##
**Reduction of branes in generalized complex geometry.**
*(English)*
Zbl 1172.53050

Let \(M\) be a manifold having a geometric structure \(\alpha,C\) a submanifold and \(\widetilde C\) is a smooth quotient manifold of \(S\). Then the reduction problem asks whether \(\alpha\) induces same geometric structure to \(\widetilde S\). The Marsden-Weinstein reduction is an example of a answer of a reduction problem. In this paper, a reduction theorem of branes in the sense of M. Gualtieri [Generalized complex geometry, arXiv:math.DG/040122], is proved without using a group action (Theorem 7.4). Then weaken the conditions in the definition of brane, at least for the time being, weak branes are defined (Def. 7.10), and show most part of Theorem 7.4 holds for weak branes (Prop. 7.11). A coisotoropic submanifold \(C\) of a symplectic manifold \(M\) is an example of a weak brane.

Gualtieri’s brane is relevant to physics [cf. A. Kapustin and D. Orlov, J. Geom. Phys. 48, 84–99 (2003; Zbl 1029.81058)]. A brane is a generalized complex submanifold \((C,L)\) of a generalized complex manifold \(M\) satisfying \({\mathcal J}(L)=L\), where \({\mathcal J}\) is the generalized complex structure of \(M\) and \(L\) is a maximal isotropic subbundle of \(E\) and an exact Courant algebroid over \(M\) (Definition 7.2). A generalized complex structure is a suitable endomorphism of \(E\). A Courant algebroid \(E\) is isomorphic to \(TM\oplus T^*M\) with a symmetric bilinear form \(\langle\cdot, \cdot\rangle\), a bilinear bracket \([\cdot,\cdot]\) and a bundle map \(\pi:E\to TM\) called anchor, with suitable conditions (Definition 2.1). If the sequence

\[ 0\to T^*M @>\pi^*>>E @>\pi>>TM\to 0, \]

is exact, \(E\) is called exact (Definition 2.2). This is explained in §2. Then, the reduction theorem for an exact Courant algebroid is presented in §3 [H. Bursztyn, C. R. Cavalcanti and M. Gualtieri, Adv. Math. 211, 726–765 (2007; Zbl 1115.53056), Theorem 3.7., hereafter refered to as [1]]. To apply this reduction theorem to the reduction of the Ševera class (characteristic class of a Courant algebroid (§5), the reduction theorem of a Dirac structure, i.e., a maximal isotropic subbundle of \(E\) which is closed under the Courant bracket, is proved in §4 (Proposition 4.1. cf.[1]).

A generalized complex structure is a vector bundle endmorphism \({\mathcal J}\) of \(E\) which preserves \(\langle\cdot,\cdot \rangle\), such that \({\mathcal J}^2=- \text{Id}_E\) and the Nijenhuis tensor \(N_{\mathcal J}\) vanishes. Two commuting generalized complex strutures \({\mathcal J}_1\), \({\mathcal J}_2\) such that the bilinear form \(\langle{\mathcal J}_1,{\mathcal J}_2,\cdot,\cdot \rangle\) is positive definite, are called a generalized Kähler structure. The reduction theorem of generalized complex structure [Proposition 6.1, cf. M. Stienon and P. Xu, Reduction of generalized complex structures, arXiv:mathDG/0509393)] and generalized Kähler structure (Proposition 6.6, cf.[1]) is proved in §6. The author says ideas and techniques of these proofs are borrowed from cited literatures, but they differ from those in the cited literature, because the generalized complex structure is reduced directly and not viewed as a Dirac structure in the complexification of \(E\).

After these preliminaries, if \((C,L)\) is a brane, then \(C\) is shown to be coisotoropic with respect to the Poisson structure induced by \({\mathcal J}\) on \(M\) and if the quotient \(\underline C\) of \(C\) by the characteristic foliation is smooth, then it is shown \(E\) and \({\mathcal J}\) induce an exact Courant algebra \(\underline E\) and a generalized complex structure \(\underline{\mathcal J}\) on \(\underline C\), and \(L\) induces the structure of a space-filling brane on \(\underline C\) and the Ševera class of \(\underline E\) is trivial (Theorem 7.4). It is also shown except the space-filling property and the statement on the Ševera class, that this holds for weak branes (Proposition 7.11). These are main theorems in this paper. The paper is concluded by showing that if \(C\) is a submanifold of a generalized complex manifold, \(L\) a maximal isotropic subbundle of \(E|_C\) with \(\pi(L)= TC\), and \({\mathcal J}(N^*C)\cap\pi^{-1}(TC)\) is contained in \(L\) and has constant rank, then \((C,\widetilde L)\) is a weak brane (Proposition 7.18). Here, \(\widetilde L\) is the pullback of \(L\) to \(\widetilde M\), a submanifold of \(M\) containing \(C\).

Gualtieri’s brane is relevant to physics [cf. A. Kapustin and D. Orlov, J. Geom. Phys. 48, 84–99 (2003; Zbl 1029.81058)]. A brane is a generalized complex submanifold \((C,L)\) of a generalized complex manifold \(M\) satisfying \({\mathcal J}(L)=L\), where \({\mathcal J}\) is the generalized complex structure of \(M\) and \(L\) is a maximal isotropic subbundle of \(E\) and an exact Courant algebroid over \(M\) (Definition 7.2). A generalized complex structure is a suitable endomorphism of \(E\). A Courant algebroid \(E\) is isomorphic to \(TM\oplus T^*M\) with a symmetric bilinear form \(\langle\cdot, \cdot\rangle\), a bilinear bracket \([\cdot,\cdot]\) and a bundle map \(\pi:E\to TM\) called anchor, with suitable conditions (Definition 2.1). If the sequence

\[ 0\to T^*M @>\pi^*>>E @>\pi>>TM\to 0, \]

is exact, \(E\) is called exact (Definition 2.2). This is explained in §2. Then, the reduction theorem for an exact Courant algebroid is presented in §3 [H. Bursztyn, C. R. Cavalcanti and M. Gualtieri, Adv. Math. 211, 726–765 (2007; Zbl 1115.53056), Theorem 3.7., hereafter refered to as [1]]. To apply this reduction theorem to the reduction of the Ševera class (characteristic class of a Courant algebroid (§5), the reduction theorem of a Dirac structure, i.e., a maximal isotropic subbundle of \(E\) which is closed under the Courant bracket, is proved in §4 (Proposition 4.1. cf.[1]).

A generalized complex structure is a vector bundle endmorphism \({\mathcal J}\) of \(E\) which preserves \(\langle\cdot,\cdot \rangle\), such that \({\mathcal J}^2=- \text{Id}_E\) and the Nijenhuis tensor \(N_{\mathcal J}\) vanishes. Two commuting generalized complex strutures \({\mathcal J}_1\), \({\mathcal J}_2\) such that the bilinear form \(\langle{\mathcal J}_1,{\mathcal J}_2,\cdot,\cdot \rangle\) is positive definite, are called a generalized Kähler structure. The reduction theorem of generalized complex structure [Proposition 6.1, cf. M. Stienon and P. Xu, Reduction of generalized complex structures, arXiv:mathDG/0509393)] and generalized Kähler structure (Proposition 6.6, cf.[1]) is proved in §6. The author says ideas and techniques of these proofs are borrowed from cited literatures, but they differ from those in the cited literature, because the generalized complex structure is reduced directly and not viewed as a Dirac structure in the complexification of \(E\).

After these preliminaries, if \((C,L)\) is a brane, then \(C\) is shown to be coisotoropic with respect to the Poisson structure induced by \({\mathcal J}\) on \(M\) and if the quotient \(\underline C\) of \(C\) by the characteristic foliation is smooth, then it is shown \(E\) and \({\mathcal J}\) induce an exact Courant algebra \(\underline E\) and a generalized complex structure \(\underline{\mathcal J}\) on \(\underline C\), and \(L\) induces the structure of a space-filling brane on \(\underline C\) and the Ševera class of \(\underline E\) is trivial (Theorem 7.4). It is also shown except the space-filling property and the statement on the Ševera class, that this holds for weak branes (Proposition 7.11). These are main theorems in this paper. The paper is concluded by showing that if \(C\) is a submanifold of a generalized complex manifold, \(L\) a maximal isotropic subbundle of \(E|_C\) with \(\pi(L)= TC\), and \({\mathcal J}(N^*C)\cap\pi^{-1}(TC)\) is contained in \(L\) and has constant rank, then \((C,\widetilde L)\) is a weak brane (Proposition 7.18). Here, \(\widetilde L\) is the pullback of \(L\) to \(\widetilde M\), a submanifold of \(M\) containing \(C\).

Reviewer: Akira Asada (Takarazuka)