{"title":"1+1d SPT phases with fusion category symmetry: interface modes and non-abelian Thouless pump","authors":"Kansei Inamura, Shuhei Ohyama","doi":"arxiv-2408.15960","DOIUrl":"https://doi.org/arxiv-2408.15960","url":null,"abstract":"We consider symmetry protected topological (SPT) phases with finite\u0000non-invertible symmetry $mathcal{C}$ in 1+1d. In particular, we investigate\u0000interfaces and parameterized families of them within the framework of matrix\u0000product states. After revealing how to extract the $mathcal{C}$-SPT invariant,\u0000we identify the algebraic structure of symmetry operators acting on the\u0000interface of two $mathcal{C}$-SPT phases. By studying the representation\u0000theory of this algebra, we show that there must be a degenerate interface mode\u0000between different $mathcal{C}$-SPT phases. This result generalizes the\u0000bulk-boundary correspondence for ordinary SPT phases. We then propose the\u0000classification of one-parameter families of $mathcal{C}$-SPT states based on\u0000the explicit construction of invariants of such families. Our invariant is\u0000identified with a non-abelian generalization of the Thouless charge pump, which\u0000is the pump of a local excitation within a $mathcal{C}$-SPT phase. Finally, by\u0000generalizing the results for one-parameter families of SPT phases, we\u0000conjecture the classification of general parameterized families of general\u0000gapped phases with finite non-invertible symmetries in both 1+1d and higher\u0000dimensions.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198770","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Quantum Games and Synchronicity","authors":"Adina Goldberg","doi":"arxiv-2408.15444","DOIUrl":"https://doi.org/arxiv-2408.15444","url":null,"abstract":"In the flavour of categorical quantum mechanics, we extend nonlocal games to\u0000allow quantum questions and answers, using quantum sets (special symmetric\u0000dagger Frobenius algebras) and the quantum functions of arXiv:1711.07945.\u0000Equations are presented using a diagrammatic calculus for tensor categories. To\u0000this quantum question and answer setting, we extend the standard definitions,\u0000including strategies, correlations, and synchronicity, and we use these\u0000definitions to extend results about synchronicity. We extend the graph\u0000homomorphism (isomorphism) game to quantum graphs, and show it is synchronous\u0000(bisynchronous) and that its perfect quantum-commuting (bi)strategies are\u0000quantum graph homomorphisms (isomorphisms). Our extended definitions agree with\u0000the existing quantum games literature, except in the case of synchronicity.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marija Dimitrijević Ćirić, Biljana Nikolić, Voja Radovanović, Richard J. Szabo, Guillaume Trojani
{"title":"Braided Scalar Quantum Electrodynamics","authors":"Marija Dimitrijević Ćirić, Biljana Nikolić, Voja Radovanović, Richard J. Szabo, Guillaume Trojani","doi":"arxiv-2408.14583","DOIUrl":"https://doi.org/arxiv-2408.14583","url":null,"abstract":"We formulate scalar electrodynamics in the braided $L_infty$-algebra\u0000formalism and study its perturbative expansion in the algebraic framework of\u0000Batalin-Vilkovisky quantization. We confirm that UV/IR mixing is absent at\u0000one-loop order in this noncommutative field theory, and that the non-anomalous\u0000Ward-Takahashi identities for the braided gauge symmetry are satisfied.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"80 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Khovanov-Rozansky homologies, Bott-Samelson spaces and twisted cohomology","authors":"Tomas Mejia-Gomez","doi":"arxiv-2409.02940","DOIUrl":"https://doi.org/arxiv-2409.02940","url":null,"abstract":"By means of Rasmussen's formulation of Khovanov-Rozansky homology originally\u0000given over $mathbb{Q}$ in arXiv:math/0607544, we compare different flavors of\u0000$mathfrak{sl}(n)$ link homology with the link invariants obtained by Kitchloo\u0000in arXiv:1910.07444 via twistings of Borel equivariant cohomology applied to\u0000the symmetry breaking spectra. In particular, we see how these geometric\u0000constructions based on Bott-Samelson varieties produce equivariant integral\u0000$mathfrak{sl}(n)$ link homology with either specialized or universal\u0000potential.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198780","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Mysterious Triality and the Exceptional Symmetry of Loop Spaces","authors":"Hisham Sati, Alexander A. Voronov","doi":"arxiv-2408.13337","DOIUrl":"https://doi.org/arxiv-2408.13337","url":null,"abstract":"In previous work, we introduced Mysterious Triality, extending the Mysterious\u0000Duality of Iqbal, Neitzke, and Vafa between physics and algebraic geometry to\u0000include algebraic topology in the form of rational homotopy theory. Starting\u0000with the rational Sullivan minimal model of the 4-sphere $S^4$, capturing the\u0000dynamics of M-theory via Hypothesis H, this progresses to the dimensional\u0000reduction of M-theory on torus $T^k$, $k ge 1$, with its dynamics described\u0000via the iterated cyclic loop space $mathcal{L}_c^k S^4$ of the 4-sphere. From\u0000this, we also extracted data corresponding to the maximal torus/Cartan\u0000subalgebra and the Weyl group of the exceptional Lie group/algebra of type\u0000$E_k$. In this paper, we discover much richer symmetry by extending the data of the\u0000Cartan subalgebra to a maximal parabolic subalgebra $mathfrak{p}_k^{k(k)}$ of\u0000the split real form $mathfrak{e}_{k(k)}$ of the exceptional Lie algebra of\u0000type $E_k$ by exhibiting an action, in rational homotopy category, of\u0000$mathfrak{p}_k^{k(k)}$ on the slightly more symmetric than $mathcal{L}_c^k\u0000S^4$ toroidification $mathcal{T}^k S^4$. This action universally represents\u0000symmetries of the equations of motion of supergravity in the reduction of\u0000M-theory to $11-k$ dimensions. Along the way, we identify the minimal model of the toroidification\u0000$mathcal{T}^k S^4$, generalizing the results of Vigu'{e}-Poirrier, Sullivan,\u0000and Burghelea, and establish an algebraic toroidification/totalization\u0000adjunction.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198775","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Logarithmic morphisms, tangential basepoints, and little disks","authors":"Clément Dupont, Erik Panzer, Brent Pym","doi":"arxiv-2408.13108","DOIUrl":"https://doi.org/arxiv-2408.13108","url":null,"abstract":"We develop the theory of ``virtual morphisms'' in logarithmic algebraic\u0000geometry, introduced by Howell. It allows one to give algebro-geometric meaning\u0000to various useful maps of topological spaces that do not correspond to\u0000morphisms of (log) schemes in the classical sense, while retaining\u0000functoriality of key constructions. In particular, we explain how virtual\u0000morphisms provide a natural categorical home for Deligne's theory of tangential\u0000basepoints: the latter are simply the virtual morphisms from a point. We also\u0000extend Howell's results on the functoriality of Betti and de Rham cohomology. Using this framework, we lift the topological operad of little $2$-disks to\u0000an operad in log schemes over the integers, whose virtual points are\u0000isomorphism classes of stable marked curves of genus zero equipped with a\u0000tangential basepoint. The gluing of such curves along marked points is\u0000performed using virtual morphisms that transport tangential basepoints around\u0000the curves. This builds on Vaintrob's analogous construction for framed little\u0000disks, for which the classical notion of morphism in logarithmic geometry\u0000sufficed. In this way, we obtain a direct algebro-geometric proof of the\u0000formality of the little disks operad, following the strategy envisioned by\u0000Beilinson. Furthermore, Bar-Natan's parenthesized braids naturally appear as\u0000the fundamental groupoids of our moduli spaces, with all virtual basepoints\u0000defined over the integers.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Homotopy transfer for L-infinity structures and the BV-formalism","authors":"James Maunder","doi":"arxiv-2408.12461","DOIUrl":"https://doi.org/arxiv-2408.12461","url":null,"abstract":"Explicit constructions for the minimal models of general and unimodular\u0000L-infinity algebra structures are given using the BV-formalism of mathematical\u0000physics and the perturbative expansions of integrals. In particular, the\u0000general formulas for the minimal model of an L-infinity algebra structure are\u0000an instance of the Homotopy Transfer Theorem and we recover the known formulas\u0000of the structure in terms of sums over rooted trees discussing their relation\u0000to Feynman diagrams.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A strange five vertex model and multispecies ASEP on a ring","authors":"Atsuo Kuniba, Masato Okado, Travis Scrimshaw","doi":"arxiv-2408.12092","DOIUrl":"https://doi.org/arxiv-2408.12092","url":null,"abstract":"We revisit the problem of constructing the stationary states of the\u0000multispecies asymmetric simple exclusion process on a one-dimensional periodic\u0000lattice. Central to our approach is a quantum oscillator weighted five vertex\u0000model which features a strange weight conservation distinct from the\u0000conventional one. Our results clarify the interrelations among several known\u0000results and refine their derivations. For instance, the stationary probability\u0000derived from the multiline queue construction by Martin (2020) and\u0000Corteel--Mandelshtam--Williams (2022) is identified with the partition function\u0000of a three-dimensional system. The matrix product operators by\u0000Prolhac--Evans--Mallick (2009) acquire a natural diagrammatic interpretation as\u0000corner transfer matrices (CTM). The origin of their recursive tensor structure,\u0000as questioned by Aggarwal--Nicoletti--Petrov (2023), is revealed through the\u0000CTM diagrams. Finally, the derivation of the Zamolodchikov--Faddeev algebra by\u0000Cantini--de Gier--Wheeler (2015) is made intrinsic by elucidating its precise\u0000connection to a solution to the Yang--Baxter equation originating from quantum\u0000group representations.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"220 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The nucleus of a $Q$-polynomial distance-regular graph","authors":"Paul Terwilliger","doi":"arxiv-2408.11282","DOIUrl":"https://doi.org/arxiv-2408.11282","url":null,"abstract":"Let $Gamma$ denote a $Q$-polynomial distance-regular graph with diameter\u0000$Dgeq 1$. For a vertex $x$ of $Gamma$ the corresponding subconstituent\u0000algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $Gamma$ and the\u0000dual adjacency matrix $A^*=A^*(x)$ of $Gamma$ with respect to $x$. We\u0000introduce a $T$-module $mathcal N = mathcal N(x)$ called the nucleus of\u0000$Gamma$ with respect to $x$. We describe $mathcal N$ from various points of\u0000view. We show that all the irreducible $T$-submodules of $mathcal N$ are thin.\u0000Under the assumption that $Gamma$ is a nonbipartite dual polar graph, we give\u0000an explicit basis for $mathcal N$ and the action of $A, A^*$ on this basis.\u0000The basis is in bijection with the set of elements for the projective geometry\u0000$L_D(q)$, where $GF(q)$ is the finite field used to define $Gamma$.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"220 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Scattering off of Twistorial Line Defects","authors":"Niklas Garner, Natalie M. Paquette","doi":"arxiv-2408.11092","DOIUrl":"https://doi.org/arxiv-2408.11092","url":null,"abstract":"The recently devised chiral algebra bootstrap computes the form factors of a\u0000special class of ``twistorial'' 4d QFTs as correlation functions of the\u0000theory's 2d celestial chiral algebra. Examples of twistorial theories include\u0000self-dual Yang-Mills theory coupled to special massless matter content, and\u0000certain form factors in these theories are equivalent to a subset of MHV\u0000amplitudes in massless QCD, coupled to the same matter. In this paper, we\u0000extend the chiral algebra bootstrap to include scattering in the presence of\u0000charged sources, using a self-dual dyon in a twistorial theory as our main\u0000example. Self-dual theories in the presence of such sources lift to holomorphic\u0000gauge theories on non-Hausdorff twistor space, and we generalize the Koszul\u0000duality construction of Costello and Paquette to this setting. With this\u0000approach, we easily reproduce a recent formula of Adamo, Bogna, Mason, and\u0000Sharma for $n$-point MHV scattering of gluons off the self-dual dyon.","PeriodicalId":501317,"journal":{"name":"arXiv - MATH - Quantum Algebra","volume":"448 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225260","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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