Geometric engineering on flops of length two
- 4 Downloads
Type IIA on the conifold is a prototype example for engineering QED with one charged hypermultiplet. The geometry admits a flop of length one. In this paper, we study the next generation of geometric engineering on singular geometries, namely flops of length two such as Laufer’s example, which we affectionately think of as the conifold 2.0. Type IIA on the latter geometry gives QED with higher-charge states. In type IIB, even a single D3-probe gives rise to a nonabelian quiver gauge theory. We study this class of geometries explicitly by leveraging their quiver description, showing how to parametrize the exceptional curve, how to see the flop transition, and how to find the noncompact divisors intersecting the curve. With a view towards F-theory applications, we show how these divisors contribute to the enhancement of the Mordell-Weil group of the local elliptic fibration defined by Laufer’s example.
KeywordsD-branes Differential and Algebraic Geometry F-Theory Brane Dynamics in Gauge Theories
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
- A. Bondal and D. Orlov, Derived categories of coherent sheaves, math/0206295.
- J. Kollár, Flops, Nagoya Math. J. 113 (1989) 15.Google Scholar
- M. Reid, Minimal models of canonical 3-folds, in Algebraic varieties and analytic varieties, S. IItaka ed., North-Holland, Amsterdam The Netherlands 1983.Google Scholar
- H.B. Laufer, On ℂℙ1 as an exceptional set, in Recent developments in several complex variables, J.E. Fornasses ed., Princeton University Press, Princeton U.S.A. (1981).Google Scholar
- Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series volume 146, Cambridge University Press, Cambridge U.K. (1990).Google Scholar
- M. van den Bergh, Non-commutative crepant resolutions, in The legacy of Niels Henrik Abel, R. Piene and A. Laudal eds., Springer, Germany (2004).Google Scholar
- M. Wemyss, private communication.Google Scholar
- M. Reineke, Quiver moduli and small desingularizations of some git quotients, arXiv:1511.08316.
- H.C. Pinkham, Factorization of birational maps in dimension 3, in Singularities, Part 2, P. Orlik ed., American Mathematical Society, Providence U.S.A. (1983).Google Scholar
- A. Collinucci, M. Fazzi, D.R. Morrison and R. Valandro, to appear.Google Scholar