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1. Fano 3-folds, reflexive polytopes and brane brick models

   https://doi.org/10.1007/JHEP08(2022)008  

   Franco, Sebastián; Seong, Rak-Kyeong (August 2022, Journal of High Energy Physics)

   A bstract Reflexive polytopes in n dimensions have attracted much attention both in mathematics and theoretical physics due to their connection to Fano n -folds and mirror symmetry. This work focuses on the 18 regular reflexive polytopes corresponding to smooth Fano 3-folds. For the first time, we show that all 18 regular reflexive polytopes have corresponding 2 d (0 , 2) gauge theories realized by brane brick models. These 2 d gauge theories can be considered as the worldvolume theories of D1-branes probing the toric Calabi-Yau 4-singularities whose toric diagrams are given by the associated regular reflexive polytopes. The generators of the mesonic moduli space of the brane brick models are shown to form a lattice of generators due to the charges under the rank 3 mesonic flavor symmetry. It is shown that the lattice of generators is the exact polar dual reflexive polytope to the corresponding toric diagram of the brane brick model. This duality not only highlights the close relationship between the geometry and 2 d gauge theory, but also opens up pathways towards new discoveries in relation to reflexive polytopes and brane brick models. 

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2. Replica topological order in quantum mixed states and quantum error correction

   https://doi.org/10.1103/PhysRevB.111.125106  

   Li, Zhuan; Mong, Roger_S K (March 2025, Physical Review B)

   Topological phases of matter offer a promising platform for quantum computation and quantum error cor- rection. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our work gives two definitions for replica topological order in mixed states, which involve n copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to n → 1 limit. 

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3. Non-invertible and higher-form symmetries in 2+1d lattice gauge theories

   https://doi.org/10.21468/SciPostPhys.18.1.008  

   Choi, Yichul; Sanghavi, Yaman; Shao, Shu-Heng; Zheng, Yunqin (January 2025, SciPost Physics)

   We explore exact generalized symmetries in the standard 2+1d lattice\mathbb{Z}_2 ${\mathbb{Z}}_2$gauge theory coupled to the Ising model, and compare them with their continuum field theory counterparts. One model has a (non-anomalous) non-invertible symmetry, and we identify two distinct non-invertible symmetry protected topological phases. The non-invertible algebra involves a lattice condensation operator, which creates a toric code ground state from a product state. Another model has a mixed anomaly between a 1-form symmetry and an ordinary symmetry. This anomaly enforces a nontrivial transition in the phase diagram, consistent with the “Higgs=SPT” proposal. Finally, we discuss how the symmetries and anomalies in these two models are related by gauging, which is a 2+1d version of the Kennedy-Tasaki transformation. 

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4. How well-conditioned can the eigenvector problem be?

   https://doi.org/10.1090/mcom/3706  

   Beltrán, Carlos; Bétermin, Laurent; Grabner, Peter; Steinerberger, Stefan (January 2022, Mathematics of Computation)

   The condition number for eigenvector computations is a well-studied quantity. But how small can it possibly be? Specifically, what matrices are perfectly conditioned with respect to eigenvector computations? In this note we answer this question for n × n n \times n matrices, giving a solution that is exact to first-order as n → ∞ n \rightarrow \infty . 

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5. Weinstein Handlebodies for Complements of Smoothed Toric Divisors

   https://doi.org/10.1090/memo/1561  

   Acu, Bahar; Capovilla-Searle, Orsola; Gadbled, Agnès; Marinković, Aleksandra; Murphy, Emmy; Starkston, Laura; Wu, Angela (May 2025, Memoirs of the American Mathematical Society)

   We study the interactions between toric manifolds and Weinstein handlebodies. We define a partially-centeredness condition on a Delzant polytope, which we prove ensures that the complement of a corresponding partial smoothing of the total toric divisor supports an explicit Weinstein structure. Many examples which fail this condition also fail to have Weinstein (or even exact) complement to the partially smoothed divisor. We investigate the combinatorial possibilities of Delzant polytopes that realize such Weinstein domain complements. We also develop an algorithm to construct a Weinstein handlebody diagram in Gompf standard form for the complement of such a partially smoothed total toric divisor. The algorithm we develop more generally outputs a Weinstein handlebody diagram for any Weinstein 4-manifold constructed by attaching 2-handles to the disk cotangent bundle of any surface  $F$, where the 2-handles are attached along the co-oriented conormal lifts of curves on  $F$. We discuss how to use these diagrams to calculate invariants and provide numerous examples applying this procedure. For example, we provide Weinstein handlebody diagrams for the complements of the smooth and nodal cubics in $\mathbb{C}{\mathbb{P}}^2$. 

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