String-Theory: Open-closed String Moduli Spaces Heidelberg, 13.10.2014
History of the Universe particular: Epoch of cosmic inflation in the early Universe
Inflation and Inflaton φ, potential V (φ) Possible approach: effective field theory Theories of quantum gravity such as string theories exist Can we consistently embed inflation into quantum gravity/string theory? Does this lead to constraints on possible effective field theories? Approach taken in this TR: Fluxbrane inflation, involving D-branes on Calabi-Yau manifolds
String Compactification Superstring theory: Defined in 10 dimensions String theory needs to be compactified: M10 = S4 I6. For example, on a 3-dimensional Calabi-Yau manifold
Scalars and Compactification Calabi-Yau manifolds come in families, parametrized by geometric moduli Physics: Massless scalars Example: Kähler and complex structure moduli of the two torus 6-d Calabi-Yau manifolds: Big moduli spaces, many massless scalars
Towards inflationary models... Many scalars: Need to find a mechanism that fixes most of them Need to find a scalar which can be a candidate for the inflaton (constrains the form of its potential) Many open problems
Open strings and D-branes Include D-branes into the setup Hypersurfaces of various dimensions where open strings can end In a string compactification, D-branes can wrap cycles of the internal manifold
D-branes and compactification Compactification on a circle two dimensional space viewed from large distance appears to be one dimensional A pointlike D0-brane is located at some position on the circle A D1 brane can wrap the compact circle or extend in the uncompactified direction
Moduli and D-branes Due to the presence of D-branes, a new additional type of modulus is introduced, open string moduli Example: D0 brane Can be located at any point of the internal manifold Possible brane configurations are parametrized by the position of the D0 brane on the manifold. The moduli space is isomorphic to the compactification manifold in this case. In general, the open string moduli space can be much more complicated. The full moduli space consists of open and closed string moduli.
Open String Inflaton TR 33: Fluxbrane Inflation Candidate for the inflaton comes from the open string sector It corresponds to the modulus describing the distance between two D7-branes. It is required to obtain a better understanding of open-closed string moduli spaces.
The worldsheet point of view Loop expansion in string theory is formulated in terms of Riemann surfaces of different genus. The diagrams are calculated by doing conformal field theory on the appropriate surface. Instead of geometrical compactifications, one can consider certain internal conformal field theories. They might or might not describe the stringy regime of a geometric object. Including D-branes means including surfaces with boundaries.
Application: Families of D5 branes on the Quintic Superpotential for D5 branes wrapping 2-cycles of a specific Calabi-Yau Consistent boundary conditions imply that the 2-cycles are holomorphic At specific points in the closed string moduli space, families of holomorphic 2 cycles exist. These can be wrapped by D5 branes (extending in 3 non-compact directions). Since there are families of cycles, there are open string moduli. Deforming the bulk (complex structure) changes the notion of holomorphicity. At generic points, there are only finitely many (counted as an application of mirror symmetry) holomorphic cycles. What happens to the brane? After perturbing the bulk, the open string moduli get lifted Controlled by a computable superpotential.
Summary Open closed string moduli spaces are relevant for embedding inflationary models into string theory Several techniques are available: Worldsheet, (algebraic) geometry, mirror symmetry... In the past, these methods were applied successfully to study D-brane moduli spaces We hope to get new insights on cosmological models.