Problem Sheet 7.

You can find Problem Sheet #7 here.

Please, submit some of your solutions by Thursday 26th at 9:00.
You can do that by email or by leaving them under my door (sala 327).

Problem Sheet 6.

You can find Problem Sheet #6 here.

Please, submit some of your solutions by Monday 23rd at 9:00.
You can do that by email or by leaving them under my door (sala 327).

Problem Sheet 5.

You can find Problem Sheet #5 here.

Please, submit some of your solutions by Thursday 12th at 9:00.
You can do that by email or by leaving them under my door (sala 327).

Problem Sheet 4.

You can find Problem Sheet #4 here.

Please, submit some of your solutions by Thursday 5th at 9:00.
You can do that by email or by leaving them under my door (sala 327).

L9a. The definition of a Courant algebroid.

We defined the Dorfman bracket in L7a. We call it bracket although it is not skew-symmetric. One may want to have a proper skew-symmetric bracket. To do so, we just have to skew-symmetrize: $$ [[ v, w ]] = \frac{1}{2}( [v,w] - [w,v] ), $$  in order to get the Courant bracket. They are indeed closely related: $$ [v,w] = [[v,w]] + D\langle v,w\rangle $$

In Problem Sheet 3 we have checked many properties that the Dorfman and the Courant bracket satisfy. They all fit in an abstract object called Courant algebroid. Let us give two definitions of this object, one using a non-skew-symmetric but well behaved Dorfman bracket, and the other one using a skew-symmetric Courant bracket.

A Courant algebroid \( (E,\langle\cdot,\cdot\rangle,[\cdot,\cdot],\pi) \) over a manifold \(M\) consists of a vector bundle \(E\to M\) together with a non-degenerate symmetric bilinear form \(\langle\cdot,\cdot\rangle\) on \(E\), a DORFMAN bracket \([\cdot,\cdot]\) on the sections \(\mathcal{C}^\infty(E)\) and a bundle map \(\pi:E\to TM\) such that the following properties are satisfied:
 (D1): \([v,[w,w']]=[[v,w],w'] + [w,[v,w']],\)
(D2): \(\pi([v,w])=[\pi(v),\pi(w)]\),
(D3): \([v,fw]=f[v,w]+(\pi(v)f)w\),
(D4): \( \pi(v)\langle w,w'\rangle = \langle [v,w], w' \rangle + \langle w, [v,w'] ,w'\rangle \),
(D5): \( [v,v]=\pi^* d\langle v, v\rangle \),
  for any \(v,w,w' \in \Gamma(E)\), \(f,g\in \mathcal{C}^\infty(M)\), where \(D: \mathcal{C}^\infty(M)\to \mathcal{C}^\infty(E)\) is defined by $$\langle D f, v \rangle=\frac{1}{2}\pi(v)(f).$$

A Courant algebroid \( (E,\langle\cdot,\cdot\rangle,[[\cdot,\cdot]],\pi) \) over a manifold \(M\) consists of a vector bundle \(E\to M\) together with a non-degenerate symmetric bilinear form \(\langle\cdot,\cdot\rangle\) on \(E\), a skew-symmetric COURANT bracket \([[\cdot,\cdot]]\) on the sections \(\mathcal{C}^\infty(E)\) and a bundle map \(\pi:E\to TM\) such that the following properties are satisfied:
 (C1): (sorry about how this looks with the double bracket) \([[v,[[w,w']]]]=[[[[v,w]],w']] + [[w,[[v,w']]]] - \frac{1}{3} D( \langle [[v,w]],w'\rangle + \langle [[w,w']], v \rangle + \langle [[w',v]],w \rangle),\)
(C2): \(\pi([[v,w]])=[[\pi(v),\pi(w)]]\),
(C3): \([[v,fw]]=f[[v,w]]+(\pi(v)f)w - \langle v,w \rangle D f\),
(C4): \( \pi(v)\langle w,w'\rangle = \langle [[v,w]]+D\langle v,w\rangle, w' \rangle + \langle w, [[v,w']] + D\langle v,w'\rangle \rangle \),
(C5): \(\pi\circ D =0\), and consequently, \(\langle D f, D g \rangle = 0\),
  for any \(v,w,w' \in \Gamma(E)\), \(f,g\in \mathcal{C}^\infty(M)\), where \(D: \mathcal{C}^\infty(M)\to \mathcal{C}^\infty(E)\) is defined by $$\langle D f, v \rangle=\frac{1}{2}\pi(v)(f).$$

L8a. The group of generalized diffeomorphisms.

Generalized diffeomorphisms are orthogonal automorphisms \( F \) of \(T+T^*\) covering a diffeomorphism \( f \) and preserving the Courant bracket.

We have proved that the group of generalized diffeomorphisms is the semidirect product
$$ \textrm{Diff}(M) \subset \Omega^2_{cl}(M), $$ where the diffeomorphisms \( f \) act by pushforward on \( T + T^* \), and a closed \(2\)-form \(B\) acts as a \( B \)-field: \( X+\xi\mapsto X+\xi+i_X B \). The \( B \)-fields are a new symmetry, not present in classical geometry, and they will play a very important role.

Note that a diffeomorphism and a \(B\)-field do not commute: $$  \exp(B) \circ g_*= g_* \exp(g^*B) .$$

If you look at automorphisms  of  \(T\) covering a diffeomorphism \( f \) and preserving the Lie bracket, you only get the differential of the diffeomorphisms, so this approach is consistent with the symmetries of classical differential geometry.

L7a. A bracket for T+T*.

The Lie bracket of two vector fields \( X, Y \in \mathcal{C}^\infty(T) \) is usually defined as the only vector field \( [X,Y] \) such that $$[X,Y](f)=X(Y(f)) - Y(X(f)).$$ This works very nicely for vector fields, since they are derivations of functions. One checks that \(f\mapsto X(Y(f)) - Y(X(f)) \) is a derivation and we call that derivation \( [X,Y] \).

The Lie bracket is also the only one such that, for any differential form $\phi$, $$ i_{[X,Y]} \phi = [ \mathcal{L}_X, i_Y] \phi = [ [d,i_X],i_Y] \phi,$$ where the bracket on the leftmost term is the Lie bracket that we are defining, while the other brackets are supercommutators ( \( [x,y]=xy-(-1)^{|x||y|}yx \) ).

We would like now to generalize this. We do want a bracket for the sections of \( T+T^* \). Our starting poing should be how they act on a differential form: $$(X+\xi)\cdot \phi = i_X \phi + \xi\wedge \phi .$$ If we only look at functions, we will not be succesful. Whereas $X(f)=i_X df$ is again a function, $(X+\xi)\cdot df $ is not. We must use the second approach. Define the bracket $[X+\xi,Y+\eta]$ as the only generalized vector field satisfying $$ i_{[X+\xi,Y+\eta]} \phi = [ [d,(X+\xi)\cdot],(Y+\eta)\cdot] \phi$$ for any differential form $\phi$. This is called a derived bracket and there is a general theory behind it.

Since the expression we are using to derive the bracket is bilinear on the fields, we can first compute the terms \( [X,Y], [X,\eta], [\xi,Y], [\xi,\eta] \) as we did in the class. The result is the Dorfman bracket. $$[X+\xi,Y+\eta] = [X,Y] + \mathcal{L}_X \eta - i_Y d\xi .$$