The development of poli­tical geography and geopolitics as an academic and research discipline in the Baltic re­gion: The historical contribution of Saint Petersburg University

In this article, we address the little-researched and complicated problems of the genesis, periodisation, and development of political geography and geopolitics as academic and re­search disciplines across the Baltic region in general and the contribution of Saint Peters­burg University in particular. The terms ‘political geography,’ ‘geopolitics’ and the corre­sponding academic disciplines, as well as the first concepts of political geography and geo­politics, emerged in the Baltic. The Russian and German schools of thought made a valuable contribution to these fields of research. Using the historical, structural-genetic, and activity-geospace approaches, we identify and analyse the major historical, research, and academic paradigms in the development of political geography. In doing so, we consider the case of Saint Petersburg University. These paradigms (state-descriptive, anthropogeographical, state-geopolitical, and activity-societal) differ in their methodological frameworks and the­matic priorities. We demonstrate that the term ‘political geography’ and the science it de­notes are of Russian origin, having been developed by German scientists during their aca­demic service for Russia. Further, we analyse the contribution of German and Russian re­searches to the development of the Saint Petersburg school of political geographic and geo­political thought and describe its current state

Geopolitical regionalisation of the Baltic area: the essence and historical dynamics

The article discusses a theoretical framework for investigating regionalisation and geopolitical regionalisation, employing the activity-geospatial approach. The main theoretical foci of this study are system-forming, or region-building, socio-geo-adaptation and geopolitical relations. The article examines various types of transboundary and transnational geopolitical regionalisation as manifestos of geopolitical relations. These types are categorised based on scale, functional area, historical and geographical characteristics, quality, legal status and geospatial features, placing particular emphasis on the Baltic region. An essential aspect of studying a region involves identifying and defining its spatial boundaries. Since determining the exact limits of the Baltic region remains problematic, this article examines various approaches to address this issue, highlighting their strengths and weaknesses, particularly in the context of geopolitical analysis. The concluding part of the article explores several centuries of the evolution of the Baltic Sea region, divided into historical geopolitical stages. It is highlighted that the geopolitical essence of the Baltic region was changing radically over time. Particular attention is paid to the current state of the Baltic regional geopolitical entity, which is classified as a conflict-ridden or confrontational geopolitical region in the 'Eurasian arc of instability' interpreted as a geopolitical macroregion

Poisson-de Rham homology of hypertoric varieties and nilpotent cones

We prove a conjecture of Etingof and the second author for hypertoric varieties, that the Poisson-de Rham homology of a unimodular hypertoric cone is isomorphic to the de Rham cohomology of its hypertoric resolution. More generally, we prove that this conjecture holds for an arbitrary conical variety admitting a symplectic resolution if and only if it holds in degree zero for all normal slices to symplectic leaves. The Poisson-de Rham homology of a Poisson cone inherits a second grading. In the hypertoric case, we compute the resulting 2-variable Poisson-de Rham-Poincare polynomial, and prove that it is equal to a specialization of an enrichment of the Tutte polynomial of a matroid that was introduced by Denham. We also compute this polynomial for S3-varieties of type A in terms of Kostka polynomials, modulo a previous conjecture of the first author, and we give a conjectural answer for nilpotent cones in arbitrary type, which we prove in rank less than or equal to 2.Comment: 25 page

Hyperkahler sigma models on cotangent bundles of Hermitian symmetric spaces using projective superspace

Kahler manifolds have a natural hyperkahler structure associated with (part of) their cotangent bundles. Using projective superspace, we construct four-dimensional N = 2 models on the tangent bundles of some classical Hermitian symmetric spaces (specifically, the four regular series of irreducible compact symmetric Kahler manifolds, and their non-compact versions). A further dualization yields the Kahler potential for the hyperkahler metric on the cotangent bundle.Comment: 47 pages, typos corrected, version accepted by JHE

Special geometry of Euclidean supersymmetry II: hypermultiplets and the c-map

We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N =2 supersymmetry. While the Minkowskian para-c-map is obtained by dimensional reduction of the Minkowskian vector multiplet lagrangian over time, the Euclidean para-c-map corresponds to the dimensional reduction of the Euclidean vector multiplet lagrangian. In both cases the resulting hypermultiplet target spaces are para-hyper-Kahler manifolds. We review and prove the relevant results of para-complex and para-hypercomplex geometry. In particular, we give a second, purely geometrical construction of both c-maps, by proving that the cotangent bundle N=T^*M of any affine special (para-)Kahler manifold M is para-hyper-Kahler.Comment: 36 pages, 1 figur

Holomorphic symplectic geometry: a problem list

A list of open problems on holomorphic symplectic, contact and Poisson manifolds

Loop operators and S-duality from curves on Riemann surfaces

We study Wilson-'t Hooft loop operators in a class of N=2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface--the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.Comment: 41 page

N = 2 supersymmetric sigma-models and duality

For two families of four-dimensional off-shell N = 2 supersymmetric nonlinear sigma-models constructed originally in projective superspace, we develop their formulation in terms of N = 1 chiral superfields. Specifically, these theories are: (i) sigma-models on cotangent bundles T*M of arbitrary real analytic Kaehler manifolds M; (ii) general superconformal sigma-models described by weight-one polar supermultiplets. Using superspace techniques, we obtain a universal expression for the holomorphic symplectic two-form \omega^{(2,0)} which determines the second supersymmetry transformation and is associated with the two complex structures of the hyperkaehler space T*M that are complimentary to the one induced from M. This two-form is shown to coincide with the canonical holomorphic symplectic structure. In the case (ii), we demonstrate that \omega^{(2,0)} and the homothetic conformal Killing vector determine the explicit form of the superconformal transformations. At the heart of our construction is the duality (generalized Legendre transform) between off-shell N = 2 supersymmetric nonlinear sigma-models and their on-shell N = 1 chiral realizations. We finally present the most general N = 2 superconformal nonlinear sigma-model formulated in terms of N = 1 chiral superfields. The approach developed can naturally be generalized in order to describe 5D and 6D superconformal nonlinear sigma-models in 4D N = 1 superspace.Comment: 31 pages, no figures; V2: reference and comments added, typos corrected; V3: more typos corrected, published versio