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Like ordinary symmetries, non-invertible symmetries can characterize Symmetry-Protected Topological (SPT) phases. In this talk, we will discuss weak SPTs protected by projective non-invertible symmetries. Projective symmetries are ubiquitous in quantum spin models and can be leveraged to constrain their phase diagram and entanglement structure, e.g., Lieb-Schultz-Mattis (LSM) theorems. We will show how, surprisingly, projective non-invertible symmetries do not always imply LSM theorems. We will first discuss a simple, exactly solvable 1+1D quantum spin model in an SPT phase protected by both translation and non-invertible symmetries forming a non-trivial projective algebra. We will then generalize this example to a class of projective non-invertible Rep(G) x G x translation symmetries. For some finite groups G, this projectivity implies an LSM theorem. When it does not, we prove it still provides a constraint through an SPT-LSM theorem: any unique and gapped ground state is necessarily a non-invertible weak SPT state with non-trivial entanglement. [This talk is based on arXiv:2409.18113]

The failure to calculate the vacuum energy remains a central problem in theoretical physics. In my talk I present a new understanding of the cosmological constant problem, grounded in the insight that vacuum energy density can be expressed in terms of phase space volume. Introduction of a UV-IR regularization implies a relationship between the vacuum energy and entropy. Combining this insight with the holographic bound on entropy then yields a bound on the cosmological constant consistent with observations. It follows that the universe is large, and the cosmological constant is naturally small, because the universe is filled with a large number of degrees of freedom. The talk is based on our papers Phys.Rev.D 107 (2023) 12, 126016; e-Print: 2212.00901 [hep-th] and Int.J.Mod.Phys.D 32 (2023) 14, 2342004; e-Print: 2303.17495 [hep-th].