We consider symmetry protected topological (SPT) phases with finite non-invertible symmetry $mathcal{C}$ in 1+1d. In particular, we investigate interfaces and parameterized families of them within the framework of matrix product states. After revealing how to extract the $mathcal{C}$-SPT invariant, we identify the algebraic structure of symmetry operators acting on the interface of two $mathcal{C}$-SPT phases. By studying the representation theory of this algebra, we show that there must be a degenerate interface mode between different $mathcal{C}$-SPT phases. This result generalizes the bulk-boundary correspondence for ordinary SPT phases. We then propose the classification of one-parameter families of $mathcal{C}$-SPT states based on the explicit construction of invariants of such families. Our invariant is identified with a non-abelian generalization of the Thouless charge pump, which is the pump of a local excitation within a $mathcal{C}$-SPT phase. Finally, by generalizing the results for one-parameter families of SPT phases, we conjecture the classification of general parameterized families of general gapped phases with finite non-invertible symmetries in both 1+1d and higher dimensions.
{"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}
In the flavour of categorical quantum mechanics, we extend nonlocal games to allow quantum questions and answers, using quantum sets (special symmetric dagger Frobenius algebras) and the quantum functions of arXiv:1711.07945. Equations are presented using a diagrammatic calculus for tensor categories. To this quantum question and answer setting, we extend the standard definitions, including strategies, correlations, and synchronicity, and we use these definitions to extend results about synchronicity. We extend the graph homomorphism (isomorphism) game to quantum graphs, and show it is synchronous (bisynchronous) and that its perfect quantum-commuting (bi)strategies are quantum graph homomorphisms (isomorphisms). Our extended definitions agree with the existing quantum games literature, except in the case of synchronicity.
{"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
We formulate scalar electrodynamics in the braided $L_infty$-algebra formalism and study its perturbative expansion in the algebraic framework of Batalin-Vilkovisky quantization. We confirm that UV/IR mixing is absent at one-loop order in this noncommutative field theory, and that the non-anomalous Ward-Takahashi identities for the braided gauge symmetry are satisfied.
{"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}
By means of Rasmussen's formulation of Khovanov-Rozansky homology originally given over $mathbb{Q}$ in arXiv:math/0607544, we compare different flavors of $mathfrak{sl}(n)$ link homology with the link invariants obtained by Kitchloo in arXiv:1910.07444 via twistings of Borel equivariant cohomology applied to the symmetry breaking spectra. In particular, we see how these geometric constructions based on Bott-Samelson varieties produce equivariant integral $mathfrak{sl}(n)$ link homology with either specialized or universal potential.
{"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}
In previous work, we introduced Mysterious Triality, extending the Mysterious Duality of Iqbal, Neitzke, and Vafa between physics and algebraic geometry to include algebraic topology in the form of rational homotopy theory. Starting with the rational Sullivan minimal model of the 4-sphere $S^4$, capturing the dynamics of M-theory via Hypothesis H, this progresses to the dimensional reduction of M-theory on torus $T^k$, $k ge 1$, with its dynamics described via the iterated cyclic loop space $mathcal{L}_c^k S^4$ of the 4-sphere. From this, we also extracted data corresponding to the maximal torus/Cartan subalgebra and the Weyl group of the exceptional Lie group/algebra of type $E_k$. In this paper, we discover much richer symmetry by extending the data of the Cartan subalgebra to a maximal parabolic subalgebra $mathfrak{p}_k^{k(k)}$ of the split real form $mathfrak{e}_{k(k)}$ of the exceptional Lie algebra of type $E_k$ by exhibiting an action, in rational homotopy category, of $mathfrak{p}_k^{k(k)}$ on the slightly more symmetric than $mathcal{L}_c^k S^4$ toroidification $mathcal{T}^k S^4$. This action universally represents symmetries of the equations of motion of supergravity in the reduction of M-theory to $11-k$ dimensions. Along the way, we identify the minimal model of the toroidification $mathcal{T}^k S^4$, generalizing the results of Vigu'{e}-Poirrier, Sullivan, and Burghelea, and establish an algebraic toroidification/totalization adjunction.
在之前的工作中，我们介绍了神秘的三重性，扩展了伊克巴尔、奈茨克和瓦法在物理学和代数几何之间的神秘二重性，以合理同调理论的形式将代数拓扑学包括在内。从4球$S^4$的有理沙利文最小模型开始，通过假说H捕捉M理论的动力学，进而发展到M理论在环$T^k$（$k ge 1$）上的维度还原，其动力学通过4球的迭代循环环空间$mathcal{L}_c^k S^4$来描述。从中，我们还提取了与最大环/卡坦次代数和例外李群/E_k$型代数的韦尔群相对应的数据。在本文中，我们通过展示一个作用，把卡尔坦子代数的数据扩展到了E_k$类型的特殊李代数的分裂实形式$mathfrak{e}_{k(k)}$的最大抛物线子代数$mathfrak{p}_k^{k(k)}$，从而发现了更丰富的对称性、在有理同调范畴中，$mathfrak{p}_k^{k(k)}$ 在比 $mathcal{L}_c^kS^4$ 略微对称的环化 $mathcal{T}^k S^4$ 上的作用。这个作用普遍地代表了超引力运动方程在把 M 理论还原到 $11-k$ 维时的对称性。在此过程中，我们确定了环化$mathcal{T}^k S^4$的最小模型，推广了Vigu'{e}-Poirrier、Sullivan和Burghelea的结果，并建立了一个代数环化/全化结点。
{"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}
We develop the theory of ``virtual morphisms'' in logarithmic algebraic geometry, introduced by Howell. It allows one to give algebro-geometric meaning to various useful maps of topological spaces that do not correspond to morphisms of (log) schemes in the classical sense, while retaining functoriality of key constructions. In particular, we explain how virtual morphisms provide a natural categorical home for Deligne's theory of tangential basepoints: the latter are simply the virtual morphisms from a point. We also extend 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 an operad in log schemes over the integers, whose virtual points are isomorphism classes of stable marked curves of genus zero equipped with a tangential basepoint. The gluing of such curves along marked points is performed using virtual morphisms that transport tangential basepoints around the curves. This builds on Vaintrob's analogous construction for framed little disks, for which the classical notion of morphism in logarithmic geometry sufficed. In this way, we obtain a direct algebro-geometric proof of the formality of the little disks operad, following the strategy envisioned by Beilinson. Furthermore, Bar-Natan's parenthesized braids naturally appear as the fundamental groupoids of our moduli spaces, with all virtual basepoints defined over the integers.
{"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}
Explicit constructions for the minimal models of general and unimodular L-infinity algebra structures are given using the BV-formalism of mathematical physics and the perturbative expansions of integrals. In particular, the general formulas for the minimal model of an L-infinity algebra structure are an instance of the Homotopy Transfer Theorem and we recover the known formulas of the structure in terms of sums over rooted trees discussing their relation to Feynman diagrams.
利用数学物理学的 BV 形式主义和积分的微扰展开，给出了一般和单模态 L 无穷代数结构的最小模型的明确构造。特别是，L-无穷代数结构的最小模型的一般公式是同调转移定理的一个实例，我们用有根树的和恢复了已知的结构公式，并讨论了它们与费曼图的关系。
{"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}
We revisit the problem of constructing the stationary states of the multispecies asymmetric simple exclusion process on a one-dimensional periodic lattice. Central to our approach is a quantum oscillator weighted five vertex model which features a strange weight conservation distinct from the conventional one. Our results clarify the interrelations among several known results and refine their derivations. For instance, the stationary probability derived from the multiline queue construction by Martin (2020) and Corteel--Mandelshtam--Williams (2022) is identified with the partition function of a three-dimensional system. The matrix product operators by Prolhac--Evans--Mallick (2009) acquire a natural diagrammatic interpretation as corner transfer matrices (CTM). The origin of their recursive tensor structure, as questioned by Aggarwal--Nicoletti--Petrov (2023), is revealed through the CTM diagrams. Finally, the derivation of the Zamolodchikov--Faddeev algebra by Cantini--de Gier--Wheeler (2015) is made intrinsic by elucidating its precise connection to a solution to the Yang--Baxter equation originating from quantum group representations.
{"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}
Let $Gamma$ denote a $Q$-polynomial distance-regular graph with diameter $Dgeq 1$. For a vertex $x$ of $Gamma$ the corresponding subconstituent algebra $T=T(x)$ is generated by the adjacency matrix $A$ of $Gamma$ and the dual adjacency matrix $A^*=A^*(x)$ of $Gamma$ with respect to $x$. We introduce a $T$-module $mathcal N = mathcal N(x)$ called the nucleus of $Gamma$ with respect to $x$. We describe $mathcal N$ from various points of view. We show that all the irreducible $T$-submodules of $mathcal N$ are thin. Under the assumption that $Gamma$ is a nonbipartite dual polar graph, we give an explicit basis for $mathcal N$ and the action of $A, A^*$ on this basis. The basis is in bijection with the set of elements for the projective geometry $L_D(q)$, where $GF(q)$ is the finite field used to define $Gamma$.
{"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}
The recently devised chiral algebra bootstrap computes the form factors of a special class of ``twistorial'' 4d QFTs as correlation functions of the theory's 2d celestial chiral algebra. Examples of twistorial theories include self-dual Yang-Mills theory coupled to special massless matter content, and certain form factors in these theories are equivalent to a subset of MHV amplitudes in massless QCD, coupled to the same matter. In this paper, we extend the chiral algebra bootstrap to include scattering in the presence of charged sources, using a self-dual dyon in a twistorial theory as our main example. Self-dual theories in the presence of such sources lift to holomorphic gauge theories on non-Hausdorff twistor space, and we generalize the Koszul duality construction of Costello and Paquette to this setting. With this approach, we easily reproduce a recent formula of Adamo, Bogna, Mason, and Sharma for $n$-point MHV scattering of gluons off the self-dual dyon.
{"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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