Properties of Some Classes of Structured Tensors

In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the nonnegative tensor theory are also discussed.Comment: arXiv admin note: text overlap with arXiv:1405.1288 by other author

Holography, Supergravity, and the Weak Gravity Conjecture

This dissertation represents my work on three different subjects relating to quantum gravity and the AdS/CFT correspondence. First, we review a holographic computation of the one-loop corrections to the Weyl anomaly on Ricci flat backgrounds in six dimensions. This allows us to determine the correction to one linear combination of the anomaly coefficients. Then, we will show that these corrections may be obtained from the six-dimensional superconformal index. The second section will cover consistent truncations on the Lunin-Maldacena (LM) background cite{Lunin:2005jy}. We show how to restore minimal supersymmetry to the model of cite{Lunin:2005jy} by determining the reduction ansatz which includes the graviton and a gauge field, which comprise the graviton multiplet of $mathcal{N} = 2$ supergravity in five dimensions. Then we discuss our attempt to construct a truncation which includes a scalar field corresponding to the $beta$-deformation parameter of the dual field theory. We show that if such a solution exists, it must differ somewhat drastically from the LM background. Finally, we discuss higher-derivative corrections to black hole solutions and the weak gravity conjecture in a few settings. We consider black holes which are charged under an arbitrary number of $U(1)$ gauge fields in four dimensional flat space. In this setting, we compute the effect of higher-derivative corrections on the extremality bound, and we discuss the constraints placed on the effective field theory coefficients by the requirements that near-extremal black holes are unstable to decay to smaller black holes. Next we consider the shifts to thermodynamic quantities due to higher-derivative corrections to charged black holes in Anti-de Sitter space. We confirm and clarify a previously noted relationship between the shift to the extremality bound and the shift to the Wald entropy. We also show that if the shift in the Wald entropy is assumed to be positive, then the coefficient of the $R_{mu nu rho sigma} R^{mu nu rho sigma}$ term in the effective Lagrangian must be positive as well.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163204/1/bmcpeak_1.pd

On-Shell Methods and Effective Field Theory

Effective field theory methods are now widely used, in both formal and phenomenological contexts, to efficiently study universal aspects of low-energy physics. In many cases, the computational complexity associated with constructing appropriate Wilsonian effective actions and calculating observables using the traditional Feynman diagram expansion, produces a barrier to what is practically calculable. In this thesis I use a variety of modern quantum field theory approaches, including on-shell methods, to efficiently calculate physical observables in EFTs in a variety of physical contexts. Results include: 1) A systematic analysis of soft theorems for photons and gravitons incorporating the effects of generic effective operators. Consistency with spacetime locality is used to prove that the recently discovered subleading soft graviton theorem is universal in generic EFTs. 2) The development of the numerical soft bootstrap algorithm incorporating Goldstone modes with spin and linearly realized supersymmetry. 3) The use of generalized unitarity methods to calculate two infinite classes of electromagnetic duality violating one-loop amplitudes in Born-Infeld electrodynamics. It is explicitly demonstrated that the duality violation can be removed by the addition of finite local counterterms, providing strong evidence that duality is unbroken by quantization. 4) The extension of the black hole Weak Gravity Conjecture to low-energy EFTs of quantum gravity with asymptotically flat boundary conditions and arbitrary numbers of U(1) gauge fields. Using on-shell methods we give a novel proof of a one-loop non-renormalization theorem in Einstein-Maxwell and use it to extend a recently given renormalization group argument for the WGC. 5) A systematic analysis of the leading higher-derivative corrections to the thermodynamic properties of charged black holes with asymptotically AdS boundary conditions in arbitrary dimensions. We generalize a recent conjecture for the positivity of the leading correction to the microcanonical entropy of thermodynamically stable black holes and demonstrate that this implies the positivity of c-a in a dual CFT.PHDPhysicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155124/1/jonescal_1.pd