Bosonic sectorized strings and the (DF)2 theory

In this work, we investigate the bosonic chiral string in the sectorized inter- pretation, computing its spectrum, kinetic action and 3-point amplitudes. As expected, the bosonic ambitwistor string is recovered in the tensionless limit. We also consider an extension of the bosonic model with current algebras. In that case, we compute the effective action and show that it is essentially the same as the action of the mass-deformed (DF )2 theory found by Johansson and Nohle. Aspects which might seem somewhat contrived in the original construction — such as the inclusion of a scalar transforming in some real representation of the gauge group — are shown to follow very naturally from the worldsheet formulation of the theory.


Introduction
When Cachazo, He and Yuan (CHY) found their celebrated formulae for the tree-level scattering amplitudes of massless particles [1,2], it seemed plausible that those expressions could be obtained from some worldsheet model. Indeed, it did not take long for Mason and Skinner to come up with such a model, dubbed ambitwistor strings [3], followed by a manifestly supersymmetric version using the pure spinor formalism [4]. The CHY formulae were later generalized to different theories [5,6] and, again, different ambitwistor strings were proposed as their underlying worldsheet model [7].
By construction, ambitwistor strings are two-dimensional chiral theories which contain no dimensionful parameter. At first, they were thought to stem from an infinite tension limit of ordinary string theory, a belief motivated in part by the fact that the spectrum of the type II version of the model is identical to that of the corresponding supergravity. However, for the bosonic and heterotic versions it is clear that no such procedure is possible, since their spectra do not match their (super)gravity counterparts.
On the other hand, as observed in [8], ambitwistor strings are equivalent to the spinor moving frame formulation of the null superstring -therefore, tensionless. This idea was supported by Siegel in [9] and other similar results followed (e.g. [10]).

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It was then noticed that the spectrum of tensionful chiral strings could contain a finite number of massive states [11], depending on the amount of spacetime supersymmetry. For the type II case, for instance, the physical spectrum is independent of the string tension. In this context, the so-called sectorized string model [12] plays an important role. It was introduced as an alternative to the above-mentioned pure spinor analogue of ambitwistor strings [4], motivated by some inconsistencies in its heterotic version and difficulties in coupling it to the N = 2 supergravity background [13]. As such, it was supposed to be a theory for massless particles only. Nevertheless, it was later shown [14] that the heterotic sectorized model actually contains the N = 1 supergravity states together with a single massive multiplet with the same quantum numbers as the first massive level of the (conventional) open superstring. This is possible thanks to a dimensionful parameter whose existence had been overlooked, since the chiral worldsheet action has no parameters. Moreover, when this parameter is taken to zero, corresponding to a tensionless limit, one recovers the heterotic ambitwistor string.
In this work, we analyze the bosonic incarnation of the sectorized model and show how the theory can be interpreted in terms of two sectors after a particular gauge-fixing is performed. As in the heterotic case, the two sectors emulate the left-and right-moving sectors of the usual string theory, but all worldsheet fields are holomorphic. Using methods similar to the ones used in [15] for the ambitwistor string -which in turn were based on [16] -, we compute its physical spectrum and the correspondent kinetic action. We also analyze the 3-point tree level functions. As expected, the bosonic ambitwistor string is recovered in the tensionless limit.
We then consider an extension of the bosonic model by including current algebras. As a main result of this work, the usual methods are shown to give rise to a worldsheet derivation of the so-called (DF ) 2 + YM theory found by Johansson and Nohle [17]. In particular, the scalar field transforming in some real representation of the gauge group, whose inclusion might seem somewhat contrived in the original construction, appears naturally in the sectorized-string formulation.
Theories whose Lagrangians include a (DF ) 2 -type kinetic term were first introduced as a way of obtaining conformal (super)gravity amplitudes (R 2 gravity, in general) from color-kinematics duality [18], and were shown to admit CHY/ambitwistor representations in [19]. Like R 2 gravity, such theories contain "ghost" states which render them nonunitary. Moreover, in the particular model studied in this paper, a tachyon is also present. It is then natural to ask what physical interest the model might have.
The answer is that scattering amplitudes computed from the (DF ) 2 theories have been recently found to play a crucial role in the double-copy construction of (ordinary) bosonic and heterotic string tree-level amplitudes [20] -see also [21,22]. Indeed, just like open superstring amplitudes with external massless states can be expressed in a basis of integrals with coefficients which are nothing but super-Yang-Mills amplitudes [23], the corresponding bosonic open string amplitudes can also be expressed in the same basis, but with coefficients which come from (DF ) 2 + YM. Note that the presence of the tachyon makes perfect sense in this context.

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This paper is organized as follows. In section 2, we introduce the sectorized description of the bosonic chiral string, having the Polyakov action in first-order form as our starting point. We then investigate the physical spectrum of the model and analyze its tensionless limit. The kinetic part of its effective action and some results on the tree-level threepoint amplitudes are also presented. In section 3, the bosonic model is extended with the inclusion of current algebras, and the effective field theory inferred from the three-point functions is shown to agree with the (DF ) 2 + YM + φ 3 theory of Johansson and Nohle. Finally, we present our conclusions and perspectives in section 4. The appendix includes further details on the CFT of current algebras that are relevant for this work.

The bosonic sectorized string
In this section we will rederive some known results for chiral bosonic strings using the sectorized description, including its physical spectrum and tensionless limit analysis.

The Polyakov action in first-order form
The Polyakov action is given by where T > 0 is the string tension, g ij is the worldsheet metric (with inverse g ij ) and g = det(g ij ), with i, j denoting the usual worldsheet coordinates τ and σ. Spacetime indices m, n, . . . are raised and lowered with the (mostly plus) Minkowski metric η mn . In the first order formulation, one can define a classically equivalent action, given bỹ where e ± denote the Weyl invariant Lagrange multipliers related to the worldsheet metric as Although not manifestly, the actionS P is invariant under worldsheet reparametrizations, generated by The corresponding gauge transformations are given by where c + and c − are local parameters. JHEP01(2020)082

The sectorized interpretation
The quantization of the action (2.2) is straightforward, and the usual conformal gauge is obtained when we choose e ± = 1. We want to discuss, instead, a particular case of the one-parameter (β) family of gauges introduced in [9], which can be cast as For β = 0, the conformal gauge is recovered. We are interested in the singular gauge β → ∞, leading to a chiral worldsheet action. In this limit, e ± = ±1. This singular gauge was proposed in the context of doubled-coordinate field theory in [24]. After a Wick rotation of the worldsheet coordinate τ , the gauge-fixed action can be written as where the gauge parameters c ± have been promoted to anticommuting ghosts with corresponding antighosts b ± . All fields in S are holomorphic and the string tension T is now hidden.
A few comments about the gauge fixing (2.6) are in order. For any finite β, a redefinition of the worldsheet coordinates can always bring the gauge fixed action to the conformal gauge. This is hardly surprising, since the physical model should be gauge independent. This was noted by Siegel in [9], but his construction of the chiral string involved another crucial ingredient related to a change in the boundary conditions of the action. At any rate, adopting the singular gauge (β → ∞) is useful since then the delta functions realizing the scattering equations become explicit.
It was later noticed that the boundary condition leading to Siegel's new propagator for the target space coordinates could in fact be described by the usual string theory in the conformal gauge (β = 0), albeit with a different choice of vacuum [25]. In the ambitwistor context, this alternative vacuum was investigated in [10] (and further in [26]) and also arises naturally from the quantization of the action (2.7). As it turns out, this seems to be the only consistent vacuum in the singular gauge β → ∞. It might look like a contradiction, but the key idea here is precisely that this is a singular gauge which effectively leads to a degenerate worldsheet metric. In other words, the action (2.7) is completely oblivious to the usual conformal gauge in string theory because this gauge choice is not invertible (hence, singular).
In spite of being chiral, the model can be interpreted in terms of two sectors, namely the "+" and the "−", which partially emulate the left and right movers of the usual bosonic string. Each sector has its own characteristic energy-momentum tensor given by

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The sectorization is manifest in the BRST charge Q: Nilpotency of Q requires the number of spacetime dimensions to be d = 26.
Note that the complete energy-momentum tensor is given by and it is BRST exact, since {Q, while the BRST charge is rewritten as 15) and the familiar Virasoro structure emerges. The tensionless limit of Q is now very clear: it is precisely the BRST operator introduced by Mason and Skinner for the bosonic ambitwistor string [3].
We will see, however, that the sectorized description is more advantageous in the cohomology analysis, for it leads to a natural splitting of the vertex operators in the different mass levels.

Physical spectrum
The BRST cohomology at ghost number zero is given by the identity operator. At ghost number one, the cohomology contains only the zero-momentum states mapped to the operators c + P + m and c − P − m . Physical states will be defined as elements of the BRST cohomology with ghost number two and annihilated by the zero mode of b. The latter follows from the usual off-shell condition (b 0 −b 0 ) = 0 on physical states, but adapted to the chiral model. The most general vertex operator with conformal weight zero satisfying these conditions can be written as D, E, F ± and G ± are the X dependent fields. This splitting of the terms appearing in the vertex operator is motivated by their mass-level, as will become clear shortly.
In order to determine the physical degrees of freedom, we will analyze each of the vertices in (2.16) separately. For V 0 , the equations of motion imposed by BRST closedness are given by (2.20) These equations become more transparent if we rewrite them in terms of the fields

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The gauge transformations, with parameters λ m and ω m , are simply It is now easy to identify the field content of the massless sector described by the vertex V 0 : φ corresponds to the dilaton, b mn is the Kalb-Ramond 2-form and g mn is the graviton, satisfying the linearized equation of motion (2.23a).
For the vertices V + and V − , the two sets of equations of motion are very similar to each other and can be displayed collectively as Again, these equations become more transparent after the field redefinitions which imply (using d = 26) that The fields f ± and c m ± are pure gauge, therefore h mn ± contain all the physical degrees of freedom, corresponding to spin 2 fields with m 2 = ±4T .
Tensionless limit. Evidently, in the tensionless limit all the physical states are massless. In fact, if we naively take the T → 0 limit of the vertex (2.16), it may seem that Mason and Skinner's results are recovered [3]. However, the analysis of such limit has to be done more carefully precisely because all the physical states become massless. In other words, the vertices (2.17), (2.18) and (2.19) should mix in the tensionless limit. Therefore, we should find a convenient combination of the fields G mn , H mn ± A m ± , B m ± , C m ± , D, E, F ± and JHEP01(2020)082 G ± in (2.16) such that the tensionless limit preserves the most general form of the vertex operator. The solution is (2.30) Here the notation for the fields was chosen so as to agree with the ambitwistor construction of [15], where it was demonstrated that the free field dynamics associated to the fields above involve higher derivative operators. This result follows naturally from our construction above. For example, we can show that the equation of motion for the fields G mn (1) , G mn (2) and G mn (3) can be obtained using (2.23a) and (2.27), and are given (in the gauge Note that, by substituting (2.31b) into (2.31c), we get an equation involving 3 G mn (1) which, in the tensionless limit, has the same form as the one found in [15]. Of course, this had to be the case since the vertex operator (2.29) preserves its form as T → 0, while the BRST operator reduces to the (bosonic) ambitwistor one, as is evident from (2.15). Indeed, all the other equations of motion can be reproduced in a similar way.

Bosonic kinetic action and 3-point amplitudes
As shown above, G mn (2) and G mn (3) can be seen as auxiliary fields 1 which effectively implement a higher derivative equation of motion for G mn (1) . This behavior can be better understood from another point of view, namely in terms of the effective action of the model and, in particular, its kinetic part.
Indeed, the kinetic terms associated to g mn and h mn ± have opposite signs. Physically, this indicates an instability of the model (ghosts), in agreement with the results of [11]. Such ghosts can usually be described in terms of higher derivative theories and this is precisely what happens here.
Bosonic kinetic action. Inspired by Zwiebach's closed string action [16], the kinetic action for ambitwistor strings was built in [15]. We will use the same prescription for the tensionful model and the kinetic action will be defined by where |V is the state associated to the vertex operator (2.16), obtained from the identity state |0 through the state-operator map and V | its BPZ conjugate. In order to simplify the calculations, we will fix the gauge f ± = c m ± = 0 -cf. equations (2.26) and (2.28) -and use the auxiliary equations of motion in (2.21) to write the vertex operator (2.16) in terms of the fields g mn , b mn , φ and h mn ± . Now, using the usual ghost measure c ± ∂c ± ∂ 2 c ± = 2, it is straightforward to show that the free action can be cast as where S 0 = 2 d 26 x{g mn g mn + ∂ p g mp ∂ q g mq + 2(g + φ)∂ m ∂ n g mn and 36) with g = g mn η mn and h ± = h mn ± η mn . As expected, the free field equations of motion derived from S 0 and S ± precisely reproduce (2.23) and (2.27). The kinetic terms for g mn and h mn ± have opposite signs, consistent with the ghost interpretation.

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3-point amplitudes. The 3-point tree level scattering amplitudes for the bosonic chiral string were obtained in [11]. However, it is instructive to redo this analysis here since our unintegrated vertex operators have a different structure and, in particular, do not give rise to a Koba-Nielsen factor. For higher point amplitudes, we would need integrated vertex operators but their definition is still unknown. It will be convenient to gauge fix the vertex operators in (2.16) and work with momentum eigenstates, such that where G mn , H mn ± are now seen as polarization tensors satisfying k m G mn = k n G mn = k m H mn ± = η mn H mn ± = 0. In order to compute the 3-point amplitudes, we have to evaluate its OPE reduction by contracting all P ± m 's with one another and with the momentum exponentials e ik·X . We also need the ghost 3-point function, which has the usual form (2.38) By virtue of the sectorized description, it is easy to show that the amplitude factorizes into a product of two open string amplitudes (where T −→ −T in the minus sector). With all this in mind, we can compute, for example, the 3-point amplitude involving only massless states. The result is andT nqs is equal to T nqs with the sign of T flipped. The amplitude does not depend on the positions of the vertex operator insertions and is, therefore, SL(2, C) invariant. This result is to some extent expected, since the vertex structure is completely analogous to the ordinary bosonic string and the Koba-Nielsen factors are just 1 for three massless vertices. However, the SL(2, C) invariance can be shown for any 3-point tree level amplitude, even though the Koba-Nielsen factor is always 1 in the chiral model (there are no contractions between the momentum exponentials since the XX OPE is trivial). The amplitudes factorize in the plus and minus sectors, and there is a precise cancelation of the poles and zeros in z ij ≡ z i − z j .

Extension of the sectorized model with current algebras
In this section we will explore the extension of the bosonic sectorized model in a target space with dimension d < 26 and the introduction of current algebras, i.e. a gauge sector.
To the action (2.7), we will add two extra pieces, S + C and S − C , describing two current algebras. The new BRST charge preserves its form in (2.10) but now with

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where T ± C denotes the energy-momentum tensor associated to different group manifolds with central charge For now we will focus on the "−" sector, which contains the tachyonic excitations. The inclusion of the "+" sector, which has an analogous structure, will be discussed in subsection 3.4.
Let us consider an affine Lie algebra associated to some group G, with structure constants f c ab (a, b, . . . = 1 to dim G) and level k. The addition of S − C to the action allows us to define currents J a which are primary conformal fields and satisfy the OPE .
Here the group generators have been orthonormalized such that the metric δ ab corresponds to a Kronecker delta, and we will make no further distinction between upper and lower indices.
The energy-momentum tensor of the algebra can be obtained using the Sugawara construction and is given by where g is the dual Coxeter number, defined through We use the ordering prescription which can be understood as the product of two operators A(z) and B(y) in the limit z → y, with singular terms removed. It is then straightforward to compute the central charge of this model, which is given by The second equality in (3.8) comes from imposing the nilpotency of the BRST operator and constrains the group G and the level k of the current algebra. For example, for a target space with d = 10 one of the solutions is G = SO(32) and k = 1, while for d = 4 we can have G = SU(5) and k = 55, and so on. Further constraints on the group should arise from the analysis of anomalies but this will not be discussed in this work.

Physical spectrum
The BRST cohomology now includes additional states with corresponding vertex operators containing the currents J a , expressed as Here F m a , G m a , S a , F a , G a and ϕ α are target space fields. The index α belongs to a tracelesssymmetric bi-adjoint representation of the group G (see appendix), with dimension J α is a primary conformal weight 2 operator defined as where J (ab) is given by the traceless-symmetric ordered product of two currents, i.e.
and C −1 αab are the inverse of the Clebsch-Gordan coefficients, C αab . The properties of these coefficients will be discussed in the next subsection and in the appendix. Observe that we could have considered also the trace contribution in the vertex, e.g. c + c − T − C ϕ. However, the field ϕ couples only to the vertex V − in subsection 2.3 and does not change the physical content of the model.
The BRST invariance of the vertex V J implies the following equations of motion ( + 4T )ϕ α = 0, (3.14a) Since S a is pure gauge, the physical states described by the vertex (3.10) correspond to a massless vector F m a and two fields with negative mass-squared m 2 = −4T namely the scalar ϕ α and the vector G m a . In parallel to subsection 2.3, we can prepare the vertex V J for the tensionless limit analysis. Considering the redefinitions of the worldsheet ghosts of (2.13), V J can be rewritten as

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Here, the fields A a , A m a , B a and B m a are defined in terms of F m a , G m a , F a and G a as (3.17) with gauge transformations δA m a = ∂ m Λ a and δB m a = T ∂ m Λ a . Their equations of motion follow from (3.14) and are given by Therefore, the physical spectrum can be described in terms of only two fields, ϕ α and A m a . The vector B m a is auxiliary, helping to implement a quartic equation of motion for A m a , which carries the degrees of freedom of both the massless and the massive vector fields, F m a and G m a . Note, in particular, the tensionless limit renders a massless spectrum with equations of motion ϕ α = 2 A m a = 0. As in the bosonic model of section 2, this behavior can be easily observed when analyzing the effective field theory associated to the model, which will be done in subsection 3.3. The first step will be to determine the 3-point amplitudes using the vertex (3.10).

3-point amplitudes
In order to compute the 3-point amplitude we need to provide further details on the current algebra CFT, in particular the OPE's involving the operator J α defined in (3.12) and the properties of the Clebsch-Gordan coefficients C αab . The operator J α satisfies the following OPE's: The first OPE states that J α is a primary operator of conformal dimension 2. The second OPE is connected to the definition of the Clebsch-Gordan coefficients (quadratic pole) and

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the group transformation of J α (simple pole). (T a ) αβ denotes the group generators in the traceless bi-adjoint representation of the group G and satisfy The OPE (3.20c) can be used to define the 2-point and 3-point functions involving only J α 's. Operators of conformal dimension 3 appear in the last line (with numerical coefficients d αβabc , d αβ[ab] and e aαβ ) but they do not contribute to A 3 . J (abc) is the totally symmetric traceless normal ordered product of J a , J b and J c , and J [ab] is the antisymmetric The Clebsch-Gordan coefficients C αab are defined in such a way that Finally, the coefficient d αβγ is defined as Although not manifestly, d αβγ is traceless, i.e. d ααγ = 0, and completely symmetric in the exchange of any pair of indices. The 2-point amplitudes involving the gauge currents can be easily determined through the OPE's (3.4), (3.20b) and (3.20c), and are given by (3.26c)

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The 3-point amplitudes are now straightforward to compute. They can be summarized as As one last step before evaluating (3.19), it will be convenient to fix the gauge degrees of freedom of V J . Using the gauge transformations (3.15), we will choose S a = 0. In this gauge, ∂ m G m a = 0 as a consequence of the equations of motion. We can use the remaining parameter to fix the transversal gauge for the massless vector, such that the vertex is simplified to Using the tree level measure for the ghosts (2.38), the 3-point amplitude (3.19) can be computed to be Observe that A 3 is at most linear in T (F m a − G m a ). If we look at the vertex (3.16), this is easy to understand because the 3-point amplitudes with two or three B m a 's vanish trivially. In principle, 4-point amplitudes can be computed using the results of Siegel in [9]. Currently, however, there is no clear definition of the integrated vertex operators and higher point amplitudes cannot be directly computed from the chiral model. This problem will be dealt with in a separate paper by one of the authors.
In the next subsection we will propose an effective field theory action for the field content of the previous subsection.

Effective field theory: (DF ) 2 + YM
As the main result of this paper, we would like to argue that the effective field theory action corresponding to this extension of the bosonic sectorized model is precisely the action of the (DF ) 2 + YM theory constructed in [17]. Indeed, we have already shown the spectrum to be the same. The action can be decomposed as where S 0 J is the kinetic part of the action and S int J corresponds to the interactions. For the kinetic part, we will proceed like in subsection 2.4. For the interaction part, we will analyze the possible vertices that give rise to the 3-point amplitudes displayed in (3.29). Next, we will require the non-linear gauge invariance of the resulting model in order to finally propose its effective action.

Kinetic action
As stated above, we will define the kinetic action as 31) up to normalization. In order to further simplify the computation, we will consider the algebraic solutions (3.14b) and (3.14c), such that It is then straightforward to show that Note that the kinetic terms of the fields F m a and G m a have opposite sign in S 0 J . Technically, the sign difference can be traced back to the OPE's of P + m and P − m with themselves. As discussed previously, this indicates an instability of the model and we can again reinterpret it in terms of a higher derivative theory. In fact, as we will now show, this behavior is more transparent if we rewrite the action in terms of the vectors A m a and B m a defined in (3.17). The kinetic action can then be cast as Ignoring for now the interaction terms, observe that the equation of motion for B m a is algebraic, given by If we replace this solution back in the action, we obtain This action can be identified with the kinetic part of the (DF ) 2 + YM theory constructed in [17]. Note that the propagator of A m a is given in momentum space by . (3.38) The pole structure of this propagator agrees with the interpretation given after equation (3.18) that A m a effectively describes the massless and the massive vector fields, F m a and G m a .

Cubic vertices and the effective action
As it turns out, the procedure of integrating B m a out can be partially extended to interactions. We say "partially" because in this paper we consider only unintegrated vertex operators, therefore only 3-point tree level amplitudes. We expect this integration to hold for higher point vertices as well.
By looking at A 3 in (3.29), it is easy to show that the 3-point vertices in terms of the vectors A m a and B m a can be schematically expressed as (3.39) The idea now is to analyze the possible gauge invariant interactions that can generate these vertices after integrating out B m a , which is at most linear in the expressions above. The equation of motion for B m a in (3.36) gets modified to where c # and d # are numerical constants and the dots contain other terms necessary to generate the correct gauge transformation for B m a (remember that the onshell 3-point amplitude A 3 was computed using gauge-fixed vertex operators). Taking this into consideration and replacing B m a in the action, we can show that all 3-point vertices come from the operators where F mn a was redefined to be the non-Abelian field strength with coupling constant g, and D m denotes the covariant derivative with respect to the vector A m a . The form of the higher point vertices (4, 5 and 6) is severely restricted by the non-linear gauge invariance of the effective action. Some contributions naturally appear after integrating out B m a and we expect them to combine with the input coming from higher-point amplitudes, which involve integrated vertex operators.
Finally, we propose the effective field theory action of the model to be where g is the coupling constant. This action describes the (DF ) 2 + YM theory of [17]. Moreover, if we include the "+" sector mentioned in the beginning of this section, the effective field theory action describes a more general model with a mirrored set of fields. In particular, if we restrict the gauge symmetry of the "+" sector to be instead a global symmetry, the effective action describes the (DF ) 2 + YM + φ 3 theory. This will be shown next.

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3.4 Including the other gauge sector: (DF ) 2 + YM + φ 3 We will consider for the "+" sector an affine Lie algebra associated to a groupĜ (with structure constantsf C AB ) and levelk. Apart from the central charge constraint (3.3), {Ĝ,k} are independent of {G, k}, from the "−" sector. The new currents,Ĵ A , are completely analogous to the ones discussed there, e.g. they satisfy the OPÊ , (3.43) when conveniently normalized. Here, δ AB is a Kronecker delta. In order to analyze the physical spectrum, we can start with the hatted version of (3.10), defined by It is easy to see that the fields appearing in this vertex operator will satisfy essentially the same equations of motion and gauge transformations as their counterparts in the "−" sector, albeit with one important difference: the replacement T → −T . By going through the same steps as in subsection 3.1, we find that the physical spectrum in this sector contains a "mirror image" of the physical spectrum in the "−" sector, but with opposite mass-squared. In addition, we can build a new type of vertex operator involving currents from both sectors. It has the form where φ aA is a bi-adjoint scalar transforming in the adjoint representation of both gauge groups. BRST closedness implies the equation of motion φ aA = 0, (3.46) whence φ aA is a massless field. Following the same method used in subsection 3.3, the kinetic part of the effective action involving the group indices can be cast as where S 0 J was given in (3.37) and S 0 J is its hatted analogue, and As for the interacting part, it clearly contains the corresponding part in (3.42) and its hatted version. Moreover, note that cubic vertices mixing the fields in V J with those in JHEP01(2020)082 V J can only appear through V φ V J V J , since the three-point functions involving JĴĴ or JJĴ vanish. The non-vanishing three-point functions with insertions of V φ are given by: and following arguments similar to the ones given in the previous subsection, we can write the effective action as where S[A, ϕ] is the right-hand side of (3.42), S[Â,φ] is its hatted version and where the covariant derivative of φ aA with respect to both gauge fields is given by Thus we have found the complete effective action in the gauge sector of the model. Now we would like to make contact with the scalar extension of the (DF ) 2 + YM theory which was introduced by Johansson and Nohle [17]. There, the groupĜ (with indices A, B, . . .) is viewed instead as a global symmetry group. 2 In the present chiral string formulation, we can turn off the gauge fieldÂ m A and the scalarφα, effectively taking S[Â,φ] → 0 and turning the groupĜ into a global symmetry at tree level. Moreover, we are free to rescale the field φ in order to eliminatek from its kinetic term. However, a factor of λ ≡ k would JHEP01(2020)082 still be present in the cubic term (with λ > 0). After performing these modifications, we can finally write the effective Lagrangian in the same form as in [17]: where m 2 = −4T .

Conclusion
In the first part of this work, we reexamined the bosonic chiral string, now in the sectorized interpretation, deriving a few novel results. By considering the action of the BRST operator on the most general vertex operator, we confirmed the physical spectrum found in [11], namely a massless level identical to that of the ordinary bosonic string and two tracelesssymmetric fields h mn ± with mass-squared m 2 = ±4T . Moreover, we showed that the extra (massive) states can be seen as auxiliary fields (cf. footnote 1) leading to a higher derivative gravity theory, which in the tensionless limit (T → 0) reduces to the recent results of [15] for the bosonic ambitwistor string. In [27] the massive spin-2 states were determined to be ghosts via a 4-point amplitude analysis based on a "twisted" Kawai-Lewellen-Tye formula. This fact is manifest in the quadratic action we construct from the vertex operator.
In the second part, we showed that the current algebra extension of the bosonic model effectively leads to the (DF ) 2 +YM+φ 3 Lagrangian of [17], with all its fields and couplings coming naturally from standard string (field) theory techniques. The emergence of the higher derivative term (DF ) 2 from two vector fields of the physical spectrum is particularly interesting. In addition, we would like to point out that the group constants C αab and d αβγ , their relations and properties emerge naturally in our model and are valid for a generic level k of the algebra. In [17], on the other hand, such relations are obtained by demanding that the gluon amplitudes satisfy the Bern-Carrasco-Johansson relations [18] and our results agree when we take k → 0. This limit corresponds to a projection to the single-trace amplitude sector, which is where we expect our results to match. The multitrace sector of the worldsheet model is "contaminated" by the gravity theory described in section 2, much like the Berkovits-Witten twistor string necessarily includes conformal gravity [28,29].
It is amusing to notice that the amplitudes of (DF ) 2 theories, which enter as doublecopy constituents of (ordinary) bosonic and heterotic string amplitudes [20], can themselves be described in terms of a worldsheet model. Since the chiral string we studied is a closed string, it is plausible that its amplitudes can be written in double-copy form. The latter would probably involve the Z-theory investigated in [30][31][32], which is related to the basis of integrals mentioned in the introduction. We plan to address this question in future work.