Relatively spin Lagrangian data

Let $(M,\omega )$ be a smooth symplectic manifold and $i:L\hookrightarrow M$ a compact smooth embedded submanifold without boundary of dimension $n=\frac{1}{2}\dim M$. Lagrangian means $i^{\ast }\omega =0$: at each $p\in L$ this says $T_{p}L\subseteq (T_{p}L{)}^{\omega }$, where $W^{\omega }=\{v\in T_{p}M:{\omega }_p(v,w)=0\text{ for all }w\in W\}$. The isomorphism ${\omega }_p^{\mathrm{♭}}:T_{p}M\to T_p^{\ast }M$ identifies $(T_{p}L{)}^{\omega }$ with the annihilator of $T_{p}L$, which has dimension $2n-n=n$; hence the inclusion is equality, the tangent-space Lagrangian condition. Choose as auxiliary data a locally finite triangulation of the pair $(M,L)$: a locally finite simplicial complex $K$, a homeomorphism $h:\mathrm{\mid }K\mathrm{\mid }\to M$, and a subcomplex $K_L\subseteq K$ with $h(\mathrm{\mid }{K}_L\mathrm{\mid })=L$. For each integer $q\ge 0$, let $K^{(q)}$ be the union of simplices of dimension at most $q$ and write $M^{(q)}=h(\mathrm{\mid }{K}^{(q)}\mathrm{\mid })$. Put $L^{(2)}=L\cap M^{(2)}=h(\mathrm{\mid }{K}_L^{(2)}\mathrm{\mid })$ and let $j_2:L^{(2)}\hookrightarrow M^{(2)}$ be the inclusion.

For a Euclidean vector space ${\mathbb{R}}^r$ with $r\ge 1$, its real Clifford algebra is generated by ${\mathbb{R}}^r$ with relations $v^2=-\parallel v{\parallel }^2$. The group $\mathrm{Spin}(r)$ consists of even products of unit vectors in the Clifford algebra, and its action on ${\mathbb{R}}^r$ by conjugation gives $\rho :\mathrm{Spin}(r)\to \mathrm{SO}(r)$; this is the standard double cover for $r\ge 2$, while for $r=1$ it is the two-to-one map $\{\pm 1\}\to \{1\}$. For $r=0$, set $\mathrm{Spin}(0)=\mathrm{SO}(0)=\{1\}$, so the rank-zero bundle has its unique identity lift. A spin structure on an oriented rank-$r$ Euclidean vector bundle $E\to X$ is a principal $\mathrm{Spin}(r)$-bundle $\tilde{P}\to X$ together with a map to the oriented orthonormal frame bundle $P_{\mathrm{SO}}(E)$ that is equivariant under $\rho$ and covers the identity on $X$.

A relative-spin datum for $L\subset M$ consists of an orientation and Euclidean bundle metric on $TL$, an oriented finite-rank Euclidean vector bundle $V\to M^{(3)}$, and a spin structure on the bundle $TL{\mathrm{\mid }}_{L^{(2)}}\oplus j_2^{\ast }(V{\mathrm{\mid }}_{M^{(2)}})$ over $L^{(2)}$, with its direct-sum orientation and metric. The submanifold is relatively spin if such a background bundle and lift exist; a chosen tuple is a relative-spin structure. This is the bundle-and-lift form of the source's relative-spin data. A spin structure on $L$ itself supplies the special case with trivial rank-zero $V$, by restriction to $L^{(2)}$.

This definition records orientation and relative-spin choices as part of Lagrangian brane data. It does not assert existence for every Lagrangian.

Symplectic form and symplectic manifold

Isotropic, coisotropic, symplectic, and Lagrangian submanifolds