Extended $5d$ Seiberg-Witten Theory and Melting Crystal

We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills on $\mathbb{R}^4 \times S^1$. We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills in the $\Omega$ background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U(1) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a $\tau$ function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann-Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg-Witten differential.

💡 Research Summary

The paper investigates an extension of five‑dimensional 𝒩=1 supersymmetric Yang‑Mills (SYM) theory formulated on the space‑time 𝑅⁴ × S¹ and placed in the Ω‑background. The authors focus on a class of observables they call “loop operators,” which are the natural five‑dimensional analogues of Wilson loops that wind once around the compact S¹ direction. After performing a topological twist, these operators become supersymmetric and their correlation functions can be evaluated exactly by supersymmetric localization.

Localization reduces the functional integral to a sum over fixed points of the toric action on the instanton moduli space. In the abelian (U(1)) case each fixed point is labeled by a Young diagram (a partition) λ, and the contribution of λ is weighted by a factor q^{|λ|} together with a product over the boxes of λ that depends on the couplings t₁, t₂, … attached to the loop operators. Consequently the generating function \