论文标题
$ b $ branes上的杨米尔斯田地
Yang-Mills fields on $B$-branes
论文作者
论文摘要
Considering the $B$-branes over a complex manifold $Y$ as objects of the bounded derived category $D^b(Y)$, we define holomorphic gauge fields on $B$-branes and the Yang-Mills functional for these fields.These definitions are a generalization to $B$-branes of concepts that are well known in the context of vector bundles.给定的$ {\ Mathscr f}^{\ bullet} \在d^b（y）$中，我们表明atiyah class $ a（{\ Mathscr f}^{\ bulter}）\ in {\ rm ext} f}^{\ bullet}））$是$ {\ mathscr f}^{\ bullet} $上量规字段的阻塞。我们确定了$ \ Mathbb {cp}^n $的$ b $ branes，允许holomorphic仪表字段。我们证明，$ b $ -brane $ {\ mathscr f}^{\ bullet} $的一组Yang-mills字段（如果是非空的）与代数子集的点相通信，则与$ {\ Mathbb c}^m $ m $ m \ cdot $ $ cdot $ pregynomial sepections $ { $ m = {\ rm dim} \，{\ rm hom}（{\ mathscr f}^{\ bulet}，\，ω^1（{\ sathscr f}^{\ buleast}）$，$ s $是非零同时组的数量f}^{\ bullet}）$。我们在他们的情况下显示出足够的条件，任何Yang-Mills田地在排名$ 1 $的反身捆上是平坦的。
Considering the $B$-branes over a complex manifold $Y$ as objects of the bounded derived category $D^b(Y)$, we define holomorphic gauge fields on $B$-branes and the Yang-Mills functional for these fields.These definitions are a generalization to $B$-branes of concepts that are well known in the context of vector bundles. Given ${\mathscr F}^{\bullet}\in D^b(Y)$, we show that the Atiyah class $a({\mathscr F}^{\bullet})\in{\rm Ext}^1({\mathscr F}^{\bullet},\,Ω^1({\mathscr F}^{\bullet}))$ is the obstruction to the existence of gauge fields on ${\mathscr F}^{\bullet}$. We determine the $B$-branes over $\mathbb{ CP}^n$ that admit holomorphic gauge fields. We prove that the set of Yang-Mills fields on the $B$-brane ${\mathscr F}^{\bullet} $, if it is nonempty, is in bijective correspondence with the points of an algebraic subset of ${\mathbb C}^m$ defined by $m\cdot s$ polynomial equations of degree $\leq 3$, where $m={\rm dim}\,{\rm Hom}({\mathscr F}^{\bullet},\,Ω^1({\mathscr F}^{\bullet}))$ and $s$ is the number of non-zero cohomology sheaves ${\mathscr H}^i({\mathscr F}^{\bullet})$. We show sufficient conditions under them any Yang-Mills field on a reflexive sheaf of rank $1$ is flat.
