Seiberg-Witten geometry of four-dimensional \(\mathcal{N} = 2\) quiver gauge theories. (English) Zbl 07727614
Summary: Seiberg-Witten geometry of mass deformed \(\mathcal{N}=2\) superconformal ADE quiver gauge theories in four dimensions is determined. We solve the limit shape equations derived from the gauge theory and identify the space \(\mathfrak{M}\) of vacua of the theory with the moduli space of the genus zero holomorphic (quasi)maps to the moduli space Bun\(_G(\mathcal{E})\) of holomorphic \(G^{\mathbb{C}}\)-bundles on a (possibly degenerate) elliptic curve \(\mathcal{E}\) defined in terms of the microscopic gauge couplings, for the corresponding simple ADE Lie group \(G\). The integrable systems \(\mathfrak{P}\) underlying the special geometry of \(\mathfrak{M}\) are identified. The moduli spaces of framed \(G\)-instantons on \(\mathbb{R}^2\times\mathbb{T}^2\), of \(G\)-monopoles with singularities on \(\mathbb{R}^2\times\mathbb{S}^1\), the Hitchin systems on curves with punctures, as well as various spin chains play an important rôle in our story. We also comment on the higher-dimensional theories.
MSC:
| 81T12 | Effective quantum field theories |
| 81T13 | Yang-Mills and other gauge theories in quantum field theory |
| 81T70 | Quantization in field theory; cohomological methods |
| 81V15 | Weak interaction in quantum theory |
| 14D21 | Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) |
| 81V60 | Mono-, di- and multipole moments (EM and other), gyromagnetic relations |
| 81Q80 | Special quantum systems, such as solvable systems |
| 32H02 | Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables |
| 14H52 | Elliptic curves |
| 70S15 | Yang-Mills and other gauge theories in mechanics of particles and systems |
| 17B45 | Lie algebras of linear algebraic groups |
| 14B05 | Singularities in algebraic geometry |
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