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Normalized Vacuum States in N = 4 Supersymmetric Yang--Mills Quantum Mechanics with Any Gauge Group

V. G. Kac, A. V. Smilga

TL;DR

This work analyzes the existence and counting of normalized vacuum states in ${ m N}=4$ supersymmetric Yang--Mills quantum mechanics for arbitrary gauge groups. It develops and deploys three complementary methods: (i) mass deformation to translate vacuum counting into solving algebraic equations tied to distinguished ${ m sl}(2)$ subalgebras, (ii) a deficit-plus-principal-term analysis of the Witten index via infrared regularization and Weyl invariance, and (iii) an asymptotic, Born--Oppenheimer-style study of effective dynamics to construct normalizable zero-energy solutions. The key results are that ${ m SU}(n)$ has a unique normalized vacuum, while non-unitary groups (starting with ${ m Sp}(6)$, ${ m SO}(8)$, and all exceptional groups) host multiple vacua, counted by distinguished ${ m sl}(2)$ embeddings; explicit counts include ${ m Sp}(2r)$ as the number of partitions of $r$ into distinct parts and ${ m SO}(n)$ as the number of partitions of $n$ into distinct odd parts, with exceptional groups yielding discrete small integers (e.g., ${ m G}_2=2$, ${ m F}_4=4$, ${ m E}_6=3$, ${ m E}_7=6$, ${ m E}_8=11$). The deficit-term analysis reveals nontrivial contributions for non-unitary groups and subvalley effects, while the asymptotic-wave-function approach demonstrates a normalizable, Weyl-invariant solution for ${ m N}=4$, reinforcing the vacua found by the other methods. Overall, the paper establishes a robust framework for vacuum counting across gauge groups, with significant implications for D-brane and M-theory perspectives.

Abstract

We study the question of existence and the number of normalized vacuum states in N = 4 super-Yang-Mills quantum mechanics for any gauge group. The mass deformation method is the simplest and clearest one. It allowed us to calculate the number of normalized vacuum states for all gauge groups. For all unitary groups, #(vac) = 1, but for the symplectic groups [starting from Sp(6) ], for the orthogonal groups [starting from SO(8)] and for all the exceptional groups, it is greater than one. We also discuss at length the functional integral method. We calculate the ``deficit term'' for some non-unitary groups and predict the value of the integral giving the ``principal contribution''. The issues like the Born-Oppenheimer procedure to derive the effective theory and the manifestation of the localized vacua for the asymptotic effective wave functions are also discussed.

Normalized Vacuum States in N = 4 Supersymmetric Yang--Mills Quantum Mechanics with Any Gauge Group

TL;DR

This work analyzes the existence and counting of normalized vacuum states in supersymmetric Yang--Mills quantum mechanics for arbitrary gauge groups. It develops and deploys three complementary methods: (i) mass deformation to translate vacuum counting into solving algebraic equations tied to distinguished subalgebras, (ii) a deficit-plus-principal-term analysis of the Witten index via infrared regularization and Weyl invariance, and (iii) an asymptotic, Born--Oppenheimer-style study of effective dynamics to construct normalizable zero-energy solutions. The key results are that has a unique normalized vacuum, while non-unitary groups (starting with , , and all exceptional groups) host multiple vacua, counted by distinguished embeddings; explicit counts include as the number of partitions of into distinct parts and as the number of partitions of into distinct odd parts, with exceptional groups yielding discrete small integers (e.g., , , , , ). The deficit-term analysis reveals nontrivial contributions for non-unitary groups and subvalley effects, while the asymptotic-wave-function approach demonstrates a normalizable, Weyl-invariant solution for , reinforcing the vacua found by the other methods. Overall, the paper establishes a robust framework for vacuum counting across gauge groups, with significant implications for D-brane and M-theory perspectives.

Abstract

We study the question of existence and the number of normalized vacuum states in N = 4 super-Yang-Mills quantum mechanics for any gauge group. The mass deformation method is the simplest and clearest one. It allowed us to calculate the number of normalized vacuum states for all gauge groups. For all unitary groups, #(vac) = 1, but for the symplectic groups [starting from Sp(6) ], for the orthogonal groups [starting from SO(8)] and for all the exceptional groups, it is greater than one. We also discuss at length the functional integral method. We calculate the ``deficit term'' for some non-unitary groups and predict the value of the integral giving the ``principal contribution''. The issues like the Born-Oppenheimer procedure to derive the effective theory and the manifestation of the localized vacua for the asymptotic effective wave functions are also discussed.

Paper Structure

This paper contains 9 sections, 130 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mass deformation of the ${\cal N}=4$ theory.
  3. Distinguished $sl(2)$ subalgebras in simple Lie algebras.
  4. Deficit term.
  5. Generalities. $SU(2)$ case.
  6. Proper deficit term.
  7. Total deficit vs. proper deficit.
  8. Asymptotic wave function.
  9. Conclusions.

Figures (5)

  • Figure 1: a) root system, b) coroot system, and c) non--trivial distinguished marked Dynkin diagram for the group $G_2$.
  • Figure 2: Non--trivial distinguished marked Dynkin diagram for the group $Sp(6)$.
  • Figure 3: Non--trivial distinguished marked Dynkin diagram for the group $SO(8)$.
  • Figure 4: Unbroken discrete subgroups $S_m$ of the exceptional groups for a supersymmetric Higgs vacuum state corresponding to a given distinguished marked Dynkin diagram.
  • Figure 5: Dynkin diagrams for some groups .