Scientific Series

1. The notion of wall-crossing structure (as defined by Maxim Kontsevich and myself in arXiv: 1303.3253)
provides the  universal framework for description of different types of wall-crossing formulas (e.g. Cecotti-Vafa in 2d or KSWCF in 4d). It also gives
a language and tools for proving algebraicity and analyticity of arising generating series (e.g. for BPS invariants).

2. Holomorphic Floer theory is the Floer theory of a pair of complex Lagrangian subvarieties  of a complex symplectic manifold (maybe infinite-dimensional).
This geometry underlies several important topics, both in mathematics and physics. Those  include  questions about analytic continuation of exponential integrals
(e.g. Feynman integrals),
deformation quantization of holomorphic symplectic manifolds (and related Riemann-Hilbert correspondences), Geometric Langlands correspondence, etc.

3. It was known for a long time that many a priori divergent series (like e.g. formal WKB series for solutions of equations with a small parameter) become analytic or meromorphic functions after taking their Laplace transform
(Borel resummation). This property was called  the resurgence property of the divergent series. The relation of the resurgence phenomenon to simplest wall-crossing formulas was realized in the early 90's in the work of French mathematicians (Ecale, Voros, Pham , Malgrange and  others).

I plan to discuss how the combination of 1 and 2 can be applied to 3 in a very general situation.
Main idea goes back to our theory of Donaldson-Thomas invariants (arXiv: 0811.2435).
Namely, analyticity of the formal series  follows from existence of a global analytic object which is glued from the local ones by means of the formal series.
This analytification of  an a priori formal variety is a by-product of the growth estimates on the data of the underlying wall-crossing structure.

In case of the Holomorphic Floer theory (which underlies e.g. the Stokes phenomenon for the WKB solutions of PDE or difference equations) one needs an estimate on the number of pseudo-holomorphic discs
with the boundary on the union of our Lagrangian subvarieties. In practice it often appears as an estimate on the number of gradient lines between two critical points
of the action or potential.

Inflation is proposed as a means of explaining why the Universe is currently so homogeneous on larger scales, solving both the horizon and flatness problems in early universe cosmology. However, if inflation itself requires homogeneous conditions to get started, then inflation is not a solution to the horizon problem. Most work up until now has focussed on a dynamical systems approach to classifying the stability of inflationary models, but recently Numerical Relativity (NR) has been used to simulate the actual evolution of the inflaton field, leading to new insights. I will describe a recent work (https://arxiv.org/abs/1608.04408) in which we used NR to consider the robustness of generic small and large field inflationary models to initial inhomogeneities in the inflaton field and the extrinsic curvature of the metric.

To build a fully functioning quantum computer, it is necessary to encode quantum information to protect it from noise. Topological codes, such as the color code, naturally protect against local errors and represent our best hope for storing quantum information. Moreover, a quantum computer must also be capable of processing this information. Since the color code has many computationally valuable transversal logical gates, it is a promising candidate for a future quantum computer architecture.

In the talk, I will provide an overview of the color code. First, I will establish a connection between the color code and a well-studied model - the toric code. Then, I will explain how one can implement a universal gate set with the subsystem and the stabilizer color codes in three dimensions using techniques of code switching and gauge fixing. Next, I will discuss the problem of decoding the color code. Finally, I will explain how one can find the optimal error correction threshold by analyzing phase transitions in certain statistical-mechanical models.

The talk is based on http://arxiv.org/abs/1410.0069, http://arxiv.org/abs/1503.02065 and recent works with M. Beverland, F. Brandao, N. Delfosse, J. Preskill and K. Svore.

In order to create ansatz wave functions for models that realize topological or symmetry protected topological phases, it is crucial to understand the entanglement properties of the ground state and how they can be incorporated into the structure of the wave function.

In this first part of this talk, I will discuss entanglement properties of models of topological crystalline insulators and spin liquids and show how to incorporate topological order, symmetry fractionalization, and lattice symmetry protected topological order into tensor network wave functions.

In the second part of this talk, I will discuss intrinsically fermionic topological phases and an exactly solvable model we built to elucidate the structure of the ground state wave functions in these phases.

References:

https://arxiv.org/abs/1507.00348

https://arxiv.org/abs/1605.06125

Condensed matter realizations of Majorana zero modes constitute potential building blocks of a topological quantum computer and thus have recently been the subject of intense theoretical and experimental investigation.  In the first part of this talk, I will introduce a new scheme for preparation, manipulation, and readout of these zero modes in semiconducting wires coated with mesoscopic superconducting islands.  This approach synthesizes recent materials growth breakthroughs with tools long successfully deployed in quantum-dot research, notably gate-tunable island couplings, charge-sensing readout, and charge pumping.  Guided by these capabilities, we map out numerous milestones that progressively bridge the gap between Majorana zero-mode detection and long-term quantum computing applications.  These include (1) detecting non-Abelian anyon ‘fusion rules’ in two complementary schemes, one based on charge sensing, the other using a novel Majorana-mediated charge pump, (2) validation of a prototype topological qubit, (3) braiding to demonstrate non-Abelian statistics, and (4) observing the elusive topological phase transition accompanying the onset of Majorana modes.  With the exception of braiding, these proposed experiments require only a single wire with as few as two islands, a setup already available in the laboratory.  In the second part of the talk, I will introduce a new class of 2D microscopic models---termed ‘Majorana-dimer models’---which generalize well-known quantum dimer models by dressing the bosonic dimers with pairs of Majorana modes.  These models host a novel interacting topological phase of matter which has the same bulk anyonic content as the chiral Ising theory, albeit with a fully gapped edge.  These seemingly contradictory statements can be reconciled by noting that our phase is inherently fermionic:  it can be understood as the product of an Ising phase with a topological p-ip superconductor.  Potential physical realizations of this exotic state via a lattice of strongly interacting Majorana modes will be discussed.

The past few years have seen a surge of interest in six-dimensional superconformal field theories (6D SCFTs).  Notably, 6D SCFTs have recently been classified using F-theory, which relates these theories to elliptically-fibered Calabi-Yau manifolds.  Classes of 6D SCFTs have remarkable connections to structures in group theory and therefore provide a physical link between two seemingly-unrelated mathematical objects.  In this talk, we describe this link and speculate on its implications for future studies of 6D SCFTs.

I will answer the question in the title. I will also describe a new quantum algorithm for Boolean formula evaluation and an improved analysis of an existing quantum algorithm for st-connectivity. Joint work with Stacey Jeffery. 

About a decade ago Eynard and Orantin proposed a powerful computation algorithm
known as topological recursion. Starting with a ``spectral curve" and some ``initial data"
(roughly, meromorphic differentials of order one and two) the topological recursion produces by induction
a collection of symmetric meromorphic differentials on the spectral curves parametrized by
pairs of non-negative integers (g,n) (g should be thought of as a genus and n as the number of punctures).
Despite of many applications of the topological recursion (matrix integrals, WKB expansions, TFTs, etc.etc.)
the nature of the recursive relations was not understood.

Recently, in a joint work with Maxim Kontsevich we found a simple underlying structure of the recursive relations of Eynard and Orantin. We call it  ``Airy structure". In this talk I am going to define this notion and explain how the recursive relations of Eynard and Orantin follow from the  quantization of
a quadratic Lagrangian subvariety in a symplectic vector space.

 Non-Fermi liquids are exotic metallic states which do not support well defined quasiparticles. Due to strong quantum fluctuations and the presence of extensive gapless modes near the Fermi surface, it has been difficult to understand universal low energy properties of non-Fermi liquids reliably.  In this talk, we will discuss recent progress made on field theories for non-Fermi liquids. Based on a dimensional regularization scheme which tunes the number of co-dimensions of Fermi surface, critical exponents that control scaling behaviors of physical observables can be computed in controlled ways. The systematic expansion also provides important insight into strongly interacting metals. This allows us find the non-perturbative solution for the strange metal realized at the antiferromagnetic quantum critical point in 2+1 dimensions and predict the exact critical exponents.

Analyzing characteristics of an unknown quantum system in a device-independent manner, i.e., using only the measurement statistics, is a fundamental task in quantum physics and quantum information theory. For example, device-independence is a very important feature in the study of quantum cryptography where the quantum devices may not be trusted. In this talk, I will discuss the ability to characterize the state that Alice and Bob share in various physical scenarios using only the correlation data. I first give a lower bound on the dimension of the underlying Hilbert spaces required by Alice and Bob to generate a given correlation in the Bell scenario. Also, I give two properties that the Schmidt coefficients of their shared state must satisfy. I’ll provide examples showing that our results can be tight, and examine when the shared pure state is characterized completely. Lastly, I will discuss these ideas in the Prepare-and-Measure scenario. 

This is joint work with Antonios Varvitsiotis and Zhaohui Wei. 

References: 

Phys. Rev. Lett. 117, 060401, 

Phys. Rev. A, to appear. (ArXiv:1606.03878), 

ArXiv:1609.01030. 

3d N=4 theories on the sphere have interesting supersymmetric sectors described by 1d QFTs and defined as the cohomology of a certain supercharge. One can define such a 1d sector for the Higgs branch or for the Coulomb branch. We study the Higgs branch case, meaning that the 1d QFT captures exact correlation functions of the Higgs branch operators of the 3d theory. The OPE of the 1d theory gives a star-product on the Higgs branch which encodes the data of these correlation functions. When the 3d theory is superconformal, the 1d theory is topological and coincides with the known construction in flat space, where the topological 1d theory lives in the cohomology of Q+S. Our construction thus generalizes it away from the conformal point. We then focus on theories constructed from vector and hypermultiplets. Using supersymmetric localization, we explicitly describe their 1d sector as the gauged topological quantum mechanics, or equivalently a gaussian theory coupled to a matrix model. This provides a very simple technique to compute the Higgs branch correlators.