This document is aimed at individuals with a graduate degree in theoretical physics who want to specialize in superstring theory. It covers the postgraduate mathematics courses needed for success in this field. Superstring theory is an advanced area of theoretical physics that has become very popular today. However, because of the complicated mathematical concepts involved, two specific postgraduate mathematics courses are necessary to stay ahead in research in this discipline: Noncommutative Geometry and K-theory.
Superstring theory, also known as supersymmetric string theory, is an attempt to explain the four fundamental forces of physics as vibrations of tiny supersymmetric strings. It is considered the most promising way to understand quantum gravity and its relatively weak force compared to other forces (“Quantum gravity”, nd). Unlike the original form called bosonic string theory, superstring theory includes fermions and supersymmetry (“Bosonic string theory”, nd; “Fermions”, nd; “Supersymmetry”, nd). The current focus of theoretical physics is understanding how gravity relates to the other three fundamental forces. However, physicists have not yet found a mathematical technique to eliminate infinite probabilities in calculations related to superstring theory like they have for electromagnetic force, strong nuclear force, and weak nuclear force (“Superstring theory”, nd). Therefore, a different approach is needed for incorporating quantum gravity into the other forces.
According to superstring theory, all physical reality consists of tiny vibrating strings, each the size of a plank’s length. The theory posits that the graviton, which carries gravitational force, is a string with zero wave amplitude. Furthermore, it suggests that there are no differences between strings wrapping around smaller dimensions and those moving along larger dimensions. In essence, a dimension with a size of R has equivalent effects to one with a size of 1/R (Superstring theory, nd para 3). This principle is upheld because current understanding indicates that a universe cannot shrink beyond the size of a string. Therefore, if a universe were to collapse inwardly, it would not self-destruct but rather need to expand again once reaching the size of a string (“Superstring theory”, nd).
According to string theory, physical space is perceived by humans as having only four dimensions. However, string theory suggests that there could be additional dimensions beyond the four known ones. It is believed that in the case of string theory, spacetime would need to have 10, 11 or 26 dimensions for consistency. These extra dimensions, though not observable due to their compact size (comparable to a Plank length), are still considered in the theory (“Superstring theory”, nd). It is challenging for humans to conceive of higher dimensions since we can only move in three spatial dimensions and perceive the world in two plus one dimensions. If we had the ability to see in three true dimensions, we would be able to observe all sides of an object simultaneously. This brings up the question of whether it is possible to design experiments that can test theories involving higher dimensions and whether human scientists can interpret the results using our limited dimensional perception. This further leads to the debate of whether models dependent on abstract concepts without experimental testing can be classified as ‘scientific’ rather than philosophical (Groleau, 2003).
Before superstring theory, Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of Harvard University described six-dimensional shapes needed to complete the equations. Figure 1 illustrates an example of one such shape. By replacing spheres with these Calabi-Yau shapes, Supersting theory achieves ten dimensions: three spatial dimensions, the six dimensions from Calabi-Yau shapes, and one dimension for time (Groleau, 2003). Figure 1 displays the six-dimensional Calabi-Yau shapes (retrieved from “Imagining Other Dimensions”, PBS.org, on August 25, 2004, from http://www.pbs.org/wgbh/nova/elegant/dimensions.html). It is almost inconceivable for humans to imagine a universe with more than four dimensions; there may never be an accurate representation that humans can comprehend without being pulled into higher dimensional space. Until the mid-1990s, it seemed there were five distinct String theories. However, M-theory introduced by the Second Superstring revolution revealed that these five string theories were interconnected and part of M-theory (“Superstring theory”, nd).
The five consistent superstring theories, which include type I, type IIA, type IIB, Heterotic E8 X E8 (HEt), and Heterotic SO(32) (HOt), each have their unique characteristics. The type I theory utilizes unoriented open and closed strings, while the other theories rely on oriented closed strings. Additionally, the type II theories possess two ten-dimensional supersymmetries, whereas the other theories only possess one. Furthermore, the IIA theory stands out for being non-chiral or parity conserving, unlike the rest of the theories which are chiral or parity violating. Chiral gauge theories can encounter inconsistencies due to certain one-loop Feynman diagrams causing a quantum mechanical breakdown of the gauge symmetry. The cancellation of these anomalies places a constraint on potential superstring theories.
Supersting theory, a highly advanced form of theoretical physics, is not the first theory to propose extra spatial dimensions. String theory relies on the “mathematics of folds, knots, and topology, which was largely developed after Kaluza and Klein, and has made physical theories relying on extra dimensions much more credible” (Witten, 1998 p. 9). In 1919, Theodor Kaluza theorized the existence of a fourth spatial dimension to link theories of general relativity and electromagnetism. Oskar Klein further refined this concept by suggesting that space consisted of both extended and curled-up dimensions (Groleau, 2003). The extended dimensions are the three spatial dimensions humans exist in, while the curled-up dimension resides deep within those extended dimensions. Although initial experiments could not reconcile general relativity and electromagnetic theory using Kaluza and Klein’s curled-up dimension, string theorists now find this idea valuable and essential.
The mathematics used in superstring theory requires a minimum of ten dimensions in order to work out equations. These equations need to incorporate additional dimensions to connect general relativity and quantum mechanics, explain particle nature, and unify forces. String theorists believe that the extra dimensions are located within the curled-up space described by Kaluza and Klein. To accommodate these additional dimensions, the Kaluza-Klein circles are substituted with spheres. Considering only the surfaces of the spheres results in two dimensions, but if the space within the spheres is taken into account, there are a total of six dimensions.
The possible relation between noncommutative geometry and string theory has been discussed. According to Madore (1999, p.16), noncommutative geometry is a field theory that is divergence-free. If the geometry in which monopole configurations are constructed can be approximated by a noncommutative geometry, monopole configurations will have finite energy because the point on which they are localized is replaced by a volume of fuzz. This is a characteristic shared by noncommutative geometry and string theory, as a throat to an adjacent D-brane replaces the point in space where certain monopoles are located, resulting in solutions with finite energy (Madore, 1999). In noncommutative geometry, the string is replaced by a finite number of elementary volumes of fuzz’, each capable of containing one quantum mode. The line joining two points, q’ and q, is quantized and characterized by creation operators aj that create longitudinal displacements. These creation operators correspond to the rigid longitudinal vibrational modes of the string due to nontrivial commutation relations. As it requires no energy to separate two points, the string tension would be zero, which diverges from traditional string theory (Madore, 1999, p.16). While noncommutative Kaluza-Klein theory shares similarities with M-theory of D-branes, it lacks a supersymmetric extension.Speculations have arisen regarding the potential for string theory to naturally generate space-time uncertainty relations and a noncommutative theory of gravity. Additionally, attempts have been made to connect a noncommutative structure of space-time with the quantization of the open string in the presence of a non-vanishing B-field (Witten, 1998).
To engage in advanced theoretical physics, such as Superstring theory, it is advisable to learn and grasp K-theory and noncommutative geometry. Superstring theory is at the forefront of explaining the universe and potentially harmonizing the four fundamental forces of physics. While comprehending the extra dimensions necessary for this may be beyond human ability, Superstring theory represents mankind’s greatest chance to understand the present and future events in the universe.
