insight - Quantum Field Theory - # Vortex Quantum Mechanics and Schwinger-Dyson Identities in 3d ${\cal N} = 4$ Gauge Theories

Core Concepts

The author derives and interprets non-perturbative Schwinger-Dyson identities satisfied by correlation functions of a certain gauge-invariant operator, the "vortex character," in 3d ${\cal N} = 4$ gauge theories. These identities are obtained from various physical perspectives, including the vortex quantum mechanics, the 3d gauge theory, a 2d q-Toda theory, and 6d little string theory.

Abstract

The author analyzes symmetries corresponding to separated topological sectors of 3d ${\cal N} = 4$ gauge theories with Higgs vacua, compactified on a circle. These symmetries are encoded in Schwinger-Dyson identities satisfied by the correlation functions of the "vortex character" observable, which is realized as the vortex partition function of the 3d gauge theory in the presence of a 1/2-BPS line defect.

The author provides four equivalent definitions of the vortex character observable:

- As the Witten index of a 1-dimensional gauged supersymmetric quantum mechanics living on the vortices of the 3d ${\cal N} = 4$ gauge theory, which includes additional chiral matter due to the defect.
- As a sum of half-indices for the 3d ${\cal N} = 4$ gauge theory, in the presence of the codimension-2 defect.
- As a deformed Wq,t-algebra correlator on an infinite cylinder, with stress tensor and higher spin current insertions, including a distinguished set of "fundamental" vertex operators.
- As the partition function of the 6-dimensional (2, 0) little string theory compactified on the cylinder, in the presence of codimension-4 and point-like D-brane defects.

The author analyzes the Schwinger-Dyson identities from each of these perspectives and establishes the dictionary between them. As an application, the author briefly discusses the action of 3d Seiberg duality on the vortex character observable.

To Another Language

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arxiv.org

Stats

The vortex character observable is graded by the vortex number k(a) = 1/(2π) ∫ TrF(a), which is the topological U(1) charge for the a-th gauge group, conjugate to the vortex counting fugacity ζ(a)_3d.

Quotes

"The vortex character is the Witten index of a 1-dimensional gauged supersymmetric quantum mechanics living on the vortices of G3d, which includes additional chiral matter due to the defect."
"The vortex character is a sum of half-indices for the 3d ${\cal N} = 4$ gauge theory, in the presence of a codimension-2 defect."
"The vortex character is a deformed Wq,t-algebra correlator on an infinite cylinder, with stress tensor and higher spin current insertions, including a distinguished set of 'fundamental' vertex operators."
"The vortex character is the partition function of the 6-dimensional (2, 0) little string theory compactified on the cylinder, in the presence of codimension-4 and point-like D-brane defects."

Key Insights Distilled From

by Nathan Haouz... at **arxiv.org** 10-01-2024

Deeper Inquiries

The Schwinger-Dyson identities associated with the vortex character observable in 3d ${\cal N} = 4$ gauge theories have profound physical implications. These identities serve as non-perturbative constraints on the correlation functions of the theory, particularly in the context of vortex configurations. They encapsulate the symmetries of the gauge theory when transitioning between different topological sectors, specifically those related to the vortex number. The vortex character, which can be interpreted as a partition function in the presence of a 1/2-BPS line defect, reflects the underlying quantum dynamics of the vortices and their interactions with the gauge fields.
The identities imply that the vortex character is not merely a mathematical construct but is deeply tied to the physical observables of the theory. They reveal how the vortex configurations contribute to the overall structure of the gauge theory, including aspects such as the moduli space of vacua and the dynamics of BPS states. Furthermore, these identities can be viewed as Ward identities that arise from the invariance of the path integral under shifts in the vortex number, thus providing a framework to understand the non-perturbative aspects of the theory, including phenomena like wall-crossing and the emergence of new vacua.

The results of this work can be generalized to other dimensions and supersymmetry settings by leveraging the underlying principles of the Schwinger-Dyson equations and the vortex character observable. For instance, in higher dimensions, such as 4d or 5d gauge theories, similar non-perturbative identities can be constructed by considering the appropriate topological sectors and their associated observables. The concept of vortex charges can be extended to these dimensions, allowing for the exploration of vortex-like solutions and their contributions to the dynamics of the gauge theories.
Moreover, the framework established in this paper can be adapted to different supersymmetry settings, such as ${\cal N} = 2$ or ${\cal N} = (4, 4)$ theories. By analyzing the corresponding moduli spaces and the associated quantum mechanics, one can derive analogous vortex characters and Schwinger-Dyson identities. The connection to representation theory, particularly in the context of quantum affine algebras, can also be explored in these generalized settings, leading to a richer understanding of the symmetries and dynamics of the theories involved.

The connection between the vortex character and the representation theory of quantum affine algebras opens up several potential applications in both mathematical physics and theoretical research. Firstly, this relationship provides a powerful tool for studying the representation theory of quantum affine algebras through physical observables. The vortex character, being a deformed character of finite-dimensional representations, can be used to derive new results in the representation theory, including character formulas and fusion rules.
Additionally, this connection can facilitate the exploration of dualities in gauge theories, particularly Seiberg duality in 3d settings. By understanding how vortex characters transform under duality, researchers can gain insights into the duality relations between different gauge theories and their respective representations. This can lead to a deeper understanding of the non-perturbative dynamics and the structure of the moduli spaces involved.
Furthermore, the vortex character's role in the context of integrable systems and mathematical physics can be significant. The insights gained from the representation theory of quantum affine algebras can be applied to study integrable models, topological field theories, and even string theory, where similar structures arise. Overall, the interplay between vortex characters and quantum affine algebras enriches the landscape of theoretical physics, providing new avenues for research and exploration.

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