Matter from geometry without resolution

Abstract

We utilize the deformation theory of algebraic singularities to study charged matter in compactifications of M-theory, F-theory, and type IIa string theory on elliptically fibered Calabi-Yau manifolds. In Ftheory, this description is more physical than that of resolution. We describe how two-cycles can be identified and systematically studied after deformation. For ADE singularities, we realize non-trivial ADE representations as sublattices of Z N , where N is the multiplicity of the codimension one singularity before deformation. We give a method for the determination of Picard-Lefschetz vanishing cycles in this context and utilize this method for one-parameter smooth deformations of ADE singularities. We give a general map from junctions to weights and demonstrate that Freudenthal's recursion formula applied to junctions correctly reproduces the structure of high-dimensional ADE representations, including the 126 of SO(10) and the 43,758 of E 6 . We identify the Weyl group action in some examples, and verify its order in others. We describe the codimension two localization of matter in F-theory in the case of heterotic duality or simple normal crossing and demonstrate the branching of adjoint representations. Finally, we demonstrate geometrically that deformations correctly reproduce the appearance of non-simply-laced algebras induced by monodromy around codimension two singularities, showing the reduction of D 4 to G 2 in an example. A companion mathematical paper will follow. IntroductionThe spectrum of particles which exist in Nature is detailed and rich. The quarks and leptons of the standard model of particle physics fill out non-trivial representations of the Lie algebra SU (3) × SU (2) × U (1), and these can be embedded into representations of higher rank groups, such as the 10 + 5 + 1 of SU (5) or the 16 of SO(10). Exotic particle representations are often introduced in phenomenologically motivated extensions of the standard model, sometimes of high dimension in grand unified theories.An important physical question is whether there exist theoretical constraints on the allowed particle representations. Though anomaly cancellation provides constraints on sets of fields in gauge theories, no individual representation is ruled out on theoretical grounds. By contrast, the possibilities 1 are more limited in four-dimensional compactifications of string theory, F-theory, and M-theory. For example, in the heterotic string matter representations typically arise from branching the adjoint of E 8 ; in weakly coupled type II string theory U (N ) gauge theories admit bifundamental, symmetric tensor, and antisymmetric tensor representations, together with their conjugates; in weakly coupled orientifold compactifications it is not possible to realize the phenomenologically relevant 16 of SO(10).Interestingly, more possibilities can be realized outside of the weakly coupled regime in M-theory [1] or F-theory [2]. In these theories massless matter representations are encoded in the structure of a singular compactificati...

Matter in transition

“…Examples that will be of interest here are the A 5 → E 6 and A 7 → E 8 enhancements described in [5]. The work of [33] presents an alternative method of determining the matter content that does not require an explicit resolution of singularities.…”

“…In F-theory, Higgs transitions occur when the coefficients in the Weierstrass model are deformed from particular values, breaking a gauge symmetry in the process. Such deformations were recently explored in [33,44]. We discuss the F-theory deformations that represent Higgsing processes where fundamental, , and multiplets obtain VEVs.…”

Dualities of deformed N = 2 $$ \mathcal{N}=2 $$ SCFTs from link monodromy on D3-brane states

“…following [36][37][38], the set R := {J ∈ H 2 (X, E p ) | (J, J) = −2, a(J) = 0} (1.3) has the structure of an ADE root lattice. In particular, we can use the intersection pairing to compute the Cartan matrix A ij := 2 (α i , α j ) (α i , α i ) , i, j = 1, .…”

“…The mathematics of string junctions has been worked out in [36,39,43] and in [23,37,38]. We will review the latter description since it makes direct contact with F-theory geometries, as will be useful for describing the seven-brane backgrounds utilized in this paper.…”

“…Applying and extending the techniques for studying singularities via deformations and string junctions introduced in [27,37,38], we construct the flavor algebra and representations of Theories f and g, but find that the deformed seven-brane background reduces or completely breaks the flavor algebra and the charged states are projected out, giving rise to a Theory h (a D3 theory at the SCFT point where the seven-brane self-intersects) consistent with approaching it from either Theory f or g (i.e. approaching the codimension two singularity along two different codimension one loci).…”