Maxime Gabella

K-decompositions and 3d gauge theories

A bstractThis paper combines several new constructions in mathematics and physics. Mathematically, we study framed flat PGL(K, ℂ)-connections on a large class of 3-manifolds M with boundary. We introduce a moduli space ℒK(M) of framed flat connections on the boundary ∂M that extend to M. Our goal is to understand an open part of ℒK(M) as a Lagrangian subvariety in the symplectic moduli space XKun∂M

Generalized Geometry in AdS/CFT and Volume Minimization

{\mathcal{X}}_K^{\mathrm{un}}\left(\partial M\right)

Exactly marginal deformations from exceptional generalised geometry

of framed flat connections on the boundary — and more so, as a “K2-Lagrangian,” meaning that the K2-avatar of the symplectic form restricts to zero. We construct an open part of ℒK(M) from elementary data associated with the hypersimplicial K-decomposition of an ideal triangulation of M, in a way that generalizes (and combines) both Thurston’s gluing equations in 3d hyperbolic geometry and the cluster coordinates for framed flat PGL(K, ℂ)-connections on surfaces. By using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of ℒK(M) is K2-isotropic as long as ∂M satisfies certain topological constraints (theorem 4.2). In some cases this easily implies that ℒK(M) is K2-Lagrangian. For general M, we extend a classic result of Neumann and Zagier on symplectic properties of PGL(2) gluing equations to reduce the K2-Lagrangian property to a combinatorial statement.Physically, we translate the K-decomposition of an ideal triangulation of M and its symplectic properties to produce an explicit construction of 3d N=2

The free energy of

\mathcal{N}=2

supersymmetric AdS

superconformal field theories TK [M] resulting (conjecturally) from the compactification of K M5-branes on M. This extends known constructions for K = 2. Just as for K = 2, the theories TK [M] are described as IR fixed points of abelian Chern-Simons-matter theories. Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between Nf = 1 SQED and the XYZ model. In the large K limit, we find evidence that the degrees of freedom of TK [M] grow cubically in K.

_4

We explicitly construct the largest dataset to date of heterotic vacua arising from stable vector bundles on Calabi-Yau threefolds. Focusing on elliptically fibered Calabi-Yau manifolds with spectral cover bundles, we show that the number of heterotic models with non-zero number of generations is finite. We classify these models according to the complex base of their Calabi-Yau threefold and to the unification gauge group that they preserve in four dimensions. This database of the order of 10^7 models, which includes potential Standard Model candidates, is subjected to some preliminary statistical analyses. The additional constraint that there should be three net generations of particles gives a dramatic reduction of the number of vacua.