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We investigate the stability of weak symmetry-protected topological phases (SPTs) in the presence of short-range interactions, focusing on the tenfold way classification. Using Atiyah's Real $\mathit{KR}$-theory and Anderson-dualized bordism, we classify free and interacting weak phases across all Altland-Zirnbauer symmetry classes in low dimensions. Extending the free-to-interacting map of Freed-Hopkins, we mathematically compute how the behavior of free weak SPTs changes when interactions are introduced as well as predict intrinsically-interacting weak phases in certain classes. Our mathematical techniques involve T-duality and the James splitting of the torus. Our results provide a mathematical framework for understanding the persistence of weak SPTs under interactions, with potential implications for experimental and theoretical studies of these phases., Comment: 48 pages. Comments welcome!

In this article, we answer two questions of Buchanan-McKean (arXiv:2312.08209) about bordism for manifolds with spin$^h$ structures: we establish a Smith isomorphism between the reduced spin$^h$ bordism of $\mathbb{RP}^\infty$ and pin$^{h-}$ bordism, and we provide a geometric explanation for the isomorphism $\Omega_{4k}^{\mathrm{Spin}^c} \otimes\mathbb Z[1/2] \cong \Omega_{4k}^{\mathrm{Spin}^h} \otimes\mathbb Z[1/2]$. Our proofs use the general theory of twisted spin structures and Smith homomorphisms that we developed in arXiv:2405.04649 joint with Devalapurkar, Liu, Pacheco-Tallaj, and Thorngren, specifically that the Smith homomorphism participates in a long exact sequence with explicit, computable terms., Comment: 15 pages; comments welcome!

Smith homomorphisms are maps between bordism groups that change both the dimension and the tangential structure. We give a completely general account of Smith homomorphisms, unifying the many examples in the literature. We provide three definitions of Smith homomorphisms, including as maps of Thom spectra, and show they are equivalent. Using this, we identify the cofiber of the spectrum-level Smith map and extend the Smith homomorphism to a long exact sequence of bordism groups, which is a powerful computation tool. We discuss several examples of this long exact sequence, relating them to known constructions such as Wood's and Wall's sequences. Furthermore, taking Anderson duals yields a long exact sequence of invertible field theories, which has a rich physical interpretation. We developed the theory in this paper with applications in mind to symmetry breaking in quantum field theory, which we study in a companion paper., Comment: 73 pages, 3 figures. Originally posted as the mathematical sections of arXiv:2309.16749. This version features new content in sections 6 and 8

We give an overview of differential cohomology from the point of view of algebraic topology. This includes a survey of several different definitions of differential cohomology groups, a discussion of differential characteristic classes, an introduction to differential generalized cohomology theory, and some applications in physics., Comment: 20 pages, article for the Encyclopedia of Mathematical Physics, 2nd ed

We provide a mathematical proposal for the anomaly indicators of symmetries of (2+1)-d fermionic topological orders, and work out the consequences of our proposal in several nontrivial examples. Our proposal is an invariant of a super modular tensor category with a fermionic group action, which gives a (3+1)-d topological field theory (TFT) that we conjecture to be invertible; the anomaly indicators are partition functions of this TFT on $4$-manifolds generating the corresponding twisted spin bordism group. Our construction relies on a bosonization construction due to Gaiotto-Kapustin and Tata-Kobayashi-Bulmash-Barkeshli, together with a ``bosonization conjecture'' which we explain in detail. In the second half of the paper, we discuss several examples of our invariants relevant to condensed-matter physics. The most important example we consider is $\mathbb{Z}/4^T\times \mathbb{Z}/2^f$ time-reversal symmetry with symmetry algebra $\mathcal T^2 = (-1)^FC$, which many fermionic topological orders enjoy, including the $\mathrm{U}(1)_5$ spin Chern-Simons theory. Using newly developed tools involving the Smith long exact sequence, we calculate the cobordism group that classifies its anomaly, present the generating manifold, and calculate the partition function on the generating manifold which serves as our anomaly indicator. Our approach allows us to reproduce anomaly indicators known in the literature with simpler proofs, including $\mathbb{Z}/4^{Tf}$ time-reversal symmetry with symmetry algebra $\mathcal T^2 = (-1)^F$, and other symmetry groups in the 10-fold way involving Lie group symmetries., Comment: 82 pages, comments welcome

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 $\mathbb{Z}_2$ automorphism swapping the two $E_8$ factors in the supersymmetric heterotic string is also non-anomalous., Comment: 67 pages + references, 10 figures, 1 table v3: correction added

We study defects in symmetry breaking phases, such as domain walls, vortices, and hedgehogs. In particular, we focus on the localized gapless excitations which sometimes occur at the cores of these objects. These are topologically protected by an 't Hooft anomaly. We classify different symmetry breaking phases in terms of the anomalies of these defects, and relate them to the anomaly of the broken symmetry by an anomaly-matching formula. We also derive the obstruction to the existence of a symmetry breaking phase with a local defect. We obtain these results using a long exact sequence of groups of invertible field theories, which we call the "symmetry breaking long exact sequence" (SBLES). The mathematical backbone of the SBLES is studied in a companion paper. Our work further develops the theory of higher Berry phase and its bulk-boundary correspondence, and serves as a new computational tool for classifying symmetry protected topological phases., Comment: 31 pages, 4 figures. The mathematical sections have been removed and now appear in 2405.04649

When $R$ is one of the spectra $\mathit{ku}$, $\mathit{ko}$, $\mathit{tmf}$, $\mathit{MTSpin}^c$, $\mathit{MTSpin}$, or $\mathit{MTString}$, there is a standard approach to computing twisted $R$-homology groups of a space $X$ with the Adams spectral sequence, by using a change-of-rings isomorphism to simplify the $E_2$-page. This approach requires the assumption that the twist comes from a vector bundle, i.e. the twist map $X\to B\mathrm{GL}_1(R)$ factors through $B\mathrm{O}$. We show this assumption is unnecessary by working with Baker-Lazarev's Adams spectral sequence of $R$-modules and computing its $E_2$-page for a large class of twists of these spectra. We then work through two example computations motivated by anomaly cancellation for supergravity theories., Comment: 41 pages, comments welcome

In the presence of a nonzero B-field, the symmetries of the $\mathrm E_8\times \mathrm E_8$ heterotic string form a 2-group, or a categorified group, as do the symmetries of the CHL string. We express the bordism groups of the corresponding tangential structures as twisted string bordism groups, then compute them through dimension 11 modulo a few unresolved ambiguities. Then, we use these bordism groups to study anomalies and defects for these two string theories., Comment: 64 pages. Comments welcome!

In this work we investigate the Swampland Cobordism Conjecture in the context of type IIB string theory geometries with non-trivial duality bundle. Quite remarkably, we find that many non-trivial bordism classes with duality bundles in Mp$(2,\mathbb{Z})$, a double cover of SL$(2,\mathbb{Z})$ related to fermions, correspond to asymptotic boundaries of well-known supersymmetric F-theory backgrounds. These include $[p,q]$-7-branes, non-Higgsable clusters, S-folds, as well as various lower-dimensional generalizations. These string theoretic objects break the global symmetries associated to the non-trivial bordism groups, providing a strong test of the Cobordism Conjecture. Further including worldsheet orientation reversal promotes the duality group to the Pin$^+$ cover of GL$(2,\mathbb{Z})$. The corresponding bordism groups require a new non-supersymmetric "reflection 7-brane" and its compactifications to ensure the absence of global symmetries, thus providing an interesting prediction of the Cobordism Conjecture for non-supersymmetric type IIB backgrounds. A major component of the present work is the explicit derivation of the involved bordism groups as well as their generators, which correspond to asymptotic boundaries of explicit string theory backgrounds. The main tool is the Adams spectral sequence, to which we provide a detailed introduction. We anticipate that the same techniques can be applied in a wide variety of settings., Comment: 143 pages + appendices + references, 35 Figures, 16 Tables, 1 Map to the Lands of IIBordia

We perform a bordism computation to show that the $E_{7(7)}(\mathbb{R})$ U-duality symmetry of 4d $\mathcal N = 8$ 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 $\eta$-invariants on the Wu manifold, which generates the bordism group., Comment: 29 pages, 4 figures

The Virasoro groups are a family of central extensions of $\mathrm{Diff}^+(S^1)$, the group of orientation-preserving diffeomorphisms of $S^1$, by the circle group $\mathbb T$. We give a novel, geometric construction of these central extensions using "off-diagonal" differential lifts of the first Pontryagin class, thus affirmatively answering a question of Freed-Hopkins., Comment: 26 pages