Type IIB supergravity enjoys a discrete non-Abelian duality group, which has potential quantum anomalies. In this paper we explicitly compute these, and present the bordism group that controls them, modulo some physically motivated assumptions. Quite surprisingly, we find that they do not vanish, which naively would signal an inconsistency of F-theory. Remarkably, a subtle modification of the standard 10d Chern-Simons term cancels these anomalies, a fact which relies on the specific field content of type IIB supergravity. We also discover other ways to cancel this anomaly, via a topological analog of the Green-Schwarz mechanism. These alternative type IIB theories have the same low energy supergravity limit as ordinary type IIB, but a different spectrum of extended objects. They could either be part of the Swampland, or connect to the standard theory via domain walls.

We point out that different choices of Gliozzi-Scherk-Olive (GSO) projections in superstring theory can be conveniently understood by the inclusion of fermionic invertible phases, or equivalently topological superconductors, on the worldsheet. This allows us to find that the unoriented Type 00 string theory with \Omega^2=(-1)^{f}Ω2=(−1)f admits different GSO projections parameterized by nn mod 8, depending on the number of Kitaev chains on the worldsheet. The presence of nn boundary Majorana fermions then leads to the classification of D-branes by KO^n(X)\oplus KO^{-n}(X)KOn(X)⊕KO−n(X) in these theories, which we also confirm by the study of the D-brane boundary states. Finally, we show that there is no essentially new GSO projection for the Type II worldsheet theory by studying the relevant bordism group, which classifies corresponding invertible phases. In two appendixes the relevant bordism group is computed in two ways.

The Arf-Brown invariant AB(Σ) is an 8th root of unity associated to a surface Σ equipped with a Pin − structure. In this note we investigate a certain fully extended, invertible, topological quantum field theory (TQFT) whose partition function is the Arf-Brown invariant. Our motivation comes from the recent work of Freed-Hopkins on the classification of topological phases, of which the Arf-Brown TQFT provides a nice example of the general theory; physically, it can be thought of as the low energy effective theory of the Majorana chain, or as the anomaly theory of a free fermion in 1 dimension.Date: March 30, 2018. Preliminaries2.1. Clifford algebras, pin groups, and pin structures. Pin structures are generalizations of spin structures to unoriented vector bundles and manifolds. In this section, we define the pin groups and state a few useful results about them. For proofs and a more detailed exposition, see [ABS].Definition 2.1. Let k be a field of characteristic not equal to 2, S be a finite set, and o : S → {±1} be a function. The Clifford algebra Cℓ(k, S, o) is defined to be the k-algebrawhere T (k[S]) denotes the tensor algebra of the space of functions S → k, and we identify s with the function equal to 1 at s and 0 elsewhere. For S := {1, . . . , m} ∪ {−1, . . . , −n} and o(x) := sign(x), we'll write Cℓ m,n (k) := Cℓ(k, S, o), as well as Cℓ n (k) := Cℓ n,0 (k) and Cℓ −n (k) := Cℓ 0,n (k). If k = C, we'll suppress C from the notation, e.g. writing Cℓ m,n , Cℓ n , and Cℓ −n .The ideal in the quotient in (2.2) contains only even-degree elements of the tensor algebra, so the Clifford algebras are Z/2-graded algebras, or superalgebras. If a is a homogeneous element in a Z/2-graded algebra or module, we will let |a| ∈ Z/2 denote its degree.Lemma 2.3 ([ABS, Proposition 1.6]). Let S 1 and S 2 be finite sets and o i :For this to be true, we must use the graded tensor product, whose multiplication contains a sign: if a, b, a ′ , b ′ are homogeneous elements, thenLet α ∈ End (Cℓ(k, S, o)) be the grading operator, whose action on a homogeneous element a is multiplication by (−1) |a| . Definition 2.5. The Clifford group is Γ(k, S, o) := {x ∈ Cℓ(k, S, o) × | α(x)yx −1 ∈ k[S] ⊂ Cℓ(k, S, o) for all y ∈ k[S]}.

give a mathematical ansatz for classifying gapped invertible phases of matter with a spatial symmetry in terms of Borel-equivariant generalized homology. We propose a slight generalization of this ansatz to account for cases where the symmetry type mixes nontrivially with the spatial symmetry, such as crystalline phases with spin-1/2 fermions. From this ansatz, we prove as a theorem a "fermionic crystalline equivalence principle," as predicted in the physics literature. Using this and the Adams spectral sequence, we compute classifications of some classes of phases with a point group symmetry; in cases where these phases have been studied by other methods, our results agree with the literature.

The generalized double semion (GDS) model, introduced by Freedman and Hastings, is a lattice system similar to the toric code, with a gapped Hamiltonian whose definition depends on a triangulation of the ambient manifold M , but whose space of ground states does not depend on the triangulation, but only on the underlying manifold. In this paper, we use topological quantum field theory (TQFT) to investigate the low-energy limit of the GDS model. We define and study a functorial TQFT Z GDS in every dimension n such that for every closed (n − 1)-manifold M , Z GDS (M ) is isomorphic to the space of ground states of the GDS model on M ; the isomorphism can be chosen to intertwine the actions of the mapping class group of M that arise on both sides. Throughout this paper, we compare our constructions and results with their known analogues for the toric code.