Based maps to Lagrangian Grassmannians, Quivers, and Bott Periodicity

Abstract: We give a quiver description of the space of based algebraic maps from $\mathbb{P}<sup>{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's ``naive algebro-geometric homotopy category'' whose topological realization (after specializing to $\mathbb{C}$) recovers the classical homotopy equivalences $Ω^{2}(Sp/U) \simeq BO\times \mathbb{Z}$ and $Ω^{2}(O/U) \simeq BSp\times \mathbb{Z}$.

• The paper introduces quiver moduli to model algebraic double loop spaces, uncovering refined geometric proofs of Bott periodicity.
• It establishes explicit isomorphisms in the algebro-geometric homotopy category, aligning stabilization with classical topological equivalences.
• The framework bridges representation theory and moduli problems, providing computational tools for flag varieties and isotropic bundles.

Quiver Descriptions of Algebraic Double Loop Spaces and Algebro-Geometric Bott Periodicity

Overview

This paper establishes an explicit algebro-geometric framework for modeling the double loop spaces of Lagrangian and orthogonal Grassmannians, introducing quiver-theoretic moduli descriptions of based algebraic maps $\mathbb{P}^1 \to X$ for such $X$. The authors leverage these constructions to provide new, geometrically refined proofs of several homotopy equivalence statements in Bott periodicity, with a focus on producing isomorphisms in the naive algebro-geometric homotopy category (HoG) whose topological realizations align with classical periodicity results.

Algebraic Double Loop Spaces and Grassmannians

The algebraic double loop space $L^2_{d,\mathrm{alg}}(X)$ is defined as the moduli space of degree-$d$ based morphisms $f:\mathbb{P}^1\to X$ fixing a chosen rational point, where $X$ is a projective variety such as a Grassmannian, a Lagrangian Grassmannian, or a maximal isotropic orthogonal Grassmannian.

A central theme is comparing the algebraic and topological double loop spaces via the inclusion

$$i: L^2_{d,\mathrm{alg}}(X) \to L^2_{d,\mathrm{top}}(X)$$

over $\mathbb{C}$, approximating topological loop space homotopy in the sense of Segal and others, under conditions that stabilize as $d\to\infty$.

Quiver Moduli as Moduli of Based Maps

The authors systematically construct equivalences between algebraic double loop spaces and GIT quotients of quiver representation spaces, providing explicit moduli-theoretic presentations:

• For $X = \mathrm{Gr}(n,n+N)$, $X$0, where $X$1 and $X$2 is a vector space of dimension $X$3. Stability conditions are articulated in terms of both surjectivity and local-freeness loci.
• For $X$4 and $X$5, analogous moduli are constructed by equipping $X$6 with a nondegenerate symmetric or symplectic form and passing to quiver data subject to adjoint compatibility conditions, with corresponding symmetry group $X$7 or $X$8.

Crucially, in the symplectic case the dimension constraint forces $X$9 even, and in the orthogonal case, the main results respect component restrictions.

Homotopical and Algebro-Geometric Bott Periodicity

The main topological consequences arise by stabilizing the above moduli spaces in both $L^2_{d,\mathrm{alg}}(X)$0 and $L^2_{d,\mathrm{alg}}(X)$1, thereby constructing objects in the category HoG. Sending these to topological realization functors recovers homotopy equivalences corresponding to Bott periodicity.

Key results include:

• Isomorphism in HoG: For the sequence $L^2_{d,\mathrm{alg}}(X)$2, there is a canonical isomorphism in HoG to $L^2_{d,\mathrm{alg}}(X)$3, and thus its topological realization yields the equivalence $L^2_{d,\mathrm{alg}}(X)$4 (complex Bott periodicity).
• Real Bott periodicity refinements: Corresponding constructions for Lagrangian and orthogonal Grassmannians yield isomorphisms in HoG to $L^2_{d,\mathrm{alg}}(X)$5 and $L^2_{d,\mathrm{alg}}(X)$6, with stabilization providing the classical real Bott periodicity homotopy types $L^2_{d,\mathrm{alg}}(X)$7 and $L^2_{d,\mathrm{alg}}(X)$8.

These isomorphisms critically depend on the increasing connectedness of stabilization inclusions, which are analyzed via codimension arguments on the unstable locus for the relevant moduli, echoing the philosophy of representation theory of quivers and the theory of (semi)stable bundles.

Technical Highlights

• The translation from the moduli of based maps to explicit stacks of quiver representations is established using Beilinson's spectral sequences and cohomology exact sequences à la Strømme.
• The critical stability conditions on quivers (seen as points modulo group action) are related to geometric properties of the underlying sheaf sequences, with duality between surjectivity and local-freeness captured precisely via adjoint operations.
• For Lagrangian and orthogonal targets, the identification and reduction through bilinear forms are achieved via a careful interplay of Serre duality, derived category techniques, and explicit matrix presentations. The equation $L^2_{d,\mathrm{alg}}(X)$9 is central for controlling adjoint symmetry.

Implications and Future Directions

This framework yields an algebro-geometric refinement of topological Bott periodicity, giving finite-dimensional algebraic varieties (as GIT quotients of quiver representation spaces) whose inductive limits have the correct homotopy type. This is significant theoretically as it bridges the gap between algebro-geometric moduli and classical homotopy-theoretic phenomena, suggesting that further refinements and spectral sequence-level comparisons may be possible.

Practically, these descriptions provide new computational tools for understanding moduli problems involving flag varieties, isotropic Grassmannians, and symplectic/orthogonal bundles, with anticipated applications in enumerative geometry and geometric representation theory. The approach also sets a precedent for modeling other algebraic loop spaces via finite-dimensional GIT quotients and handling stabilization phenomena via the HoG framework.

Given the duality to linear control systems alluded to, as well as the analogies to Nakajima quiver varieties, this work points towards a deeper unification of moduli theory, homotopy theory, and representation theory, with potential for explicit calculations and categorifications in algebraic topology and algebraic geometry.

Conclusion

The paper presents a comprehensive algebro-geometric perspective on double loop spaces of Lagrangian and orthogonal Grassmannians, giving explicit quiver moduli interpretations, proving their stabilization yields the expected Bott periodicity homotopy types, and establishing new finite-dimensional, algebro-geometric realizations of these classical topological spaces. This provides both new proof techniques and a concrete computational framework for the study of periodic phenomena in the topology of homogeneous spaces, with broad implications for the interaction between algebraic geometry, representation theory, and stable homotopy theory (2607.10956).