Why String Theory? Two theories dominate modern physics - general relativity and the Standard Model of particle physics. General relativity describes gravity and acts over large distances, and the Standard Model describes elementary particles and their interactions over small distances. To study the small scale physics of the early universe, there is a need for a quantum theory of gravity. String theory describes both gravity and particle interactions. It postulates that matter is made of strings and requires that our universe is composed of ten dimensions (nine space + one time). However, only four spacetime dimensions are observed, leaving the rest curled up, or compactified. String Theory, Calabi-Yau Threefolds and the Expanding Universe Herbie Smith – University of New Hampshire, Prof. Per Berglund – University of New Hampshire, The Importance of Calabi-Yau Manifolds Calabi-Yau manifolds are three-complex dimensional manifolds that meet the string theoretic requirements for models of extra- dimensional space [1]. The specific shape and size of a Calabi-Yau, given in terms of various types of holes that the manifold contains, have significant effects on string interactions and the evolution of the universe. The potential energy of the universe depends on the volume as well as the shape of the Calabi-Yau manifold, with the latter fixed by generalized magnetic fluxes. Knowing the potential energy allows predictions about the fate of the universe, and gives us a better understanding of the early universe, inflation, and the current accelerated expansion of the universe due to dark energy. Figure 1: Projection of a Calabi-Yau manifold. Courtesy of String Theory, Inflation, and Dark Energy Inflation is an assumed period of very rapid expansion in the early universe. It provides explanations for recent observations of the universe, including the cosmic microwave background. There is little fundamental understanding of how the expansion was triggered. String theory offers a fundamental explanation in terms of the string vacuum energy and the geometry of Calabi-Yau manifolds. Calabi-Yau manifolds can be smoothly deformed, changing the size and shape of their holes without affecting the topology, i.e., the number of holes. This changes the string vacuum energy and the potential energy of the universe. A positive potential energy has a repulsive effect on the fabric of spacetime itself, which accounts for observations of dark energy. String theory posits that inflation occurred because the string vacuum energy was very high. In addition, the universe is currently expanding at an accelerating rate because the string vacuum energy has a small, positive value, see Figure 2. Algorithmic Analysis of Calabi-Yau Manifolds Estimates predict about different mathematically acceptable Calabi-Yau compactifications, including the various ways in which generalized magnetic fluxes influence the shape of the extra dimensions. Our research aims to find realistic cosmological models using Calabi-Yau compactifications, focusing on the dependence on the size of the manifold. This requires the ability to investigate Calabi-Yau manifolds with information that is readily available. Calabi-Yau manifolds are hypersurfaces in an ambient space constructed from 4-dimensional reflexive polytopes. Fortunately, all possible 4-dimensional reflexive polytopes have been classified and a great deal of information about them is known [2]. This allows an algorithmic approach to studying Calabi-Yau manifolds. As a first step, we developed an algorithm for constructing Calabi-Yau manifolds from 4-dimensional polyhedra. This allows the analysis of any Calabi-Yau manifold. Next, we introduced a method to compute the volume of a Calabi-Yau manifold in terms of its two-cycles, or holes. From here, the potential energy function of the universe can be calculated. Future Plans The next steps in this research program is to perform detailed analysis on those models which are determined to be Swiss Cheese manifolds. First, the algorithmic search for Swiss Cheese Calabi-Yau manifolds will be completed, see also related work [4]. The cosmological models which can be obtained from these Swiss Cheese manifolds will be examined in detail. This will be followed by a detailed analysis of del Pezzo divisors, a particular mathematical surface, which are embedded in some Swiss Cheese manifolds. The physics associated with del Pezzo divisors admit particle physics into string theory, which would bring together semi-realistic particle and cosmological models in string theory. Ongoing Research Because of the vast number of possible models, explicitly constructing and analyzing all models is unfeasible. Current research is focused on finding a more limited set of promising manifolds for cosmological applications. In 2005, Berglund et al., showed that the volume of a special class of Calabi-Yau manifolds, known as Swiss Cheese manifolds, can be used to compute the potential energy of the universe explicitly [3]. The volume of a generic Swiss Cheese manifold is not written in terms of the volumes of its two-cycles, t i, but in terms of the volume of its four-cycles, which are related to scalar fields, Φ i. Current work is focused on performing this change of variables on any volume given in terms of two- cycles. Figure 2: A schematic plot of the potential of the universe. References 1) B. Greene, String Theory on Calabi-Yau Manifolds, Proceedings TASI-96, World Scientific (1997). 2) M. Kreuzer, H. Skarke, Complete Classification of Reflexive Polyhedra in Four Dimensions, Adv. Theor. Math. Phys. 4 (2002) ) V. Balasubramanian, P. Berglund, J. P. Conlon, F. Quevedo, Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactifications, JHEP 0503 (2005) ) J. Gray, Y. H. He, V. Jejjala, B. Jurke, B. D. Nelson, J. Simón, Calabi-Yau Manifolds with Large-Volume Vacua, Phys. Rev. D86 (2012) The Volume Calculation The simplest example of a Calabi- Yau manifold is one with only two two-cycles. The volume of such a manifold is given as V = t t 1 2 * t 2 + t 1 *t t 2 3 t 1 ∫ t t 1 2 * t 2 + t 1 * t t 2 3 Acknowledgements We thank E. Ebrahim for collaborations, and the Hamel Center for Undergraduate Research and National Science Foundation grant PHY for financial support. Calculating the Potential The volume of a Calabi-Yau manifold is given in terms of the volume of its two- cycles, one class of holes. A simple example is the Calabi-Yau manifold with two two-cycles, with volume where the t i are the volumes of the two-cycles. We can describe the potential using the Kähler potential, K(Φ i, ), and the Kähler metric,, which are then used to determine the scalar potential of the universe, V, given by the equation with where W is the superpotential, which depends on the volume.