Quantization of anomaly coefficients in 6D N = 1 , 0 $$ \mathcal{N}=\left(1,\;0\right) $$ supergravity

Abstract

We obtain new constraints on the anomaly coefficients of 6D N = (1, 0) supergravity theories using local and global anomaly cancellation conditions. We show how these constraints can be strengthened if we assume that the theory is well-defined on any spin space-time with an arbitrary gauge bundle. We distinguish the constraints depending on the gauge algebra only from those depending on the global structure of the gauge group. Our main constraint states that the coefficients of the anomaly polynomial for the gauge group G should be an element of 2H 4 (BG; Z) ⊗ Λ S where Λ S is the unimodular string charge lattice. We show that the constraints in their strongest form are realized in F-theory compactifications. In the process, we identify the cocharacter lattice, which determines the global structure of the gauge group, within the homology lattice of the compactification manifold.

Remarks on the Green–Schwarz Terms of Six-Dimensional Supergravity Theories

“…One naturally wonders whether all consistent six-dimensional supergravities must be such that a is a characteristic element, but thus far, such a condition has not been derived from the low energy point of view: in fact, there are 6d supergravity theories satisfying all known low energy consistency conditions but violating (2.12). An example was given in Section 5 of [3], involving 244 neutral hypermultiplets, no gauge symmetry, a single tensor multiplet, Λ " Z 2 with bilinear form˜0 1 1 0¸ ( 2.13) and a " p4, 1q. In this paper we will show that our construction of the Green-Schwarz term requires (2.12).…”

“…By considering U " CP 3ˆS1 , W " CP 3ˆD2 and suitable bundles over CP 3 , we can recover the constraints of [3] on the anomaly coefficients b i and b IJ . As explained there, b i and b IJ can be seen as the coefficients of an element b in H 4 pBG 1 ; Λ R q, where G 1 is the connected component of the identity of G. The constraints of [3] read b P H 4 pBG; Λq , (7.15) showing at the same time that the fundamental object encoding the gauge anomaly coefficients is 1 2 b and not b. (The factor 1 2 now matters as 2-torsion may be present in H 4 pBG; Λq.)…”

“…This property is automatically satisfied in F-theory constructions of 6d supergravity theories (see Section 4.1 of [3] for the argument), but up to now it was unclear how it should arise from the point of view of the 6d supergravity. In Section 5 of [3], we gave an example of a 6d supergravity theory that looks completely consistent, except for the fact that a is not characteristic. It consists of a single tensor multiplet, a string lattice…”

Global Tensor‐Matter Transitions in F‐Theory

“…The parameters n[R], n[q i ], and n[R, q i ] take into account the multiplicity of fields transforming in representation R of G and with charge q i under U (1) i . The letters H, V , and T denote the overall number of hyper-, vector-, and tensor 2 See also [31] for constraints due to the global realization of the gauge algebra. 3 We mainly follow the notational conventions of [27,28].…”

Nongeometric heterotic strings and dual F-theory with enhanced gauge groups

“…Birational maps from an elliptic fibration with a type II * fiber and a type III * fiber to an elliptic fibration with a 7 F-theory compactifications on genus-one fibrations without a section have recently been studied, e.g., in [43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65]. 8 See, for example, [67,68,69,70,71,72,73,74,75,76,77,78,79,80,81] for recent studies on F-theory compactifications on elliptic Calabi-Yau 3-folds. F-theory compactifications on Calabi-Yau 3-folds with a terminal singularity are discussed in [43,82,83].…”