What Bordism-Theoretic Anomaly Cancellation Can Do for U

We perform a bordism computation to show that the E7(7)(R) U-duality symmetry of 4d N=8 supergravity could have an anomaly invisible to perturbative methods; then we show that this anomaly is trivial. We compute the relevant bordism group using the Adams and Atiyah–Hirzebruch spectral sequences, and we show the anomaly vanishes by computing η-invariants on the Wu manifold, which generates the bordism group

Taking topological field theory at phase value

In this thesis, we use methods of topological field theory to model and study topological phases of matter. This includes computing TFTs that capture low-energy information for the GDS model and for the Majorana chain with time-reversal symmetry. We then investigate phases of matter with spatial symmetries that mix with the internal symmetry type; we provide a mathematical model for these phases and prove a “fermionic crystalline equivalence principle” theorem as predicted in the physics literature. Some of our computations lead to a bonus theorem on the classification of some unorientable 4-manifolds up to stable diffeomorphism.Mathematic

Toric 2-group anomalies via cobordism

2-group symmetries arise in physics when a 0-form symmetry G[0] and a 1-form symmetry H[1] intertwine, forming a generalised group-like structure. Specialising to the case where both G[0] and H[1] are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such ‘toric 2-group symmetries’ using the cobordism classification. As a warm up example, we use cobordism to study various ’t Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of B|| where is any 2-group whose 0-form and 1-form symmetry parts are both U(1), and || is the geometric realisation of the nerve of the 2-group . By leveraging a variety of algebraic methods, we show that ΩSpin5(B|G|)≅Z/m where m is the modulus of the Postnikov class for , and we reproduce the expected physics result for anomalies in 2-group symmetries that appear in 4d QED. Moving down two dimensions, we recap that any (anomalous) U(1) global symmetry in 2d can be enhanced to a toric 2-group symmetry, before showing that its associated local anomaly reduces to at most an order 2 anomaly, when the theory is defined with a spin structure

Toric 2-group anomalies via cobordism

Abstract 2-group symmetries arise in physics when a 0-form symmetry G [0] and a 1-form symmetry H [1] intertwine, forming a generalised group-like structure. Specialising to the case where both G [0] and H [1] are compact, connected, abelian groups (i.e. tori), we analyse anomalies in such ‘toric 2-group symmetries’ using the cobordism classification. As a warm up example, we use cobordism to study various ’t Hooft anomalies (and the phases to which they are dual) in Maxwell theory defined on non-spin manifolds. For our main example, we compute the 5th spin bordism group of B|| where is any 2-group whose 0-form and 1-form symmetry parts are both U(1), and || is the geometric realisation of the nerve of the 2-group . By leveraging a variety of algebraic methods, we show that Ω 5 Spin B G ≅ ℤ / m {\varOmega}_5^{\textrm{Spin}}\left(B\left|\mathbbm{G}\right|\right)\cong \mathbb{Z}/m where m is the modulus of the Postnikov class for , and we reproduce the expected physics result for anomalies in 2-group symmetries that appear in 4d QED. Moving down two dimensions, we recap that any (anomalous) U(1) global symmetry in 2d can be enhanced to a toric 2-group symmetry, before showing that its associated local anomaly reduces to at most an order 2 anomaly, when the theory is defined with a spin structure

Global anomalies & bordism of non-supersymmetric strings

Abstract The three tachyon-free non-supersymmetric string theories in ten dimensions provide a handle on quantum gravity away from the supersymmetric lamppost. However, they have not been shown to be fully consistent; although local anomalies cancel due to versions of the Green-Schwarz mechanism, there could be global anomalies, not cancelled by the Green-Schwarz mechanism, that could become fatal pathologies. We compute the twisted string bordism groups that control these anomalies via the Adams spectral sequence, showing that they vanish completely in two out of three cases (Sugimoto and SO(16)2) and showing a partial vanishing also in the third (Sagnotti 0’B model). We also compute lower-dimensional bordism groups of the non-supersymmetric string theories, which are of interest to the classification of branes in these theories via the Cobordism Conjecture. We propose a worldvolume content based on anomaly inflow for the SO(16)2 NS5-brane, and discuss subtleties related to the torsion part of the Bianchi identity. As a byproduct of our techniques and analysis, we also reprove that the outer ℤ2 automorphism swapping the two E 8 factors in the supersymmetric heterotic string is also non-anomalous