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Special Kähler geometries of $\mathcal{N}=4$ superYang-Mills

Philip C. Argyres, Antoine Bourget, Julius F. Grimminger, Matteo Lotito, Mitch Weaver

TL;DR

This work provides a complete framework to classify the SK moduli-space geometries of 4d ${ m N}=4$ sYM theories with simple gauge algebras by translating SK data into integral symplectic representations of Weyl groups. The central achievement is proving a one-to-one correspondence between SK structure orbits (for principal Dirac pairing) and S-duality orbits of global gauge structures, thereby enabling a low-energy probe of S-duality from moduli-space geometry alone. The authors develop a concrete algorithm—involving lattice invariants, Ext groups, and intertwiners—to construct all admissible SK data $(S, au)$, determine their fixed loci, and compute self-duality groups, with explicit results for ${ m su}(N)$ and ${ m so}(12)$ (and ${ m so}(5) ight)$ examples. They also analyze and reconcile several discrepancies with field-theoretic S-duality, distinguishing IR accidental identifications from genuine mismatches, and they extend the discussion to non-principal Dirac pairings and twisted product SK geometries, including a notable ${ m u}(N)$ example. Overall, the paper strengthens the link between low-energy moduli-space geometry and the global structure/S-duality data of ${ m N}=4$ sYM theories, while also raising intriguing questions about exotic SK geometries and the role of higher-form symmetries.

Abstract

The low energy effective theory on the moduli space of vacua of 4d superYang-Mills (sYM) theory defines a special Kähler geometry. For simple sYM gauge algebras, $\mathfrak{g}$, we classify all compatible special Kähler structures by showing that they are in one-to-one correspondence with certain equivalence classes of integral symplectic representations of the Weyl group of $\mathfrak{g}$. We further demonstrate that, for principal Dirac pairing, these equivalence classes are in one-to-one correspondence with the S-duality orbits of the global structures of the corresponding $\mathfrak{g}$ sYM gauge theory, after a mistake in the field theory literature is corrected. This provides a low-energy test of S-duality. We also discuss twisted product geometries made from factors with special Kähler structures with non-principal Dirac pairings.

Special Kähler geometries of $\mathcal{N}=4$ superYang-Mills

TL;DR

This work provides a complete framework to classify the SK moduli-space geometries of 4d sYM theories with simple gauge algebras by translating SK data into integral symplectic representations of Weyl groups. The central achievement is proving a one-to-one correspondence between SK structure orbits (for principal Dirac pairing) and S-duality orbits of global gauge structures, thereby enabling a low-energy probe of S-duality from moduli-space geometry alone. The authors develop a concrete algorithm—involving lattice invariants, Ext groups, and intertwiners—to construct all admissible SK data , determine their fixed loci, and compute self-duality groups, with explicit results for and (and examples. They also analyze and reconcile several discrepancies with field-theoretic S-duality, distinguishing IR accidental identifications from genuine mismatches, and they extend the discussion to non-principal Dirac pairings and twisted product SK geometries, including a notable example. Overall, the paper strengthens the link between low-energy moduli-space geometry and the global structure/S-duality data of sYM theories, while also raising intriguing questions about exotic SK geometries and the role of higher-form symmetries.

Abstract

The low energy effective theory on the moduli space of vacua of 4d superYang-Mills (sYM) theory defines a special Kähler geometry. For simple sYM gauge algebras, , we classify all compatible special Kähler structures by showing that they are in one-to-one correspondence with certain equivalence classes of integral symplectic representations of the Weyl group of . We further demonstrate that, for principal Dirac pairing, these equivalence classes are in one-to-one correspondence with the S-duality orbits of the global structures of the corresponding sYM gauge theory, after a mistake in the field theory literature is corrected. This provides a low-energy test of S-duality. We also discuss twisted product geometries made from factors with special Kähler structures with non-principal Dirac pairings.

Paper Structure

This paper contains 58 sections, 176 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction and summary
  2. Outline of the paper.
  3. Further directions.
  4. Note added:
  5. Integer representations of Weyl and complexified Weyl groups
  6. Ingredients of an ${\mathcal{N}}{=}4$ SK structure
  7. Classifying integral symplectic representations of a Weyl group
  8. Weyl group invariant lattices
  9. Symplectic representations of Weyl groups and their fixed points
  10. Classifying bindings by Ext groups
  11. Practical computation.
  12. Integral equivalence of symplectic representations
  13. Lemma 1.
  14. Lemma 2.
  15. Lemma 3.
  16. ...and 43 more sections

Figures (1)

  • Figure 1: The S-duality orbits of ${\mathfrak g}=\mathfrak{su}(4)$${\mathcal{N}}{=}4$ sYM global structures. These are depicted by intertwiners $\textsf{S}$ and $\textsf{T}$ in \ref{['PSL ST']} acting on the symplectic representations $S_{(d,z)}$ associated to the global forms by \ref{['magnetic S']}.