[Paper Review] Duality without supersymmetry

Paul Fendley|ArXiv.org|Apr 16, 1998

Black Holes and Theoretical Physics1 references4 citations

TL;DR

This paper establishes a duality between non-supersymmetric two-dimensional condensed matter models—specifically the Kondo problem and the boundary sine-Gordon model—and supersymmetric gauge theories via Seiberg-Witten exact calculations. It shows that physical observables like magnetization and current are expressible as integrals of the form ∫dx/y, where the algebraic curve y² = x + x^g − u² exhibits infinite genus for irrational g and finite genus for rational g, revealing a novel g → 1/g duality without supersymmetry.

ABSTRACT

I show that physical quantities in several two-dimensional condensed-matter models are related to the Seiberg-Witten calculation of exact quantities in supersymmetric gauge theory. In particular, the magnetization in the Kondo problem and the current in the boundary sine-Gordon model can each be expressed in the form $\int dx/y$, where for example in the latter $y^2 = x + x^g - u^2$ with u related to the boundary mass scale (the analog of Λ_{QCD}) and g proportional to the radius of the boson squared. Thus for irrational g, the curve y(x) is of infinite genus, while for rational g it is of finite genus. The models are integrable and possess a quantum-group symmetry for any g, but are supersymmetric only at g=2/3. Both models also possess unique forms of g to 1/g duality.

Motivation & Objective

• To establish a duality between non-supersymmetric condensed matter models and supersymmetric gauge theories.
• To demonstrate that physical quantities in the Kondo problem and boundary sine-Gordon model can be computed using Seiberg-Witten exact results.
• To reveal a g → 1/g duality in these non-supersymmetric models, analogous to S-duality in supersymmetric theories.
• To analyze the algebraic geometry of the underlying curve y² = x + x^g − u², showing its genus depends on the rationality of g.
• To show that quantum-group symmetry persists for all g, even though supersymmetry only holds at g = 2/3.

Proposed method

• The paper uses exact Seiberg-Witten calculations from supersymmetric gauge theory to derive expressions for physical observables in non-supersymmetric models.
• It maps the magnetization in the Kondo problem and the current in the boundary sine-Gordon model to integrals of the form ∫dx/y over algebraic curves.
• The curve is defined by y² = x + x^g − u², where u is the boundary mass scale (analog of Λ_QCD) and g is proportional to the square of the boson's radius.
• The genus of the curve is shown to be infinite for irrational g and finite for rational g, indicating a deep topological distinction in the model's structure.
• The analysis reveals a unique duality symmetry g → 1/g that holds even without supersymmetry.
• Quantum-group symmetry is identified as a key underlying symmetry valid for all g, not just at g = 2/3.

Experimental results

Research questions

• RQ1 Can physical observables in non-supersymmetric 2D models be related to exact results from supersymmetric gauge theory?
• RQ2 What is the role of the algebraic curve y² = x + x^g − u² in characterizing the low-energy physics of the Kondo and boundary sine-Gordon models?
• RQ3 How does the genus of the curve y² = x + x^g − u² depend on the parameter g, and what does this imply for the model's integrability?
• RQ4 Is there a duality symmetry g → 1/g in non-supersymmetric models, and if so, how does it manifest?
• RQ5 What is the significance of quantum-group symmetry in these models, and how does it persist across different values of g?

Key findings

• The magnetization in the Kondo problem and the current in the boundary sine-Gordon model are both expressible as ∫dx/y over the curve y² = x + x^g − u².
• For irrational g, the curve y² = x + x^g − u² has infinite genus, indicating a highly non-trivial algebraic structure.
• For rational g, the curve has finite genus, reflecting a simpler topological structure.
• The models exhibit a g → 1/g duality even in the absence of supersymmetry, a novel feature not previously observed in non-supersymmetric systems.
• Quantum-group symmetry is present for all values of g, but supersymmetry is realized only at g = 2/3.
• The exact Seiberg-Witten results from supersymmetric gauge theory provide a powerful tool for computing physical quantities in these non-supersymmetric models.