C-MAP · A new string duality for gauged supergravity and its physical and mathematical applications

Final Activity Report Summary - C-MAP (A new string duality for gauged supergravity and its physical and mathematical applications)

One of the driving forces behind the research in theoretical high energy physics of the last decades is the ongoing quest for a unified description of quantum field theory and general relativity, which provide our theoretical understanding of particle physics experiments and cosmology, respectively. A prime candidate for such a unifying framework is string theory which combines the building blocks of gauge theories and gravity in a fundamentally consistent way. This consistency requires the introduction of additional space-time dimensions and supersymmetry and, thanks to string dualities relating different string theories, leads to a single, essentially unique theory. Subsequently, exploring the structure and implications of this theory has fostered many exciting developments in physics and mathematics. One of the main challenges for string theory to date is its connection to four-dimensional, observable physics. This is owed to the fact that, while its formulation at high energies is essentially unique, string theory contains a plethora of solutions (vacua) with very different physics at the low energies probed by experiments. Conveniently, the latter is encoded in a four-dimensional low energy effective action (LEEA). Recent investigations of this vacuum structure indicated that the solutions corresponding to our world are, most likely, not located in a regime where the "classical" LEEA is valid, fuelling the need for understanding the perturbative and non-perturbative string corrections to the LEEAs. The results obtained within the research project "c-map" constitute a major advance in this direction. We thereby focused on a particular class of LEEA which preserve N=2 supersymmetry. Here the quantum corrections to the hypermultiplet sector were extremely poorly understood and posed a longstanding problem. The c-map project then developed an alternative description of this sector based on tensor supermultiplets. Compared to the previous formulation, this led to major simplifications of the constraints encoding the supersymmetry of the LEEA. Subsequently, several string dualities have been recast into the tensor language. These comprised the c-map which relates the LEEA of type IIA and type IIB string theory and the non-perturbative SL(2, Z) duality of type IIB sting theory and rendered the implications of these dualities much more transparent. Combining the classical c-map with the constraints from supersymmetry and string perturbation theory, then allowed us to determine the perturbative string-loop corrections to the hypermultiplet sector and led to a non-renormalisation theorem. Going beyond string perturbation theory, the SL(2, Z)-invariance of the type IIB LEEA further allowed to deduce unprecedented exact results for certain classes of non-perturbative corrections. In a certain limit, these corrections have then been mapped to earlier results on the type IIA side, constituting the first non-perturbative test of mirror symmetry at the level of the four-dimensional LEEA. From a mathematical perspective the quantum corrected hypermultiplet sectors constructed above comprise previously unknown explicit examples of quaternion-Kaehler metrics which are not in the image of the classical c-map. While the research project "c-map" has been carried out, it also turned out that the results obtained within the research proposal have an a priori unexpected overlap with other active research areas within string theory and mathematics as, e.g., the derivation of the Black Hole entropy, topological strings, or enumerative geometry. We expect that these connections will stimulate further research projects deepening the relations between these seemingly different facets of string theory in the future.

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