Background constraints in the infinite tension limit of the heterotic string

In this work we investigate the classical constraints imposed on the supergravity and super Yang-Mills backgrounds in the $\alpha'\to 0$ limit of the heterotic string using the pure spinor formalism. Guided by the recently observed sectorization of the model, we show that all the ten-dimensional constraints are elegantly obtained from the single condition of nilpotency of the BRST charge.


Introduction
About three years ago, Cachazo, He and Yuan (CHY) proposed a compact formula for computing tree-level amplitudes in both Yang-Mills and gravity theories [1]. There was an increasing interest then to find a string origin of those results given their known connection to string amplitudes at the low-energy limit.
Soon after that work, Mason and Skinner introduced the so-called ambitwistor string [2], which could be viewed as an α ′ → 0 limit of the usual string and provided a clear derivation of the CHY formulae for D = 10 Yang-Mills and NS-NS supergravity.
Taking advantage of the pure spinor formalism's manifest supersymmetry, Berkovits proposed its ambitwistor version in [3], which was explicitly shown in [4] to provide the supersymmetric version of the CHY amplitudes.
When extended to curved backgrounds, one would expect that consistency of the ambitwistor string should put the target space fields on-shell. In [5], Adamo et al demonstrated that the nonlinear equations of motion of the NS-NS background arise as anomalies of the worldsheet supersymmetry algebra. In the pure spinor case, Chandia and Vallilo investigated the type II background [6] and realized that Berkovits' original proposal for the infinite tension string was incomplete and had to be modified in order to obtain the usual background constraints coming from the pure spinor formalism. By performing a semi-classical analysis, they were able to reproduce the known results of [7] with the introduction of the extra condition of BRSTclosedness of H, a generalized particle-like Hamiltonian.
The ideas in [6] were further explored by one of the authors in [8] and it was shown that the new model, although still chiral, could be interpreted in terms of two sectors resembling the usual left and right-movers of the superstring. This construction was also extended to the heterotic case, providing a sensible description of the massless heterotic spectrum in this α ′ → 0 limit. This was achieved by incorporating the observed sectorization in the heterotic BRST charge, which was then redefined to be Q =˛{λ α d α +cT + −bc∂c}, (1.1) where λ α is the pure spinor ghost, d α is the improved worldsheet realization of the superderivative introduced in [6], (b,c) are the reparametrization ghosts and T + accounts for one of the sectorized energy-momentum-like tensors, which are defined in terms of H and the full energy-momentum tensor T as As we show in the present work, the problem of finding the constraints on the heterotic background is somewhat more natural than in type II, in that H enters the BRST charge Q itself, cf. (1.1), and the background constraints all come from the sole requirement that Q be nilpotent. In a general heterotic background, the action for the sectorized model and the generalized particle-like Hamiltonian will be cast as The vielbein appears through Π A = ∂Z M E A M , mapping the curved superspace coordinates Z M , to the generalized superspace invariants with flat (super) indices A. The Lorentz connection Ω C AB , enters the covariant derivative ∇. The super Kalb-Ramond field is denoted by B AB , while A I A , W αI and U βI α represent the super Yang-Mills background. All the worldsheet fields above will be detailedly introduced in section 2.
By performing a classical analysis and computing the generalized Poisson brackets associated to S, we will show that classical nilpotency of the BRST charge (1.1) imposes some constraints on the torsion T C AB , the 3-form field strength H ABC , the curvature tensor R D ABC , and the super Yang-Mills field strength F I AB , given by in addition to the so-called holomorphicity constraints 1 and All together, the constraints in (1.5) imply the supergravity and super Yang-Mills equations of motion of the heterotic background, as explained in [7].
This work is organized as follows. Section 2 presents the sectorized model introduced in [

The free heterotic string with infinite tension
The heterotic pure spinor string is described in the α ′ → 0 limit by the chiral action S = 1 2πˆd 2 z{P a∂ X a + p α∂ θ α + w α∂ λ α +b∂c} + S C .
(2.1) X a and θ α are the N = 1 superspace coordinates with conjugate momenta P a and p α , with a = 0, . . . , 9 and α = 1, . . . , 16 denoting the flat vector and spinor indices respectively. The ghost sector is represented by the usual reparametrization ghosts,b andc, the pure spinor λ α , satisfying (λγ a λ) = 0, and its conjugate w α . The gamma matrices satisfy {γ a , γ b } = 2η ab , where η ab is the SO(1, 9) metric. The gauge sector is encoded in S C . Note that S has no conformal anomaly and its energy-momentum tensor is given by where T C is the gauge sector energy-momentum tensor with central charge c = 16.
In [3], the action (2.1) was provided with the BRST charge However, it does not correctly describe the expected massless heterotic spectrum, in particular it fails to reproduce the gauge transformations of the supergravity states, which are directly related to the invariance of the theory under general coordinate transformations.
Following the ideas of [6], an alternative BRST charge was proposed in [8] by one of the authors. We will review this construction now.

Review: sectorization and BRST cohomology
Perhaps the first observation hinting at the inadequacy of the BRST charge (2.3) is the existence of an extra nilpotent symmetry of the action (2.1), also linear in λ α , generated by To consistently absorb K in the BRST charge, the supersymmetry charges have to be redefined to q α ≡˛{p α + 1 2 (P a − ∂X a )(γ a θ) α − 1 12 (θγ a ∂θ)(γ a θ) α }, (2.5) which in turn brings forth the new invariants: Note that the operators P ± a of [8] would be written here as P ± a = P a ± Π a . The action and its energy-momentum tensor can be expressed in terms of the above invariants as Although not manifestly, S is invariant under supersymmetry. Consider a transformation with constant parameter ξ α , then Using the property (γ a αβ γ b γλ + γ a αγ γ b βλ + γ a αλ γ b γβ )η ab = 0, the integrand in the last line can be rewritten as which proves the invariance of the action S up to boundary terms.
We will also define the operator which is the heterotic analogous of the generalized particle-like Hamiltonian for the type II case of [6]. Using these operators, it was shown in [8] that the chiral action S can be interpreted in terms of two sectors (+) and (−) with characteristic energy-momentum-like tensors such that The new BRST charge makes the sectorization of the theory explicit and is given by Q λ is very similar to the usual (left-moving) pure spinor BRST charge while Q + is composed by the familiar BRST charge coming from the reparametrization symmetry plus an analogous contribution with the operator H, cf. equation (2.12).
The massless spectrum of the heterotic string consists of non-abelian super Yang-Mills and N = 1 supergravity, respectively described by the vertex operators where J I corresponds to (holomorphic) generators of the SO(32) or E(8) × E(8) current algebra, with I denoting the adjoint representation of the gauge group. BRST-closedness of U SY M and U SG with respect to (2.14) provides the known superfield equations of motion at the linearized level, The gauge transformations of the superfields, given by can be written in terms of BRST-exact expressions, as expected. More details can be found in [8].
Next, we will discuss the classical equations associated to the nilpotency of the BRST-charge (2.14) to establish the basis for the curved background analysis of section 3.

Classical analysis
In order to determine the classical conditions to be imposed on the background, it might be useful to understand their meaning in the flat case. Recall that the heterotic action can be cast with P a and d α being supersymmetric invariants defined in terms of the conjugate momenta of X a and θ α respectively, cf. equation (2.6). It is convenient, however, to treat them as independent variables. The above action is just one step behind the curved space one that we will define in the next section.
The BRST symmetry is described by the charge displayed in (2.14). To compute the classical BRST transformations of the worldsheet variables, we will rewrite Q in terms of the fields {X a , θ α , λ α ,c}, collectively denoted by φ, and their canonical conjugates, which are given in terms of {P a , d α , w α ,b}. The latter will be denoted byP φ and are usually defined with respect to τ , the worldsheet time. We will use the Minkowski parametrization with z = σ − τ and denotes the spatial coordinate. The derivatives can then be cast as With this convention, the canonical momenta will be defined to bê leading to the following identifications: The fundamental Poisson brackets are simply given by Therefore, the BRST transformations of the worldsheet fields are easily computed when the BRST charge is written in terms ofP and φ. For example, Q λ in (2.15) is expressed as Concerning the nilpotency of the BRST charge Q, it can be stated as Because Q λ is independent of the reparametrization ghosts, each term in the equation above should vanish separately. Therefore, following the classical construction just presented, it is easy to demonstrate that Q is nilpotent if and only if In flat space, it is straightforward to see that all these relations are satisfied. In the next section they will be our guidelines for nontrivial backgrounds. The difference then will be how the background manifests itself in the definition of the conjugate momenta, in particular (2.22a) and (2.22b), which contain the fundamental ingredients of the BRST charge, P a and d α .

Classical consistency of the heterotic background
In this section we will show how the nilpotency conditions discussed above ultimately impose constraints on the heterotic background, providing the expected supergravity and super Yang-Mills equations of motion in superspace detailedly presented in [7] for the pure spinor superstring.
After understanding how the infinite tension string couples to the heterotic background, we will be able to build the operator set necessary for our analysis. The supergravity sector is presented alone beforehand for two reasons. First, to the best of our knowledge, there is no good description for N = 1 (heterotic) supergravity in any ambitwistor string so far. So this will be a good test for the modifications discussed in [8] for the sectorized string. Second, the generalization from flat space is straightforward and it will help establish the curved superspace language that is extensively used. Next, we will turn on the super Yang-Mills background and extend the results.

Supergravity background and constraints
The curved superspace generalization of (2.7) is given by The vielbein E A M , and the Lorentz connection Ω C AB , enter the action through the generalized superspace invariants and the covariant derivative 2 , given bȳ with This form is more suitable to show the gauge invariance of the action with respect to the trans- More details on the conventions used here can be found in Appendix A. Concerning the dilaton superfield, it plays no role in the classical description and this can be seen from the fact that its coupling to the action naively vanishes in the Following the analysis of subsection 2.2, P a and d α can be viewed as independent objects invariant under supesymmetry, and the flat space limit of S is recovered when we express them in terms of regular variables, cf. (2.6), together with the non-vanishing components of E and B in that limit: (3.5) The energy-momentum tensor of the curved space action is given by The BRST-charge in the curved background has the same structure of (2.14) and the presence of the background can be seen through the canonical conjugates of the superspace coordinates Z M , denoted byP M . Using the definition (2.21), one obtainŝ which enables us to rewrite P a and d α as we obtain the following transformations 3 Here ǫ is a constant anticommuting parameter and we have defined Now we can compute the transformation of λ α d α , whose vanishing is equivalent to the first condition displayed in (2.26): (3.14) Hence the first set of constraints required for the nilpotency of Q is: Nilpotency of the BRST charge also requires δT + to vanish. This is just another way of stating the condition (2.26b). The operator T + is obtained from the definition (2.12) and the curved versions of T and H, respectively (3.6) and (3.7). It can be cast as Now, to compute δT + we just have to use the transformations of P a and Π a in (3.12) and the result is For this expression to vanish, we need to impose another set constraints: In the usual pure spinor superstring, this set comes from the holomorphicity of the BRST charge [7].
Finally, rewriting T + in terms of the canonical conjugates and using the Poisson brackets of (3.10) together with we can show that

Turning on the super Yang-Mills background
In order to find the remaining heterotic background constraints, we need to consider the case in which the super Yang-Mills fields are present. We will introduce the minimal coupling between the gauge potential A I M and the currents J I , such that the action has the form which transforms covariantly under (3.23), The coupling to the super Yang-Mills background changes the energy-momentum tensor to and we also expect the operator H to be modified accordingly. It was suggested in [6] that fluctuations of the background would be manifested through H. Therefore, inspired by the superstring integrated vertex, we propose where W αI and U βI α are background superfields 4 which will be related to (3.26). Again, using (2.12), T + can be cast as and we now have all the ingredients to analyze the BRST symmetry in this background.
The modification of the action entails a change in the classical BRST transformations of the worldsheet fields. This is clearly seen from the canonical conjugates of the superspace coordinates, which now have a linear dependence on the gauge field: To compute these transformations we will use the fundamental brackets of (3.10), (3.11), (3.19) and (3.20), together with which are derived in Appendix B.
Considering first Q λ , it is clear that most of the transformations displayed in (3.12) remain unchanged, except for δP a and δd α , which are now given by We can now easily compute the transformation of λ α d α and the result is Thus, together with the constraints displayed in (3.15) we also need to impose in order to satisfy the first nilpotency condition, cf. equation (2.26a).
Next, to compute δT + and evaluate the condition (2.26b) it is worth noting that T C and J I now have nonvanishing transformations with respect to Q λ , given by where we have defined the gauge-like parameter Σ I ≡ ǫλ α A I α . The introduction of Σ I is convenient when we look at the variations of the background superfields, which can be interpreted in terms of a gauge-like transformation with parameter Σ, a Lorentz-like transformation with parameter Λ, and a superspace translation, with ∇ α denoting the covariant derivative with respect to the local symmetries, e.g.
Gathering all these results and using the supergravity constraints of (3.18), we obtain Hence, we have to further impose the following constraints: Note that the last line of (3.40) vanishes automatically after the identification in (3.41c).
As a final consistency check, it is not difficult to show that T + satisfies which demonstrates the last necessary condition for the nilpotency of the BRST charge at the classical level.

Discussion
It is possible to show that the constraints displayed in (3.15), (3.18), (3.36) and (3.41) imply the ten-dimensional supergravity and super Yang-Mills equations of motion. Instead of presenting these results, which for the pure spinor superstring were originally obtained and detailedly studied in [7], we will discuss the particularities of the infinite tension string model.
As presented in subsection 2.2, there are in principle three independent conditions to check in order to ensure classical nilpotency of the BRST charge, The first one is identical to the condition on the left-moving BRST charge of the usual pure spinor superstring and not surprisingly provides the so-called nilpotency constraints, given by exactly as in [7]. The second condition can be stated as and leads to the constraints of (3.18) and (3.41), which were obtained in [7] by requiring holomorphicity of the BRST current. In the present model, holomorphicity plays no role when it comes to imposing constraints. This is so because the heterotic background coupled action, given by is still chiral. Therefore the remaining constraints should manifest themselves through the condition (4.3). To interpret it, it might be useful to recall that conformal symmetry is preserved at the classical level, such that [Q λ , T ] = 0. We are then left with cf. the definition 2.12. This is precisely the ad hoc condition used in [6] for the type II construction. Here, however, it is naturally embedded in the BRST operator. From the sectorized point of view, the condition above is equivalent to the conservation of the BRST charge separately in each sector.
Concerning the third condition in (4.1), it was verified to hold independently of the background constraints. This is partially connected to the classical conformal symmetry but we do not have a clear understanding so far. It implies, for example, that We expect this relation to hold in the type II case as well.
An interesting observation is that the background can be absorbed by a field redefinition in the action, such that where We still do not have a simple proposal for such operators. However, there are interesting hints pointing out that the holomorphic sectorization can be extended the bosonic string and to the Ramond-Neveu-Schwarz and Green-Schwarz formalisms [10]. We hope that understanding this construction will provide a better basis to approach the problem of the integrated vertex operator in the infinite tension limit using pure spinors. Once this step is taken, we will finally be able to compute the tree level amplitudes to compare them with the Cachazo-He-Yuan formulae [1] and investigate the modular invariance of the theory at 1-loop, for example, as done in [11].
Acknowledgements: TA acknowledges financial support from Conselho Nacional de De-

A Superspace notation and conventions
In this work, we use the following sets of indices: Our conventions concerning differential forms are the same as those in [12]. In particular, given a manifold with vielbein E and connection Ω, we define the torsion to be or, in components, where the graded symmetrization is defined as and the indices appearing at the exponents should be replaced by their gradings, i.e. +1 for spinorial indices and 0 otherwise. As usual, one can write the torsion in terms of tangent space indices by contracting it with vielbeins: Another important quantity is the curvature tensor, defined in terms of the connection as or, in components,

B SO(32) realization of the gauge sector
Here we will present a realization of the action S C describing the gauge sector of the heterotic string, focusing on the SO(32) group, which has a simpler construction.
The generators of the SO(32) group will be denoted by the anti-Hermitian operators T I .
The algebra can be cast as where f IJK = f L IJ δ LK are real and totally antisymmetric structure constants constrained to satisfy the Jacobi identity The action S C consists of a (free) set of 32 real worldsheet fermions, ψ i , such that 3) The associated energy-momentum tensor is Using the simple OPE ψ i (z)ψ j (y) ∼ δ ij (z − y) , (B.5) one can easily compute showing that the central charge of the system is 16, as required by the vanishing conformal anomaly in the heterotic string.
The SO(32) group structure in the worldsheet theory can be seen through the current Observe that J I is conserved,∂J I = 0, which follows from the classical equation of motion for ψ i . For completeness, we can mention that the currents J I define an Affine Lie algebra at the quantum level, which can be read from the OPE: . (B.8) Following the notation of subsection 2.2, we can see that the canonical conjugate of ψ i is identified with ψ i itself, as usual for fermionic systems. Therefore we have to use Dirac's procedure to deal with this constraint in order to obtain the Dirac brackets for ψ i , given by It is interesting to observe that this coupling has a very natural symmetry at the classical level.
Consider the transformation δĀ I =∂Σ I − f I JK Σ JĀK , (B.14) where Σ I is a generic parameter in the adjoint representation of SO(32). It is straightforward to show that S int C transforms as