Molien--Weyl Singlet Counting and BFSS2--Factorization in Gaussian Matrix QM

https://arxiv.org/abs/2605.04621 

We study the singlet-sector structure of mass-deformed BFSSd+1 matrix quantum mechanics by combining the large--\(d\) Gaussian reduction with the Molien--Weyl projection. The Gaussian reduction captures the bulk matrix dynamics through a gauged harmonic oscillator, while the Molien--Weyl integral imposes the Gauss law and reorganizes the physical Hilbert space into holonomy-projected singlet excitations.
We show that the very-low-temperature bosonic singlet spectrum is universally controlled by the quadratic Gram operators \(\Tr(X_aX_b)\), whose number is \(d(d+1)/2\). For \(N=2\), this result is established by explicit residue computations and character methods; for \(N>2\), it is supported by the character analysis. Thus the infrared spectrum begins as a collection of BFSS2--like Gram towers, although higher invariant structures generally modify the full partition function.
We also give a Hamiltonian derivation of the exceptional exact factorization at \((d,N)=(2,2)\), where the BFSS3 singlet partition function equals the cube of the BFSS2 one for all temperatures. This rigidity is special to the \(SU(2)\) invariant tensor structure and explains why \(d=1\) and \(N=2\) are exceptional regimes without a deconfinement crossover. Finally, we extend the Gram-counting picture to supersymmetric BFSS/BMN models and indicate how the Molien--Weyl formulation can benchmark Monte Carlo simulations in both \(X_a\)-space and holonomy space.