Mathematicians unravel a thread of string theory

Date:
August 18, 2020
Source:
Utah State University
Summary:
Mathematicians are exploring a string duality between F-theory
and heterotic string theory in eight dimensions.



FULL STORY ========================================================================== 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.


========================================================================== Story Source: Materials provided by Utah_State_University. Original
written by Mary-Ann Muffoletto. Note: Content may be edited for style
and length.


========================================================================== Journal Reference:
1. Adrian Clingher, Thomas Hill, Andreas Malmendier. The duality
between F-
theory and the heterotic string in $$D=8$$ with two
Wilson lines. Letters in Mathematical Physics, 2020; DOI:
10.1007/s11005-020-01323-8 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2020/08/200818094037.htm

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