{"title":"Conjugate Frobenius Manifold and Inversion Symmetry","authors":"Zainab Al-Maamari, Yassir Dinar","doi":"10.1007/s11040-022-09436-3","DOIUrl":"10.1007/s11040-022-09436-3","url":null,"abstract":"<div><p>We give a conjugacy relation on certain type of Frobenius manifold structures using the theory of flat pencils of metrics. It leads to a geometric interpretation for the inversion symmetry of solutions to Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48415836","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Frobenius Manifolds on Orbits Spaces","authors":"Zainab Al-Maamari, Yassir Dinar","doi":"10.1007/s11040-022-09434-5","DOIUrl":"10.1007/s11040-022-09434-5","url":null,"abstract":"<div><p>The orbits space of an irreducible linear representation of a finite group is a variety whose coordinate ring is the ring of invariant polynomials. Boris Dubrovin proved that the orbits space of the standard reflection representation of an irreducible finite Coxeter group <span>({mathcal {W}})</span> acquires a natural polynomial Frobenius manifold structure. We apply Dubrovin’s method on various orbits spaces of linear representations of finite groups. We find some of them has non or several natural Frobenius manifold structures. On the other hand, these Frobenius manifold structures include rational and trivial structures which are not known to be related to the invariant theory of finite groups.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09434-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43518701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Equivariant Spectral Triples for Homogeneous Spaces of the Compact Quantum Group (U_q(2))","authors":"Satyajit Guin, Bipul Saurabh","doi":"10.1007/s11040-022-09432-7","DOIUrl":"10.1007/s11040-022-09432-7","url":null,"abstract":"<div><p>In this article, we study homogeneous spaces <span>(U_q(2)/_phi mathbb {T})</span> and <span>(U_q(2)/_psi mathbb {T})</span> of the compact quantum group <span>(U_q(2),,qin {mathbb {C}}setminus {0})</span>. The homogeneous space <span>(U_q(2)/_phi mathbb {T})</span> is shown to be the braided quantum group <span>(SU_q(2))</span>. The homogeneous space <span>(U_q(2)/_psi mathbb {T})</span> is established as a universal <span>(C^*)</span>-algebra given by a finite set of generators and relations. Its <span>({mathcal {K}})</span>-groups are computed and two families of finitely summable odd spectral triples, one is <span>(U_q(2))</span>-equivariant and the other is <span>(mathbb {T}^2)</span>-equivariant, are constructed. Using the index pairing, it is shown that the induced Fredholm modules for these families of spectral triples give each element in the <span>({mathcal {K}})</span>-homology group <span>(K^1(C(U_q(2)/_psi mathbb {T})))</span>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134878272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Gauge Symmetries and Renormalization","authors":"David Prinz","doi":"10.1007/s11040-022-09423-8","DOIUrl":"10.1007/s11040-022-09423-8","url":null,"abstract":"<div><p>We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (Commun Math Phys 276:773–798, 2007) that the quantum counterparts of gauge symmetries—the so-called Ward–Takahashi and Slavnov–Taylor identities—correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (Ann Phys 323:49–60, 2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang–Mills theory without reference to a particular renormalization scheme.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09423-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49009761","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lattice Gauge Theory and a Random-Medium Ising Model","authors":"Mikhail Skopenkov","doi":"10.1007/s11040-022-09430-9","DOIUrl":"10.1007/s11040-022-09430-9","url":null,"abstract":"<div><p>We study linearization of lattice gauge theory. Linearized theory approximates lattice gauge theory in the same manner as the loop O(n)-model approximates the spin O(n)-model. Under mild assumptions, we show that the expectation of an observable in linearized Abelian gauge theory coincides with the expectation in the Ising model with random edge-weights. We find a similar relation between Yang-Mills theory and 4-state Potts model. For the latter, we introduce a new observable.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09430-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4257427","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Levi–Civita Connections on Quantum Spheres","authors":"Joakim Arnlind, Kwalombota Ilwale, Giovanni Landi","doi":"10.1007/s11040-022-09431-8","DOIUrl":"10.1007/s11040-022-09431-8","url":null,"abstract":"<div><p>We introduce <i>q</i>-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with <i>q</i>-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. On the quantum 3-sphere, a <i>q</i>-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi–Civita connection for a general class of metrics. We also give metric connections on a class of projective modules over the quantum 2-sphere. Finally, we outline a generalization to any Hopf algebra with a (left) covariant calculus and associated quantum tangent space.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09431-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4255091","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Prolongations of Convenient Lie Algebroids","authors":"Patrick Cabau, Fernand Pelletier","doi":"10.1007/s11040-022-09429-2","DOIUrl":"10.1007/s11040-022-09429-2","url":null,"abstract":"<div><p>We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a fibred manifold. Then we show that this construction is stable under projective and direct limits under adequate assumptions.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4828523","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Zeros of the Partition Function of the Pinning Model","authors":"Giambattista Giacomin, Rafael L. Greenblatt","doi":"10.1007/s11040-022-09428-3","DOIUrl":"10.1007/s11040-022-09428-3","url":null,"abstract":"<div><p>We aim at understanding for which (complex) values of the potential the pinning partition function vanishes. The pinning model is a Gibbs measure based on discrete renewal processes with power law inter-arrival distributions. We obtain some results for rather general inter-arrival laws, but we achieve a substantially more complete understanding for a specific one parameter family of inter-arrivals. We show, for such a specific family, that the zeros asymptotically lie on (and densely fill) a closed curve that, unsurprisingly, touches the real axis only in one point (the critical point of the model). We also perform a sharper analysis of the zeros close to the critical point and we exploit this analysis to approach the challenging problem of Griffiths singularities for the disordered pinning model. The techniques we exploit are both probabilistic and analytical. Regarding the first, a central role is played by limit theorems for heavy tail random variables. As for the second, potential theory and singularity analysis of generating functions, along with their interplay, will be at the heart of several of our arguments.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"52479927","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Theodoros Assiotis, Mustafa Alper Gunes, Arun Soor
{"title":"Convergence and an Explicit Formula for the Joint Moments of the Circular Jacobi (beta )-Ensemble Characteristic Polynomial","authors":"Theodoros Assiotis, Mustafa Alper Gunes, Arun Soor","doi":"10.1007/s11040-022-09427-4","DOIUrl":"10.1007/s11040-022-09427-4","url":null,"abstract":"<div><p>The problem of convergence of the joint moments, which depend on two parameters <i>s</i> and <i>h</i>, of the characteristic polynomial of a random Haar-distributed unitary matrix and its derivative, as the matrix size goes to infinity, has been studied for two decades, beginning with the thesis of Hughes (On the characteristic polynomial of a random unitary matrix and the Riemann zeta function, PhD Thesis, University of Bristol, 2001). Recently, Forrester (Joint moments of a characteristic polynomial and its derivative for the circular <span>(beta )</span>-ensemble, arXiv:2012.08618, 2020) considered the analogous problem for the Circular <span>(beta )</span>-Ensemble (C<span>(beta )</span>E) characteristic polynomial, proved convergence and obtained an explicit combinatorial formula for the limit for integer <i>s</i> and complex <i>h</i>. In this paper we consider this problem for a generalisation of the C<span>(beta )</span>E, the Circular Jacobi <span>(beta )</span>-ensemble (CJ<span>(beta text {E}_delta )</span>), depending on an additional complex parameter <span>(delta )</span> and we prove convergence of the joint moments for general positive real exponents <i>s</i> and <i>h</i>. We give a representation for the limit in terms of the moments of a family of real random variables of independent interest. This is done by making use of some general results on consistent probability measures on interlacing arrays. Using these techniques, we also extend Forrester’s explicit formula to the case of real <i>s</i> and <span>(delta )</span> and integer <i>h</i>. Finally, we prove an analogous result for the moments of the logarithmic derivative of the characteristic polynomial of the Laguerre <span>(beta )</span>-ensemble.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09427-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45779003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Remark on the Spherical Bipartite Spin Glass","authors":"Giuseppe Genovese","doi":"10.1007/s11040-022-09426-5","DOIUrl":"10.1007/s11040-022-09426-5","url":null,"abstract":"<div><p>Auffinger and Chen (J Stat Phys 157:40–59, 2014) proved a variational formula for the free energy of the spherical bipartite spin glass in terms of a global minimum over the overlaps. We show that a different optimisation procedure leads to a saddle point, similar to the one achieved for models on the vertices of the hypercube.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"25 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2022-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09426-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45456044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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