arXiv subjects

Spencer Tamagni

Publications and source records attributed to Spencer Tamagni.

7 recordsLinked to original sources

arXiv2026

Quasimap critical cohomology, Coulomb branches, and quantum groups

This article presents a systematic study of the critical cohomology of $\QM^ξ(\bbP^1, X)$, the moduli space of based quasimaps from $\bbP^1$ to a Nakajima quiver variety $X$. We develop two derived equivalent presentations of this moduli space as a global critical locus. Accordingly, we obtain two equivalent explicit realizations of the canonical DT sheaf of the moduli space. Using each critical presentation, we describe canonical actions of shifted Yangians and quantized Coulomb branches on the quasimap critical cohomologies. We produce, by geometric means, a canonical morphism from an appropriate Hall algebra to the Coulomb branch, and show that the two actions we describe are compatible via this morphism under an additional hypothesis. We conjecture the compatibility holds generally, and reduce the claim to a sheaf-theoretic statement that may be approached via Joyce conjecture.

math.AG↗

arXiv2026

Quasimaps to Nakajima varieties as critical loci

$X$ is a Nakajima quiver variety. In this note we report the observation that the space parameterizing quasimaps from $\mathbb{P}^1$ to $X$, sending infinity to a given point, may be presented globally as the critical locus of an explicit function.

math.AG↗

arXiv2026

A Scattering Transform for Noncommutative Instantons

We give a detailed and mathematically rigorous analysis of the path integrals of chiral fermions supported on holomorphic curves on $T^* \mathbb{C}$ in a general noncommutative instanton background. It is shown that such path integrals can be interpreted as computing instanton analogs of matrix coefficients of monopole scattering matrices. Generalizing the known relation between monopole scattering matrices and $R$-matrices of (shifted) Yangians $\mathsf{Y}(\mathfrak{gl}_r)$, our formalism gives rise to a novel geometric method to calculate $R$-matrices of (shifted) affine Yangians $\mathsf{Y}(\widehat{\mathfrak{gl}}_r)$. This may also be viewed as an explicit description of double affine Grassmannian slices by $\infty \times \infty$ matrices, compatible with factorization. Our approach unifies a number of earlier results in the literature, and also leads to interesting new results and conjectures.

hep-th↗

arXiv2024

Stable Envelopes, Vortex Moduli Spaces, and Verma Modules

We explicitly construct K-theoretic and elliptic stable envelopes for certain moduli spaces of vortices, and apply this to enumerative geometry of rational curves in these varieties. In particular, we identify the quantum difference equations in equivariant variables with quantum Knizhnik-Zamolodchikov equations, and give their monodromy in terms of geometric elliptic R-matrices. A novel geometric feature in these constructions is that the varieties under study are not holomorphic symplectic, yet nonetheless have representation-theoretic significance. In physics, they originate from 3d supersymmetric gauge theories with $\mathcal{N} = 2$ rather than $\mathcal{N} = 4$ supersymmetry. We discuss an application of the results to the ramified version of the quantum q-Langlands correspondence of Aganagic, Frenkel, and Okounkov.

hep-th↗

arXiv2024

Coulomb Branches in 3d $\mathcal{N} = 4$ Revisited

Using ideas from the gauge theory approach to the geometric Langlands program, we revisit supersymmetric localization with monopole operators in 3d $\mathcal{N} = 4$ supersymmetric gauge theories subject to $Ω$-deformation. The key novel feature of our setup is a pair of dual boundary conditions, which drastically simplify the dynamics of the theory and the nature of the localization loci. From a careful calculation with these boundary conditions, the mathematical definition of Coulomb branches proposed by Braverman, Finkelberg and Nakajima emerges naturally. It is straightforward to incorporate codimension two defects in the setup, and in this way we gain insight into Webster's construction of tilting bundles on Coulomb branches.

hep-th↗

arXiv2024

Nonabelian shift operators and shifted Yangians

We introduce nonabelian analogs of shift operators in the enumerative theory of quasimaps. We apply them on the one hand to strengthen the emerging analogy between enumerative geometry and the geometric theory of automorphic forms, and on the other hand to obtain results about quantized Coulomb branch algebras. In particular, we find a short and direct proof that the equivariant convolution homology of the affine Grassmannian of $GL_n$ is a quotient of a shifted Yangian.

math.AG↗

arXiv2020

Electrostatics and Riemann Surfaces

Using techniques from geometry and complex analysis in their simplest form, we present a derivation of electric fields on surfaces with non-trivial topology. A byproduct of this analysis is an intuitive visualization of elliptic functions when their argument is complex-valued. The underlying connections between these techniques and the theory of Riemann surfaces are also explained. Our goal is to provide students and instructors a quick reference article for an extraordinary topic that is not included in the standard books.

physics.gen-ph↗