Five-dimensional gauge theories and local mirror symmetry. (English) Zbl 1043.81714
Summary: We study the dynamics of 5-dimensional gauge theory on \(M_4\times S^1\) by compactifying type II/M theory on degenerate Calabi-Yau manifolds. We use the local mirror symmetry and show that the prepotential of the 5-dimensional SU(2) gauge theory without matter is given exactly by that of the type II string theory compactified on the local \(F_2\), i.e., Hirzebruch surface \(F_2\) lying inside a non-compact Calabi-Yau manifold. It is shown that our result reproduces the Seiberg-Witten theory at the 4-dimensional limit \(R\rightarrow0\) (\(R\) denotes the radius of \(S^1\)) and also the result of the uncompactified 5-dimensional theory at R\(\rightarrow\infty\). We also discuss SU(2) gauge theory with \(1\leq N_f\leq4\) matter in vector representations and show that they are described by the geometry of the local \(F_2\) blown up at \(N_f\) points.
MSC:
| 81T60 | Supersymmetric field theories in quantum mechanics |
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
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