Saturday, 27 January 2018

Lattice Quantum Chromodynamics (Lattice QCD) is a first principle non-perturbative method to describe the fundamental strong interactions of nature where quantitative answers with controlled systematics can be obtained through numerical path integration. Over the last 40 years Lattice QCD has been extensively utilized and numerous results have been obtained, some of which are difficult to obtain even experimentally. This includes understanding the origin of most of the masses of the visible universe, various decay constants and form factors of subatomic particles, quark-gluon-plasma of the early universe, the topological structure of the QCD vacuum, multi-hadron states in nuclear physics and many more. In this talk, with an introduction to Lattice QCD, various results will be presented. Current challenges and future prospects of Lattice QCD will also be discussed.

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Sunday, 28 January 2018

Introduction. Twisted formulations. Construction of lattice N=2 SYM in D=2

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In this introductory lecture we review basic CFT tools, which include representation theory of the conformal algebra and the operator product expansion. We prove that two- and three-point functions are completely fixed by conformal symmetry, while four-point functions depend on two conformal invariants. The non-trivial structure of four-point functions sets the stage for the bootstrap analysis of the next lecture.

One of the challenges in understanding quantum many body systems is computational in origin. Often the underlying dynamics is well known but in order to compute a quantity one needs to sum over an exponentially large number of terms with varying signs and phases. While in the absence of these signs and phases one can use Monte Carlo sampling to perform the relevant sum, in their presence the sampling becomes inefficient. The origin of the signs and phases are at the heart of quantum mechanics and cannot be ignored. In this lecture I will outline various ideas that have been explored over the years and those that have recently emerged. In the end I will argue that the challenge is an interesting research problem at the cross roads of mathematical and computational physics.

Monday, 29 January 2018

Construction of lattice N=4 SYM in D=4. Renormalization. Sign problems.

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We give a review of the modern conformal bootstrap program and outline the main results. We show how constraints on the operator spectrum of conformal field theories can be obtained numerically from the consistency requirements allied with unitarity. We discuss what changes when considering superconformal field theories and briefly review some of the recent results.

Tuesday, 30 January 2018

Supersymmetric lattice quivers. Super QCD on a lattice. Holographic applications.

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Wednesday, 31 January 2018

The lattice formulation of supersymmetric theories is important for understanding some important problems in Physics beyond standard model. Certain supersymmetric theories are now possible to study on the lattice using ideas of topological twisting. We will present results from lattice simulations of four-supercharge SYM theory in two dimensions and discuss supersymmetry breaking in this theory. We will then discuss some preliminary results from lattice simulations of finite-temperature sixteen-supercharge SYM in three dimensions at strong couplings and large N and compare it to predictions from its dual supergravity theory.

Building on the pedagogical lectures introducing supersymmetric lattice field theories, I will discuss ongoing numerical investigations of maximally supersymmetric Yang--Mills (N=4 SYM), the only known four-dimensional theory for which there exists a lattice formulation that exactly preserves a subset of the supersymmetry algebra. I will focus on ongoing lattice studies of the N=4 SYM static potential and conformal operator scaling dimensions, comparing preliminary non-perturbative results with analytic predictions. Time permitting I will also present recent results for the thermodynamics of the dimensional reduction of the theory to N=(8,8) SYM in two dimensions, which holography relates to properties of certain black hole solutions in supergravity.

Using complex Langevin dynamics and stochastic quantization we examine the phase structure of a large N unitary matrix model at low temperature with finite quark chemical potential. This model is obtained as the low temperature effective theory of QCD with N number of colors and N_f number of quark flavors defined on the manifold S^1 X S^3. We simulate several observables of the model, including Polyakov lines and quark number density, for large N and N_f. The action is manifestly complex and thus the dominant contributions to the path integral come from the space of complexified gauge field configurations. For this reason, the Polyakov line eigenvalues lie off the unit circle and out in the complex plane. A distinct feature of this model, the occurrence of a series of Gross-Witten-Wadia transitions, as a function of the quark chemical potential, is reproduced using complex Langevin simulations.

Questions about quantum field theories at non-zero chemical potential and/or real-time correlators are often impossible to investigate numerically due to the notorious sign problem. A possible solution to this problem is to deform the integration domain for the path integral in the complex plane. We describe a family of such deformations, built using the holomorphic gradient flow, that interpolate between the original integration domain (where the sign problem is severe) and the union of relevant thimbles (where the sign problem is mild but a multimodal probability distribution complicates the Monte Carlo sampling). We show how this works in a fermionic model and for computing real time correlators for a simple thermal quantum field theory.

We study massless fermions interacting through a particular four-fermion term in four dimensions. Exact symmetries prevent the generation of bilinear fermion mass terms. We determine the structure of the low energy effective action for the auxiliary field needed to generate the four-fermion term and find it has a novel structure that admits topologically non-trivial defects with non-zero Hopf invariant. We show that fermions propagating in such a background pick up a mass without breaking symmetries. Furthermore, pairs of such defects experience a logarithmic interaction. We argue that a phase transition separates a phase where these defects proliferate from a broken phase where they are bound tightly. We conjecture that by tuning one additional operator the broken phase can be eliminated with a single BKT-like phase transition separating the massless from massive phases.

The Hamiltonian formulation of lattice Dirac fermions offers new opportunities to study strongly coupled quantum critical points in four-fermion models with enhanced symmetries. Unfortunately traditional quantum Monte Carlo methods to study these critical points encounter difficulties at large lattices sizes. Typical lattice sizes that are usually explored with exactly massless fermions are of the order of 600-1000 sites, with the largest sizes being of the order of 2500 sites. Going to larger sizes, while in principle possible, requires one to overcome numerical instabilities especially at low temperatures. We have recently extended the fermion bag idea to Hamiltonian methods and show that we can study lattice sizes of the order of 10000 sites without encountering any numerical instabilities. We have used this new approach to compute critical exponents in the 3D Gross-Neveu Ising model with one flavor of massless Dirac fermions using lattice sizes as large as 4096 sites.

I will give a brief overview of the uses and applications of resurgence and transseries within string theoretic contexts, with emphasis on topological and non-critical strings and their (matrix model) large N duals.

The celebrated Veneziano amplitude marks the birth of string theory. In this talk I will present a relation between large-N QCD and the Veneziano ampltiude. The derivation relies on the assumption of an area law, which is valid in flat space string theory, but not in real QCD. I will show how to incorporate effects due to holography that take into account deviations from an area law for small Wilson loops. Finally, I will also comment on the relation between the Veneziano amplitude and the lattice strong coupling expansion.

Physicists have beautiful theories for the microphysical interactions between particles, but face a number of problems when trying to compute predictions for the properties of matter. Many of these obstacles cannot be surmounted simply by using bigger computers and better code, but require the development of new theoretical insights. The speaker will discuss the particular examples from lattice quantum chromodynamics, including chiral symmetry and so-called sign problems, which lead us naturally to consider extra dimensions, topological insulators, and quantum computers.

Thursday, 01 February 2018

The speaker will describe the modern understanding for how to realize global chiral symmetry in lattice field theory. This work leads us to a view of the world as a topological insulator in five dimensions, with us living mostly on its four-dimensional surfaces.

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In this talk I discuss supersymmetric SU(N) Yang-Mills gauge theories at finite fermion density which, in general, suffer from a fermion sign problem. One approach to the sign problem is to use the canonical formulation where the fermion number is fixed. Using Wilson fermions on the lattice the canonical sectors of the theories can be obtained directly from transfer matrices explicitly defined through the fermion Dirac operator. As an example, I consider supersymmetric SU(N) Yang-Mills theories in 0+1 dimensions and present some results concerning the moduli space of these theories. In particular, I discuss the appearance of flat directions and the corresponding divergence of the moduli in some of the fermion sectors and for thermal boundary conditions.12

In this talk I briefly summarize the main obstacles in the formulation of supersymmetry on a space-time lattice show how the fine tuning is resolved in the case of N=1 supersymmetric Yang-Mills theory.

I will provide our most recent results for the bound state spectrum and the Ward identities of this theory obtained in simulations of the DESY-Münster collaboration.

With an analysis of lattice results for SU(N), Sp(2N) and SO(N) gauge groups in two and three spatial dimensions, numerical evidence is presented that the ratio between the squared mass of the scalar glueball and the string tension in a Yang-Mills theory is proportional to the Casimir of the adjoint over the Casimir of the fundamental representation of the gauge group. The proportionality constant between the ratio of the relevant dynamical quantities and the quadratic Casimir eigenvalues is universal, in the sense that it depends only on the dimensionality of the system. In order to get a physical handle on this universality, an argument based on the saturation of the scale anomaly by the lightest scalar glueball is discussed. In addition, it is observed that the ratio of the lightest spin-two glueball mass over the scalar global mass is compatible with the square root of two in a confining and chirally broken non-Abelian gauge theory, while it becomes significantly larger as the onset of the conformal window is approached. Both these findings provide powerful insights on the dynamics of non-Abelian gauge theories that go beyond standard large-N arguments.

I introduce an effective field theory which can be used to quantitatively understand the long distance physics of warm QCD. I show that it works surprisingly well in quantitatively reproducing results obtained from lattice QCD.

Lattice discretization allows nonperturbative study of QCD. For many interesting questions in strong interaction physics, however, it is not possible/very difficult to do first principles QCD calculation. In some cases, a combination of lattice techniques and effective field theory may allow us to get a better handle on the problem.

I will discuss the interesting problem of heavy quarkonia in quark-gluon plasma. I will discuss the problems encountered in doing a first-principles study, and how we can use insights from effective field theory to get better control on the problem.

I will give an overview of some recent results obtained by our group concerning the Twisted Eguchi-Kawai model and other twisted reduced models. I will explain what are the main goals of this program and what are some of the unknown aspects that have to be elucidated in the near future.

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Friday, 02 February 2018

An embarrassing fact about modern particle physics is that there is no known way to define the Standard Model nonperturbatively. The problem is a general one with chiral gauge theories. A possible starting point for a solution inspired by the solution for global chiral symmetries anticipated the Quantum Spin Hall Effect discovered by condensed matter physicists over a decade later. However, from there the path is obscure: one direction requires invoking dynamical behavior for which there is no evidence, while another leads to a world with exotic particles that interact with ordinary matter non-locally through topological fluctuations.

We use the Berkooz-Douglas model, which adds a D4-brane multiplet to the BFSS model, to probe the conjectured 10-dimensional dual black hole geometry.

An idea to formulate string theory or M-theory by a gauge theory attracts theorists and has been extensively studied. The gauge theory should be lower dimensional so that a geometry in string or M-theory, which has higher dimensions, must emerge from it. This suggests that there should be a phase transition in the gauge theory and that the geometry would appear as its temperature decreases. In this talk, we focus on the BMN matrix model, which is considered as a non-perturbative formulation of M-theory on the pp-wave geometry and also conjectured to have a gauge/gravity duality, which relates the vacua on the gauge theory side to bubbling geometries in the type IIA supergravity. Our preliminary results of Monte Carlo simulations show two phase transitions and one of them looks related with emergent geometry.

We consider the matrix theoretical description of transverse M5-branes in M-theory on the 11-dimensional maximally supersymmetric pp-wave background. We apply the localization to the plane wave matrix model (PWMM) and show that transverse spherical fivebranes emerge as the distribution of low energy moduli of the scalar fields in PWMM.

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D0-brane QM is a 0+1D large-N supersymmetric gauged matrix quantum mechanics which is known to reduce to 10+1D supergravity in the low-temperature limit, and which has been proposed as a nonperturbative definition of M-theory. I will discuss the direct tests of gauge/gravity duality that can be made by studying the black hole internal energy with lattice calculations.

We perform a tree-level O(a) improvement of two-dimensional N=(2,2) super Yang-Mills theory on the lattice, motivated by a fast convergence in numerical simulations. The improvement preserves one supercharge Q and an U(1)_R symmetry, which are exactly realized on the lattice in the original formulation. This is needed to obtain the correct continuum limit without parameter fine-tuning. We find that the improved lattice action satisfies a milder locality condition, in which the interactions decay as the exponential of the distance on the lattice.

We propose a new lattice field theory formulation which has exact symmetries of the corresponding continuum theory including supersymmetry and avoids the chiral fermion problem. We can find an explicit projection map from a continuum field theory to the corresponding lattice theory and vice versa. The formulation is proposed by a momentum representation and defines a new star product which has non-local nature in the coordinate space. However the locality is recovered in the continuum limit. Since this lattice formulation is equivalent to a continuum theory a regularization is still needed. Once a regularization is introduced associativity of the star product is mildly broken. Thus gauge invariance can be broken but assured to be recovered in the continuum limit. The lattice supersymmetry is exactly kept even with the broken associativity. I will give an overview of the formulation.

I will describe a new approach to formulating 3d N = 4 super-Yang-Mills on a lattice. The strategy is to complexify the Donaldson-Witten twist of 4d N = 2 super-Yang-Mills and apply geometric discretization. Lattice gauge invariance then forces the model to live in at most three dimensions and uncomplexified 3d N = 4 super-Yang-Mills can be reached in the continuum limit by supplementing the lattice action with mass terms. This talk is based on recent work in collaboration with Joel Giedt.

Saturday, 03 February 2018

Even if we can discover a nonperturbative formulation of chiral gauge theories like the Standard Model, we will be faced with a hard computational problem whose difficulty scales exponentially with the size of the system. This so-called sign problem is ubiquitous in the study of many-body quantum systems and is one of the greatest obstacles currently to making conceptual and technological breakthroughs in the study of quantum matter, from understanding the cores of neutron stars, to high temperature superconductors, topological field theories and M-theory. I discuss the nature of the sign problem, how quantum computers have the potential to solve it, and the first steps taken in this direction.

We use numerical bootstrap methods to search for nonlocal conformal field theories in three dimensions having three relevant scalars. One such example is the critical point of the 3D long-range Ising model. The Hamiltonian for this system has a tunable parameter which controls how quickly the interactions fall off. As a function of this parameter, the critical exponents interpolate between those of the (short-range) Ising model and those of a suitable mean-field theory. In the space of scaling dimensions carved out by demanding crossing symmetry, we find a series of kinks close to the latter regime. Their positions show good agreement with the scaling dimensions obtained by perturbing the mean-field theory with the epsilon expansion. Our approach relies on a recently discovered relation between OPE coefficients in the long-range Ising model, which may be efficiently imposed on a system of six correlators.

We use the chiral algebra description of 4d N=2 SCFTs in order to obtain analytic central charge bounds that severely constrain the landscape of N=2 theories. We then identify a very special class of models that saturate these bounds, and attempt to solve the simplest of them using modern numerical bootstrap techniques.

In this talk we discuss the bootstrap program applied to four-dimensional N>1 superconformal field theories, with focus on analytical results. We show how a protected subsector captured by a two-dimensional chiral algebra is obtained from the four-dimensional SCFT and discuss some of its main features.

I will discuss the notion of a complete basis of functionals for the conformal bootstrap equation. I will then describe how to analyticall y construct two particularly convenient bases for the 1D bootstrap. The two choices manifest crossing symmetry in the theories of the generalized free boson and fermion respectively and allow us to study deformations of these theories. Assuming no new operators appear in the OPE, I will show the fermion theory admits no deformations while the boson theory has a one-parameter deformation coinciding with the AdS_2 four-point contact interaction at the leading order.

A non-perturbative method to investigate conformal field theories on curved manifolds is proposed as an extension of Regge Calculus (RC) and Finite Element methods (FEM) on simplicial lattices. For Bosonic, Fermionic and Gauge field, it reproduces the classical equation of motion. To converge to the full Euclidean quantum path integral, new simplicial counter terms are require to restore isometries as the UV cut-off is removed. Tests in for d = 2,3 Ising CFT limit of phi 4th theory are formulated for maximally symmetric de Sitter and Anti de Sitter geometries.

We perform the lattice simulations of two-dimensional N=(8,8) SYM to test the gauge gravity duality. We employ Sugino lattice action that keeps two of sixteen supercharges exactly on the lattice.

The lattice results clearly show that the thermodynamics of the gauge theory reproduce the behavior of dual black-brane.

A Toy Model for Time Evolving QFT on Lattice with Controllable Chaos Abstract: I will describe a class of toy models with a dynamics is that of generalized quantum cat maps on a product of quantum tori. These tori are defined by an algebra of clock-shift matrices of dimension N.

I will show examples of the dynamics for initial product states. Some of these entangle at rates determined by Lyapunov exponents of the system at large N when the initial states are gaussian. I will also show examples where this intuition fails.