Supersymmetry’s Vacuum Geometry Revealed Using Algebraic Techniques and Symmetry.

The fundamental nature of the vacuum, often perceived as space, is a complex and dynamic entity central to our understanding of particle physics. Recent research explores the geometrical structure of this vacuum within the minimal supersymmetric extension of the Standard Model. This theoretical framework aims to resolve inconsistencies within the established Standard Model and provide a candidate for dark matter. This investigation, utilising advanced techniques from algebraic geometry, reveals a surprisingly intricate landscape of possible vacuum states. Yang-Hui He of the London Institute for Mathematical Sciences, Royal Institution, and Merton College, University of Oxford, collaborates with Vishnu Jejjala from the Mandelstam Institute for Theoretical Physics, University of the Witwatersrand, and Brent D. Nelson of Northeastern University, alongside Hal Schenck from Auburn University and Michael Stillman from Cornell University, to present these findings in their article, “Vacuum Geometry of the Standard Model”. Their work employs Gröbner bases, a computationally intensive method for solving polynomial equations, to characterise the vacuum moduli space, revealing three distinct geometrical components with complex dimensions 1, 2, and 3.

Researchers investigate the vacuum structure of field theories, with a particular focus on supersymmetric gauge theories. These theories, extensions of the Standard Model of particle physics, posit a symmetry between bosons and fermions, potentially resolving inconsistencies within established physics and offering candidates for dark matter. The vacuum, in this context, does not represent empty space, but rather the lowest energy state of the theory, a complex landscape determining the behaviour of particles.

Currently, they employ Gröbner bases, a powerful computational technique originating in algebraic geometry, to analyse the geometry of the vacuum within the Minimal Supersymmetric Standard Model (MSSM). The MSSM introduces a minimal set of new particles and interactions to address shortcomings of the Standard Model, and understanding its vacuum structure is crucial for predicting its observable consequences. Gröbner bases allow researchers to systematically solve polynomial equations defining the vacuum, revealing its properties and stability.

A significant obstacle to this approach lies in the computational complexity inherent in these calculations. The number of equations and variables grows rapidly with the complexity of the theory, quickly exceeding the capabilities of conventional computing methods. To mitigate this, researchers exploit the symmetries present within the MSSM, reducing the dimensionality of the problem and simplifying the calculations. Multigrading, a technique assigning multiple degrees to each variable, further refines the computational process, enabling more efficient analysis of the polynomial system defining the vacuum. This allows for a more thorough exploration of the parameter space and a better understanding of the possible vacuum configurations.