Mathematics Inspired by String Theory | Science

We are in a small office, poorly lit and cluttered with white boards leaning against walls, doors, and windows. Two theoretical physicists stare intently at a complicated handwritten equation on one of them, while it plays Eye of the Tiger. In this scene from the series Big Bang TheoryWhere a string theorist and a cosmologist collaborate to unravel the mysteries of matter, an important ingredient is missing: a mathematician, not only providing the proper language, but also taking good notes.

Indeed, mathematics is the main working tool for theoretical physicists to advance their knowledge of nature. An example is the use of the differential calculus in Newtonian physics or, more recently, the formulation of Einstein’s theory of general relativity, where space and time merge into a single entity, through the complicated geometry of the curved spaces. However, the influence happens in both directions and a physical theory, emerged about 40 years ago, has managed to capture the attention and interest of the most abstract mathematicians. We talk about string theory.

This fundamental theory, proposed in the 1970s by Jöel Scherk and John Henry Schwarz, unifies gravity with the other forces and could solve the problem of combining gravity with quantum theory. For this, it is based on one-dimensional objects or “strings”, instead of point particles, whose modes of vibration distinguish the different constituents of the known universe. In parallel, Pierre Ramond, Andrè Neveu and John Henry Schwarz proposed a modification of the theory that harbors supersymmetry and that allows the presence of certain particles, known as fermions. This version of superstrings requires the existence of six dimensions in addition to the fourth-dimensional spacetime we observe, rolled into compact shapes of a tiny size.

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The geometric shape of this internal space must be very particular, since it must allow good vibration of the fundamental strings in the total space of ten dimensions. In the 1980s Philip Candelas, Gary T. Horowitz, Andrew Strominger, and Edward Witten characterized the interesting shape of these internal spaces. And they came up with precisely some geometries whose theoretical existence had been demonstrated ten years earlier by the mathematician and medalist fields S.-T. Yau. They are the so-called Calabi-Yau spaces or varieties.

Based on this relationship, the prism of string theory has made it possible to tackle various mathematical problems; the first, a question of enumerative geometry. This type of problem dates back to the Greek Apollonius of Perga (262 BC – 190 BC), who asked himself: given a configuration of three disjoint circles in the plane, how many circles are tangent to these three dice?

Given three circles (those marked in black), how many circles are tangent to them?  In the image they are represented with different colors.  Image: Melchoir.
Given three circles (those marked in black), how many circles are tangent to them? In the image they are represented with different colors. Image: Melchoir.

A problem of a similar nature has interested algebraic geometricians since the nineteenth century. In the previous problem, the plane is replaced by an algebraic space, the circles given by points and the circles we count by spheres with a certain degree of d associated. The question then is: How many spheres of fixed degree d, passing through the 3d -1 points, does this space contain? When this space is Calabi-Yau, the problem is known as the Clemens conjecture. In the simplest Calabi-Yau variety there are a total of 609,250 grade 2 spheres, as calculated by mathematician Sheldon Katz in 1986.

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To the complete surprise of the mathematical community, in 1991 a team of four physicists – Philip Candelas, Xenia De La Ossa, Paul Green, and Linda Parkes – carried out a calculation that predicted the number of spheres of different degree d in a Calabi space. -Yau. The complicated series of quantities of spheres, which had remained inaccessible to mathematical calculations, could be encoded into an elegant function that measured the probability of propagation of a string.

The further development of these ideas has given rise to a whole field of study in mathematics known as mirror symmetry, with important open problems such as the homological conjectures of mirror symmetry, formulated by the mathematician and medalist Fields Maxim Kontsevich.

Another mathematical problem in whose solution tools from string theory were key is the Poincaré conjecture, solved by the Russian mathematician Grigori Perelman, which earned him the distinction of the Fields Medal in 2006 (which he rejected). The problem deals with the classification of compact three-dimensional spaces. To do this, Perelman developed a new technique, based on previous ideas by Richard Hamilton and known as Ricci flow with surgery, which allowed modifying these spaces, while implementing certain cuts and pastes, to break them down into more elementary ones and thus be able to classify them.

Both the space modification equation used by Perelman and the energy that made it possible to detect the precise moment in which to perform each surgery were well known in string theory: they corresponded to the so-called renormalization flow and the effective action, which allows describing the physics observed in four dimensions, from a ten-dimensional string model.

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These ideas continue to develop in the so-called generalized Ricci flow, collected in a book recently published by the American Mathematical Society. According to a proposal by the mathematician Jeffrey Streets, this theory may have future applications in the classification of an important class of four-dimensional spaces, complex surfaces.

These are just some examples of how, despite the fact that over the years the physical interest in string theory has diminished – mainly due to its lack of predictive character and because some of its postulates, such as supersymmetry, have not been observed experimentally – the theory continues to be of great value to mathematics. Either as a source of inspiration for new problems, the solution of which requires the development of new lines of mathematical thought or even complete theories, or because of its ability to make exact predictions about the most abstract mathematical problems.

Mario Garcia Fernandez He is a researcher at the Autonomous University of Madrid and a member of the Institute of Mathematical Sciences (CSIC-UAM-UCM-UC3M)

Timon G. Longoria Agate is coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences

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George Holan

George Holan is chief editor at Plainsmen Post and has articles published in many notable publications in the last decade.

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