Dhagash Mehta

Tumbling through a landscape: Evidence of instabilities in high-dimensional moduli spaces

We argue that a generic instability afflicts vacua that arise in theories whose moduli space has large dimension. Specifically, by studying theories with multiple scalar fields we provide numerical evidence that for a generic local minimum of the potential the usual semiclassical bubble nucleation rate,

Numerical algebraic geometry: a new perspective on gauge and string theories

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Finding all the stationary points of a potential-energy landscape via numerical polynomial-homotopy-continuation method.

, increases rapidly as function of the number of fields in the theory. As a consequence, the fraction of vacua with tunneling rates low enough to maintain metastability appears to fall exponentially as a function of the moduli space dimension. We discuss possible implications for the landscape of string theory. Notably, if our results prove applicable to string theory, the landscape of metastable vacua may not contain sufficient diversity to offer a natural explanation of dark energy.

Finding all flux vacua in an explicit example

A bstractThere is a rich interplay between algebraic geometry and string and gauge theories which has been recently aided immensely by advances in computational algebra. However, symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these shortcomings. The so-called ‘embarrassing parallelizability’ allows us to solve many problems and extract physical information which elude symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

Numerical Polynomial Homotopy Continuation Method and String Vacua

The stationary points (SPs) of a potential-energy landscape play a crucial role in understanding many of the physical or chemical properties of a given system. However, unless they are found analytically, no efficient method is available to obtain all the SPs of a given potential. We present a method, called the numerical polynomial-homotopy-continuation method, which numerically finds all the SPs, and is embarrassingly parallelizable. The method requires the nonlinearity of the potential to be polynomial-like, which is the case for almost all of the potentials arising in physical and chemical systems. We also certify the numerically obtained SPs so that they are independent of the numerical tolerance used during the computation. It is then straightforward to separate out the local and global minima. As a first application, we take the XY model with power-law interaction, which is shown to have a polynomial-like nonlinearity, and we apply the method.

Phase transitions detached from stationary points of the energy landscape.

A bstractWe explicitly construct all supersymmetric flux vacua of a particular Calabi-Yau compactification of type IIB string theory for a small number of flux carrying cycles and a given D3-brane tadpole. The analysis is performed in the large complex structure region by using the polynomial homotopy continuation method, which allows to find all stationary points of the polynomial equations that characterize the supersymmetric vacuum solutions. The number of vacua as a function of the D3 tadpole is in agreement with statistical studies in the literature. We calculate the available tuning of the cosmological constant from fluxes and extrapolate to scenarios with a larger number of flux carrying cycles. We also verify the range of scales for the moduli and gravitino masses recently found for a single explicit flux choice giving a Kähler uplifted de Sitter vacuum in the same construction.

Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian

Finding vacua for the four-dimensional effective theories for supergravity which descend from flux compactifications and analyzing them according to their stability is one of the central problems in string phenomenology. Except for some simple toy models, it is, however, difficult to find all the vacua analytically. Recently developed algorithmic methods based on symbolic computer algebra can be of great help in the more realistic models. However, they suffer from serious algorithmic complexities and are limited to small system sizes. In this paper, we review a numerical method called the numerical polynomial homotopy continuation (NPHC) method, first used in the areas of lattice field theories, which by construction finds all of the vacua of a given potential that is known to have only isolated solutions. The NPHC method is known to suffer from no major algorithmic complexities and is embarrassingly parallelizable, and hence its applicability goes way beyond the existing symbolic methods. We first solve a simple toy model as a warm-up example to demonstrate the NPHC method at work. We then show that all the vacua of a more complicated model of a compactified M theory model, which has an SU(3) structure, can be obtained by using a desktop machine in just about an hour, a feat which was reported to be prohibitively difficult by the existing symbolic methods. Finally, we compare the various technicalities between the two methods.

Enumerating Gribov copies on the lattice

The stationary points of the potential energy function V are studied for the ϕ4 model on a two-dimensional square lattice with nearest-neighbor interactions. On the basis of analytical and numerical results, we explore the relation of stationary points to the occurrence of thermodynamic phase transitions. We find that the phase transition potential energy of the ϕ4 model does in general not coincide with the potential energy of any of the stationary points of V. This disproves earlier, allegedly rigorous, claims in the literature on necessary conditions for the existence of phase transitions. Moreover, we find evidence that the indices of stationary points scale extensively with the system size, and therefore the index density can be used to characterize features of the energy landscape in the infinite-system limit. We conclude that the finite-system stationary points provide one possible mechanism of how a phase transition can arise, but not the only one.

Energy-landscape analysis of the two-dimensional nearest-neighbor φ4 model

Abstract We study the stationary points of what is known as the lattice Landau gauge fixing functional in one-dimensional compact U(1) lattice gauge theory, or as the Hamiltonian of the one-dimensional random phase XY model in statistical physics. An analytic solution of all stationary points is derived for lattices with an odd number of lattice sites and periodic boundary conditions. In the context of lattice gauge theory, these stationary points and their indices are used to compute the gauge fixing partition function, making reference in particular to the Neuberger problem. Interpreted as stationary points of the one-dimensional XY Hamiltonian, the solutions and their Hessian determinants allow us to evaluate a criterion which makes predictions on the existence of phase transitions and the corresponding critical energies in the thermodynamic limit.

On the sign problem in 2D lattice super Yang-Mills

Abstract In the modern formulation of lattice gauge fixing, the gauge-fixing condition is written in terms of the minima or stationary points (collectively called solutions) of a gauge-fixing functional. Due to the non-linearity of this functional, it usually has many solutions, called Gribov copies. The dependence of the number of Gribov copies, n [ U ] , on the different gauge orbits plays an important role in constructing the Faddeev–Popov procedure and hence in realising the BRST symmetry on the lattice. Here, we initiate a study of counting n [ U ] for different orbits using three complimentary methods: (1) analytical results in lower dimensions, and some lower bounds on n [ U ] in higher dimensions, (2) the numerical polynomial homotopy continuation method, which numerically finds all Gribov copies for a given orbit for small lattices, and (3) numerical minimisation (“brute force”), which finds many distinct Gribov copies, but not necessarily all. Because n for the coset SU ( N c ) / U ( 1 ) of an SU ( N c ) theory is orbit independent, we concentrate on the residual compact U(1) case in this article, and establish that n is orbit dependent for the minimal lattice Landau gauge and orbit independent for the absolute lattice Landau gauge. We also observe that, contrary to a previous claim, n is not exponentially suppressed for the recently proposed stereographic lattice Landau gauge compared to the naive gauge in more than one dimension.