arXiv subjects

Reese Lance

Publications and source records attributed to Reese Lance.

5 recordsLinked to original sources

arXiv2026

Higher order double point formulas via SSM-Thom polynomials

We study the geometry of double point loci of maps $F:M\to N$ of complex manifolds through the lens of Segre-Schwartz-MacPherson (SSM) classes. Classical double point formulas express the fundamental class of the closure of the double point locus of $F$ in terms of global invariants of source and target spaces, as well as $F$. In this paper we extend these results by computing a one-parameter cohomological deformation of the double point formula given by the SSM class. We compute the SSM class of the double point locus in a large cohomological degree range. The leading term in our new formulas recovers the classical double point formula of Fulton and Laksov, while higher-degree terms provide explicit universal corrections. Our approach uses interpolation techniques for SSM-Thom polynomials of multisingularities, recently developed by Koncki, Nekarda, Ohmoto and Rim\'anyi. We also compute SSM-Thom polynomials for the singularities $A_0$ and $A_1$ in the same range. As an application, we show how the deformed formulas yield refined geometric information about those singularity loci through a theorem of Aluffi and Ohmoto, including constraints on when such loci can arise as complete intersections.

math.AG↗

arXiv2025

Integrals of stable envelopes for cotangent bundles to Grassmannians

We consider cohomological stable envelopes for a natural torus action $\mathsf{T}$ on $X=T^*Gr(k,n)$, introduced by Maulik-Okounkov. We define the $\mathbb{C}^*_\hbar$-equivariant integral of the stable envelope using equivariant localization over the subtorus $\mathbb{C}^*_\hbar\subset\mathsf{T}$, and compute the integral as a non-equivariant limit of the localization over the full torus, $\mathsf{T}$. The integral of such a class is an integer times a power of $\hbar$, and the main result of this paper is a combinatorial formula for these integers. In 3d mirror symmetry, these non-equivariant limits are expected to reflect some curve counting phenomena on the 3d mirror dual, $X^\vee$. When $k=1$, we obtain the binomial coefficients, and we study some of the combinatorics of the integers for higher $k$, which haven't appeared in the literature before. We give some conjectures and interpretations on extending this phenomena to type A quiver and bow varieties.

math.AG↗

arXiv2025

Quantum Difference Equations for Grassmannians

We consider quantum difference equation (QDE) for equivariant quantum K-theory of the Grassmannian. In this paper we obtain a solution to the QDE and use the solution to asymptotically derive the Bethe ansatz equations. In the limit, we obtain similar results for the cohomological analogue. For both cases, we describe the nonequivariant solutions as well. As an application, we identify the quantum K-theory ring of $\mathrm{Gr}(k,n)$ with a quantum 5 vertex XXZ integrable spin chain.

math-ph↗

arXiv2025

Slant sums of quiver gauge theories

We define the slant sum of quiver gauge theories, a gluing on the underlying quivers that identifies a gauge vertex with a framing vertex. Under some mild assumptions, we relate torus fixed points on the corresponding Higgs branches, which are Nakajima quiver varieties. Then we prove a ``branching rule" relating the quasimap vertex functions before and after a slant sum and deduce a number of ``factorization" corollaries. Our construction is motivated by a factorization conjecture for the vertex functions of zero-dimensional quiver varieties, which can be approached inductively using the branching rule. In special cases, it also shows that vertex functions can be written as sums over reverse plane partitions, even outside ADE type. We make some conjectures for Coulomb branches reflecting what can be seen on the Higgs side and prove them in ADE type. In particular, we obtain refined character formulas for the so-called ``extremal'' irreducible modules over shifted Yangians. We also study slant sums of Coulomb branches and their quantizations. We observe that for one-dimensional framing, the slant sum of Coulomb branches is the same as the product.

math.RT↗

arXiv2023

Homotopy groups of quasi-spheres and applications to indefinite orthogonal groups

In this note, we present a new proof of the isomorphism $\pi_1(SO^+(p,q)) \cong \pi_1(SO(p))\times \pi_1(SO(q))$ using the long exact sequence associated to a fibration. While this formula is already known, the method of proof presented here fills an existing hole in the literature to naturally generalize an approach following the classical reduction of this formula for $q=0$, involving the homotopy groups of the $(p-1)$-sphere and a long exact sequence arising from a fibration constructed from these spaces. To generalize for $q \ne 0$, we upgrade spheres to the corresponding quasi-spheres, then analyze the resulting long exact sequence to obtain the isomorphism for all $p,q$. Some low-dimensional inductive steps are delicate in this approach, but it is more direct than other standard methods and involves some interesting calculations with fundamental groups. The question of whether or not this approach would work is absolutely natural, and in this paper we show that the answer is yes. This particular approach avoids some difficult technical steps in other standard proofs, which we review at the end. The family of spaces we consider are also of interest in physical applications. We remark about this direction to provide some context, but do not explore it in depth in this paper.

math.AT↗