Discussion of quantum mechanics and field theory is primarily targeted at mathematicians, discussing subjects from classical and quantum mechanics and field theory. Topological quantum field theory captures the formal aspects of the integral path and illustrates how these principles can impact branches of mathematics that might not appear related at first glance.

In the relation between geometry and the laws of nature, there is something almost magical. A source of wonder can be an encounter at the seashore with the perfection of a logarithmic spiral in a seashell. A wonder which is multiplied in telescopic images of distant galaxies when the self-same spirals appear. It has been a rich theme in science that nature is, at different stages, amenable to a geometrical understanding. It has consistently yielded beautiful observations, albeit little more than a conviction. But it is maybe safe to say that geometry has never played a role as pro-found at any time in the past, as it currently does in the attempts of string theory to explain quantum gravity. It shouldn’t be too shocking here. We have learned to think of space-time not as a passive arena for activities, but as a complex force since the time when Einstein told us that gravity is geometry. The quantum fluctuations of geometry must therefore be represented by any quantum gravity theory, such as string theory.

String theory is a label applied to a number of various kinds and purposes of formulations. Some were related to subatomic particles and others to cosmology and astronomical structures. For presenting its history, a timeline structure is appropriate, even necessary.

In 1867, an essay was written by Lord Kelvin (Sir William Thomson) in the Proceedings of the Royal Society of Edinburgh, Vol. VI, 1867, pp. 94-105, in which he suggested that in the ether of space, atoms consist of vortex rings. To form complex structures, such vortex rings could be connected together.

Due to the constant proportions seen in chemical reactions, atoms have been recognized to exist. This paper, however, was from a period before the presence of electrons was understood. J.J.’s experiments Thomson contributed to the discovery of the electron as a basic building block of matter in 1897. Experiments by others brought them to the same conclusion and someone else coined the word electron.

The German mathematician Theodor Kaluza proposed a theory in 1919, based on the General Theory of Relativity, but involving four rather than three spatial dimensions. Kaluza derived electromagnetism from his theory as an inference. The prospect of uniting gravity and electromagnetism was highlighted.

**How strings see space–time**

Extended objects are strings. This is the critical difference with the notion of fundamental particles as point-like, localized entities. A localized entity ‘sees’ space-time in a much more constrained way compared to an ex- tended one. We know that a cylindrical tube is topologically distinct from a plane layer, to give a clear example. There are paths (winding around the circular direction) on the cylinder that can not be constantly shrunk to zero scale. This property of the cylinder can not be separated from the plane by a small ball by its very local probe. But a band of rubber can. The circular path of the cylinder can be wrapped by the rubber band and we know that such a wrapped band varies qualitatively from an unwrapped one. While we can lift the un-wrapped one off the cylinder quickly, without breaking the bank, we can not do so with the wrapped one. Closed strings, which can be thought of as rubber bands, see more of the space-time topology in a similar way than do point particles. Thus, string theory instantly provides new ways of probing space-time by assuming that the familiar particles are excitations of a more fundamental extended entity.

** How strings make space–time**

It is therefore time to take a look at how string theory can explain the dynamics of space-time itself. Like a rubber band, a closed relativistic string in flat space-time has several oscillation modes in which to vibrate. One which turns out to be massless is among these oscillations. The surprise is that the quantum numbers are borne by this massless state and have the symmetry of the quantum mediating gravitational interaction, the graviton. The result is unavoidable that a quantum theory of strings involves gravity since the leading interactions of this particle are also those of the graviton.

**Strings and the dimension of space–time**

A pause We have so far discussed novel elements of string theory linked to the extended form of strings and their gravity relationship. However, all that has been said so far is reasonably well known in the sense of what is called classical string theory. We have shown that the ex-tense existence of strings contributes to strings viewing space-time differently even at the classical level. We also mentioned how classical gravity, including Einstein’s equations, arises from string theory. The observed spectrum of light particles in nature can be reproduced broadly by Kaluza-Klein compactifications of classical theory. But we have not yet talked about effects that are genuinely quantum gravitational. Quantum string theory is a subject in which, in recent years, we have made some headway, but still little is understood. Never, unless the little we have understood so far is enormously encouraging in terms of the rich and consistent structure of the tent that is shown. As we will see, in this context, geometry plays an ever more novel role. But let us take a digression into some of the fruits of the interaction between classical string theory and mathematics before we continue to describe that.

**classical strings and mathematics**

In the middle of the 20th century, mathematics developed along lines that were somewhat removed from physics’ motivations or applications. However, modern ideas in mathematics, specifically in topology and geometry, were seen to be crucial in the understanding of different non-perturbative aspects of quantum field theory in the seventies and eighties. The topological nature of different solitons, such as vortices and monopoles, now informs any discussion of the subject. Quantum string theory is only slowly uncovering its new notions of geometry, as the reader must have realized. The questions raised in the introduction have not yet been completely answered by us. But we have found a lot of fascinating threads that are sure to go into our quantum gravity picture. That final picture may not be anything like what we now imagine, but whatever it is, the way we view space-time will change.

1968: For about a year at CERN, Gabriele Veneziano worked to develop a theory of nuclear force as manifested in the interaction of hadron particles. A function he discovered that played an important part in his theory was:

**B(x, y) = ∫01tx-1(1 – t)y-1dt**

He found a name for this function in a book on mathematical functions. It was called the beta function of Euler. This story was somehow transformed into the notion that the beta function was discovered as a function that would explain the strong force.

**The Old String Theory is dead! Long live the New String Theory!**

The new string theory had the advantage of not needing the presence of a tachyon and a universe of 26 dimensions for compatibility with Special Relativity and quantum theory. It only needed a world of ten dimensions. This modified string theory became known as the theory of superstrings.

Three spatial dimensions and one temporal dimension appear to consist of the world. For the universe to be finite, however, a fourth spatial dimension would have to be curved. It is also possible that time is curved, and a second temporal dimension would be required. So it is possible that six dimensions actually involve our four-dimensional world. But this is not how additional unseen dimensions are rationalized by string theorists.

A world that is a narrow tube would appear to be one-dimensional, but a second dimension would constitute a position around the circular cross-section of the tube. String theorists envisage some multi-dimensional space compaction at every point in space.

References

https://www.researchgate.net/publication/237512757_Geometry_and_string_theory

https://www.ias.edu/sns/witten/anomalies-string-theory

https://www.ias.edu/math/new-connections

https://www.worldscientific.com/doi/abs/10.1142/9789812799630_0004

https://www.worldscientific.com/doi/abs/10.1142/S0217751X89000972