Fourier transform of irregular connections on $\mathbb P^1$ and classification of Argyres-Douglas theories

Abstract

We give a mathematical interpretation of the dualities between type $A$ Argyres-Douglas theories recently obtained by Beem, Martone, Sacchi, Singh and Stedman, building on work of Xie. Using the fact that, via the wild nonabelian Hodge correspondence, the data defining such a theory amount to singularity data for irregular connections on $\mathbb P^1$ of a specific form, we show that these dualities can all be realized as compositions of two types of more basic operations acting on such irregular connections: the Fourier transform and a Möbius transformation exchanging zero and infinity. The proof relies on the stationary phase formula giving explicit expressions for the singularity data of the Fourier transform. We also clarify the relation between the quivers describing the 3d mirrors of type $A$ Argyres-Douglas theories and the nonabelian Hodge diagrams defined in work of Boalch-Yamakawa and of the author: the 3d mirror corresponds to the unique nonabelian Hodge diagram with no negative edges/loops among those of singularity data in the corresponding orbit under basic operations.

Figures (6)

• Figure 1: Effect of an elementary AD-$A$ operation on the Young diagrams of a type I AD-$A$ parameter $\mathcal{T}=(m,k, [Y^0], [Y^\infty])$: it removes the column of one of the diagrams (represented in blue), and add its 'complement' to the other diagram (represented). Which one of $[Y^0], [Y^\infty]$ is the blue/red diagram depends on the case.
• Figure 2: Dictionary between parameters for type $A$ Argyres--Douglas theories in the notations of beem2025simplifying and the parameter of the irregular curve with boundary data of standard AD-$A$ type (possibly type I) in our notation here, in both directions. The unipotent conjugacy class $\mathcal{C}_0$ is given by $\mathcal{C}_0=\{(1, [Y])\}$.
• Figure 3: Structure of the orbit $\mathcal{O}(\mathcal{T})$, for $s>1$. The vertices correspond to the elements of the orbit, and the arrows correspond to elementary type I AD-$A$ operations. The blue arrows correspond to the steps where the Young diagram coming from $[Y^0]$ decreases and the one coming from $[Y^\infty]$ increases, and the red ones to the opposite steps. On the bottom row, the Fourier transform exchanges the Young diagrams at 0 and $\infty$. The slopes are all of the form $k'=\frac{s}{r'}$, with $r'\equiv \pm \rho \mod s$. The oriented arrows correspond to applying an elementary AD-operation. The left column corresponds to the case $r'\equiv \rho \mod s$, while the right one corresponds to the case $r'\equiv -\rho \mod s$. The bottom row corresponds to the only two slopes satisfying $k'>1$.
• Figure 4: Structure of the orbit $\mathcal{O}(\mathcal{T})$, for $s=1$. At $\mathcal{T}_1$, applying $F$ yields an irregular curve with boundary data with $m+2$ regular singularities.
• Figure 5: The duality transforming the Young diagram $[Y]$ into its complement $[Y^c]$, and the corresponding intermediate steps, drawn for the example $[Y]=[5,3, 2^2, 1]$, $m=1$, $s=7$. The diagram $[Y]$ is represented in blue on the left part and the diagram $[Y^c]$ is represented in red, up to reading the columns from right to left. The right part of the figure shows an intermediate step: at each step one column of the blue diagram is removed, and its complement is added to the red diagram. The intermediate parameter thus features two nontrivial Young diagrams $[Y_l]$ and $[\widetilde{Y}_l]$.

Theorems & Definitions (78)

• Theorem 1.1: Prop. \ref{['prop:transformation_parameter_elementary_operation_type_I']}
• Theorem 1.2: Prop. \ref{['prop:duality_I']}, \ref{['prop:duality_II_complement_diagram']}, \ref{['prop:duality_III_gen_case']}
• Theorem 1.3: Prop. \ref{['prop:nonnegative_diagram_type_I']}, \ref{['prop:3d_mirrors_are_nonneg_diagrams_type_I']}, \ref{['prop:nonnegative_diagram_general_case']}, \ref{['prop:3d_mirrors_are_nonneg_diagrams_gen_case']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Remark 2.6
• Definition 2.8