Bulk reconstruction in moduli space holography

Abstract

It was recently suggested that certain UV-completable supersymmetric actions can be characterized by the solutions to an auxiliary non-linear sigma-model with special asymptotic boundary conditions. The space-time of this sigma-model is the scalar field space of these effective theories while the target space is a coset space. We study this sigma-model without any reference to a potentially underlying geometric description. Using a holographic approach reminiscent of the bulk reconstruction in the AdS/CFT correspondence, we then derive its near-boundary solutions for a two-dimensional space-time. Specifying a set of Sl(2, ℝ) boundary data we show that the near-boundary solutions are uniquely fixed after imposing a single bulk-boundary matching condition. The reconstruction exploits an elaborate set of recursion relations introduced by Cattani, Kaplan, and Schmid in the proof of the Sl(2)-orbit theorem. We explicitly solve these recursion relations for three sets of simple boundary data and show that they model asymptotic periods of a Calabi-Yau threefold near the conifold point, the large complex structure point, and the Tyurin degeneration.

Asymptotic String Compactifications

“…This leads us to a contradiction, where the initially large baryonic charge seems to have disappeared completely, and is thus in particular not conserved. 7 For our Standard Model this suggests the B-L symmetry must be broken at some higher energy scale, for instance by a scattering process.…”

“…As final formulation, we briefly want to mention the so-called period mapping. While in our review of asymptotic Hodge theory here we will not utilize this perspective often, these object play an important role in the holographic description of the moduli space [7,45,125] (for related work see [40,126]). The period mapping can then be viewed as a set of matter fields living in the bulk, and in fact in [125] it was shown to be described by a deformed Wess-Zumino-Witten model.…”

“…where the T k is the matrix generator of the monodromy that lies in the symmetry group G of the bilinear pairing (3.13). It follows from [120] that these generators 7 To be more precise, when the moduli space is not simply connected -there are singularities forbidding the contraction of loops to a point -we have to take the monodromy group Γ around these loci into account. This restricts h(t, t) to the double coset Γ\G/V .…”

“…Note that this Hodge star operator could have equivalently followed from the period mapping in (3.20). Below we give this description, where we primarily give the form of the various boundary and bulk objects, and refer to [7] for their explicit computation.…”

“…For the moment we forget about the signing of these diagrams, and study conjugacy classes of N under G C . 7 By Jacobson-Morosov we know that we can think of the nilpotent matrix N as a lowering operator N − = N , and complete it into an sl(2, C)-triple with a weight operator N 0 (the raising operator N + is then fixed by commutation relations (2.9)). Under this sl(2, C)-triple we can perform a decomposition of the vector space H into highest-weight components…”