Ricci tensor in graded geometry

We define the notion of the Ricci tensor for NQ symplectic manifolds of degree 2 and show that it corresponds to the standard generalized Ricci tensor on Courant algebroids. We use an appropriate notion of connections compatible with the generalized metric on the graded manifold.


Generalized metric
In what follows, we shall use E to denote an NQ symplectic manifold of degree 2 [15]. This means that E is an N-graded manifold 1 , equipped with a symplectic form of degree 2, and a degree 1 symplectic vector field Q E , satisfying Q 2 E = 0. There is an associated sequence of fibrations which corresponds to the subsheafs generated by coordinates of degrees up to 2, up to 1, and up to 0, respectively. In particular, the last arrow gives a vector bundle.
A generalized metric is, in the graded language, a symplectic involution ι on E, which preserves the basis E 0 , i.e. it is a diffeomorphism of E satisfying Given a generalized metric, it is always possible to locally choose the coordinates x i , e a , eȧ, p i on E of degrees 0, 1, 1, 2, respectively, such that ω = dp i dx i + de a de a + deȧdeȧ, where e a := g ab e b , eȧ := gȧ˙be˙b for some constants g ab , gȧ˙b. We will sometimes denote the coordinates e a , eȧ collectively as e α . 2 It is easy to see that Q E is always Hamiltonian, i.e. it comes from a degree 3 function H. The most general such function has the form where, in setting e α := g αβ e β , we extended g ab , gȧ˙b to g αβ by adding g aḃ = gȧ b = 0. The condition Q 2 E = 0 translates to the classical master equation {H, H} = 0. Since ι preserves the degree, it induces an involution on E 1 . Because of the basefixing condition, ι is in turn fully determined by the corresponding fixed point set V 1 ⊂ E 1 . Seeing V 1 as a vector bundle over E 0 , we can pull it back along E → E 0 to obtain a bundle V → E. The latter bundle is locally described by coordinates x i , e a , ξ a , eȧ, p i , with one copy of ξ a , deg ξ a = 1, corresponding to each e a . Finally, notice that E 1 , V 1 , and V are all symplectic vector bundles.
Remark. The relation to Courant algebroids is as follows [11,13]. The Courant algebroid is given by the ordinary vector bundle E = E 1 [−1] over E 0 . (In other words, x i are coordinates on the base and e α correspond to the linear coordinates on the fibers of the algebroid.) The coefficients c and ρ give the structure functions of the bracket and the anchor, respectively, while g encodes the fiberwise inner product. Finally, the involution ι corresponds to the usual viewpoint of generalized metric as a fiberwise reflection on the Courant algebroid, or equivalently, as a subbundle an ordinary manifold E 0 together with a sheaf of N-graded commutative algebras, locally of the form C ∞ (U) ⊗ S(V ), with S(V ) the free graded commutative algebra generated by a finite-dimensional vector space V = m i=1 Vm, and U an open subset of E 0 2 The proof goes as follows: Using the (graded) Darboux theorem we first find coordinates x i , e α , p i such that ω = dp i dx i + g αβ de α de β , for g αβ a diagonal matrix with only 1 and −1 on the diagonal. It follows that the (x-dependent) matrix R α β defined by ι * e α = R α β (x)e β is idempotent and orthogonal w.r.t. g, and thus can be made into the form diag(1, . . . , 1, −1, . . . , −1) using Making an appropriate shift of p i , the form of ω is preserved in the new coordinates.

Tautological section and contraction
Let us now denote the vector bundle morphism V → V 1 by ϕ. There is a unique section τ ′ : E → V such that ϕ • τ ′ coincides with the map E → E 1 → V 1 (the last arrow is the orthogonal projection). Since the bundle V is symplectic, we get an induced section τ on the dual bundle V * → E. We call τ the tautological section [1]. More concretely, identifying sections of V * with functions on V which are linear in the fiber coordinates, we get τ = e a ξ a . The involution on C ∞ (V) (induced by ι) allows us to split any vector field on V into the sum of its self-dual and anti-self-dual part. We will denote the anti-selfdual part of a vector field D by πD. Let us now consider a special subspace End 2 of the space of vector fields, given by degree 2 bundle endomorphisms of V * . Locally we have . We define the contraction map C : End 2 → Γ(V ⊥ * 1 ⊗V * 1 ) as the projection π followed by the contraction of the first and third factor in the last expression. Explicitly,

Connections, torsion and curvature
Following [1], a connection on a V * → E is a degree 1 vector field Q on V, which projects to Q E , and which preserves the space of sections Γ(V * ) ⊂ C ∞ (V). The torsion is then a particular section of V * , defined as Qτ . The curvature of Q is the vector field Q 2 ≡ 1 2 [Q, Q] on V. One easily sees that Q 2 ∈ End 2 . We will say that a connection Q on V is Levi-Civita if its torsion is invariant under the induced involution on Γ(V * ). Finally, we define the Ricci tensor Ric by Ric := CQ 2 ∈ Γ(V ⊥ * 1 ⊗ V * 1 ). Remark. The contraction corresponds to the usual procedure for obtaining the Ricci tensor from the Riemann tensor. The insertion of the projection π keeps only the part of the tensor which can be identified, via Γ(V ⊥ * 1 ⊗ V * 1 ) ∼ = Hom E0 (V 1 , V ⊥ 1 ), with an infinitesimal deformation of the generalized metric. 3 This is the generalized Ricci flow [17]. ⊳ Passing again to a coordinate description, a general connection on V * has the form

Its torsion is
Since the involution preserves the coordinates ξ a , the invariance of torsion is equivalent to the constraint (1) ψ a bȧ = c a bȧ . In particular, the coefficients ψ a bc are left unrestricted. However, as we will see, in the Levi-Civita case the curvature only depends on ψ a bc through the trace Concretely, a short calculation (see Appendix) reveals that for a Levi-Civita connection Q, This recovers exactly the formula for the generalized Ricci tensor from [16].
Remark. More precisely, using the notation of [16] we have where the divergence operator is given by div(e a ) = −λ a . ⊳

Connection with Courant algebroid connections
From the definition it follows that connections are in one-to-one correspondence with linear maps 4Q : . we can understandQ as a map Dually, we have a map ∇ : Γ(V + ) → Γ(E * ⊗ V + ) (satisfying the same Leibniz identity). This is known in the literature as the Courant algebroid (or generalized) connection [5].

The exact case
Let us now consider the following example [13,14]. 5 First, In particular, we have a vector-bundle isomorphism T E 0 ∼ = V + . Combining this with the result of the previous section, a Levi-Civita connection on V * is now given by an ordinary connection on T E 0 . Let us now take Q given by the Levi-Civita connection (in the usual sense) on T E 0 w.r.t. the metric on E 0 . Choose a local frame E a on T E 0 satisfying E a , E b = ±δ ab . We define the frame F a on T * E 0 by F a := E a , · and we denote by E a , F a the induced fiber coordinates on where Ric η is the (usual) Ricci tensor on T E 0 for the metric connection (w.r.t. ·, · ) with torsion given by η. For the proof of this fact we refer the reader to [16]. It can also be verified by a direct calculation.