Monte Carlo studies of supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature (0707.4454v1)

Abstract: We present the first Monte Carlo results for supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature. The recently proposed non-lattice simulation enables us to include the effects of fermionic matrices in a transparent and reliable manner. The internal energy nicely interpolates the weak coupling behavior obtained by the high temperature expansion, and the strong coupling behavior predicted from the dual black hole geometry. The Polyakov line takes large values even at low temperature suggesting the absence of a phase transition in sharp contrast to the bosonic case. These results provide highly non-trivial evidences for the gauge/gravity duality.

• The paper demonstrates the use of a novel non-lattice Monte Carlo method to explore energy dynamics and support gauge/gravity duality.
• It employs the Rational Hybrid Monte Carlo algorithm to effectively handle fermionic determinants, maintaining maximal supersymmetry in a dimensionally reduced 1D model.
• Results reveal a smooth transition in internal energy from weak to strong coupling, aligning with predictions from dual black hole geometries and a deconfined phase.

Monte Carlo Studies of Supersymmetric Matrix Quantum Mechanics at Finite Temperature

The paper presented in the paper focuses on supersymmetric matrix quantum mechanics with sixteen supercharges at finite temperature, utilizing non-lattice Monte Carlo simulations. The paper aims to provide evidence supporting gauge/gravity duality by exploring the dynamics of this model, which plays a crucial role in non-perturbative formulations of string/M theories and large-$N$ gauge theories.

The authors apply a novel non-lattice simulation method to avoid the complex issues often presented by traditional lattice techniques when dealing with supersymmetric systems. The research examines the internal energy dynamics of the models and their correspondence to theoretical predictions from string theory, notably interpolating between weak coupling behaviors and those expected from strong coupling dual black hole geometries.

Simulation Techniques and Theoretical Framework

The model under investigation is derived by dimensionally reducing a 10-dimensional super Yang-Mills theory to 1-dimensional quantum mechanics. It comprises $N \times N$ Hermitian matrices for both bosonic and fermionic components, preserving maximal supersymmetry of the system. The simulations employ the Rational Hybrid Monte Carlo (RHMC) algorithm, which is particularly adept at handling the fermionic determinants crucial for maintaining accurate dynamical simulations of supersymmetric systems.

Utilizing a Fourier mode cutoff $\Lambda$, the authors circumvent challenges associated with lattice gauge theory, such as the preservation of supersymmetry and gauge invariance. The gauge symmetry is trivially managed in 1D simulations, allowing for complete gauge fixing, which considerably simplifies the computational approach.

Results and Observations

Key findings illustrate that the internal energy behavior transitions smoothly from weak to strong coupling regimes, aligning well with anticipated results from dual geometry perspectives. This serves as a non-trivial evidence supporting gauge/gravity duality, a fundamental tenet in string theory connecting strongly coupled gauge theories with weakly coupled supergravity. The absence of a phase transition, as evidenced by the behavior of the Polyakov loop, contrasts with expectations from purely bosonic models and further corroborates theoretical predictions concerning the deconfined nature of the phase even at low temperatures.

Moreover, the authors offer a comprehensive analysis of potential instabilities inherent in the model due to non-trivial commuting matrix directions, presenting a robust methodology to manage these through scaling arguments and empirical adjustment of model parameters.

Implications and Future Directions

The results provide substantial insights into the dynamics of string/M-theoretical models beyond perturbative regimes. They underscore the significance of the gauge/gravity duality, presenting deep connections between quantum mechanical properties of gauge systems and classical gravitational theories. These findings might guide future studies in exploring non-perturbative regimes of gauge theories and refining theoretical models related to the holographic principle.

Future investigations could expand on this work by exploring higher-dimensional analogues and examining different couplings within the models or extending the temperature range and observing the effect on system parameters. Additionally, continued development and refinement of non-lattice simulation techniques can offer further clarity in probing these complex systems and their broader implications in theoretical physics.