Toric geometry of G2-manifolds

TY - JOUR

T1 - Toric geometry of G2-manifolds

AU - Madsen, Thomas Bruun

AU - Swann, Andrew Francis

PY - 2019/12

Y1 - 2019/12

N2 - We consider G 2 –manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of T 3 –actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three-and four-form, we derive a Gibbons–Hawking type ansatz giving the geometry on an open dense set in terms a symmetric 3 ☓ 3 matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to G 2 . We prove that the multimoment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.

AB - We consider G 2 –manifolds with an effective torus action that is multi-Hamiltonian for one or more of the defining forms. The case of T 3 –actions is found to be distinguished. For such actions multi-Hamiltonian with respect to both the three-and four-form, we derive a Gibbons–Hawking type ansatz giving the geometry on an open dense set in terms a symmetric 3 ☓ 3 matrix of functions. This leads to particularly simple examples of explicit metrics with holonomy equal to G 2 . We prove that the multimoment maps exhibit the full orbit space topologically as a smooth four-manifold containing a trivalent graph as the image of the set of special orbits and describe these graphs in some complete examples.

U2 - 10.2140/gt.2019.23.3459

DO - 10.2140/gt.2019.23.3459

M3 - Journal article

SN - 1465-3060

VL - 23

SP - 3459

EP - 3500

JO - Geometry & Topology

JF - Geometry & Topology

IS - 7

ER -