Gauge origami on broken lines (2502.07149v1)

Abstract: In analogy to Nekrasov's theory of gauge origami on intersecting branes, we introduce the gauge origami moduli space on broken lines. We realize this moduli space as a Quot scheme parametrising zero-dimensional quotients of a torsion sheaf on two intersecting affine lines, and describe it as a moduli space of quiver representations. We construct a virtual fundamental class and virtual structure sheaf, by which we define $K$-theoretic invariants. We compute its associated partition function for all ranks, and show that it reproduces the generating series of equivariant $\chi_{y}$-genus when the moduli space is smooth. Finally, we relate our partition function with the virtual invariants of the Quot schemes of the affine plane and Nekrasov's partition function.

• The paper introduces a novel gauge origami formulation on broken lines by modeling zero-dimensional quotients with Grothendieck’s Quot schemes and quiver representations.
• It derives a closed formula for partition functions and constructs virtual cycle formalism that underpins equivariant K-theoretic invariants.
• The study demonstrates framing independence in the partition function, linking its results to classical Nekrasov functions and advancing intersecting brane dynamics in string theory.

Insights into Gauge Origami on Broken Lines

This paper, authored by Sergej Monavari, explores an innovative extension of Nekrasov's theory of gauge origami to "broken lines," an approach designed to better fit mathematical frameworks associated with coupled vortex systems in string theory. By focusing specifically on zero-dimensional quotients of a torsion sheaf on two intersecting affine lines, Monavari lays out a comprehensive approach to interpret these under quiver representations, offering deep insights into moduli spaces and gauging dynamics in intersecting brane systems.

Summary and Key Results

In formulating the gauge origami moduli space on broken lines, Monavari employs Grothendieck's Quot schemes to define the zero-dimensional quotient parameters and subsequently models them as quiver representations. A significant theoretical development in the paper is the realization of the moduli space as a zero locus of a section of a vector bundle descended from the non-commutative Quot scheme's space of representations, thus leveraging quiver theory for computational purposes.

Key results include:

1. Closed Formula for Partition Function: The paper derives a closed expression for the partition function, crucial for $K$-theoretic invariants, computing associated partition functions for all ranks and aligning these with equivariant $\chi_{y}$-genus when the moduli space is smooth.
2. Virtual Cycle Formalism: The author constructs a virtual fundamental class and a virtual structure sheaf, providing tools for the definition of virtual invariants and enabling the computation of the derived partition function.
3. Framing Independence: A striking result is the demonstration of the independence of the partition function from the framing parameters—an insight that hints at deeper symmetries or stabilizing constraints within the mathematical model, potentially reflective of physical invariances.
4. Relational Mapping: Monavari effectively relates the newly-derived partition function to previously-known theories, such as the classical Nekrasov partition function with fundamental and anti-fundamental matters and the virtual invariants of Quot schemes on the affine plane, consolidating this work's theoretical significance.

Implications and Future Directions

Theoretically, the paper demonstrates that description within this framework preserves a broad set of structures, drawing a tension between complex geometric properties and comparatively elegant algebraic manipulations through the use of quivers and partition functions. Practically, these foundations have deep implications for string theory's exploration of dynamics in intersecting branes, potentially illuminating new physical phenomena within those frameworks. Specifically, the formalism developed could inform the paper of D1-branes probing intersecting D3-branes, amongst other configurations.

Additionally, this line of research opens further inquiry into the computation and interpretation of refined partition functions, such as those involving the elliptic genus within the same geometric context. This challenges current computational practices and emphasizes the need for an optimized approach considering the higher complexities introduced by rank variations.

In conclusion, Monavari's work represents a sophisticated and valuable contribution to the paper of gauge origami and related systems, enriching our understanding of moduli spaces and their automatic symmetry-preserving transformations for future mathematical and physical applications. The engagement with both classical and contemporary mathematical tools highlights a quintessential synergy in the field of modern theoretical physics.