Supersymmetric field theories geometric structures and dualities
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General references for supergravity include refs. Being positiveâ€” ab for the inverse metrics. F+ has manifestly the right transformation under H, Eq. Relevant geometric concepts are introduced and described in detail, providing a self-contained toolkit of useful techniques, formulae and constructions. From the discussion in Section 2.

In the above discussion no supersymmetry is implied. We consider, at the linearized level, the superspace formulation of lower-dimensional F-theory. The existence of a set of parallel complex structures implies quite strong constraints on the geometry of M. In the textbooks one finds also other numbers typically 1. The following articles are written for advanced students and researchers in quantum field theory, string theory and mathematical physics, our goal being to familiarize these readers with the forefront of current research. Here the story is continued by showing how various branes are Kaluza-Klein monopoles of these higher dimensional theories.

For the application to physics, the following result is also important: Proposition 11. Our construction reproduces all previously considered cases of double and exceptional field theories. We discuss the necessary modifications of the symmetries defined in supersymmetric gauged double field theory. These include foundational studies of the structure of Calabi-Yau flux vacua, as well solutions involving strings on spaces of negative curvature. It is suitable for use as a textbook in graduate courses on modern string theory and theoretical particle physics, and will also be an indispensable reference for seasoned practitioners. Sijkl Ï† should be a covariant 4â€” tensor on M made out of the metric gij and its derivatives. We propose a boundary action to complement the recently developed duality manifest actions in string and M-theory using generalized geometry.

We shall not prove this theorem, although at this point we have all the necessary tools. By the same argument, if on M there are n linearly independent parallel 2â€”forms, the corresponding skewâ€”symmetric endomorphisms f a should generate a Cl n algebra as in Chapter 2. Let us recall our findings for this case: the group Spin 4 decomposes into the product Spin 3 1 Ã— Spin 3 2 ; correspondingly, the scalar manifold M splits into the product of two hyperKÂ¨ahler spaces, M1 Ã— M2. The O g terms in the Yukawa couplings A2 , A3 have the general structure m A2 m ÏˆÂ¯ Î¼ Î³ Î¼ Î»m + A3 Â¯Ä± m Ï‡ Ä±Â¯ Î»m + A3 mn Î» Î»n + H. The invariant physical requirement is that the vectors entering into the Lagrangian are mutually local fields.

Writing E as in Eq. By the argument in Eqs. Remarkably, ideas from quantum theory turn out to carry tremendous mathematical power too, even though we have little intuition dealing with elementary particles. We construct a supersymmetric extension of double field theory that realizes the ten-dimensional Majorana-Weyl local supersymmetry. String vacua correspond to models which have both conformal and 2D Lorentz symmetry at the quantum level.

The spinor parametrisation of the momenta naturally solves simultaneously both the mass-shell condition and the weak section condition. Note that the frame {eÏƒ r } is unique up to local Spin N Ã— C rotations. This is a very strong constraint on the geometry of M, and â€” as we shall see in Chapter 3 â€” it implies its metric to be flat. It is a deep physical property. For a different, recently identified class of bumpy black holes, we find evidence that this family ends in solutions with a localized singularity that exhibits apparently universal properties, and which does not seem to allow for transitions to any known class of black holes.

Now assume we have two anticommuting complex structures f 1 , f 2. We do this in the next subsection, exploiting the Cartanâ€”Kostant isomorphism. Cl n stands for the universal Clifford algebra in dimension n, Cl0 n for its even subalgebra. Is this definition well posed? We may rephrase the situation as a cohomology problem Spencer cohomology. We show that the worldsheet boundary conditions of the doubled formalism describe in a unified way a T-dual pair of D-branes, which we call double D-branes.

We pay particular attention to their arithmetic aspects, which are crucial for the quantum theory and have never been discussed previously, to the best of our knowledge. Double field theory and exceptional field theory are formulations of supergravity that make certain dualities manifest symmetries of the action. Can you see the magic of this result? Note, however, that the conical geometries are everywhere smooth only when they are flat. Instead, it is constructed using the Courant bracket. In doing so we provide suggestions as to how Ramond fields can be incorporated into the double field theory. This is perfectly in agreement with our findings, and solves quite elegantly our little paradox. For higher spin fields there is more structure.