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Math-Physics Joint Seminar

Tuesday, October 29, 2019 - 4:30pm

Herb Clemens

Ohio State University


University of Pennsylvania

DRL 4c2

In joint work with Stuart Raby, we propose a phenomenilogically consistent global model for Heterotic/F-theory duality. Our principal challenge was to overcome the seeming contradiction that a Z_2 action used to break symmetry to that of the Standard Model in the Heterotic model and still yield a Calabi-Yau fourfold on the F-theory side seemed to break F-theory symmetry in a way that is incompatible with that of the Standard Model. With considerable guidance from Dave Morrison, Sakura Schäfer-Nameki, and Tony Pantev, we were finally led to overcome this difficulty by re-examining two peices of the puzzle: 1) the Narasihham-Seshadri equivalence that equates flat E_8(real) bundles with flat complex holomorphic complex E_8(complex) vector bundles on a Riemann surface, 2) the properties of the Tate form that govern the evolution of symmetry-breaking from the initial E_8 x E_8 state that allow the Tate form (with the help of the Jacobson-Morosov theorem) to uniquely determine an explicit deformation of the E_8 rational double-point singularity. We notice in 1) that there are actually two (conjugate) complex structures that can be assigned to the given real bundle, With repect to 2) we apply the Brieskorn-Grothendieck equivariant crepant resolution of the E_8 rational double-point singularity. This resolution is built intrinsically from E_8(complex) and therefore there are two resolutions that differ by the outer automorphism of the real group E_8(complex) given by complex conjugation. (We are tempted to call the passage between the two resolutions an ’equivariant semi-universal flop.’)  Incorporating 1) and 2) into the  Z_2-action allow it to preserve initial E_8(real) symmetry but act as complex conjugation on the associated complex algebraic group, thereby avoiding the seeming contradiction mentioned at the outset.