## Final Report Summary - STRING GUTS (Phenomenological implications and signatures of Grand Unified Theories in String Theory)

The project objectives were to develop a general understanding of, and construct explicit models exhibiting, phenomenological features in F-theory Grand Unified Theories models. In particular regarding aspects of flavour physics, neutrino physics, dark matter, and supersymmetry breaking. This required a development of the necessary mathematical tools and formal understanding, as well as phenomenological investigation of particular explicit models. A common theme of the phenomenological objectives is the required control over operators in theory. In flavour physics controlling the Yukawa couplings, in order to recreate the observed patterns, in neutrino physics controlling the the masses and mixing, in dark matter understanding the operatros that can couple it to the standard model and the mass of the dark matter, and in supersymmetry breaking, patterns and features of soft supersymmetry breaking terms,

The work done during the project attacked these requirements from two perspectives: calculating operator coefficients from the wavefunction overlaps of fields in the internal dimensions, and using U(1) symmetries, both global and local, to control the operator structures. The results of the project were a significant advancement in both these approaches: wavefunction overlap techniques were developed that allow us to calculate coupling of massless fields to massive fields, which in turn, after integrating out the massive states, allows the determination of the coefficients of couplings in the theory. This was applied in particular to flavour physics, both of quark and leptons and neutrinos, to show how using the techniques quite precise calculations of the relevant observable parameters can be made, with quite surprising results in some cases showing a substantial suppression or enhancement when compared with naive expectations. This presents a challenge for future string theory model building of ensuring that any suppression or enhancement coming from wavefunctions is appropriately accounted for in extracting precise predictions, as well as an opportunity to explain some deep puzzles in beyond-the-standard-model physics using string theory physics, for example the absence of proton decay. From the perspective of U(1) symmetries, the work was primarily on the development of the mathematical and formal techniques necessary to build models which exhibit such symmetries. In particular elliptic fibrations that can be used in F-theory were developed which exhibit a non-trivial Mordell-Weil group thereby exhibiting a number of U(1) symmetries. The matter charged under the symmetries as well as its Yukawa couplings were calculated explicitly for these models.

Overall, the work carried out during the fellowship, has led to two types of important advancements in our understanding of the outlined phenomenological themes from the perspective of F-theory: the formal understanding and technical ability to i) Explicitly construct models with U(1) symmetries and the associated selection rules and ii) Calculate to increased accuracy the coefficients of operators in the theory using wavefunction overlap techniques.The second advancement was a direct extraction of phenomenological results from explicit models, for example, soft supersymmetry breaking terms induced by integrating out massive modes were calculated and shown to induce flavour changing processes that are constrained experimentally. Overall the work carried out has advanced the subject by actually exploring the details of how to induce U(1) symmetries or how to calculate coefficients precisely, themes which had before been left relatively unexplored and which are clearly crucial for extracting phenomenological features of the models that can be tested against experiments.

The work done during the project attacked these requirements from two perspectives: calculating operator coefficients from the wavefunction overlaps of fields in the internal dimensions, and using U(1) symmetries, both global and local, to control the operator structures. The results of the project were a significant advancement in both these approaches: wavefunction overlap techniques were developed that allow us to calculate coupling of massless fields to massive fields, which in turn, after integrating out the massive states, allows the determination of the coefficients of couplings in the theory. This was applied in particular to flavour physics, both of quark and leptons and neutrinos, to show how using the techniques quite precise calculations of the relevant observable parameters can be made, with quite surprising results in some cases showing a substantial suppression or enhancement when compared with naive expectations. This presents a challenge for future string theory model building of ensuring that any suppression or enhancement coming from wavefunctions is appropriately accounted for in extracting precise predictions, as well as an opportunity to explain some deep puzzles in beyond-the-standard-model physics using string theory physics, for example the absence of proton decay. From the perspective of U(1) symmetries, the work was primarily on the development of the mathematical and formal techniques necessary to build models which exhibit such symmetries. In particular elliptic fibrations that can be used in F-theory were developed which exhibit a non-trivial Mordell-Weil group thereby exhibiting a number of U(1) symmetries. The matter charged under the symmetries as well as its Yukawa couplings were calculated explicitly for these models.

Overall, the work carried out during the fellowship, has led to two types of important advancements in our understanding of the outlined phenomenological themes from the perspective of F-theory: the formal understanding and technical ability to i) Explicitly construct models with U(1) symmetries and the associated selection rules and ii) Calculate to increased accuracy the coefficients of operators in the theory using wavefunction overlap techniques.The second advancement was a direct extraction of phenomenological results from explicit models, for example, soft supersymmetry breaking terms induced by integrating out massive modes were calculated and shown to induce flavour changing processes that are constrained experimentally. Overall the work carried out has advanced the subject by actually exploring the details of how to induce U(1) symmetries or how to calculate coefficients precisely, themes which had before been left relatively unexplored and which are clearly crucial for extracting phenomenological features of the models that can be tested against experiments.