# Root systems of torus graphs and automorphism groups of torus manifolds

# Root systems of torus graphs and automorphism groups of torus manifolds

A torus manifold is a compact oriented 2n-dimensional T^n-manifold with fixed points. We can define a labelled graph from given torus manifold as follows: vertices are fixed points; edges are invariant 2-spheres; edges are labelled by tangential representations around fixed points. This labelled graph is called a torus graph (this may be regarded as the special class of generalized GKM graphs). It is known that we can compute the equivariant cohomology of torus manifold by using combinatorial data of torus graphs. In this talk, we study when torus actions of torus manifolds can be induced from non-abelian compact connected Lie group (i.e. when torus actions can be extended to non-abelian compact Lie group actions). To do this, we introduce root systems of torus graphs. By using this root system, we can characterize what kind of compact connected non-abelian Lie group (whose maximal torus is T^n) acts on the torus manifold. This is a joint work with Mikiya Masuda.