu-Schwarz and Ramond sectors through a of higher-form potentials that encode the symmetries of the superconformal algebra. The formulation ensures gauge invariance under BRST transformations and picture-changing operations, with the WZW term arising from the integration over the supergroup manifold, providing a geometric interpretation of string interactions. Hybrid formulations extend covariant open superstring field theory, achieving manifest while preserving Lorentz invariance. These approaches, developed in the hybrid formalism, combine with by treating the string fields as sections of bundles over , allowing for consistent inclusion of both open and closed sectors without auxiliary fields. For instance, the action is built using superfields that couple open string to closed string moduli, facilitating computations of dynamics and in a unified manner. Non-commutative geometry provides another covariant avenue, where the star product deforms the in superstring field theory, capturing effects from B-field backgrounds on D-branes. This approach reformulates the string field action using a Moyal-like star product in the deformed target space, enabling background-independent descriptions of non-commutative solitons and gauge theories on branes. The formulation aligns superstring interactions with non-commutative Yang-Mills theories, where the star product ensures associativity and encodes higher-order corrections from string scattering. In the , level-truncation methods emerged as practical variants for solving covariant superstring field equations on systems, approximating solutions by truncating the at fixed mass levels. These techniques, applied to cubic superstring field theory, yield accurate vacuum energies for condensation and kink solutions representing lower-dimensional branes, with convergence improving at higher truncation levels. Compared to the cubic and non-polynomial formulations building on Witten and Berkovits, these alternatives excel in handling the Ramond sector through natural incorporation of fermionic currents and supergroup symmetries, avoiding ad hoc picture-changing insertions while maintaining full covariance.

Covariant Closed String Field Theories

Bosonic Closed Strings

The covariant formulation of string field theory for closed bosonic strings addresses the interactions of closed strings in 26 spacetime dimensions, building on the structure of open string field theory by incorporating left- and right-moving sectors. The closed string field Φ\Phi is defined as a state in the tensor product of two copies of the open string Hilbert space, HL⊗HR\mathcal{H}_L \otimes \mathcal{H}_R, where HL\mathcal{H}_L and HR\mathcal{H}_R represent the left- and right-moving Fock spaces, respectively, including both matter and ghost sectors. This construction ensures the natural incorporation of the closed string topology, with Φ\Phi carrying a total ghost number of 2 to facilitate BRST invariance. Additionally, Φ\Phi satisfies level-matching and ghost-number subsidiary conditions: (L0−Lˉ0)Φ=0(L_0 - \bar{L}_0) \Phi = 0 and (b0−bˉ0)Φ=0(b_0 - \bar{b}_0) \Phi = 0, where L0L_0 and Lˉ0\bar{L}_0 are the left- and right-moving Virasoro generators, and b0b_0, bˉ0\bar{b}_0 are the corresponding antighost zero modes. The action for this theory, proposed by Zwiebach, is a non-polynomial functional designed to capture all orders in the string coupling constant while satisfying the Batalin-Vilkovisky (BV) master equation for quantum consistency at the classical level: S[Φ]=12⟨Φ∣QΦ⟩+∑n=3∞1n!⟨Φ∗n⟩n,S[\Phi] = \frac{1}{2} \langle \Phi | Q \Phi \rangle + \sum_{n=3}^\infty \frac{1}{n!} \langle \Phi^{*n} \rangle_n, where QQ is the BRST charge, ⟨⋅⟩n\langle \cdot \rangle_n denotes the nn-point correlation function on the appropriate Riemann surface, and ∗* is the closed string star product. The kinetic term 12⟨Φ∣QΦ⟩\frac{1}{2} \langle \Phi | Q \Phi \rangle governs free propagation, while the interaction terms begin with the cubic contribution 13⟨Φ∗Φ∗Φ⟩3\frac{1}{3} \langle \Phi * \Phi * \Phi \rangle_3 on the three-punctured sphere and extend to higher-point vertices ⟨Φ∗n⟩n\langle \Phi^{*n} \rangle_n on genus-zero surfaces with nn punctures. The star product ∗* is modified from the open string case to enforce level matching and closed string geometry: it is constructed via conformal sewing of Riemann surfaces using minimal area metrics in the moduli space and closed geodesics of length 2π2\pi, ensuring the product maps back to the tensor product space HL⊗HR\mathcal{H}_L \otimes \mathcal{H}_R. These vertices satisfy recursion relations derived from the L∞L_\infty algebra structure, allowing systematic computation. A quartic term, for instance, arises as 14!⟨Φ∗Φ∗Φ∗Φ⟩4\frac{1}{4!} \langle \Phi * \Phi * \Phi * \Phi \rangle_4, but higher-order terms become increasingly complex due to the growing dimensionality of the moduli space. Gauge invariance is a cornerstone of the theory, protecting against overcounting in the infinite-dimensional field space. The gauge transformations are given by δΦ=QΛ+Φ∗Λ1+Λ2∗Φ,\delta \Phi = Q \Lambda + \Phi * \Lambda_1 + \Lambda_2 * \Phi, where Λ=ΛL⊗I+I⊗ΛR\Lambda = \Lambda_L \otimes I + I \otimes \Lambda_R parameterizes the deformations, with ΛL\Lambda_L and ΛR\Lambda_R being independent ghost fields of ghost number 1 in the respective left and right open Hilbert spaces. Here, Λ1\Lambda_1 and Λ2\Lambda_2 correspond to the left- and right-acting components, ensuring the transformation respects the tensor product structure and level matching. This double gauge structure reflects the independent diffeomorphism invariances of the left and right movers, and the full symmetry closes under the nonlinear algebra defined by the star product, consistent with the L∞L_\infty relations. Fixing the gauge, such as via the modified Siegel gauge b0+Φ=bˉ0+Φ=0b_0^+ \Phi = \bar{b}_0^+ \Phi = 0, simplifies computations but requires careful handling of the Fadeev-Popov determinant. Despite these advances, the theory encounters significant challenges in its interacting sector. While the cubic and low-order terms are well-defined, higher-point interactions are limited by the intricate construction of vertices, which becomes computationally prohibitive beyond tree level due to the need for explicit minimal area prescriptions. Moreover, no fully consistent quantum formulation of closed bosonic string field theory exists, primarily owing to anomalies that arise in the measure of the path integral at higher genera, obstructing a complete loop expansion. These issues stem from subtleties in the BV quantization, where the antighost insertions and conformal anomaly cancellations in D=26D=26 do not fully extend to off-shell quantum corrections. Perturbatively, however, the theory successfully reproduces the Polyakov path integral at tree level: the string field action's Feynman diagrams, constructed from the local vertices, integrate over the moduli space of punctured spheres to yield the exact on-shell scattering amplitudes of the first-quantized bosonic string, confirming equivalence in the weak-coupling regime.

Superstring Closed Strings

Covariant formulations of closed superstring field theory extend the open superstring framework to incorporate both left- and right-moving modes, representing the super closed string field as a of independent left and right movers. This structure accommodates the Neveu-Schwarz/Neveu-Schwarz (NS-NS), Neveu-Schwarz/Ramond (NS-R), Ramond/Neveu-Schwarz (R-NS), and Ramond/Ramond (R-R) sectors of type II superstrings, with picture numbers assigned to handle the fermionic and ghost contributions in each sector—typically picture number -1 for NS sectors and -1/2 for R sectors to ensure conformal invariance. The action for closed superstring field theory builds on attempts to double the open superstring formulation, such as the non-polynomial action proposed by Berkovits for open strings, adapted via a to capture closed string interactions while preserving supersymmetry. Non-polynomial terms are essential for consistency, as cubic actions alone fail to reproduce the full and interactions due to the need for higher-order vertices in the Ramond sectors; recent constructions yield Lorentz- and gauge-invariant effective actions that satisfy the classical Batalin-Vilkovisky (BV) master equation, equivalent to type II at the classical level. Gauge invariance relies on an extended BRST operator that incorporates superconformal ghosts for both left and right movers, ensuring the correct representation of transformations and the GSO projection. Level-matching conditions between left and right oscillators are imposed to maintain the closed string constraint L0−Lˉ0=0L_0 - \bar{L}_0 = 0, which is more restrictive in the superstring context than in the purely bosonic case, where no fermionic sectors complicate the . Challenges in closed superstring field theory are amplified compared to the bosonic version due to the GSO projection's role in eliminating unphysical states across fermionic sectors, leading to difficulties in defining consistent interactions without introducing anomalies or incomplete spectra; nonetheless, partial successes have been achieved in type II theories through L∞L_\infty-structured actions that reproduce tree-level S-matrices matching first-quantized results. These formulations facilitate explorations of open-closed string duality in superstring theory, enabling the consistent coupling of closed strings to open string sectors via D-branes and orientifold planes, with the effective action satisfying quantum BV equations to support perturbative computations.

Covariant Heterotic String Field Theories

Formulation and Gauge Structure

Covariant heterotic string field theory addresses the intrinsic asymmetry of heterotic strings, where the left-moving sector is supersymmetric with N=1 worldsheet supersymmetry and the right-moving sector is purely bosonic (following conventions common in SFT literature; note that left/right labels vary across references), all in the critical dimension D=10. The fundamental object is the string field Ψ, constructed as a tensor product Ψ = Ψ_left ⊗ Φ_right, with Ψ_left belonging to the Neveu-Schwarz and Ramond sectors of the superstring Hilbert space (ghost number 2, appropriate picture numbers) and Φ_right to the bosonic string Hilbert space, subject to constraints like b_0^- Φ_right = 0 and L_0^- Φ_right = 0 to ensure level matching. This structure captures the distinct conformal field theories for each sector, with the left sector incorporating fermionic partners and the right sector generating gauge symmetries upon compactification. The BRST operators are defined separately for the two sectors: Q_left acts on the supersymmetric left movers as the N=1 super BRST charge, while Q_right is the standard bosonic BRST charge for the right movers, both satisfying nilpotency Q_left^2 = 0 and Q_right^2 = 0. The classical action takes the simple quadratic form S = \frac{1}{2} \langle \Psi | Q | \Psi \rangle, where Q = Q_left + Q_right and the inner product \langle \cdot | \cdot \rangle is the bilinear pairing from the BRST-invariant vacuum, often using the BPZ inner product in the Siegel gauge. This action is gauge invariant under transformations of the form \delta \Psi = Q \Lambda for infinitesimal gauge parameters \Lambda in the appropriate ghost sector, with higher-order nonlinear terms such as \delta \Psi = Q \Lambda + [\Psi, Q \Lambda] + \mathcal{O}(\kappa^2) ensuring full consistency when interactions are included. A key feature of the left supersymmetric sector is the presence of an anomalous U(1) symmetry associated with the superghost current, which shifts the picture number and requires careful treatment to maintain , often via picture-changing operators or constraints on the . To resolve issues with auxiliary fields and ghosts-for-ghosts in the super sector, formulations such as the non-polynomial approach in the RNS framework reformulate the theory while preserving the separate Q_left structure. The achieves consistency through a structure adapted for heterotic strings, where the full action includes higher-order L_\infty products, ensuring gauge invariance and reproducing known scattering amplitudes in the open heterotic string limit. This basis extends naturally to the heterotic case by factorizing the left and right contributions.

Interactions and Quantization

In covariant heterotic string field theory, the interacting action extends the free quadratic term with higher-order interactions that incorporate the non-trivial structure of the left-moving supersymmetric and right-moving bosonic sectors. A key is the non-polynomial action in the style of Berkovits, expanded as S=2α′∑n=2∞κn−2SnS = \frac{2}{\alpha'} \sum_{n=2}^\infty \kappa^{n-2} S_n, with the cubic term S3=13!⟨ηV,[V,QV]⟩S_3 = \frac{1}{3!} \langle \eta V, [V, Q V] \rangle and higher terms ensuring consistency without picture-changing operators. This embeds the SO(32) or E8×E8E_8 \times E_8 gauge groups through the left-moving sector, where the gauge bosons arise from the bosonic string coordinates, while the right-moving superghosts handle the supersymmetry. Quantization of heterotic string field theory proceeds via the path integral formalism, treating the string field as a field in a second-quantized theory, but with a mixed measure reflecting the asymmetric left-right structure: the left-moving supersymmetric sector incorporates fermionic integration over superghosts, while the right-moving bosonic sector uses a standard measure. This approach ensures consistency at one-loop level, reproducing known heterotic amplitudes without anomalies in the massless sector. The heterotic Green-Schwarz mechanism, which cancels gauge and gravitational anomalies through a shift in the BB-field, appears in the low-energy effective action of , with the required Chern-Simons terms modifying the anomaly polynomial. This resolution confirms the consistency of the theory beyond tree level, aligning with the one-loop torus diagram contributions in the first-quantized formulation. Applications of heterotic string field theory extend to non-perturbative regimes, providing a framework to explore strong-coupling limits that connect to heterotic via dualities, such as the identification of heterotic five-branes with M-theory orbifolds. It also facilitates studies of with type I string theory, where the SO(32) heterotic sector maps to open string configurations, offering insights into dynamics and orientifold projections. Numerically, level truncation approximates solutions by restricting the string field to components below a fixed mass level, enabling iterative computations of vacua such as tachyon-free configurations in heterotic backgrounds; convergence is observed at low truncation levels for the energy spectrum, mirroring successes in bosonic and superstring cases.

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