Heterotic supersymmetry, anomaly cancellation and equations of motion

We show that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimension six this reduces the choice of that connection to the unique SU(3) instanton on a manifold with stable tangent bundle of degree zero.

The bosonic fields of the ten-dimensional supergravity which arises as low energy effective theory of the heterotic string are the spacetime metric g, the NS three-form field strength H, the dilaton φ and the gauge connection A with curvature F A . The bosonic geometry considered in this paper is of the form R 1,9−d × M d where the bosonic fields are non-trivial only on M d , d ≤ 8. One considers the two connections ∇ ± = ∇ g ± 1 2 H, where ∇ g is the Levi-Civita connection of the Riemannian metric g. Both connections preserve the metric, ∇ ± g = 0 and have totally skew-symmetric torsion T ± ijk = g sk (T ± ) s ij = ±H ijk , respectively.
The bosonic part of the ten-dimensional supergravity action in the string frame is [1] where R is the curvature of a connection ∇ on the tangent bundle and F A is the curvature of a connection A on a vector bundle E.
The string frame field equations (the equations of motion induced from the action (1.1)) of the heterotic string up to two-loops [2] in sigma model perturbation theory are (we use the notations in [3]) ∇ g i (e −2φ H i jk ) = 0; (1.2) ∇ + i (e −2φ (F A ) i j ) = 0, The field equation of the dilaton φ is implied from the first two equations above.
The instanton equation, the last equation in (1.3) means that the curvature 2-form F A is contained in the Lie algebra of the Lie group which is the stabilizer of the spinor ǫ. It is known that in dimension 5,6,7 and 8 the stabilizer is the group SU (2), SU (3), G 2 and Spin (7), respectively. An instanton (a solution to the last equation in (1.3)) in dimension 5,6,7 and 8 is a connection with curvature 2-from which is contained in the lie algebra su(2), su(3), g 2 and spin (7), respectively [5,4,6,7,8,9,10].
The Green-Schwarz anomaly cancellation mechanism requires that the three-form Bianchi identity receives an α ′ correction of the form A class of heterotic-string backgrounds for which the Bianchi identity of the three-form H receives a correction of type (1.4) are those with (2,0) world-volume supersymmetry. Such models were considered in [11]. The target-space geometry of (2,0)-supersymmetric sigma models has been extensively investigated in [11,4,12]. Recently, there is revived interest in these models [13,14,15,9,3] as string backgrounds and in connection to heterotic-string compactifications with fluxes [16,17,18,19,20,21,22,23].
In writing (1.4) there is a subtlety to the choice of connection ∇ on M d since anomalies can be cancelled independently of the choice [24]. Different connections correspond to different regularization schemes in the two-dimensional worldsheet non-linear sigma model. Hence the background fields given for the particular choice of ∇ must be related to those for a different choice by a field redefinition [25]. Connections on M d proposed to investigate the anomaly cancellation (1.4) are ∇ g [4,9], ∇ + [14], ∇ − [1,16,3,26], Chern connection ∇ c when d = 6 [4,20,21,22,23].
It is known [27,15] ( [3] for dimension d = 6), that the equations of motion of type I supergravity (1.2) with R = 0 are automatically satisfied if one imposes, in addition to the preserving supersymmetry equations (1.3), the three-form Bianchi identity (1.4) taken with respect to a flat connection on T M, R = 0.
According to no-go (vanishing) theorems (a consequence of the equations of motion [28,27]; a consequence of the supersymmetry [29,30] for SU(n)-case and [9] for the general case) there are no compact solutions with non-zero flux and non-constant dilaton satisfying simultaneously the supersymmetry equations (1.3) and the three-form Bianchi identity (1.4) if one takes flat connection on T M , more precisely a connection satisfying T r(R ∧ R) = 0. Therefore, in the compact case one necessarily has to have a non-zero term T r(R ∧ R). However, under the presence of a non-zero curvature 4-form T r(R ∧ R) the solution of the supersymmetry equations (1.3) and the anomaly cancellation condition (1.4) obeys the second and the third equations of motion but does not always satisfy the Einstein equation of motion (the first equation in (1.2)) [3]. A quadratic expression for R which is necessary and sufficient condition in order that (1.3) and (1.4) imply (1.2) in dimension five, six, seven and eight are presented in [31,32,33]. In particular, if R is an instanton the supersymmetry equations together with the anomaly cancellation condition imply the equations of motion.
In this note we show that the converse statement holds. The main goal of the paper is to prove In the compact case in dimension six, it is shown in [32, Theorem 1.1b] that the no-go theorems in [29,30] force the flux H to vanish and the dilaton φ to be a constant for any compact solution to the heterotic supersymmetry (1.3) such that the (-)-connection on the tangent bundle is an SU (3)-instanton, i.e. such a solution is a Calabi-Yau manifold. This result combined with Theorem 1.1 leads to  This suggests that in order to find compact heterotic supersymmetric solutions to the equations of motion (1.2) in dimension six one needs to start with a conformally balanced hermitian six manifold admitting holomorphic complex volume form with stable tangent bundle of degree zero and take the corresponding unique SU (3)-instanton in (1.4) and (1.1).
Six dimensional compact supersymmetric solutions with non-zero flux H and constant dilaton of this kind are presented in [32].
In the context of perturbation theory the curvature R − of the (-)-connection is an one-loop-instanton due to the well known identity R + ijkl − R − klij = 1 2 dT ijkl , the first equation in (1.3) and (1.4) taken with respect to the (-)-connection. We thank the referee reminding this point to us. In this case, according to Theorem 1.1, the supersymmetry (1.3) together with the anomaly cancellation (1.4) imply the heterotic equations of motion (1.2) up to two loops. In fact the SU(3) case in dimension six has originally been dealt in [3]. The G 2 case in dimension seven has been investigated in [37, Section 6] when the anomaly cancellation has no zeroth order terms in α ′ . Compact up to two loops solutions in dimension six with non-zero flux H and non-constant dilaton involving the (-)-connection are constructed in [38].
If the anomaly cancellation has zeroth order term in α ′ (for example in heterotic near horizons associated with AdS 3 investigated in the very recent paper [39]) then R − is no longer one-loop instanton. In particular, in dimension six, Corollary 1.2 and Remark 1.3 is applicable suggesting a possible lines for further investigations.
One can take the anomaly contribution which appears at order α ′ as exact. Suppose that (1.4) is exact in the first order in α ′ . Then, in dimension six Corollary 1.2 applies and arguments in Remark 1.3 could be helpful in further developments.
Conventions: We choose a local orthonormal frame e 1 , . . . , e d , identifying it with the dual basis via the metric and write e i1i2...ip for the monomial e i1 ∧ e i2 ∧ · · · ∧ e ip .
We rise and lower the indices with the metric and use the summation convention on repeated indices. For example, The Hodge star operator on a d-dimensional manifold is denoted by * d .
A consequence of the gravitino and dilatino Killing spinor equations is an expression of the Ricci tensor Ric + mn = R + imnj g ij of the (+)-connection, and therefore an expression of the Ricci tensor Ric g of the Levi-Civita connection, in terms of the suitable trace of the torsion three-form T = H (the Lee form) and the exterior derivative of the torsion form dT = dH (see [40] for dimensions 5 and 7, [29] for dimension 6 (more precisely for any even dimension) and [44] for dimension 8 as well as [31,32,33]).
We recall that the Ricci tensors of ∇ g and ∇ + are connected by (see e.g. [40,33]) The existence of ∇ + -parallel spinor in dimension 5 determines an almost contact metric structure whose properties as well as solutions to gravitino and dilatino Killing-spinor equations are investigated in [40,41,31].
We recall that an almost contact metric structure consists of an odd dimensional manifold M 2k+1 equipped with a Riemannian metric g, vector field ξ of length one, its dual 1-form η as well as an endomorphism ψ of the tangent bundle such that ψ(ξ) = 0, The Reeb vector field ξ is determined by the equations η(ξ) = η s ξ s = 1, (ξ dη) i = dη si ξ s = 0, where denotes the interior multiplication. The fundamental form F is defined by F (., .) = g(., ψ.), F ij = g is ψ s j and the Nijenhuis tensor N of an almost contact metric structure is given by An almost contact metric structure is called normal if N = 0; contact if dη = 2F ; quasi-Sasaki if N = dF = 0; Sasaki if N = 0, dη = 2F . The Reeb vector field ξ is Killing in the last two cases [49].

2.2.
Dimension six. Proof of Theorem 1.1 in d = 6. The necessary and sufficient condition for the existence of solutions to the first two equations in (1.3) in an even dimension were derived by Strominger [4] and investigated by many authors since then. Solutions are complex conformally balanced manifold with non-vanishing holomorphic volume form satisfying an additional condition.
In dimension six any solution to the gravitino Killing spinor equation reduces the holonomy group Hol(∇ + ) ⊂ SU (3). This defines an almost hermitian structure (g, J) with non-vanishing complex volume form [4] which is preserved by the torsion connection. We adopt for the Kähler form Ω ij = g is J s j . The Lee form θ 6 is defined by An almost hermitian structure admits a (unique) linear connection ∇ + with torsion 3-form preserving the structure, i.e. ∇ + g = ∇ + J = 0, if and only if the Nijenhuis tensor is totally skew-symmetric [40]).
In addition, the dilatino equation forces the almost complex structure to be integrable and the Lee form to be exact determined by the dilaton. The torsion (the NS three-form H) is given by [4] (2.10) Since ∇ + g = ∇ + J = 0 the restricted holonomy group Hol(∇ + ) of ∇ + contains in U (k) and Hol(∇ + ) ⊂ SU (k) is equivalent to the next curvature condition found in [29, Proposition 3.1] In addition to these equations, the vanishing of the gaugino variation requires the 2-form F A to be of instanton type. In dimension six, an SU (3)-instanton (or a hermitian-Yang-Mils connection) is a connection A with curvature two form F A ∈ su(3). The SU (3)-instanton condition is In complex coordinates the condition (2.12) reads F A µν = F Ā µν = 0, F A µν Ω µν = 0 which is the well known Donaldson-Uhlenbeck-Yau instanton.

Theorem 1.1 in dimension 6.
Proof. We need to investigate the Einstein equation of motion in dimension 6. Substitute the second equation of (2.10) into (2.11) and the obtained equality insert into (2.1) and use 2∇ g = ∇ + − T to get [29] where we used that on a complex manifold dT = 2 √ −1∂∂Ω is a (2,2)-form and therefore J s j dT islm Ω lm is symmetric in i and j.
We briefly recall the notion of a G 2 structure. Consider the three-form Θ on R 7 given by Θ = e 127 − e 236 + e 347 + e 567 − e 146 − e 245 + e 135 .
The subgroup of GL(7, R) fixing Θ is the Lie group G 2 of dimension 14. The 3-form Θ corresponds to a real spinor and therefore, G 2 can be identified as the isotropy group of a non-trivial real spinor. The Hodge star operator supplies the 4-form * 7 Θ given by * 7 Θ = e 3456 + e 1457 + e 1256 + e 1234 + e 2357 + e 1367 − e 2467 .
We have the well known formula (see e.g. [53,9,54,55]) A 7-dimensional Riemannian manifold M is called a G 2 -manifold if its structure group reduces to the exceptional Lie group G 2 . The existence of a G 2 -structure is equivalent to the existence of a global nondegenerate three-form which can be locally written as (2.3).
The 4-form Φ is self-dual and the 8-form Φ ∧ Φ coincides with the volume form of R 8 . The subgroup of GL(8, R) which fixes Φ is isomorphic to the double covering Spin (7) of SO (7). The 4-form Φ corresponds to a real spinor and therefore, Spin (7) can be identified as the isotropy group of a non-trivial real spinor. We have the well known formula (see e.g. [9]) A Spin (7)-structure on an 8-manifold M is by definition a reduction of the structure group of the tangent bundle to Spin(7). This can be described geometrically by saying that there exists a nowhere vanishing global differential 4-form Φ on M which can be locally written as (2.29).