arXiv subjects

Eugen Rogozinnikov

Publications and source records attributed to Eugen Rogozinnikov.

9 recordsLinked to original sources

arXiv2026

Symmetric spaces for groups over involutive algebras and applications to Higgs bundles

We study symplectic groups and indefinite orthogonal groups over involutive, possibly noncommutative, algebras $(A, σ)$. In the case when the algebra $(A, σ)$ is Hermitian, or the complexification $(A_{\mathbb C}, σ_{\mathbb C})$ of a Hermitian involutive algebra, one can identify maximal compact subgroups of such groups, and consider their associated Riemannian symmetric spaces. This new perspective allows for the realization of various geometric models for the symmetric space. We describe explicitly the complexified tangent space for each of the models, as well as the diffeomorphisms between them and their differentials. In the second part of the article, we give a number of applications of this theory. The geometric realizations of the Riemannian symmetric spaces described in the first part provide new geometric interpretations of Higgs bundle data that can be used for the study of fundamental group representations into symplectic or into indefinite orthogonal groups over Hermitian involutive algebras. We give an exact component count for the moduli spaces of $\rm{Sp}_2(A_{\mathbb C}, σ_{\mathbb C})$-Higgs bundles and of $\rm O(A_{\mathbb C}, σ_{\mathbb C})$-Higgs bundles, using the topology of the corresponding maximal compact subgroups rather than Morse-Bott theory techniques. Furthermore, we use the noncommutative symmetric-space models to construct a factorization of the Hitchin morphism for $\rm{Sp}_2(A_{\mathbb C},σ_{\mathbb C})$-Higgs bundles, together with analogous factorizations for the real groups $\rm{Sp}_2(A,σ)$ and $\rm O_{(1,1)}(A,σ)$. These factorizations are induced by quadratic norm maps from the corresponding tangent models to Jordan-algebraic targets and pass through intermediate affine GIT quotients. As a consequence, they reduce the algebraic complexity required in order to characterize the Hitchin base explicitly.

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arXiv2026

Spectral Networks: Bridging higher-rank Teichmüller theory and BPS states

This book offers a comprehensive introduction to spectral networks from a unified viewpoint that bridges geometry with the physics of supersymmetric gauge theories. It provides the foundational background needed to approach the frontiers of this rapidly evolving field, treating geometric and physical aspects in parallel. After surveying fundamental topics in algebra and geometry, a detailed introduction to higher-rank Teichmüller theory is developed, including Fock-Goncharov theory for Hitchin representations, maximal representations and the more recent notion of $Θ$-positivity. Spectral networks are subsequently introduced, emphasizing their utility in the study of character varieties via the abelianization and non-abelianization maps they define. In parallel, key aspects of four-dimensional gauge dynamics with eight supercharges are explored, including electric-magnetic duality, Seiberg-Witten theory, and class $\mathcal S$ theories. The role of spectral networks as a framework for determining and analyzing BPS spectra in class $\mathcal S$ theories is then examined. The final chapter outlines recent applications of spectral networks across a range of contemporary research areas. This volume is intended for researchers and advanced students in either mathematics or physics who wish to enter the field.

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arXiv2025

Symplectic groups over Lie subgroups of involutive algebras

We introduce the symplectic group $\mathrm{Sp}_2(G, σ)$ associated to a Lie subgroup $G$ of a (possibly noncommutative) associative algebra $A$ equipped with an anti-involution $σ$. Our construction recovers several classical Lie groups as special cases, and in particular provides new realizations of spin groups as instances of $\mathrm{Sp}_2(G, σ)$ for suitable subgroups $G$ of the Clifford algebra. This case is not covered by the framework, which focuses on the specific situation $G = A^\times$, and is thus of particular interest. We construct and study geometric spaces on which $\mathrm{Sp}_2(G, σ)$ acts. In particular, we define the space of $G$-isotropic elements and the corresponding space of $G$-isotropic lines, which generalize the classical projective line. We analyze the group action on these spaces and introduce natural invariants, such as the notion of positive triples and quadruples of $G$-isotropic lines and a generalized cross-ratio of positive quadruples of $G$-isotropic lines. Finally, when the Lie algebra of $G$ is Hermitian, we define the associated Riemannian symmetric space of $\mathrm{Sp}_2(G,σ)$ and provide several models for it.

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arXiv2024

On double brackets for marked surfaces

We propose a construction of a double quasi-Poisson bracket on the group algebra associated to the twisted fundamental group of a marked oriented surface $(S,P)$ with boundary, where $P$ is a finite set of marked points on the boundary of the surface $S$ such that on every boundary component there is at least one point of $P$. We show that this double bracket is a noncommutative generalization of the well-known Goldman bracket, defined on the space of free homotopy classes of loops on $S$. For an algebra $A$ without polynomial identities, we construct a double bracket on the space of decorated twisted $\mathrm{GL}_n(A)$-, symplectic and indefinite orthogonal local systems.

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arXiv2023

On a parametrization of spaces of maximal framed representations

This article is an extended version of the talk given by the author in the seminar Théorie Spectrale et Géométrie at the Institut Fourier in March 2022. We present some results from the author's doctoral thesis, extended by several results from other papers. We give a parametrization of the space of maximal framed representations of the fundamental group of a punctured surface into a Hermitian Lie group of tube type that can be seen as $\mathrm{Sp}_2(A,σ)$ for a Hermitian algebra $(A,σ)$. Using this parametrization, we count connected components of the space of maximal framed representations as well as the space of maximal (non-framed) representations.

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arXiv2022

On partial abelianization of framed local systems

D.~Gaiotto, G.~W.~Moore and A.~Neitzke introduced spectral networks to understand the framed $G$-local systems over punctured surfaces for $G$ a split Lie group via a procedure called abelianization. We generalize this construction to groups $G$ of the form $\mathrm{GL}_2(A)$, where $A$ is a unital associative ring, and to some of its subgroups. This relies on a precise analysis of the degree 2 ramified coverings associated with spectral networks and triangulations and on a matrix reinterpretation of their path lifting rules; along the way we provide another proof of the Laurent phenomenon brought to light by A.~Berenstein and V.~Retakh. The partial abelianization enables us to gives parametrizations of the moduli spaces of decorated $G$-local systems and of framed $G$-local systems over punctured surfaces. For $(A, σ)$ a Hermitian involutive $\mathbf{R}$-algebra the group $G=\mathrm{Sp}_2(A, σ)$ is a classical Hermitian Lie group of tube type, and we are able to identify and parametrize the moduli space of maximal framed $G$-local systems.

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arXiv2022

Parametrizing spaces of positive representations

Using Lusztig's total positivity in split real Lie groups V. Fock and A. Goncharov have introduced spaces of positive (framed) representations. For general semisimple Lie groups a generalization of Lusztig's total positivity was recently introduced by O. Guichard and A. Wienhard. They also introduced the associated space of positive representations. Here we consider the corresponding spaces of positive framed representations of the fundamental group of a punctured surface. We give several parametrizations of the spaces of framed positive representations. Using these parametrizations, we describe their topology and their homotopy type. We show that the number of connected components of the space of framed positive representations agrees with the number of connected components of the space of positive representations, and determine this number for simple Lie groups. Along the way, we also parametrize, for an arbitrary semisimple Lie group, the space of representations of the fundamental group of a punctured surface which are transverse with respect to a fixed ideal triangulation of the surface.

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arXiv2022

Noncommutative coordinates for symplectic representations

We introduce coordinates on the spaces of framed and decorated representations of the fundamental group of a surface with nonempty boundary into the symplectic group $Sp(2n,\mathbf R)$. These coordinates provide a noncommutative generalization of the parametrizations of the spaces of representations into $SL(2,\mathbf R)$ or $PSL(2,\mathbf R)$ given by Thurston, Penner, Kashaev, and Fock-Goncharov. On the space of decorated symplectic representations the coordinates give a geometric realization of the noncommutative cluster-like structures introduced by Berenstein-Retakh. The locus of positive coordinates maps to the space of framed maximal representations. We use this to determine an explicit homeomorphism between the space of framed maximal representations and a quotient by the group $O(n)$. This allows us to describe the homotopy type and, when $n=2$, to give an exact description of the singularities. Along the way, we establish a complete classification of pairs of nondegenerate quadratic forms.

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arXiv2021

Symplectic groups over noncommutative algebras

We introduce the symplectic group $\mathrm{Sp}_2(A,σ)$ over a noncommutative algebra $A$ with an anti-involution $σ$. We realize several classical Lie groups as $\mathrm{Sp}_2$ over various noncommutative algebras, which provides new insights into their structure theory. We construct several geometric spaces, on which the groups $\mathrm{Sp}_2(A,σ)$ act. We introduce the space of isotropic $A$-lines, which generalizes the projective line. We describe the action of $\mathrm{Sp}_2(A,σ)$ on isotropic $A$-lines, generalize the Kashiwara-Maslov index of triples and the cross ratio of quadruples of isotropic $A$-lines as invariants of this action. When the algebra $A$ is Hermitian or the complexification of a Hermitian algebra, we introduce the symmetric space $X_{\mathrm{Sp}_2(A,σ)}$, and construct different models of this space. Applying this to classical Hermitian Lie groups of tube type (realized as $\mathrm{Sp}_2(A,σ)$) and their complexifications, we obtain different models of the symmetric space as noncommutative generalizations of models of the hyperbolic plane and of the three-dimensional hyperbolic space. We also provide a partial classification of Hermitian algebras in Appendix A.

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