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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!

Interacting crystalline SPT phases were first classified by Thorngren and Else through the crystalline equivalence principle, suggesting that a spatial group symmetry can be treated as if it were an internal symmetry group. Using techniques from topology we elucidate how this principle holds for one of the proposals for the interacting bosonic case and yet fails for weak free fermion phases. Last we show how a variant of the principle does hold for strong crystalline free fermion phases., Comment: 3 pages, comments welcome!

There is a lack of knowledge about the topological invariants of non-linear $d$-dimensional systems with a periodic potential. We study these systems through a classification of the linearized NLS/GP equation around their soliton solutions. Stability conditions under linearized (mode) adiabatic evolution can be interpreted topologically and we can use equivariant $\mathit{K}$-theory and cohomology for their classification. On a lattice with crystallographic point group $P$, modes around stable, $P$-symmetric solitons are coarsely classified by the groups $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)\oplus H^2(BP;\mathbb{Z})$. Similarly, for $P$-symmetric gap solitons that are oscillatory stable, we have $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)\oplus \tilde{R}(P)$ instead. If we include a boundary, we can replace $\bar{\mathit{K}}_{P}^{0,\tau}(\mathbb{T}^d)$ with $\mathit{K}^{-1,\tau}_{P}(\mathbb{T}^{d-1})$. Finally, we mention how to use these, and the spaces of soliton solutions $M_{D}(E_{gap})$ and $M_{O}(E_{gap})$ to provide global invariants for the system., Comment: 6 pages, 1 figure

For symmorphic crystalline interacting gapped systems we derive a classification under adiabatic evolution. This classification is complete for non-degenerate ground states. For the degenerate case we discuss some invariants given by equivariant characteristic classes. We do not assume an emergent relativistic field theory nor that phases form a topological spectrum. We also do not restrict to systems with short-range entanglement, stability against stacking with trivial systems nor assume the existence of quasi-particles as is done in SPT and SET classifications respectively. Using a slightly generalized Bloch decomposition and Grassmanians made out of ground state spaces, we show that the $P$-equivariant cohomology of a $d$-dimensional torus gives rise to different interacting phases, where $P$ denotes the point group of the crystalline structure. We compare our results to bosonic symmorphic crystallographic SPT phases and to non-interacting fermionic crystallographic phases in class A. Finally we discuss the relation of our assumptions to those made for crystallographic SPT and SET phases., Comment: Stronger results. Included examples in section 6. Accepted for publication in JHEP

We construct a topological space to study contextuality in quantum mechanics. The resulting space is a classifying space in the sense of algebraic topology. Cohomological invariants of our space correspond to physical quantities relevant to the study of contextuality. Within this framework the Wigner function of a quantum state can be interpreted as a class in the twisted $K$-theory of the classifying space., Comment: 30 pages, 1 figure

We derive a rigorous classification of topologically stable Fermi surfaces of non-interacting, discrete translation-invariant systems from electronic band theory, adiabatic evolution and their topological interpretations. For systems on an infinite crystal it is shown that there can only be topologically unstable Fermi surfaces. For systems on a half- space and with a gapped bulk, our derivation naturally yields a $\mathit{K}$-theory classification. Given the $d-1$-dimensional surface Brillouin zone $\mathrm{X}_{s}$ of a $d$-dimensional half-space, our result implies that different classes of globally stable Fermi surfaces belong in $\mathit{K^{-1}}\mathrm{(X_{s})}$ for systems with only discrete translation-invariance. This result has a chiral anomaly inflow interpretation, as it reduces to the spectral flow for $d = 2$. Through equivariant homotopy methods we extend these results for symmetry classes $AI,\,AII,\, C$ and $D$ and discuss their corresponding anomaly inflow interpretation., Comment: Removed Born-von Karman boundary conditions for $\mathbb{R}^{d}$ and $\mathbb{R}^{d-1}\times [0,\infty)$ and includes the 'weak' topological phase found by Kitaev for $\Xi^2 = I, d= 2$