Thomas Hill and Andreas Malmendier of Utah State University check out a string duality in between F-theory and heterotic string theory in 8 measurements.
Simply put, string theory is a proposed technique of discussing whatever. Actually, there’s absolutely nothing easy about it. String theory is a theoretical structure from physics that explains one-dimensional, vibrating fibrous items called “strings,” which propagate through space-time and engage with each other. Piece by piece, energetic minds are finding and understanding essential strings of the physical universe utilizing mathematical designs. Among these brave explorers are Utah State University mathematicians Thomas Hill and his professors coach, Andreas Malmendier.
With coworker Adrian Clingher of the University of Missouri-St. Louis, the group released findings about 2 branches of string theory in the paper, “The Duality Between F-theory and the Heterotic String in D=8 with Two Wilson Lines,” in the August 7, 2020, online edition of Letters in Mathematical Physics.
“One important feature of string theories is that these theories require extra dimensions of space-time for mathematical consistency,” states Malmendier, associate teacher in USU’s Department of Mathematics and Statistics. “However, not every way of handling these extra dimensions, also called ‘compactification’, produces a model with the right properties to describe nature.”
For so-called eight-dimensional compactification of a string theory design described as F-theory, he states, the additional measurements should be formed like a K3 surface area.
“We studied a special family of K3 surfaces – compact, connected complex surfaces of dimension 2, which are important geometric tools for understanding the symmetries of physical theories,” states Hill, who finished from USU’s Honors Program with a bachelor’s degree in mathematics in 2018 and finished a master’s degree in mathematics this previous spring. “In this case, we were examining a string duality between F-theory and heterotic string theory.”
Using an abstract chart, Utah State University scientists determine divisors within each K3 surface area to analyze different balances. The various Jacobian elliptic fibrations represent particular colors of a linked subset of the nodes of the chart. The balances of the chart and the possible colorings of the nodes are vital to comprehending the balances of the underlying physical theories. Credit: Malmendier/Hill, USU
Hill states the group discovered 4 distinct methods to slice the K3 surface areas in an especially helpful method, called Jacobian elliptic fibrations. Fibrations, he states, are developments of torus-shaped fibers supported by the K3 surface areas.
“You can think of this family of surfaces as a loaf of bread and each fibration as a ‘slice’ of that loaf,” Malmendier states. “By examining the sequence of slices, we can visualize, and better understand, the entire loaf.”
Hill states a vital part of this research study includes recognizing specific geometric foundation, called “divisors”, within each K3 surface area.
“Using these divisors, crucial geometric information about the K3 surface is then encoded in an abstract graph,” he states. “This process enables us to investigate symmetries of the underlying physical theories demonstrated by the graph.”
The endeavor explained in the paper, Malmendier states, represents hours of painstaking “paper and pencil” work to show theorems of each of the 4 fibrations, followed by pressing each theorem through tough algebraic solutions.
“For the latter part of this process, we used Maple Software and the specialized Differential Geometry package developed at USU, which streamlined our computational efforts,” he states.
The group’s efforts were boosted by specialized Maple Software libraries established by USU mathematics teacher Ian Anderson and USU physics teacher Charlie Torre, in addition to efforts by Hill in broadening this library, beginning throughout his undergraduate years.
“This paper represents the ‘grand finale’ of Thomas (Hill)’s achievements here at Utah State,” Malmendier states. “As an undergrad, he honed his software and programming skills and carried this expertise into his master’s program, where he developed expertise in K3 surfaces. Finally, in this publication, he tackles a very complex classification problem. It’s truly exceptional for a master’s student.”
Hill advances this fall to a doctoral program in mathematics, with a focus in algebraic geometry, at the University of Utah.
“During his undergraduate career, Thomas was awarded an Undergraduate Research and Creative Opportunities (URCO) grant and was also named a 2017 Goldwater Scholar,” Malmendier states. “In 2018, he got the Honors Program’s Joyce Kinkead Award for Outstanding Honors Capstone Thesis. This previous spring, Thomas got a respectable reference throughout the 2020 National Science Foundation Graduate Research Fellowship Program search.”
In addition to his scholastic distinctions, Malmendier applauds Hill’s mentor achievements.
“Thomas began working as a teaching assistant for Calculus III with me during his undergrad years,” Malmendier states. “He made short videos explaining vector calculus which, unbeknownst to us at the time, were the perfect preparation for our current pandemic situation.”
In addition, he states, Hill was entirely accountable, as a master’s trainee, for mentor classes in trigonometry and calculus, in addition to establishing the curriculum for the university’s innovative calculus test.
“All of Thomas’ research and teaching experiences give him a head start as he advances in his academic career,” Malmendier states.
Reference: “The duality between F-theory and the heterotic string in D=8 with two Wilson lines” by Adrian Clingher, Thomas Hill and Andreas Malmendier, 7 August 2020, Letters in Mathematical Physics.
DOI: 10.1007/s11005-020-01323-8
