Heterotic Quantum Cohomology

We reexamine the massless spectrum of a heterotic string vacuum at large radius and present two results. The first result is to construct a vector bundle $\mathcal{Q}$ and operator $\overline{\mathcal{D}}$ whose kernel amounts to deformations solving `F-term' type equations. This resolves a dilemma in previous works in which the spin connection is treated as an independent degree of freedom, something that is not the case in string theory. The second result is to utilise the moduli space metric, constructed in previous work, to define an adjoint operator $\overline{\mathcal{D}}^\dag$. The kernel of $\overline{\mathcal{D}}^\dag$ amounts to deformations solving `D-term' type equations. Put together, we show there is a vector bundle $\mathcal{Q}$ with a metric, a $\overline{\mathcal{D}}$ operator and a gauge fixing (holomorphic gauge) in which the massless spectrum are harmonic representatives of $\overline{\mathcal{D}}$. This is remarkable as previous work indicated the Hodge decomposition of massless deformations were complicated and in particular not harmonic except at the standard embedding.


Introduction
We study heterotic vacua that have large radius limits. In this limit, the vacua are solutions of α -corrected supergravity. This effective field theory is fixed up to and including α 2 by supersymmetry and the result is also consistent with string scattering amplitudes. We refer to the resulting equations of motion as the Hull-Strominger system [1,2]. The vacua we study have a smooth limit α → 0 which forces H → 0 and a ten-dimensional spacetime of the form R 3,1 ×X with X a complex 3-fold with c 1 (X ) = 0. With an appropriate gauge fixing the dilaton is constant up to order α 3 [3] and the background is topologically a CY manifold with a holomorphic vector bundle E admitting a connection A that satisfies the hermitian Yang-Mills equation. These are the only compact supergravity solutions with a valid α → 0 limit, which guarantees the supergravity solutions are also solutions of string theory up to non-perturbative corrections. There are solutions in which as α → 0, the three-form H is non-trivial. However, one must study either non-compact manifolds, be non-perturbative in the dilaton via, e.g. non-geometric solutions, or consider solutions without a convergent α expansion. We do not consider such solutions here because the higher order perturbative α corrections to the Hull-Strominger equations (and likely worldsheet instantons) will play an important role modifying the quadratic order α behaviour, and these α corrections have not been completely determined yet. We expand upon our justification for restricting to solutions with a smooth α → 0 limit in Appendix B.
The moduli space is a finite dimensional complex Kähler manifold M . Each point y ∈ M corresponds to a heterotic vacuum and there is a Kuranishi map which relates tangent vectors δy on M to deformations of fields. Small gauge transformations correspond to deforming the Kuranishi map and there is a choice in which the map is holomorphic [4]. We call this holomorphic gauge. In holomorphic gauge, there is a relation between deformations of the connection on the tangent bundle T X and the moduli of X [5] Here we have introduced coordinates on M as y a = (y α , y β ) using the complex structure inherited from the heterotic theory. On X real coordinates are denoted x m , complex coordinates (x µ , x ν ) and ∇ σ is the usual Levi-Civita connection on X . We denote J the complex structure on X and (in holomorphic gauge) a holomorphic deformation is a (0, 1)-form valued in the holomorphic tangent bundle. We write this in the notation of [6] viz. δy α ∆ α µ ν .
Similarly, if ω is the hermitian form on X then (∂ α ω) σµ is a holomorphic deformation of the hermitian form restricted to its (1, 1) component. In holomorphic gauge this is equal (up to a factor of i) to a gauge invariant deformation of the B-field, denoted B (1,1) α [4]. The analogue of complexified Kähler deformations for heterotic theories is the combination The key point is that δΘ (0,1) is fixed in terms of ∆ α and Z (1,1) α . In contrast, in [7][8][9] the deformation δΘ (0,1) is treated a degree of freedom independent of the other moduli with its own set of parameters. That is, δΘ (0,1) is an arbitrary element of H 1 (X , End T X ) and with these spurious degrees of freedom one finds a D-operator on a double extension bundle Q in which D 2 = 0 and its cohomology is related to the deformations of heterotic theories plus the spurious degrees of freedom. A natural question is to ask what happens to the operator D and bundle Q when we eliminate the spurious degrees of freedom? Our initial goal is to answer this question. What we find is that in holomorphic gauge [4] there is an extension bundle Q without the spurious degrees of freedom with a D operator acting on sections of this bundle. Its kernel amounts to 'F-term' type equations. Our next goal is to use the moduli space metric constructed in [5,6] 1 to construct an adjoint operator D † and show that the co-kernel of D describe D-term type equations. D-term equations here refer to first order deformations of the Hermitian-Yang-Mills equation and balanced equations. 2 In other words, the physical massless moduli of the heterotic supergravity at large radius are harmonic representatives of the cohomology of the D-operator.
In the next section we review some results setting up our notation. In §3 we review how extension bundles describe deformations of complex manifolds. In §4 we revist and refine the calculation in [7]. We find a family of operator D on a double extension, and using the metric in [6] show its adjoint describes deformations of the Hermitian-Yang-Mills equation and balanced equation. In §5, we construct an extension Q and D-operator whose cohomology describe the F-terms and the harmonic representatives the D-terms.

α -corrected Supergravity
The heterotic action is fixed by supersymmetry up to and including α 2 corrections. There is nice basis of fields in which the action, equations of motion and supersymmetry variations are particularly compact. This was constructed in [10] to order α by supersymmetrising the Lorentz-Chern-Simons terms in H and extended to α 2 in [11]. The action and Bianchi identity for H is unique up to field redefinitions; there is no ambiguity in the theory. The action, in the field basis of [3], is given by Our notation is such that µ, ν, . . . are holomorphic indices along X with coordinates x; m, n, . . . are real indices along X . The 10D Newton constant is denoted by κ 10 , g 10 = − det(g M N ), Φ is the 10D dilaton, R is the Ricci scalar evaluted using the Levi-Civita connection and F is the Yang-Mills field strength with the trace taken in the adjoint of the gauge group.
We take the p-form norm as |T | 2 = 1 p! T M 1 ···Mp T M 1 ···Mp and the curvature squared terms correspond to where the Riemann curvature is evaluated using a twisted connection with Θ M is the Levi-Civita connection and A, B are the tangent space indices. The threeform H satisfies a Bianchi-identity while at the same time it is related to the hermitian form as H = d c ω. With respect to a fixed complex structure this corresponds to and so the Bianchi identity in this complex structure is The decomposition of the connections into type is where A = A (0,1) and θ = Θ (0,1) and we are using conventions in which A and Θ are antihermitian.
All we care about is that the Bianchi identity (2.2), equations of motion and supersymmetry variations match calculations from string scattering amplitudes [12] which has been checked [13]. This data is fixed, up to the usual caveat of field redefinitions. Indeed, if one were to perform a field redefinition, so that for example the curvature tensor R in the Bianchi identity is evaluated with a different connection, then the field redefinitions will propagate through the supersymmetry variations and equations of motion, likely losing their simple closed form. For example, if one evaluated R using the Chern connection then this requires a field redefinition in which resulting in a metric g that is no longer a tensor -it is charged under gauge transformations like the B-field -and this field redefinition is likely modify the supersymmetry variations already at order α . 3 We choose to use the usual conventions in which g is a metric tensor and R is evaluated with Θ + .
From now on we work to first order in α . For emphasis, we will sometimes include + O(α 2 ) in our equations, but this will mostly be suppressed.

Holomorphic gauge and F-term equations
In the notation of [5,6] the field deformations, parameterised by the real coordinate y a , obey the following equations to first order in α : We assume h (0,2) = 0 and so the last line is Z (0,2) a = ∂β (0,1) a and this means left hand side of the third line is ∂-exact.
There are equations coming from the HYM equation ω 2 F = 0 and the balanced equation d(ω 2 ) = 0 which are analysed below in §2.3. Both these equations and (2.4) are invariant under small gauge transformations [4]: where Z a = B a +i∂ a ω and Z a = B a −i∂ a ω. A convenient choice of gauge fixing is holomorphic gauge: In this gauge, we have, for example, where ∆ α µ is a (0, 1)-form valued in T (1,0) X and D α A = 0. The gauge simplifies other equations such as and determines While the field Z (0,2) α vanishes in this gauge fixing, the field Z (0,2) α does not. Hence, it is a physical degree of freedom whose role in a heterotic vacuum is not yet clear. The role of Z (1,1) α is analogous to the complexified Kähler modulus for a CY manifold.

D-term equations
An analysis of deformations in holomorphic gauge of the balanced equation and hermitian Yang-Mills equation was presented in [4]. This gives a relation between deformations in terms of adjoint operators. In this section we revist this calculation massaging the equations with a prescenice of results to come.
The top-form Ω on X has a norm It is related to the dilaton d log Ω = −2dφ. We gauge fix as in [3,4] so that the dilaton is a constant on X and so in this gauge Ω is also a constant. Furthermore, in holomorphic gauge (2.6), (∂ α Ω) (3,0) = k α Ω, where k α is a constant on X. Using this, a first order variation of the norm is As Ω and k α are constants over X and it follows ω λσ ∂ α ω λσ = 1 2i ω λσ Z α λσ is a constant on
(2.13) where ∇ Ch/H µ is the Chern or Hull connection (see [4] for discussion on this ambiguity) and because H = O(α ) and ∆ α[ρλ] = O(α ) the last term is O(α 2 ) and so is dropped from hereon. A good consistency check is to derive this equation by differentiating g µν H µνρ = 0.
A first order variation of the Hermitian Yang-Mills equation can be written as A similar equation is satisfied for the connection Θ on the tangent bundle to this order in α . That is In holomorphic gauge, we declare 'F-term' type equations to be (2.7)-(2.8). We do not have a good definition of string field theory, so we use this terminology with care. The motivation for the nomenclature is that these are the equations, together with an appropriate definition of holomorphy, that derive from a superpotential type construction. The equations (2.13)-(2.15) deriving from the balanced and HYM equation we declare to be D-term equations.
A main outcome of this paper is to show that there is a D-operator acting on first order fluctuations whose kernel corresponds to F-term equations, and co-kernel to the D-term equations. This is helps justifies this terminology.

The Hodge decomposition
The balanced and HYM equations play a role in determining the exact and co-exact terms in the Hodge decomopsition of fields. We present this to emphasise the fact that deformations of a heterotic theory are not simply the harmonic representatives. Taking into account the F-term equations, the Hodge decomposition is (2.16) The role of the cohomology we aim to compute is to tell us what linear combinations of harmonic terms in (2.16) correspond to unobstructed deformations. The exact and co-exact terms in the Hodge decomposition are physical and so are important and are determined by substituting into the HYM and balanced equations and solving the corresponding Poission equations: The point here is that deformations of fields are not only highly coupled but they are not harmonic representatives. A natural question to ask is: can I change the gauge to find a harmonic decomposition? The answer is not likely without significant sacrifice. For example, one easily loses the holomorphic dependence of deformations on parameters. This is unlike the study of CY manifolds in which one studies holomorphic deformations that are also harmonic, viz. ∂-harmonic (1, 1) forms and (2, 1)-forms. This derives from H = 0, an additional constraint that we do not have in the more general situation.
One point of this paper is to point out that without sacrificing holomorphy, the complicated Hodge decomposition beautifully reorganises itself into harmonic representatives of a Doperator on a certain extension bundle.

Warm-up: Extension bundles for complex manifolds
One approach to understanding (2.4)-(2.6) is inspired from the work of Atiyah [15] who described the deformations of a holomorphic bundle on a complex manifold. Its instructive to review this with an eye towards the full heterotic story.
Consider a holomorphic bundle on a complex manifold. Atiyah, in 1955, pointed out that not all complex structure deformations are allowed as some may introduce a non-trivial F (0,2) component destroying holomorphy of the bundle. Those that are allowed satisfy Intuitively such complex structure deformations may introduce F (0,2) but can compensated by a simultaneous deformation of the bundle. A way to realise these constraints is to define a vector bundle Q 1 as a short exact sequence where π 1 : Q 1 → T X is the canonical projection and i 1 the inclusion map. 4 One can then study (0, p)-forms valued in Q. These are vectors, schematically There is a holomorphic structure defined by an operator D 1 given by where, for example, Ω (p,q) (X , End E) denotes (p, q)-forms on X valued in End E. The action of D 1 on a typical fibre of Q 1 vanishes if the field deformations obey the requisite equations of motion: Under a small gauge transformation, which includes a small diffeomorphisms for generality, where ε α m is a vector valued (0, p − 1)-form parameterising the small diffeomorphism, and φ α is a (0, p − 1)-form valued in End E parameterising the small gauge transformation.
The bundle Q and operator D 1 are defined with respect to holomorphic deformations only. The operator D 1 is equivariant with respect to gauge transformations: For p = 1 we enter the case of interest, which are deformations of a complex manifold with bundle. The notation for deformations The space of deformations is the cohomology H (0,1) (X , Q 1 ). However, as can be seen from (3.2) together with the bundle being stable, so ∂ A φ α = 0 has only trivial solutions, any gauge transformation will introduce a non-holomorphic dependence on parameters. i.e. D α A † = 0. This gauge choice is already made in many physical calculations such as the moduli space metric [6] or results deriving from the superpotential, e.g. [16]. So the cohomology is to be treated with care: we are always gauge fixed, and so always fixed a representative. This becomes more important in the situation to come. 5 Setting this issue aside one can use a long exact sequence in cohomologies inherited from the short exact sequence to show The Hodge decomposition of an element of H 1 Equation (3.3) implies the tangent space to the moduli space is a direct sum. So there is a choice of coordinates for the moduli space in which parameters decompose into two components y α = (y α 1 , y α 2 ). The first component is one-to-one with H 1 (X , End E): , which has a solution provided we can invert F µ and solve for κ α 1 µ . The terms κ α 1 and Φ α 1 are completely undetermined at this point and so there is an infinite dimensional space of solutions. In the full heterotic theory, there are additional equations of motion, and these fix κ α 1 and Φ α 1 .
The second component of solutions is one-to-one with ker F ⊂ H 1 (X , T (1,0) X ): where δy α 2 is a vector in ker F. That is, the cohomology tells us what harmonic repre- The lesson here is that in writing we do not mean that the field deformations decompose into a direct product. Instead, it is telling us about what combinations of harmonic forms can appear and that they have a Hodge decomposition. This example, we have some undetermined exact terms; in the full heterotic theory these are all fixed by the equations of motion and gauge fixing.

F-terms
The situation in the Hull-Strominger system is more complicated as we have hermitian deformations, the B-field and a set of equations in (2.4)-(2.6) together with a Bianchi identity. The approach of [7] is to include deformations of the tangent bundle δΘ (0,1) = δy α D α θ as independent degrees of freedom. The tangent bundle T X is holomorphic with R (0,2) = 0, and the curvature two-form obeys the HYM equation ω 2 R = 0. 6 Hence, D α θ obeys an Atiyah equation, of the same form as the second equation of (2.4). These we refer to as spurious degrees of freedom because in the physical theory the deformations D α θ are determined in terms of the other deformations ∆ α , D α A and D α ω. Nonetheless, they lead to a nice mathematical result which we review and refine here. Define an extension Q 2 of Q 1 : which acts on forms valued in Q 2 . The operator R is analogous to F: where R µ is the curvature two-form for θ. The operator D 2 2 = 0 provided R (0,2) = 0 and its Bianchi identity holds.
To incorporate hermitian deformations define a bundle Q by the short exact sequence The first row is a (0, p)-form valued in T * (1,0) X , the middle two rows are (0, p)-forms valued in EndT X and EndE respectively, while the last line is We will need this more general construction in later sections and when we construct the adjoint operator.
In the appendix §C we find a family of operators that satisfy D 2 = 0 off-shell and reproduce the F-term equations. The operator in [7] is an example of this family. In this paper we use a D-operator in this family whose presentation is simpler: where H : and we show in §C.2 that D 2 = 0 off-shell. We see that corresponds to the F-term equations (2.7)-(2.8).
We note that while under a small gauge transformation (2.5) It is important that the gauge symmetry has an action on the complexified hermitian term Here b α is a one-form associated to gauge transformations of the B-field. At first sight, we seem to have a problem: DY There are no residual gauge transformations that preserve holomorphic gauge which means any small gauge transformation is going to violate the gauge fixing condition. Instead of (2.8) we need to use is the third line of (2.4) and this involves Z (0,2) α , which is no longer zero.
Indeed, we need to pair the transformation of Z (1,1) Then we find that under a small gauge transformation As h (0,2) = 0, the equations of motion imply Z (0,2) where β α is a gauge dependent quantity. In holomorphic gauge it vanishes. A gauge invariant formulation likely involves this combination. It would be interesting to relate these observations to the complexified gauge transformations studied in [20], particularly in light of the D-term calculations we perform below.
The lesson is that the cohomological property is subtle. A better understanding of complexified gauge symmetries and GIT would probably help.
Setting this issue aside, it is shown in [7] that there is a long exact sequence in cohomology to describe first order deformations of D: where There are the hermitian Yang-Mills and balanced equations to take into account. We show their role appears in the adjoint operator D † .

The D-terms and the adjoint operator D †
A metric on Q describes how to pair field deformations to produce a real number. In string theory, a natural metric to use is the moduli space metric. As we preserve N = 1 supersymmetry in spacetime, the metric is Kähler. With the spurious degrees of freedom, this was first derived in [6] using a dimensional reduction of the α -corrected supergravity theory considered here. The result in our notation is We have written the antihermitian connections as We take the metric on a pair of sections Y (p) α and Y (p) β to be the natural generalisation of the moduli space metric: where we hermitian conjugate in the appropriate way so the metric is real.
The metric allows us to define an adjoint operator D † .
It is instructive to explicitly compute: where our notation is, for example, Z α µν (p+1) = 1 p! Z α µν λ 1 ···λp dx λ 1 ···λp . Comparing with (4.5) we identify where 9) where we use that R and F are antihermitian and in the last line, we used the balanced equation to derive an identity We then find that is precisely the D-term equations (2.13)-(2.15).
Hence, in holomorphic gauge with spurious degrees of freedom we find that This is despite the fact the individual deformations such as D α A or ∆ α µ are not harmonic representatives as demonstrated explicitly in (2.16), (2.17a)-(2.17c). Instead, if we work (0, 1)-forms valued in Q then all we need to do is study harmonic representatives of this D-operator. However, we are not yet at the physical case yet: we still need to eliminate the spurious degrees of freedom.

Eliminating the spurious degrees of freedom
The space of deformations of D on the bundle Q is not the moduli space of string theory. String scattering amplitudes and/or supersymmetry tell us that D α Θ is not an independent degree of freedom, but determined in terms of the remaining fields. In [3,5] this is calculated to lowest order in α : The Γ are the Levi-Civita symbols, which because this result is derived for solutions which have a smooth limit α → 0 with H → 0, are the same as the Chern connection. The dilaton is constant to α 3 [3] by a choice of gauge fixing; see also appendix C of [4] for the calculation in the notation of this paper.

Constructing the bundle Q
We now turn to constructing an extension bundle Q whose sections describe the physical degrees of freedom. Recall that to first order in α , the first order deformations satisfy an Atiyah equation ∂ θ (D α θ) = ∆ α µ R µ . We check (1.1) satisfies this explicitly. First, note where R µ λρν is the Riemann curvature tensor on a complex manifold. Second, where R λ σνρ = R λ ρνσ is used in the penultimate line. We use that R (0,2) = 0 and ∂Z α = O(α ). In fact because θ is an instanton (to first order in α ), D α Θ obeys an Atiyah equation and so we conclude that this result actually holds up to and including first order in α . Hence, if there are any α corrections to (1.1) then we know that in (5.2) that they must cancel.

A corollary of (5.2) is that the expression (5.1) is a solution of DY
(1) α = 0 and so bona-fide element of the cohomology H 1 D (X , Q). This is serves as another consistency check and is part of our intuition that the true moduli space is some subspace of H 1 D (X , Q). Using the symmetry D α Θ µ ν σ = −g νλ D α Θ µ ρ λ g ρσ we can write where R ν µ = R ρσ ν µ dx ρσ is the Riemann tensor and we have used R µν = −R νµ . Now, using (5.2) with the Bianchi identity for R, (A.1) we find Using (5.3) we see that a consequence of (5.4) is that and so it should be ∂-exact. Indeed, this is the case and we now derive an explicit expression. Start by recalling that R ν µ obeys a hermitian Yang-Mills equation ∇ µ R ν µρσ = 0 to first order in α and so As the Levi-Civita symbols are pure to this order in α we find the term in parenthesis can be written as Using that ∂(Z (0,1) α µ ) = O(α ) and that R (0,2) = 0 we find α λ and so does not become ∂. Using these results (2.8) becomes: (5.5) It is natural to perform a field redefinition which is well-defined in α -perturbation theory, and in particular, invertible. It is not overly surprising that the complexified hermitian form needs to be corrected in α . The field redefinition will propagate into other relevant equations, such as deformations of the HYM and balanced equations and the moduli space metric. But it is the natural thing to do to realise a holomorphic structure on the extension bundle.
There is a cohomology that captures these field deformations. To describe it, we need to generalise some of the above results above to (0, p)-forms. First, we ansatz that (1.1) generalises to Second we need the generalisation of the D-operator for the spurious case, written in components in (C.10). The only non-trivial part is to evaluate Tr δ α θ (p) R ν using (5.7) and substitute into the first line of (C.10) giving A consistency check is that the relation (5.7) satisfies the second line of (C.10). We also have the field redefinition Now construct the extension bundle. The extension Q 1 defined in (3.1) is unchanged. Q 2 is now not necessary. Consider a short exact sequence where T * X (1,0) are holomorphic co-vectors (forms).
p-forms valued in Q are denoted as in the previous subsection We have abused notation in using the same letter Y (p) α as for the spurious case, but it should be clear from context what we mean.
Define the operator D on sections valued in this bundle as where the map H new : This is due to the appearance of a holomorphic derivative in the new extension map H new ν . 8

The D-terms and the adjoint operator D †
The moduli space metric, after including (1.1), is [5,6] The operator D squares to zero on the nose if one modifies the Hull-Strominger system to use the curvature of the zeroth order Ricci-flat Calabi-Yau metric in the Tr R 2 term in the heterotic Bianchi identity.
Otherwise, this only holds modulo α 2 corrections. 8 We are grateful to Mario Garcia Fernandez for pointing this out.
where we have used holomorphic gauge (2.6) in writing Z (1,1) α = 2iD α ω (1,1) . We need to write down g αβ in the new variables Z (1,1) α via the field redefinition (5.6). First note that The second term is The commutator involves both the Riemann curvature and Ricci tensor Ric δτ = −R δ µ µτ : The Ricci tensor vanishes to this order in α . So The third term follows in the same way. Hence, the moduli space metric in these variables is It is straightforward to generalise this calculation to a metric on (0, p)-forms: For p = 0, the third and last terms are not present. We have also used Using the derivative operator in (5.10) and the metric above, we compute the adjoint following the logic of §4.2. Many of the terms follow through identically. For ease of notation, we now drop the tilde on Z (p) α . The adjoint is We compare with the metric (5.14) and work to first order in α to deduce the action of the adjoint operator on sections The case D † Y (1) α = 0 amounts to precisely the D-terms.

Conclusion and Outlook
So we have shown that physical field deformations in holomorphic gauge are given by harmonic representatives of D operator on the new Q bundle with the spurious degrees of freedom eliminated. These are harmonic with respect to the moduli space metric in [4]. We see Along the way we also demonstrated that the non-physical calculation, in which one has additional spurious degrees of freedom, are captured by a D-operator with an analogous relation between the F-terms and D-terms and D-closure and D-coclosure. That means that with respect to the D-operator we could equally well call holomorphic gauge, harmonic gauge.
It would be interesting to reconcile these results with the complexified gauge transformations and moment maps in a GIT-type analysis of these solutions studied in [20,21]. There are also examples of mathematical interest, such as that related to the Hopf surface which do not have an obvious valid α -expansion, yet appear to be governed by a (0, 2) superconformal algebra [22]. Nonetheless, one could still try to construct the adjoint operator D † and compare with the space of deformations obtained. It would also be interesting to see to what extent the structure discovered in this paper extends to next order in α . We expect the generalised geometry approach of [9,20,21,23,25] to be relevant for these scenarios.
One could also ask about higher orders in deformation theory. Presumably at second and third order in deformation theory there are relations with the L 3 algebra described in [16]. It is known in the study of CY manifolds that special geometry involves relations between the moduli and matter sectors. It would be interesting to extend these results to the charged matter sector utilising the matter metric [26]. The results here are reminiscent of the study of the tt * equations of the special geometry of CY manifolds [27], in which one studies a fiber bundle whose base manifold is the moduli space and fibres are harmonic elements of the cohomologies H (2,1) ⊕ H (1,1) . Until now, we haven't been able to demonstrate a cohomology with harmonic representative capturing the deformations of the full heterotic theory. This might be a new door worth opening.

A.1. Curvatures and Bianchi identity
The Bianchi identity for a curvature R evaluated with a connection Θ is For now, we take R to be the Riemann curvature tensor. It has symmetry properties We have have need for commutators of derivatives to lowest order in α e.g.
It involves both the Riemann curvature and Ricci tensor Ric δτ = −R δ µ µτ : The Ricci tensor vanishes to this order in α .
It is useful to note that where we omit the Tr R 2 term as it follows in the obvious way with a minus sign.

A.2. Hodge dual relations
It is useful to recount from the appendix of [4] adjoint differential operators and some Hodge dual relations for forms on X.
Given k-forms η, ξ and a metric ds 2 = g mn dx m ⊗ dx n on X, the Hodge dual defines an inner product ( · , · ) : Now consider two forms η k and ξ l , where the subscript denotes their degree and k ≤ l,.

Contraction is
: and acts as follows An interesting feature of this operator is that it is the adjoint of the wedge product Recall, the de Rahm operator d can be written

B. A supergravity no-go theorem
No-go theorems are well-known in the context of type II and M-theory flux vacua. Classical supergravity equations of motion, when integrated over a compact manifolds, force all nontrivial fluxes to vanish. The evasion of these theorems comes in the form of α -corrections to supergravity, interpreted as the contribution of orientifold planes, objects with negative stress-energy tensor. Here we briefly recount an analogous result in heterotic string theory; this demonstrates that if one wants solutions of supergravity then H = O(α ).
In this section only we work to α 3 , as it is not complicated to include the α 2 correction. The equations of motion are given by , with ∇ − computed with respect to the Θ − connection, and R M N is the Ricci tensor. We restrict to the internal manifold X directions and by tracing the second equation, subtracting form the first we find We we want to remain in the regime of perturbative string theory, one in which we might hope to have a worldsheet description via a (0, 2)-sigma model or even better a (0, 2)-CFT, then we require where Φ 0 is a constant scalar on X. As we see below this is related to the string coupling constant g s . It is possible there are non-perturbative solutions involving either 5-branes or duality, however these are beyond the scope of what we study. They require tools beyond the Hull-Strominger system to determine if the quantum corrections, in both α and g s , are convergent.
Integrating both sides over X we see From the previous paragraph, we see that for g s -perturbative solutions and because |H| 2 is positive definite, that H = O(α ). So if we want to study solutions of string theory with supergravity limits then H = O(α ).
If we didn't have the contribution of the last term there would be no solution at all -this is the classical supergravity no-go theorem. We evade this because of the Green-Schwarz anomaly cancellation condition between the classical supergravity contributions and the αcorrection involving the curvature squared terms. In certain backgrounds heterotic theories are dual to type IIB theories, and this contribution can be understood as coming from the contribution of O7-planes in type IIB, which carry negative energy density. This explains physically why the contribution from the Tr R 2 term in the moduli space metric appears with the opposite sign from all the other terms. That is [6], This does not mean the moduli space metric has indefinite signature: it is constructed in supergravity and so necessarily α 1 meaning the last term, no matter its value, cannot change the signature of the metric. Even after substituting for the spurious degrees of freedom, the same logic will apply.
If one wanted to study solutions where α is large, then if we want it to be relevant at all to string theory (along with all its beautiful results such as mirror symmetry, special geometry and sigma models) then we need to include the higher order α -corrections. It is already known that there is an α 2 correction to the hermitian sector of the metric [3]. As pointed out in that paper, there are going to be α 3 corrections, such as the α 3 correction to the moduli space metric calculated using mirror symmetry [24] at the standard embedding where V is the volume of X, ζ is the Riemann zeta function and χ the Euler number. Presumably, if we understood (0, 2)-mirror symmetry, we could determine all the α -corrections to the moduli space metric.
With all deformations turned on, the Kähler potential for the moduli space metric g αβ was first shown to be [6] It is remarkable that this also encodes the deformations of the vector bundle despite involving quantities which naively belong to X. 9 In the gauge in which the dilaton is constant [3], the norm Ω is constant over X and so the compatibility relation between Ω and ω is There is additional term coming from the universal axio-dilaton which in the tradition of special geometry we do not write as it is completely decoupled from all the other parameters. We discuss it below.
Substituting into (B.4) we see that where we have dropped irrelevant numerical constants.
We now show this is equivalent to the four-dimensional dilaton as written in Einstein frame. Indeed, in Einstein frame of the dimensional reduction of heterotic supergravity to R 3,1 , the ten-dimensional dilaton is written 10 where X e = (X 0 , · · · , X 3 ) are the R 3,1 coordinates and x m are the coordinates of X. The first term is the zero mode and so a constant. There is a purely d = 4 fluctuation ϕ(X) and the next non-zero term is Φ 3 which is cubic order in α . The canonical definition of the four-dimensional dilaton, even for non-Kähler manifolds is (see for example [28]): where g s = e Φ 0 is the zero-mode of the dilaton and is the string coupling constant. For the Hull-Strominger system to be a good approximation to string theory, we require g s → 0. V 0 is a reference volume of X, measured at any point in moduli space, and appears in order for the dilaton to be dimensionless.
A consequence of supersymmetry is that where d is the ten-dimensional exterior derivative. We therefore get Supersymmetry here is important. The four-dimensional dilaton appears in a combination with the universal axion -the B-field whose legs span R 3,1 : S = b + ie −2φ 4 . In the language of N = 1, d = 4 supersymmetry S is a component of a linear multiplet.
We we see that, up to irrelevant constants, The second term we often do not include as the universal axio-dilaton is not coupled to any other fields at this order in α and g s and so its contribution to the moduli space metric is somewhat trivial [6]. 10 We again use that modulo an appropriate gauge fixing the dilaton is constant up to order α 3 [3].
As promised, we have shown that K = 2φ 4 and the four-dimensional dilaton is the Kähler potential (B.4) first written in [6]. All we demanded here were the physically reasonable assumptions that supergravity is a good approximation, viz. α , g s → 0.
Supersymmetry does not afford the dilaton functional any protection from α corrections. The linear multiplet is a fully-fledged D-term and so will receive corrections at all orders in α . 11 That being so, we expect the first correction already at α 2 . Many issues arise. It is not even clear, for example, if the four-dimensional dilaton and the Kähler potential for the moduli space metric at α 2 are the same thing. Given the supergravity action and supersymmetry variations are known at this order, determining this correction is an interesting future direction. It would be very interesting to determine the α 3 corrections as well, though the supersymmetry variations and equations of motion are not completely known at this order.
Recently, [21] took (B.7) as the Kähler potential for the moduli space metric to deformations of the first order α Hull-Strominger system assuming the complex structure is fixed. They then related this to the Aeppli and Bott-Chern cohomology. In the limit α → 0 with H → 0, in which the Hull-Strominger system accurately describes string backgrounds, it reproduces [6] as we demonstrated above. Their metric also applies to formal solutions of the Hull-Strominger system one allow for α to be large and for non-trivial dilaton with torsion H = O(1). However, as discussed above, such solutions are not obviously string solutions because of the non-trivial dilaton behaviour and because the α 2 , α 3 , · · · corrections are not included. They do not find, for example, the α 3 correction to the metric predicted from mirror symmetry in the special case of the standard embedding cf. (B.3). This term will dominate of the lower order contributions in such solutions. Furthermore, interesting pathologies can also arise in the large α limit, such as the moduli space metric being degenerate and so not Kähler as required by N = 1 d = 4 supersymmetry. This reinforces the importance of including perturbative α 2 , α 3 , · · · -corrections to the Hull-Strominger system -they will dominate over the lower order Hull-Strominger equations at string scale volumes. No doubt α -corrections in the form of worldsheet instantons will play an even more important role as well -it is these that encode geometric invariants of interest in pure mathematics.
So if we want to apply the formalism of the Hull-Strominger system to examples where H = O(1) then we must necessarily include the α -corrections and g s -corrections to the supersymmetry variations, equations of motion and in turn the moduli space metric. Even if the dilaton is constant, determining how the α -corrections modify the Kähler potential beyond the first order result in [5,6] for heterotic theories remains an open question. These higher order terms will dominate and qualitatively change the behaviour of the metric. For example, such terms should presumably force the metric to remain positive definite even though we have a contribution at order α with a negative sign. We know this must be the case as there are strong arguments from [29,30] that at least in α -perturbation theory these geometries define sigma models that flow to unitary conformal field theories with positive definite kinetic terms for the moduli fields, and so, positive definite metric.
The role of the dilaton is even more mysterious: if there are string solutions on compact manifolds, then they are not described by conventional sigma models. As pointed out in [31], the (0, 2) sigma model for the Hopf surface -a simple geometric example in which the dilaton is non-trivial at zeroth order in α -is not unitary and so the sigma model does not describe a physical string theory. In fact, the dilaton scalar is not even globally well-defined on the Hopf surface, so it is not clear how it solves the supergravity equations of motion (B.1) globally. Finding a suitable modification of this worldsheet theory and Hopf surface geometry so that we have a unitary theory flowing to a unitary (0, 2) conformal field theory would be very interesting. Recently, [22] conjectured some of the algebraic structures such a theory should have -it would be very interesting to try to turn this into a description of a unitary conformal field theory.

C. Constructing a family of D operators
We construct a family of D-operators on the Q bundle of (0, p)-forms that square to zero and DY .
We define D 1 to be with λ p , µ p arbitrary coefficients. We see that D We use the appropriate restriction of moduli space metric (4.5) to compute the adjoint operator Tr F λν δ β A † As a matrix we write this as where H p (∆ (p) α , δ α A (p) ) = 2iε p ∆ (p) µ α (∂ω) µν + α 2 λ p Tr (δ α A (p) F ν ) − Tr (δ α θ (p) R ν ) and D 2 is the natural generalisation of D 1 taken to act on both δA (p) and δθ (p) .
(C.8) these satisfy the consistency equations (C.7). Taken together these consistency checks are non-trivial.
We list two convenient solutions. The first is µ p = λ p = µ p = 1 and ε p = λ p = (−1) p . This corresponds to the D-operator [7] in our conventions. A second choice is µ p = −1 and λ = µ p = λ p = ε p = 1, which avoids any explicit (−1) p factors. In components this second choice is The non-spurious case follows in an identical manner to this case after making the appropriate substitution for δ α θ (p) and redefinition in Z α discussed in the main text. 12 Its worth noting the last term of (2.13) does not appear as it would come from the inner product of a pair of (0, 2)-deformations, Z (0,2) α , which is O(α 2 ) and so dropped.