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In this talk I describe a holographic perspective to study field spaces that arise in string compactifications. The constructions are motivated by a  general description of the asymptotic, near-boundary regions in complex structure moduli spaces of Calabi-Yau manifolds using asymptotic Hodge theory. For real two-dimensional field spaces, I introduce an auxiliary bulk theory and describe aspects of an associated sl(2) boundary theory. The bulk reconstruction from the boundary data is provided by the sl(2)-orbit theorem of Schmid and Cattani, Kaplan, Schmid, which is a famous and general result in Hodge theory. I then apply this correspondence to the flux landscape of Calabi-Yau fourfold compactifications and discuss how this allows us, in work with C. Schnell, to prove that the number of self-dual flux vacua is finite.

Zoom: https://harvard.zoom.us/j/91780604388?pwd=d3BqazFwbDZLQng0cEREclFqWkN4UT09

In this talk, I will present a new way to look at symmetry. We show that symmetry can be viewed as a non-invertible gravitational anomaly, and a non-invertible gravitational anomaly is classified by topological order in one higher dimension. This leads to a holographic view of symmetry:  symmetry is a shadow of topological order in one higher dimension. This point of view allows us to see the duality (i.e. the equivalence) between symmetries that look very different. It also gives rise to a more general symmetry – algebraic higher symmetry, which is beyond higher group description.

Zoom: https://harvard.zoom.us/j/779283357?pwd=MitXVm1pYUlJVzZqT3lwV2pCT1ZUQT09

Moduli spaces of curves admit finite covers by moduli spaces which parametrize curves together with so-called level structures. In my talk, I will discuss how the cohomology of these spaces at infinite level is related to a profinite property of the mapping class group. I will then explain why tools from p-adic geometry yield vanishing statements for these cohomologies in high degree.

Zoom: https://harvard.zoom.us/j/91794282895?pwd=VFZxRWdDQ0VNT0hsVTllR0JCQytoZz09

The temperature versus doping phase diagram of the cuprate high-T_c superconductors features an enigmatic pseudogap region whose microscopic origin remains a subject of intensive study. Experimentally resolving its symmetry properties is imperative for narrowing down the list of possible explanations. In this talk I will give an overview of how optical second harmonic generation (SHG) can be used as a sensitive probe of symmetry breaking, and recap the ways it has been used to solve outstanding problems in condensed matter physics. I will then describe how we have been applying SHG polarimetry and spectroscopy to interrogate the cuprate pseudogap. In particular, I will discuss our data on YBa₂Cu₃O_y [1], which show an order parameter-like increase in SHG intensity below the pseudogap temperature T* across a broad range of doping levels. I will then focus on our more recent results on a model parent cuprate Sr₂CuO₂Cl₂ [2], where evidence of anomalous broken symmetries surprisingly also exists. Possible connections between these observations will be speculated upon.

[1] L. Zhao, C. A. Belvin, R. Liang, D. A. Bonn, W. N. Hardy, N. P. Armitage and D. Hsieh, “A global inversion-symmetry-broken phase inside the pseudogap region of YBa₂Cu₃O_y,” Nature Phys. 13, 250 (2017).

[2] A. de la Torre, K. L. Seyler, L. Zhao, S. Di Matteo, M. S. Scheurer, Y. Li, B. Yu, M. Greven, S. Sachdev, M. R. Norman and D. Hsieh. “Anomalous mirror symmetry breaking in a model insulating cuprate Sr₂CuO₂Cl₂,” Preprint at https://arxiv.org/abs/2008.06516.

Zoom: https://harvard.zoom.us/j/977347126

There are several ways to specify a number field. One can provide the minimal polynomial of a primitive element, the multiplication table of a $\bf Q$-basis, the traces of a large enough family of elements, etc. From any way of specifying a number field one can hope to deduce a bound on the number $N_n(H)$ of number fields of given degree $n$ and discriminant bounded by $H$. Experimental data suggest that the number of isomorphism classes of number fields of degree $n$ and discriminant bounded by $H$ is equivalent to $c(n)H$ when $n\geqslant 2$ is fixed and $H$ tends to infinity. Such an estimate has been proved for $n=3$ by Davenport and Heilbronn and for $n=4$, $5$ by Bhargava. For an arbitrary $n$ Schmidt proved a bound of the form $c(n)H^{(n+2)/4}$ using Minkowski’s theorem. Ellenberg et Venkatesh have proved that the exponent of $H$ in $N_n(H)$ is less than sub-exponential in $\log (n)$. I will explain how Hermite interpolation (a theorem of Alexander and Hirschowitz) and geometry of numbers combine to produce short models for number fields and sharper bounds for $N_n(H)$.

Zoom: https://harvard.zoom.us/j/96767001802

Password: The order of the permutation group on 9 elements.

In this talk we will discuss the application of Machine Learning techniques to obtain numerical approximations to various metrics of SU(3) structure on six manifolds. More precisely, we will be interested in SU(3) structures whose torsion classes make them suitable backgrounds for various string compactifications. A variety of aspects of this topic will be covered. These will include learning moduli dependent Ricci-Flat metrics on Calabi-Yau threefolds and obtaining numerical approximations to torsional SU(3) structures.

Zoom: https://harvard.zoom.us/j/96047767096?pwd=M2djQW5wck9pY25TYmZ1T1RSVk5MZz09

In this talk, I will introduce an analytic bootstrap approach for two-point correlation functions in CFTs on real projective space, and CFTs with a conformal boundary. We will use holography as a kinematical tool to derive universal results. By examining the conformal block decomposition properties of exchange diagrams in AdS space, we identify a useful new basis for decomposing correlators. The dual basis gives rise to a basis of functionals, whose actions we can compute explicitly via holography. Applying these functionals to the crossing equations, we can systematically extract constraints on the CFT data in the form of sum rules. I will demonstrate this analytic method in the canonical example of \phi^4 theory in d=4-\epsilon, fixing the CFT data to \epsilon^2.

Zoom: https://harvard.zoom.us/j/977347126