Abstract. String theory on singular spaces is a rich playground to study QFT's. On the one hand, D3-branes probing singularities give rise to quiver gauge theories. On the other hand, M-theory on ALE-fibered spaces gives rise to gauge theories, whereby M2-branes wrapping exceptional curves correspond to roots of Lie algebras.

So far, physicists have mainly studied probes of non-compact toric Calabi-Yau spaces on the one hand, and threefolds that give rise to simple flops "of length one" on the M-theory side.

Recently, two important pieces of mathematics have fruitfully been combined: The simultaneous Grothendieck resolution of families of ADE singularities, and representations of Non-commutative Crepant Resolutions of singular threefolds. This opens the door to a whole new world of singular threefolds that were impossible to study before from either point of view. From the D3-probe perspective, it means new non-Abelian quiver gauge theories. From the M-theory side, it means gauge theories with matter fields of higher electric charge.

In the work that I will present, I will pedagogically introduce these notions. I will then work out an explicit example, and show how to see a flop transition in the quiver representation language. So far, such flop transitions were only understood abstractly. I will then show how such geometries can be applied to geometric engineering in M-theory.
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