Inequivalence between the Euclidean and Lorentzian versions of the type IIB matrix model from Lefschetz thimble calculations
Chien-Yu Chou, Jun Nishimura, Tripathi, Ashutosh · arXiv (Cornell University) · 2025
The type IIB matrix model is conjectured to describe superstring theory nonperturbatively in terms of ten $N \times N$ bosonic traceless Hermitian matrices $A_μ$ ($μ=0, \ldots , 9$), whose eigenvalues correspond to $(9+1)$-dimensional space-time. Quite often, this model has been investigated in its Euclidean version, which is well defined although the ${\rm SO}(9,1)$ Lorentz symmetry of the original model is replaced by the ${\rm SO}(10)$ rotational symmetry. Recently, a well-defined model respecting the Lorentz symmetry has been proposed by gauge-fixing the Lorentz symmetry nonperturbatively using the Faddeev-Popov procedure. Here we investigate the two models by Monte Carlo simulations, overcoming the severe sign problem by the Lefschetz thimble method, in the case of matrix size $N=2$ omitting fermionic contributions. We add a quadratic term $γ\, \mathrm{tr} (A_μA^μ)$ in the action and calculate the expectation values of rotationally symmetric (or Lorentz symmetric) observables as a function of the coefficient $γ$. Our results exhibit striking differences between the two models around $γ=0$ and in the $γ>0$ region, associated with the appearance of different saddle points, clearly demonstrating their inequivalence against naive expectations from quantum field theory.