<p>We investigate the twisted Eguchi-Kawai (TEK) reduced model of four-dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge theory in the presence of two-flavor adjoint fermions (adjoint TEK model). Using Monte Carlo simulations with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>121</mn></math>, twist parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>=</mo><mn>1</mn></math>, hopping parameter <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>κ</mi><mo>=</mo><mn>0.01</mn><mi>–</mi><mn>0.03</mn></mrow></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo>≪</mo><msub><mi>κ</mi><mi>c</mi></msub></math>) and inverse ’t Hooft coupling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo>=</mo><mn>0.30</mn><mi>–</mi><mn>0.45</mn></math>, we show that heavy adjoint fermions stabilize the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> center-symmetric vacuum even for the minimal twist satisfying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo>/</mo><msqrt><mi>N</mi></msqrt><mo>&lt;</mo><mn>1</mn><mo>/</mo><mn>9</mn></math>, where the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub><msup><mo stretchy="false">)</mo><mn>4</mn></msup></math> symmetry is spontaneously broken in the absence of adjoint fermions. This result also suggests that the adjoint TEK model with the minimal twist is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></math> gauge theory over a broader <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo stretchy="false">(</mo><mi>κ</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></math> parameter region than the adjoint EK model without twist. We further extend our analysis to a partially reduced model to realize a geometry akin to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi mathvariant="double-struck">R</mi><mn>3</mn></msup><mo>×</mo><msup><mi>S</mi><mn>1</mn></msup></math> and study the theory under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup></math> compactification with periodic adjoint fermions. Numerical simulations with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>N</mi><mo>=</mo><mn>16</mn><mi>–</mi><mn>49</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>b</mi><mo>=</mo><mn>0.30</mn><mi>–</mi><mn>0.46</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>κ</mi><mo>=</mo><mn>0.03</mn><mi>–</mi><mn>0.16</mn></math> are consistent with the adiabatic continuity conjecture; with periodic adjoint fermions, the theory remains in a center-symmetric (confined) phase as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>S</mi><mn>1</mn></msup></math> circle size is reduced, in contrast to the deconfining transition observed in the pure TEK model or in the TEK model with antiperiodic adjoint fermions. We present the Polyakov loop measurements and consistency checks supporting these findings.</p>

Japan

Yudai Hamada

Tatsuhiro Misumi

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi mathvariant="double-struck">Z</mi><mi>N</mi></msub></math> stability and continuity in the twisted Eguchi-Kawai model with two-flavor adjoint fermions

We consider the twistor description of classical self-dual Einstein gravity in the presence of a defect operator wrapping a certain ℂℙ1. The backreaction of this defect deforms the flat twistor space to that of Eguchi-Hanson space. We show that the celestial chiral algebra of self-dual gravity on the Eguchi-Hanson background is likewise deformed to become the loop algebra of a certain scaling limit of the family of W(μ)-algebras, where the scaling limit is controlled by the radius of the Eguchi-Hanson core. We construct this algebra by computing the Poisson algebra of holomorphic functions on the deformed twistor space, and check this result with a space-time calculation of the leading contribution to the gravitational splitting function. The loop algebra of a general W(μ)-algebra (away from the scaling limit) similarly arises as the celestial chiral algebra of Moyal-deformed self-dual gravity on Eguchi-Hanson space. We also obtain corresponding results for self-dual Yang-Mills.

Canada

Roland Bittleston

United Kingdom

Simon Heuveline

David Skinner

The celestial chiral algebra of self-dual gravity on Eguchi-Hanson space

Geodesic equations are solved when at least two of  $$\theta $$  <math> <mi>θ</mi> </math>  ,  $$\phi $$  <math> <mi>ϕ</mi> </math>   and  $$\psi $$  <math> <mi>ψ</mi> </math>   are constant, or r is constant, on scalar flat metrics of Eguchi–Hanson type. They can also be solved also on Eguchi–Hanson metrics which are Ricci flat if only  $$\phi $$  <math> <mi>ϕ</mi> </math>   is constant. However, the explicit solution of the geodesic equations is not available yet if only  $$\psi $$  <math> <mi>ψ</mi> </math>   is constant.

China

Yekun Yang

Xiao Zhang

Geodesics on metrics of Eguchi–Hanson type

The Eguchi-Hanson-AdS5 family of spacetimes are a class of static, geodesically complete asymptotically locally AdS5 soliton solutions of the vacuum Einstein equations with negative cosmological constant. They have negative mass and are parameterized by an integer p ≥ 3 with a conformal boundary with spatial topology L(p, 1). We investigate mode solutions of the scalar wave equation on this background and show that, similar to AdS5, the geometry admits a normal mode spectrum (i.e. solutions that neither grow or decay in time). In addition, we also discuss other geometric properties of these soliton spacetimes, including the behaviour of causal geodesics and their thermodynamic properties. We also point out a surprising connection with the AdS soliton.

Turkuler Durgut

Spain

Robie Hennigar

Hari Kunduri

Robert Mann

Phase transitions and stability of Eguchi-Hanson-AdS solitons

We construct higher-dimensional generalizations of the Eguchi-Hanson gravitational instanton in the presence of higher-curvature deformations of general relativity. These spaces are solutions to Einstein gravity supplemented with the dimensional extension of the quadratic Chern-Gauss-Bonnet invariant in arbitrary even dimension <math><mi>D</mi><mo>=</mo><mn>2</mn><mi>m</mi><mo>≥</mo><mn>4</mn></math>, and they are constructed out of nontrivial fibrations over (<math><mrow><mn>2</mn><mi>m</mi><mo>−</mo><mn>2</mn></mrow></math>)-dimensional Kähler-Einstein manifolds. Different aspects of these solutions are analyzed; among them, the regularization of the on-shell Euclidean action by means of the addition of topological invariants. We also consider higher-curvature corrections to the gravity action that are cubic in the Riemann tensor and explicitly construct Eguchi-Hanson type solutions for such.

Chile

Cristóbal Corral

Mexico

Daniel Flores-Alfonso

United States

Argentina

Gastón Giribet

Julio Oliva

Higher-curvature generalization of Eguchi-Hanson spaces

The motivation and construction of the original quenched Eguchi-Kawai (QEK) model are reviewed, providing much greater detail than in the first, 1982 QEK paper. A 2008 article announced that QEK fails as a reduced model because the average over permutations of eigenvalues stays annealed. It is shown here that the original quenching logic naturally leads to a formulation with no annealed average over permutations.

Herbert Neuberger

Quenched Eguchi-Kawai model revisited

We present the results of an exploratory study of the numerical stochastic perturbation theory (NSPT) applied to the four dimensional twisted Eguchi-Kawai (TEK) model. We employ a Kramers type algorithm based on the Generalized Hybrid Molecular Dynamics (GHMD) algorithm. We have computed the perturbative expansion of square Wilson loops up to O(g 8). The results of the first two coefficients (up to O(g 4)) have a high precision and match well with the exact values. The next two coefficients can be determined and even extrapolated to large N, where they should coincide with the corresponding coefficients for ordinary Yang-Mills theory on an infinite lattice. Our analysis shows the behaviour of the probability distribution for each coefficient tending to Gaussian for larger N. The results allow us to establish the requirements to extend this analysis to much higher order.

Antonio González-Arroyo

Issaku Kanamori

Ken-Ichi Ishikawa

Kanata Miyahana

Masanori Okawa

Ryoichiro Ueno

Numerical stochastic perturbation theory applied to the twisted Eguchi-Kawai model

The double copy is a map from non-abelian gauge theories to gravity, that has been demonstrated both for scattering amplitudes and exact classical solutions. In this study, we reconsider the double copy for exact solutions that are self-dual in either the gauge or gravity theory. In this case, one may formulate a general double copy in terms of a certain differential operator, which generates the gauge and gravity solutions from a harmonic function residing in a biadjoint scalar theory. As an illustration, we examine the single copy of the well-known Eguchi-Hanson instanton in gravity. The gauge field thus obtained represents an abelian-like object whose field is dipole-like at large distances, and which has no magnetic or electric charge.

David Berman

Erick Chacón

Andrés Luna

Chris White

The self-dual classical double copy, and the Eguchi-Hanson instanton

Holographic theories with a local gravitational dual have a number of striking features. Here I argue that many of these features are controlled by the Eguchi-Kawai mechanism, which is proposed to be a hallmark of such holographic theories. Higher-spin holographic duality is presented as a failure of the Eguchi-Kawai mechanism, and its restoration illustrates the deformation of higher-spin theory into a proper string theory with a local gravitational limit. AdS/CFT is used to provide a calculable extension of the Eguchi-Kawai mechanism to field theories on curved manifolds and thereby introduce “topological volume independence.” Finally, I discuss implications for a general understanding of the extensivity of the Bekenstein-Hawking-Wald entropy.

Edgar Shaghoulian

Emergent gravity from Eguchi-Kawai reduction

I first recall the last occasion I met the late Professor Yoichiro Nambu, in a hospital in Osaka. I then present a brief introduction to a moonshine phenomenon in string theory that have recently come under investigation.

Tohru Eguchi

Professor Nambu, string theory, and the moonshine phenomenon