Planckian Axions in String Theory

We argue that super-Planckian diameters of axion fundamental domains can naturally arise in Calabi-Yau compactifications of string theory. In a theory with $N$ axions $\theta^i$, the fundamental domain is a polytope defined by the periodicities of the axions, via constraints of the form $-\pi<Q^{i}_{j} \theta^j<\pi$. We compute the diameter of the fundamental domain in terms of the eigenvalues $f_1^2\le\...\le f_N^2$ of the metric on field space, and also, crucially, the largest eigenvalue of $(QQ^{\top})^{-1}$. At large $N$, $QQ^{\top}$ approaches a Wishart matrix, due to universality, and we show that the diameter is at least $N f_{N}$, exceeding the naive Pythagorean range by a factor $>\sqrt{N}$. This result is robust in the presence of $P>N$ constraints, while for $P=N$ the diameter is further enhanced by eigenvector delocalization to $N^{3/2}f_N$. We directly verify our results in explicit Calabi-Yau compactifications of type IIB string theory. In the classic example with $h^{1,1}=51$ where parametrically controlled moduli stabilization was demonstrated by Denef et al. in [1], the largest metric eigenvalue obeys $f_N \approx 0.013 M_{pl}$. The random matrix analysis then predicts, and we exhibit, axion diameters $>M_{pl}$ for the precise vacuum parameters found in [1]. Our results provide a framework for achieving large-field axion inflation in well-understood flux vacua.Comment: 42 pages, 4 figure

The Singularity Structure of Scale-Invariant Rank-2 Coulomb Branches

We compute the spectrum of scaling dimensions of Coulomb branch operators in 4d rank-2 $\mathcal{N}{=}2$ superconformal field theories. Only a finite rational set of scaling dimensions is allowed. It is determined by using information about the global topology of the locus of metric singularities on the Coulomb branch, the special K\"ahler geometry near those singularities, and electric-magnetic duality monodromies along orbits of the $\rm\, U(1)_R$ symmetry. A set of novel topological and geometric results are developed which promise to be useful for the study and classification of Coulomb branch geometries at all ranks.Comment: 2 references added, submitted to JHE

An Examination of Students\u27 Perceptions of Civic Issues: A Comparison of 1957 and 2011

Illuminated by a historical review of trends in educating for citizenship in American social studies classroom, the purpose of this study was to investiagte students\u27 perceptions of civic principles across three domains- democratic values, economic principles, and constitutional rights and responsibilities. To this end, a purposive, nonrandom sample of two hundred 12th grade students from eight high schools in a suburban school district in a Sour=theastern state was drawn and subsequently surveyed using an instrument originally developed during the 1950\u27s by Purdue University researchers. To determine whether there were generational differences in perceptions, the responses of the contemporary sample of 12th grade students were statistically compared to the norms obtained for a national sample of 12th graders surveyed over 50 years ago. In keeping with the emphasis on diversity in today\u27s social studies classrooms, also conducted were analyses of responses by gender, ethnicity, overall grade point average, level of class discussion, political orientation, and confidence in current administration\u27s policies to determine whether such factors influenced current student perceptions with regard to one or more of the issues investigated.Employing the one-way or goodness of fit chi-square test, statistical analyses of contemporary responses versus historical norms indicated generational differences across all five items within the domain of democratic values, all three items within the domain of economic principles, and five of the 10 items in the domain of constitutional rights and resonsibilities. Especially robust differences were observed with respect to items referencing affirmative action laws (c^2 (2, N = 200)= 41.37, p \u3c .001, w= 0.70), universal voting rights (c^2 (3, N = 200) = 93.72, p\u3c .001, w = 0.68, and the legal roght to face one\u27s accuser (c^2 (3, N = 200) = 112.52, p \u3c .001, w = 0.75).. However, when the two-way or test of independence chi-square was employed to identify differences in item responses by student characteristics, statistically signicant results were much less commonly observed and only systematically emerged with respect to the issue of limiting and controlling immigration. When levels of agreement and disagreement to this item were compared, differences among students in the contemporary sample were observed by ethnicity (c^2 (2, N =200)= 17.19, p \u3c .001, V = 0.29), political orientation (c^2 (2, N =195) = 14.85, p \u3c .001, V = 0.28), and confidence in the current U.S. administration\u27s policies (c^2 (2, N = 200) = 3.96, p \u3c .05, V = 0.14). To help clarify the generational findings, reference to the historical record is made, while more current events are evoked to help make the subgroup differences in contemporary student responses more interpretable