The Algebra of Grand Unified Theories

The Standard Model of particle physics may seem complicated and arbitrary, but it has hidden patterns that are revealed by the relationship between three "grand unified theories": theories that unify forces and particles by extending the Standard Model symmetry group U(1) x SU(2) x SU(3) to a larger group. These three theories are Georgi and Glashow's SU(5) theory, Georgi's theory based on the group Spin(10), and the Pati-Salam model based on the group SU(2) x SU(2) x SU(4). In this expository account for mathematicians, we explain only the portion of these theories that involves finite-dimensional group representations. This allows us to reduce the prerequisites to a bare minimum while still giving a taste of the profound puzzles that physicists are struggling to solve.Comment: 73 pages, 20 ps figure

Mutual information and the F-theorem

Mutual information is used as a purely geometrical regularization of entanglement entropy applicable to any QFT. A coefficient in the mutual information between concentric circular entangling surfaces gives a precise universal prescription for the monotonous quantity in the c-theorem for d=3. This is in principle computable using any regularization for the entropy, and in particular is a definition suitable for lattice models. We rederive the proof of the c-theorem for d=3 in terms of mutual information, and check our arguments with holographic entanglement entropy, a free scalar field, and an extensive mutual information model.Comment: 80 pages, 16 figure

A New Approach to Ramanujan\u27s Partition Congruences

MacMahon provided Ramanujan and Hardy a table of values for p(n) with the partitions of the first 200 integers. In order to make the table readable, MacMahon grouped the entries in blocks of five. Ramanujan noticed that the last entry in each block was a multiple of 5. This motivated Ramanujan to make the following conjectures, p(5n+4) ≡ 0 (mod 5); p(7 n+5) ≡ 0 (mod 7); p(11n+6) ≡ 0 (mod 11) which he eventually proved. The purpose of this thesis is to give new proofs for Ramanujan\u27s partition congruences. This would be done by using theta functions to construct certain vector spaces of modular forms. Computations within these vector spaces result in new proofs for Ramanujan\u27s partition congruences modulo five and seven. Similar techniques will use to derive congruences for a wider class of generating functions

Importance of including small body spin effects in the modelling of extreme and intermediate mass-ratio inspirals

We explore the ability of future low-frequency gravitational wave detectors to measure the spin of stellar mass and intermediate mass black holes that inspiral onto super-massive Kerr black holes (SMBHs). We develop a kludge waveform model based on the equations of motion derived by Saijo et al. [Phys Rev D 58, 064005, 1998] for spinning BH binaries, augmented with spin-orbit and spin-spin couplings taken from perturbative and post-Newtonian (PN) calculations, and the associated conservative self-force corrections, derived by comparison to PN results. We model the inspiral phase using accurate fluxes which include perturbative corrections for the spin of the inspiralling body, spin-spin couplings and higher-order fits to solutions of the Teukolsky equation. We present results of Monte Carlo simulations of parameter estimation errors and of the model errors that arise when we omit conservative corrections from the waveform template. For a source 5000+10^6 solar mass observed with an SNR of 1000, LISA will be able to determine the two masses to within a fractional error of ~0.001, measure the SMBH spin magnitude, q, and the spin magnitude of the inspiralling BH to 0.0001, 10%, respectively, and determine the location of the source in the sky and the SMBH spin orientation to within 0.0001 steradians. For a 10+10^6 solar mass system observed with SNR of 30, LISA will not be able to determine the spin magnitude of the inspiralling BH, although the measurement of the other waveform parameters is not significantly degraded by the presence of spin. The model errors which arise from ignoring conservative corrections become significant for mass-ratios above 0.0001, but including these corrections up to 2PN order may be sufficient to reduce these systematic errors to an acceptable level.Comment: 24 pages, 11 figures. v2 mirrors published version in PRD. Edits in Sections V and VI in response to comments from refere

Apacible brevedad de los renglones, abreviada vida de monarcas: Ana de Castro Egas, Francisco de Quevedo y la escritura del panegírico regio

Quevedo compuso una serie de paratextos, prólogos a obras ajenas, en un momento por un lado, de especial beligerancia en su revalidación de un «habla castellana pura» y por otro, de reflexión sobre la escritura de «vidas de monarcas». De tales textos quevedianos destaca por diversas razones —una ellas, el estar dedicado a una mujer— el laudatorio escrito que prologa la Eternidad del rey nuestro señor don Felipe III de Ana de Castro Egas, donde elogia este panegírico escrito por una docta dama próxima al entorno del Cardenal Infante y de la que se desconoce casi todo. El presente trabajo estudia panorámicamente dicho paratexto quevediano e investiga aspectos del contexto —histórico, literario, político— en que don Francisco redacta el escrito. Quevedo wrote a series of paratextual writings, of prologues to different authors’ works, at a time in which, on the one hand, he actively vindicated a «pure Spanish language» and, on the other hand, he reflected on the writing of kings’ biographies. Among those paratextual works, the encomiastic prologue to Ana de Castro Egas’ Eternidad del rey nuestro señor don Felipe III (1629) is particularly interesting for several reasons. Ana de Castro, from whom we ignore almost everything, was an erudite lady close to Cardinal Infante’s entourage. The present article studies Quevedo’s prologue to Eternidad and examines aspects of the historical, literary and political contexts in which don Francisco composed his writing

Importance of including small body spin effects in the modelling of intermediate mass-ratio inspirals. II Accurate parameter extraction of strong sources using higher-order spin effects

We improve the numerical kludge waveform model introduced in [1] in two ways. We extend the equations of motion for spinning black hole binaries derived by Saijo et al. [2] using spin-orbit and spin-spin couplings taken from perturbative and post-Newtonian (PN) calculations at the highest order available. We also include first-order conservative self-force corrections for spin-orbit and spin-spin couplings, which are derived by comparison to PN results. We generate the inspiral evolution using fluxes that include the most recent calculations of small body spin corrections, spin-spin and spin-orbit couplings and higher-order fits to solutions of the Teukolsky equation. Using a simplified version of this model in [1], we found that small body spin effects could be measured through gravitational wave observations from intermediate-mass ratio inspirals (IMRIs) with mass ratio eta ~ 0.001, when both binary components are rapidly rotating. In this paper we study in detail how the spin of the small/big body affects parameter measurement using a variety of mass and spin combinations for typical IMRIs sources. We find that for IMRI events of a moderately rotating intermediate mass black hole (IMBH) of ten thousand solar masses, and a rapidly rotating central supermassive black hole (SMBH) of one million solar masses, gravitational wave observations made with LISA at a fixed signal-to-noise ratio (SNR) of 1000 will be able to determine the inspiralling IMBH mass, the central SMBH mass, the SMBH spin magnitude, and the IMBH spin magnitude to within fractional errors of ~0.001, 0.001, 0.0001, and 9%, respectively. LISA can also determine the location of the source in the sky and the SMBH spin orientation to within ~0.0001 steradians. We show that by including conservative corrections up to 2.5PN order, systematic errors no longer dominate over statistical errors for IMRIs with typical SNR ~1000.Comment: 21 pages, 7 figures. v2: three references added, edits in Sections II-V, including additional results in Section V to address comments by the referee. v3: mirrors version accepted to PR