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We propose a renormalization group flow with emergent supersymmetry in two dimensions from a non-Lagrangian theory. The ultraviolet theory does not have supersymmetry while the infrared theory does. The flow is constrained analytically by topological defect lines including a new spin constraint, and further supported by numerics from the truncated conformal space approach.

We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure [..] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [..] or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.

Let U be a connected non-singular quasi-projective variety and f : A → U a family of abelian varieties of dimension g. Suppose that the induced map U → Ag is generically finite and there is a compactification Y with complement S = Y \U a normal crossing divisor such that Ω1 Y (log S) is nef and ωY (S) is ample with respect to U. We characterize whether U is a Shimura variety by numerical data attached to the variation of Hodge structures, rather than by properties of the map U →Ag or by the existence of CM points. More precisely, we show that f : A → U is a Kuga fibre space, if and only if two conditions hold. First, each irreducible local subsystem V of R1f∗CA is either unitary or satisfies the Arakelov equality. Second, for each factor M in the universal cover of U whose tangent bundle behaves like that of a complex ball, an iterated Kodaira-Spencer map associated with V has minimal possible length in the direction of M. If in addition f : A → U is rigid, it is a connected Shimura subvariety of Ag of Hodge type.

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