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- Type of Document: M.Sc. Thesis
- Language: Farsi
- Document No: 58483 (04)
- University: Sharif University of Technology
- Department: Physics
- Advisor(s): Abolhasani, Ali Akbar; Rastgar, Arash
- Abstract:
- At first glance, two-dimensional gravity theories may appear trivial. For example, the Einstein–Hilbert action reduces to the Euler characteristic of a two-dimensional surface, which is a topological invariant. However, two-dimensional quantum gravity is in fact a remarkably rich theory. Critical two-dimensional gravity constitutes a central building block of string theory, while non-critical two-dimensional gravity is a more difficult theory with profound applications in string theory and quantum gauge theories. Regarding gravity on Riemann surfaces, there are three main approaches. The first, introduced by Polyakov, is related to quantum field theory. The second, studied through triangulation of surfaces, connects to matrix models. The third employs ideas from topological quantum field theory and can be reformulated in terms of the cohomology of the moduli space of Riemann surfaces. Various arguments show that theories constructed by these three approaches are equivalent. In precise mathematical terms, the proposal that topological gravity is equivalent to the one-matrix model leads us to a remarkable conjecture about intersection theory on the moduli space of Riemann surfaces. This is Witten’s conjecture concerning the equivalence of these perspectives on two-dimensional quantum gravity. Its formulation relies on the mathematical tools of intersection theory, moduli spaces of Riemann surfaces, and abelian and Jacobian varieties. A generalized form of Witten’s conjecture replaces the moduli space of Riemann surfaces with the moduli space of maps from a Riemann surface into Kähler manifolds. In this thesis, beyond studying Witten’s conjecture and its generalizations, we discuss the proofs by Kontsevich and Mirzakhani and ultimately extend Witten’s conjecture and its generalization to abelian and Jacobian varieties, constructing an analogue of the moduli space of Riemann surfaces in this new setting. Another important contribution of this work is the formulation of a profound trinity. This trinity consists of three analogues of one theory in three distinct perspectives. Mathematicians have developed a framework known as the axioms of quantum field theory, which specify conditions under which any collection of correlation functions—constructed from mathematical objects such as intersection numbers—that satisfy these axioms can be regarded as a quantum field theory. Studying an action that produces such correlation functions is therefore of fundamental importance. One aspect of the trinity is precisely such a collection of intersection numbers satisfying the axioms of QFT, forming the first leg of the correspondence. The second leg states that corresponding to these intersection numbers defined on a moduli space, there exists an action whose correlation functions reproduce exactly those numbers, thereby forming a quantum field theory. Here, the action and correlation functions emerge explicitly: the physical counterpart to the mathematical data of the first leg. This provides a striking application, since knowing the correlation functions in the second (physical) perspective—which may be easier to compute—allows one to evaluate intersection numbers in the first (mathematical) perspective, which are typically more difficult. Constructing a QFT from the mathematical side is comparatively easier, whereas identifying an action describing its physical side remains an open problem. Witten conjectured that the theory describing the unbroken phase of string theory with its fundamental symmetries constitutes the physical analogue of the intersection numbers of the moduli space of maps from Riemann surfaces to Kähler manifolds (the generalized Witten conjecture), a remarkable proposal. The third leg of the trinity states that the intersection numbers, which form a QFT (sometimes coupled to gravity) and correspond to an action in the second perspective, also admit a generating function (partition function) that is directly related to the partition function in the second leg. This partition function realizes a highest weight vector representation of a conformal field theory (CFT). Indeed, the string equation and another equation governing the QFT and its correlation functions can be expressed as differential operators acting on the partition function. These operators and their generalizations satisfy the commutation relations of the Virasoro algebra, the fundamental symmetry algebra of CFT. Thus, the third leg consists of a conformal field theory whose partition function serves as a highest weight vector, equivalent to the QFT described in the first two legs. Within this perspective, new CFTs emerge from the intersection numbers, opening avenues for further research concerning their rationality, central charge, and scaling dimensions of primary fields. Accordingly, the threefold correspondence is as follows: the mathematical formulation of intersection numbers (Witten’s contribution), the equivalent action (Kontsevich’s proof of Witten’s conjecture), and the associated CFT (Mirzakhani’s proof). A further major contribution of this thesis is the extension of this trinity and the formulation of intersection numbers on the moduli spaces of abelian and Jacobian varieties. In other words, the mathematical side (the first leg) is constructed in this new setting, and identifying the second and third analogues—namely the QFT action and the conformal field theory—becomes a significant research direction, aiming to discover what new physical theories and CFTs arise in correspondence with these moduli spaces.
- Keywords:
- Two Dimensional Gravity ; Intersection Theory ; Abelian Varieties ; Abelian-Jacobi Map ; Matrix Models ; Conformal Fields Theory ; Jacobi Equations ; Witten Techniques
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