Topological Recursion in the Seiberg-Witten partition function via AGT correspondence

In the vast landscape of modern physics, there exists a persistent challenge: understanding how the fundamental forces of nature behave when they are pushed to their limits. While scientists have successfully described how particles interact at high energies using standard equations, these equations often break down when the interactions become incredibly strong. This is the realm of non-perturbative physics, a domain where the usual tools of approximation fail, and the behavior of the universe becomes deeply complex and difficult to predict. For decades, researchers have sought a way to map this difficult terrain, particularly for theories involving Yang-Mills fields, which describe the strong nuclear force holding atomic nuclei together. The key to unlocking these mysteries often lies in finding hidden symmetries or dualities—connections between two seemingly different descriptions of the same physical reality. One such connection, known as the AGT correspondence, suggests a surprising link between four-dimensional theories of particle physics and two-dimensional theories of conformal field theory, a branch of mathematics used to describe shapes and patterns that remain unchanged under scaling.

This thesis, authored by Bastam Tajik at Sapienza University of Rome, sets out to test the strength and validity of this AGT correspondence. The central goal is to demonstrate that a specific set of recursive relations, originally discovered by the mathematician Alexander Zamolodchikov in the context of two-dimensional conformal field theories, give rise to the recursive relations governing the Seiberg-Witten partition function in four-dimensional supersymmetric gauge theories, and vice versa. The Seiberg-Witten partition function is a complex mathematical object that encodes the non-perturbative properties of these gauge theories, essentially acting as a master key to understanding their behavior. By establishing that the recursive patterns found in the two-dimensional world generate the exact same results as those calculated in the four-dimensional world, the work aims to provide strong evidence that these two distinct areas of physics are indeed two sides of the same coin.

To achieve this, the author navigates through a dense forest of advanced mathematical concepts, starting with the geometry of the spaces where these theories live. The research relies heavily on the ADHM construction, a method for describing instantons—special, stable solutions to the equations of motion that represent tunneling events between different vacuum states. These instantons are not just abstract points; they form a geometric structure known as a moduli space, which can be visualized as a landscape with hills, valleys, and singular points where the geometry becomes sharp or undefined. The thesis explores how this landscape is constructed using hyper-Kähler quotients, a sophisticated geometric technique that slices and reshapes higher-dimensional spaces to reveal the underlying structure of the instanton solutions. The author carefully details how these spaces behave, including how they change when the theory is placed on a non-commutative spacetime, a theoretical framework where the coordinates of space and time do not commute, much like how the order of operations matters in certain algebraic systems.

A significant portion of the work involves overcoming the computational difficulties associated with calculating the partition function. Traditional methods for integrating over these complex moduli spaces are often plagued by singularities and divergences that make precise calculation nearly impossible. The thesis investigates the use of localization techniques, a powerful mathematical strategy that simplifies these integrals by showing that the entire contribution to the result comes from a few specific, isolated points rather than the whole space. The author demonstrates how introducing a non-commutative parameter effectively "resolves" the singularities in the moduli space, smoothing out the sharp edges and allowing for a clean, well-defined calculation. This resolution is crucial because it allows the recursive relations to be applied without the mathematical breakdowns that usually occur at these critical points.

The core objective of the thesis is to derive the Nekrasov partition function, which describes the instanton contributions to the Seiberg-Witten theory, directly from the recursive relations of the two-dimensional conformal block. By meticulously matching the terms generated by Zamolodchikov's recursion with the terms derived from the instanton calculus, the work intends to confirm that the two approaches yield identical results. This equivalence is not merely a coincidence; it aims to validate the AGT correspondence as a robust duality. The work seeks to show that the complex, high-dimensional physics of four-dimensional gauge theories can be fully captured by the simpler, recursive structures of two-dimensional conformal field theories. The thesis further explores the implications of this duality for the prepotential, a function that determines the low-energy behavior of the theory, showing that the non-commutative resolution of the moduli space leads to a consistent and finite result that aligns perfectly with the expectations from the dual theory.

Ultimately, this research aims to provide a concrete bridge between two vast and seemingly disconnected areas of theoretical physics. It proposes that the recursive patterns governing the behavior of fields in two dimensions are sufficient to reconstruct the intricate, non-perturbative dynamics of four-dimensional gauge theories. By navigating the geometric complexities of instanton moduli spaces and utilizing the power of localization and non-commutative geometry, the thesis offers a clear and rigorous verification of a profound theoretical link. The work stands as a testament to the power of mathematical duality, showing that by looking at a problem from the right angle, the most difficult calculations in quantum field theory can be reduced to elegant, recursive steps.