Quantization of continuum Kac-Moody algebras

Continuum Kac-Moody algebras have been recently introduced by the authors and O. Schiffmann. These are Lie algebras governed by a continuum root system, which can be realized as uncountable colimits of Borcherds-Kac-Moody algebras. In this paper, we prove that any continuum Kac-Moody algebra is canonically endowed with a non-degenerate invariant bilinear form. The positive and negative Borel subalgebras form a Manin triple with respect to this pairing, inducing on the continuum Kac-Moody algebra a topological quasi-triangular Lie bialgebra structure. We then construct an explicit quantization, which we refer to as a continuum quantum group, and we show that the latter is similarly realized as an uncountable colimit of Drinfeld-Jimbo quantum groups.Comment: Final version. Minor change

Fock space representation of the circle quantum group

In [arXiv:1711.07391] we have defined quantum groups $\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ and $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$, which can be interpreted as continuous generalizations of the quantum groups of the Kac-Moody Lie algebras of finite, respectively affine type $A$. In the present paper, we define the Fock space representation $\mathcal{F}_{\mathbb{R}}$ of the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(\mathbb{R}))$ as the vector space generated by real pyramids (a continuous generalization of the notion of partition). In addition, by using a variant of the "folding procedure" of Hayashi-Misra-Miwa, we define an action of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ on $\mathcal{F}_{\mathbb{R}}$.Comment: 25 pages; v2: 29 pages, Final version published in IMR

The circle quantum group and the infinite root stack of a curve (with an appendix by Tatsuki Kuwagaki)

In the present paper, we give a definition of the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ of the circle $S^1\colon =\mathbb{R}/\mathbb{Z}$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ associated to the rational circle $S^1_\mathbb{Q}\colon= \mathbb{Q}/\mathbb{Z}$ as a direct limit of $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(n))$'s, where the order is given by divisibility of positive integers. The quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $X_\infty$ over a fixed smooth projective curve $X$ defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $g_X$, of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. Moreover, we show that $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(+\infty))$ and $\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(\infty))$ are subalgebras of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. As proved by T. Kuwagaki in the appendix, the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $S^1$.Comment: 63 pages, Latex; Introduction largely rewritten, a new section comparing $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ to other known infinite quantum groups is added, as well as an appendix by T. Kuwagaki giving a mirror dual construction of $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$; v3: 64 pages, Final version published in Selecta Mathematic

Two-dimensional categorified Hall algebras

In the present paper, we introduce two-dimensional categorified Hall algebras of smooth curves and smooth surfaces. A categorified Hall algebra is an associative monoidal structure on the stable $\infty$-category $\mathsf{Coh}^{\mathsf{b}}(\mathbb{R}\mathcal{M})$ of complexes of sheaves with bounded coherent cohomology on a derived moduli stack $\mathbb{R}\mathcal{M}$. In the surface case, $\mathbb{R}\mathcal{M}$ is a suitable derived enhancement of the moduli stack $\mathcal M$ of coherent sheaves on the surface. This construction categorifies the K-theoretical and cohomological Hall algebras of coherent sheaves on a surface of Zhao and Kapranov-Vasserot. In the curve case, we define three categorified Hall algebras associated with suitable derived enhancements of the moduli stack of Higgs sheaves on a curve $X$, the moduli stack of vector bundles with flat connections on $X$, and the moduli stack of finite-dimensional local systems on $X$, respectively. In the Higgs sheaves case we obtain a categorification of the K-theoretical and cohomological Hall algebras of Higgs sheaves on a curve of Minets and Sala-Schiffmann, while in the other two cases our construction yields, by passing to $\mathsf K_0$, new K-theoretical Hall algebras, and by passing to $\mathsf H_\ast^{\mathsf{BM}}$, new cohomological Hall algebras. Finally, we show that the Riemann-Hilbert and the non-abelian Hodge correspondences can be lifted to the level of our categorified Hall algebras of a curve.Comment: 54 page

AGT relations for abelian quiver gauge theories on ALE spaces

We construct level one dominant representations of the affine Kac-Moody algebra $\widehat{\mathfrak{gl}}_k$ on the equivariant cohomology groups of moduli spaces of rank one framed sheaves on the orbifold compactification of the minimal resolution $X_k$ of the $A_{k-1}$ toric singularity $\mathbb{C}^2/\mathbb{Z}_k$. We show that the direct sum of the fundamental classes of these moduli spaces is a Whittaker vector for $\widehat{\mathfrak{gl}}_k$, which proves the AGT correspondence for pure $\mathcal{N}=2$ $U(1)$ gauge theory on $X_k$. We consider Carlsson-Okounkov type Ext-bundles over products of the moduli spaces and use their Euler classes to define vertex operators. Under the decomposition $\widehat{\mathfrak{gl}}_k\simeq \mathfrak{h}\oplus \widehat{\mathfrak{sl}}_k$, these vertex operators decompose as products of bosonic exponentials associated to the Heisenberg algebra $\mathfrak{h}$ and primary fields of $\widehat{\mathfrak{sl}}_k$. We use these operators to prove the AGT correspondence for $\mathcal{N}=2$ superconformal abelian quiver gauge theories on $X_k$.Comment: 58 pages; v2: typos corrected, reference added; v3: Introduction expanded, minor corrections and clarifying remarks added throughout, references added and updated; Final version published in Journal of Geometry and Physic

Some topics in the geometry of framed sheaves and their moduli spaces

This dissertation is primarily concerned with the study of framed sheaves on nonsingular projective varieties and the geometrical properties of the moduli spaces of these objects. In particular, we deal with a generalization to the framed case of known results for (semi)stable torsion free sheaves, such as (relative) Harder-Narasimhan filtration, Mehta-Ramanathan restriction theorems, Uhlenbeck-Donaldson compactification, Atiyah class and Kodaira-Spencer map. The main motivations for the study of these moduli spaces come from physics, in particular, gauge theory, as we shall explain in the following

Intraoperative neurophysiology in pediatric neurosurgery: a historical perspective

Introduction: Intraoperative neurophysiology (ION) has been established over the past three decades as a valuable discipline to improve the safety of neurosurgical procedures with the main goal of reducing neurological morbidity. Neurosurgeons have substantially contributed to the development of this field not only by implementing the use and refinement of ION in the operating room but also by introducing novel techniques for both mapping and monitoring of neural pathways. Methods: This review provides a personal perspective on the evolution of ION in a variety of pediatric neurosurgical procedures: from brain tumor to brainstem surgery, from spinal cord tumor to tethered cord surgery. Results and discussion: The contribution of pediatric neurosurgeons is highlighted showing how our discipline has played a crucial role in promoting ION at the turn of the century. Finally, a view on novel ION techniques and their potential implications for pediatric neurosurgery will provide insights into the future of ION, further supporting the view of a functional, rather than merely anatomical, approach to pediatric neurosurgery

Organization and structure of the chain in the Integrated Projects of Food Chain in Basilicata region: the effects on the new rural dynamics

The introduction of the Integrated Projects of Food Chain requires the development of models capable of interpreting the dynamics of vertical and horizontal coordination between agents and the definition of the issues that most affect the ability of professionals to provide value added to goods and products to acquire in exchange a competitive advantage. With reference to the Basilicata region, the production structure of the region and the recent development of the Integrated Projects of Food Chain, this research has developed a new model of territorial organization of rural development. Now connect a new food chain model that combines theories of productivity, typical of contract economic, with those of social welfare and environmental economics: multifunctionality and biodiversity related to the needs of income and efficiency of companies in various stages of the food chain classic, in a context in which planning consultation is major determinant of local and regional development.Food Chain, Rural Development, Integrated Project of Food Chain., Agribusiness, Agricultural and Food Policy, Community/Rural/Urban Development, Food Consumption/Nutrition/Food Safety, Labor and Human Capital,