Why String Theory?

1
String Theory, Calabi-Yau Threefolds and the
Expanding Universe Herbie Smith University of
New Hampshire, hlk25_at_wildcats.unh.edu Prof. Per
Berglund University of New Hampshire,
per.berglund_at_unh.edu

• 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, 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.

• 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, ti, but in terms of the volume of its
  four-cycles, which are related to scalar fields,
  Fi. Current work is focused on performing this
  change of variables on any volume given in terms
  of two-cycles.

• 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 t13
  t12t2 t1t22 t23 t1 ? t13 t12t2
  t1t22 t23

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 ti are
the volumes of the two-cycles. We can describe
the potential using the Kähler potential, K(Fi,
), 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.
Figure 1 Projection of a Calabi-Yau manifold.
Courtesy of www.math.sjsu.edu/simic/Spring11/Math
213B/213B.html.
Figure 2 A schematic plot of the potential of
the universe.

• 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.

• Algorithmic Analysis of Calabi-Yau Manifolds
• Estimates predict about 10500 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.

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)
1209. 3) V. Balasubramanian, P. Berglund, J. P.
Conlon, F. Quevedo, Systematics of Moduli
Stabilisation in Calabi-Yau Flux
Compactifications, JHEP 0503 (2005) 007. 4) 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) 101901.
Acknowledgements We thank E. Ebrahim for
collaborations, and the Hamel Center for
Undergraduate Research and National Science
Foundation grant PHY-1207895 for financial
support.