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I have recently been reading literature on geometric constructions of representations of affine Lie algebras by Nakajima and others. In particular, the representations arise as cohomologies of moduli spaces of sheaves on a surface.

For a smooth surface, $S$, if we consider the Hilbert scheme of $n$-points $S^{[n]}$, for a suitably chosen cohomology theory, the vector space formed by,

$$\bigoplus_{n=0}^\infty H^*(S^{[n]})$$

is a highest weight representation of the Heisenberg algebra on $H^*(S)$. A recent paper has shown if $S$ is an ADE surface, like those studied in the context of gauge theory, one obtains a larger action of an affine Lie algebra of corresponding ADE type.

Since Verma modules appear in conformal field theory, is there also a physical context or motivation for such geometric constructions of representations (including string theory and more broadly gauge theory)?

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The idea is that within string theory any field theoretical symmetry can be engineered via some brane configuration. This is true for a wide class of finite groups, simply-laced Lie group symmetries, and even affine symmetries. See Geometric Engineering of Quantum Field Theories for details.

I will summarize two brane configurations that give rise to gauge theories with affine Lie group symmetries and some applications of those constructructions.

  1. Type IIA strings over an $A_{n}$ singularity are dual to a set of $n$ adjacent $NS$-$5$ branes of the type IIB theory (page 17 of Branes and Toric Geometry and Geometric Singularities and Enhanced Gauge Symmetries for details). Now the point is that an $A_{n}$ singularity can be though locally as a $\mathbb{C}^{2}/\mathbb{Z}_{n}$ geometry and intuitively you can take the $n \rightarrow \infty$ limit to construct the $A_{\infty}^{\infty}$ quiver. The brane configuration dual to this $\mathbb{C}^{2}/\mathbb{Z}_{\infty}$ geometry is an array of adjacent $NS$-$5$ branes following the pattern of the $\tilde{A}_{n}$ affine Dynkin digram. Notice that this related to your actual question because a $\mathbb{C}^{2}/\mathbb{Z}_{n}$ geometry can be thought as an $A_{n}$ singularity inside a $K3$ surface and the $n \rightarrow \infty$ limit can be taken if this $K3$ is non-compact.

Applications

  • The reference Crystal Melting and Black Holes (page 7) use the latter construction to compute the Donaldson-Thomas invariants of a local $\mathbb{C}^{2}/\mathbb{Z}_{\infty} \times \mathbb{C}$ geometry.
  • In the page 15 of F-theory and the Classification of Little Strings an affine $\tilde{A}_{n}$ geometry is explicitly engineered and used to obtain theories with sixteen superchargers in six dimensions.
  • You can learn in the page 72 of the famous Strong Coupling Test of S-duality about the relationship between the latter construction, the moduli space of instantons on $ALE$ singularities and two dimensional rational conformal field theory.
  • An interesting application for the geometric Langlands duality in the context of complex surfaces emerges in Five-Branes in M-Theory and a Two-Dimensional Geometric Langlands Duality by considering the lift of the construction from above to $M$-theory on a Taub-Nut space $TN_{n-1}$ and recalling that $TN_{n}$ is locally $\mathbb{C}^{2}/\mathbb{Z}_{n}$.
  1. $N=2$ quiver gauge theories in $4d$ with affine $ADE$ symmetry: Here the trick is nearly the same as above. Embbed an array of singularities following an affine $ADE$ pattern into a $K3$ surface or a Calabi-Yau threefold. Deformations with $N=1$ supersymmetry can also be constructed allowing deformations by superpotential terms. Reference: A Geometric Unification of Dualities.
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