Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology

We present a model of (modified) gravity on spacetimes with fractal structure based on packing of spheres, which are (Euclidean) variants of the Packed Swiss Cheese Cosmology models. As the action functional for gravity we consider the spectral action of noncommutative geometry, and we compute its expansion on a space obtained as an Apollonian packing of 3-dimensional spheres inside a 4-dimensional ball. Using information from the zeta function of the Dirac operator of the spectral triple, we compute the leading terms in the asymptotic expansion of the spectral action. They consist of a zeta regularization of a divergent sum which involves the leading terms of the spectral actions of the individual spheres in the packing. This accounts for the contribution of the points 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). There is an additional term coming from the residue at the additional point in the real dimension spectrum that corresponds to the packing constant, as well as a series of fluctuations coming from log-periodic oscillations, created by the points of the dimension spectrum that are off the real line. These terms detect the fractality of the residue set of the sphere packing. We show that the presence of fractality influences the shape of the slow-roll potential for inflation, obtained from the spectral action. We also discuss the effect of truncating the fractal structure at a certain scale related to the energy scale in the spectral action.Comment: 38 pages LaTe

Celestial Locality and the Jacobi Identity

We show the equivalence of several different tests of the Jacobi identity for celestial currents at tree level, in particular finding a simple, practical condition on hard momentum space 4-point amplitudes in any EFT. Along the way we clarify the role of the order of soft and collinear limits in obstructing Jacobi for soft insertions and we argue that, despite their current-algebra-like properties, soft insertions as formulated in this paper cannot be interpreted as local operators in celestial conformal field theory.Comment: 18 pages, 1 figur

Teaching TAs To Teach: Strategies for TA Training

"The only thing that scales with undergrads is undergrads". As Computer Science course enrollments have grown, there has been a necessary increase in the number of undergraduate and graduate teaching assistants (TAs, and UTAs). TA duties often extend far beyond grading, including designing and leading lab or recitation sections, holding office hours and creating assignments. Though advanced students, TAs need proper pedagogical training to be the most effective in their roles. Training strategies have widely varied from no training at all, to semester-long prep courses. We will explore the challenges of TA training across both large and small departments. While much of the effort has focused on teams of undergraduates, most presenters have used the same tools and strategies with their graduate students. Training for TAs should not just include the mechanics of managing a classroom, but culturally relevant pedagogy. The panel will focus on the challenges of providing "just in time", and how we manage both intra-course training and department or campus led courses

SCET sum rules for heavy-to-light form factors

We consider a sum rule for heavy-to-light form factors in soft-collinear effective theory (SCET). Using the correlation function given by the time-ordered product of a heavy-to-light current and its hermitian conjugate, the heavy-to-light soft form factor zeta_P can be related to the leading-order B meson shape function. Using the scaling behavior of the heavy-to-light form factor in Lambda_QCD/m_b, we put a constraint on the behavior of the $B$ meson shape function near the endpoint. We employ the sum rule to estimate the size of zeta_P with the model for the shape function and find that it ranges from 0.01 to 0.07.Comment: 11 pages, 5 figure

Multicollinear Singularities in Celestial CFT

The purpose of this paper is to study the holomorphic multicollinear limit of (celestial) amplitudes and use it to further investigate the double residue condition for (hard celestial) amplitudes and the celestial operator product expansion. We first set up the notion of holomorphic multicollinear limits of amplitudes and derive the 3-collinear splitting functions for Yang-Mills theory, Einstein gravity, and massless $\phi^3$ theory. In particular, we find that in $\phi^3$ theory the celestial 3-OPE contains a term with a branch cut. This explicit example confirms that branch cuts can obstruct the double residue condition for hard celestial amplitudes, which is the underlying cause of the celestial Jacobi identities not holding for certain theories. This addresses an ongoing debate in the literature about associativity of the celestial OPEs and concretely demonstrates a new (multi-particle) term in the celestial OPE coming from the multi-particle channel in the amplitudes.Comment: 19 page