On Curvature and Torsion in Courant Algebroids

Abstract

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are conventional tensors on the body. Finally, we compute their Ricci and scalar curvature.

Appendices

Generalized Connection on ${{\mathbb {T}}}[1]M$ and Torsion Components

In this appendix, we collect our conventions on the component form of a generalized connection on $E = {{\mathbb {T}}}[1]M$ which are used in the main text, in particular in Sect. 5 for the torsion. A generalized connection D is split into two parts, one along a vector field and another along a one form. More precisely, $D = \nabla + V$ where

$$\begin{aligned} \nabla = \left( \begin{array}{c@{\quad }c} \nabla ^{TT} &{} \nabla ^{TT^*}\\ \nabla ^{T^*T}&{}\nabla ^{T^*T^*} \end{array}\right) , \end{aligned}$$

where $\nabla ^{TT}$ and $\nabla ^{T^*T^*}$ are the connections on TM and $T^*M$ defined by the coefficients¹$\Gamma _\mu {}^\nu {}_\rho \equiv \Gamma _\mu {}^{s_\nu }{}_{s_\rho }$ and $\tilde{\Gamma }_{\mu }{}_\nu {}^\rho \equiv \Gamma _{\mu }{}^{s^\nu }{}_{s^\rho }$, respectively, while $\nabla ^{TT^*}\in \Gamma (T^*M\otimes TM^{\otimes 2})$ and $\nabla ^{T^*T}\in \Gamma (T^*M{}^{\otimes 3})$ are defined by $\Gamma _{\mu }{}^\nu {}^\rho \equiv \Gamma _{\mu }{}^{s_\nu }{}_{s^\rho }$ and $\Gamma _{\mu }{}_{\nu }{}_\rho \equiv \Gamma _{\mu }{}^{s^\nu }{}_{s_\rho }$. Analogously, the component V decomposes as

$$\begin{aligned} V = \left( \begin{array}{c@{\quad }c} V^{TT} &{} V^{TT^*}\\ V^{T^*T}&{}V^{T^*T^*} \end{array}\right) , \end{aligned}$$

so that we define the components of $V^{T^*T}$ by $V^{\mu }{}_{\nu }{}_\rho \equiv V^{\mu }{}^{s^\nu }{}_{s_\rho }$, respectively of $V^{T^*T^*}$ by $V^{\mu }{}_\nu {}^\rho \equiv V^{\mu }{}^{s^\nu }{}_{s^\rho }$. Similarly, for $V^{TT}$ by $\tilde{V}^{\mu }{}^\nu {}_\rho \equiv V^{\mu }{}^{s_\nu }{}_{s_\rho }$ and for $V^{TT^*}$ by $V^{\mu }{}^\nu {}^{\rho }\equiv V^{\mu }{}^{s_\nu }{}_{s^\rho }$.

With these conventions, the direct evaluation of the torsion $T(Q_{{{\mathcal {E}}}}) = Q_{{{\mathcal {E}}}}(\tau _{{\mathcal {M}}})$ gives, as mentioned in chapter 5, three contributions

$$\begin{aligned} T(Q_{{\mathcal {E}}}) = T(\Gamma ) + T(V) + T^{(1,1)}(\Gamma ,V) \end{aligned}$$

(52)

which we now give explicitly in components. They are

$$\begin{aligned} T(\Gamma )= & {} \Gamma _\mu {}^\rho {}_\nu \psi ^\mu \psi ^\nu s_\rho + (\tfrac{1}{2}H_{\rho \mu \nu } + \Gamma _{\mu \rho \nu })\psi ^\mu \psi ^\nu s^\rho \in \Gamma (\Lambda ^2 T^*M\otimes {{\mathbb {T}}}M), \\ T(V)= & {} V^{\mu }{}^\rho {}^\nu b_\mu b_\nu s_\rho + V^\mu {}_\rho {}^\nu b_\mu b_\nu s^\rho \in \Gamma (\Lambda ^2 TM\otimes {{\mathbb {T}}}M), \end{aligned}$$

and

$$\begin{aligned} T^{(1,1)}(\Gamma ,V)= ({\tilde{V}}^{\mu }{}^\rho {}_\nu - \Gamma _\nu {}^\rho {}^\mu )b_\mu \psi ^\nu s_\rho + (V^\mu {}_\rho {}_\nu - {\tilde{\Gamma }}_\nu {}_\rho {}^\mu )b_\mu \psi ^\nu s^\rho + p_\mu s^{\mu }\;. \end{aligned}$$

(53)

It is clear that the first term appearing in $T(\Gamma )$ is the usual torsion of the affine connection $\nabla ^{TT}$. As in the discussion of the (1, 1) component of the curvature, we can introduce an affine connection on TM in order to get a covariant $\tilde{p}$ so that $T^{(1,1)}(\Gamma ,V)-\tilde{p}_\mu s^\mu \in \Gamma (T^*M\otimes TM\otimes {{\mathbb {T}}}M)$. This gives the K-dependent form of the (1, 1) component of the torsion, as described and used in (36) of the main text.

Lagrangian Submanifolds of $T^*[2]T[1]M$

It is known that lagrangian submanifolds of $T^*[2]T[1]M$ invariant under the homological vector field correspond to Dirac structures of the Courant algebroid associated to it (e.g., section 4 of [27]). In this appendix, we work out explicitly the details of the characterization of Dirac structures in this language which is particularly useful for the main text. Let $L \subset {{\mathbb {T}}}M$ be a Dirac structure; we are going to describe the corresponding $d_{{\mathcal {M}}}$-invariant lagrangian submanifold ${{\mathcal {L}}}_L {\mathop {\hookrightarrow }\limits ^{\phi }} T^*[2]T[1]M$.

As a Lie algebroid, L is described by the NQ-manifold L[1] of degree 1. Locally, let us choose for the latter coordinates $(y^\mu ,\lambda ^A)$ for L[1] of degrees (0, 1) and for the Courant algebroid $(x^\mu , \psi ^\mu , b_\mu , p_\mu )$ as in the main text. Finally, let us denote by $\omega = dx^\mu dp_\mu + d\psi ^\mu db_\mu$ the canonical symplectic structure of $T^*[2]T[1]M$. We describe the embedding $\phi$ by the following formulas:

$$\begin{aligned} y^\mu = x^\mu ,\;\;\; \psi ^\mu = \rho ^\mu _A \lambda ^A ,\quad b_\mu = \rho _{\mu A} \lambda ^A , \quad p_\mu = \frac{1}{2}\phi _{\mu A B}\lambda ^A\lambda ^B, \end{aligned}$$

(54)

where $\rho ^\mu _A$ and $\rho _{\mu A}$ denote the components of the maps $\rho _{T^*}$ and $\rho$ introduced in Sect. 4 and $\phi _{\mu AB}$ to be determined. For a basis of vector fields, using (54) this means

$$\begin{aligned} \phi _*\left( \frac{\partial }{\partial y^\mu }\right)&=\frac{\partial }{\partial x^\mu } + \frac{\partial \rho ^\kappa _A}{\partial x^\mu } \lambda ^A \frac{\partial }{\partial \psi ^\kappa } + \frac{\partial \rho _{\kappa A}}{\partial x^\mu }\lambda ^A \frac{\partial }{\partial b_\kappa } + \frac{\partial \phi _{\kappa A B}}{\partial x^\mu } \lambda ^A\lambda ^B \frac{\partial }{\partial p_\kappa }, \end{aligned}$$

(55)

$$\begin{aligned} \phi _*\left( \frac{\partial }{\partial \lambda ^A}\right)&= \rho ^\mu _A \frac{\partial }{\partial \psi ^\mu } + \rho _{\mu A}\frac{\partial }{\partial b_\mu } + \phi _{\mu A B}\lambda ^B \frac{\partial }{\partial p_\mu }. \end{aligned}$$

(56)

${{\mathcal {L}}}_L$ being lagrangian means $\phi ^* \omega = 0$, which gives the following two conditions:

$$\begin{aligned} 0&= (\phi ^*\omega )\Bigl (\frac{\partial }{\partial y^\mu }, \frac{\partial }{\partial \lambda ^A}\Bigr ) = \,\Bigl (\phi _{\mu A B} - \rho ^\kappa _A \frac{\partial \rho _{\kappa B}}{\partial x^\mu } - \frac{\partial \rho ^\kappa _B}{\partial x^\mu } \rho _{\kappa A}\Bigr )\lambda ^B, \end{aligned}$$

(57)

$$\begin{aligned} 0&= (\phi ^*\omega )\Bigl (\frac{\partial }{\partial \lambda ^A},\frac{\partial }{\partial \lambda ^B}\Bigr ) = \,\rho ^\mu _A \rho _{\mu B} + \rho ^\mu _{B}\rho _{\mu A} = 0\;. \end{aligned}$$

(58)

The first condition fixes $\phi _{\mu AB}$ in the transformations (54), and the second one is satisfied thanks to the fact that L is lagrangian.

It is now a direct computation the check that $d_{{\mathcal {M}}}$ restricts to ${{\mathcal {L}}}_L$, indeed $\phi _*(d_{L[1]})=d_{{\mathcal {M}}}$ follows from involutivity of L.