USU mathematicians unravel a thread of string theory

August 17, 2020

LOGAN, UTAH, USA - Simply put, string theory is a proposed method of explaining everything. Actually, there's nothing simple about it. String theory is a theoretical framework from physics that describes one-dimensional, vibrating fibrous objects called "strings," which propagate through space and interact with each other. Piece by piece, energetic minds are discovering and deciphering fundamental strings of the physical universe using mathematical models. Among these intrepid explorers are Utah State University mathematicians Thomas Hill and his faculty mentor, Andreas Malmendier.

With colleague Adrian Clingher of the University of Missouri-St. Louis, the team published findings about two 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.' The USU researchers' work is supported by a grant from the Simons Foundation.

"We studied a special family of K3 surfaces - compact, connected complex surfaces of dimension 2 - which are important geometric tools for understanding symmetries of physical theories," says Hill, who graduated from USU's Honors Program with a bachelor's degree in mathematics in 2018 and completed a master's degree in mathematics this past spring. "In this case, we were examining a string duality between F-theory and heterotic string theory in eight dimensions."

Hill says the team proved the K3 surfaces they investigated admit four unique ways to slice the surfaces as Jacobian elliptic fibrations, formations of torus-shaped fibers. The researchers constructed explicit equations for each of these fibrations.

"An important part of this research involves identifying certain geometric building blocks, called 'divisors,' within each K3 surface," he says. "Using these divisors, crucial geometric information is then encoded in an abstract graph."

This process, Hill says, enables researchers to investigate symmetries of underlying physical theories demonstrated by the graph.

"You can think of this family of surfaces as a loaf of bread and each fibration as a 'slice' of that loaf," says Malmendier, associate professor in USU's Department of Mathematics and Statistics. "By examining the sequence of slices, we can visualize, and better understand, the entire loaf."

The undertaking described in the paper, he says, represents hours of painstaking "paper and pencil" work to prove theorems of each of the four fibrations, followed by pushing each theorem through difficult algebraic formulas.

"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," Malmendier says.
-end-


Utah State University

Related Mathematics Articles from Brightsurf:

A new method for boosting the learning of mathematics
How can mathematics learning in primary school be facilitated? UNIGE has developed an intervention to promote the learning of math in school.

Could mathematics help to better treat cancer?
Impaired information processing may prevent cells from perceiving their environment correctly; they then start acting in an uncontrolled way and this can lead to the development of cancer.

People can see beauty in complex mathematics, study shows
Ordinary people see beauty in complex mathematical arguments in the same way they can appreciate a beautiful landscape painting or a piano sonata.

Improving geothermal HVAC systems with mathematics
Sustainable heating, ventilation, and air conditioning systems, such as those that harness low-enthalpy geothermal energy, are needed to reduce collective energy use and mitigate the continued effects of a warming climate.

How the power of mathematics can help assess lung function
Researchers at the University of Southampton have developed a new computational way of analyzing X-ray images of lungs, which could herald a breakthrough in the diagnosis and assessment of chronic obstructive pulmonary disease (COPD) and other lung diseases.

Mathematics pushes innovation in 4-D printing
New mathematical results will provide a potential breakthrough in the design and the fabrication of the next generation of morphable materials.

More democracy through mathematics
For democratic elections to be fair, voting districts must have similar sizes.

How to color a lizard: From biology to mathematics
Skin color patterns in animals arise from microscopic interactions among colored cells that obey equations discovered by Alan Turing.

US educators awarded for exemplary teaching in mathematics
Janet Heine Barnett, Caren Diefenderfer, and Tevian Dray were named the 2017 Deborah and Franklin Tepper Haimo Award winners by the Mathematical Association of America (MAA) for their teaching effectiveness and influence beyond their institutions.

Authors of year's best books in mathematics honored
Prizes for the year's best books in mathematics were awarded to Ian Stewart and Tim Chartier by the Mathematical Association of America (MAA) on Jan.

Read More: Mathematics News and Mathematics Current Events
Brightsurf.com is a participant in the Amazon Services LLC Associates Program, an affiliate advertising program designed to provide a means for sites to earn advertising fees by advertising and linking to Amazon.com.