This is an extended version of our short report [1], where a holographic interpretation of entanglement entropy in conformal field theories is proposed from AdS/CFT correspondence. In addition to a concise review of relevant recent progresses of entanglement entropy and details omitted in the earlier letter, this paper includes the following several new results: We give a more direct derivation of our claim which relates the entanglement entropy with the minimal area surfaces in the AdS3/CFT2 case as well as some further discussions on higher dimensional cases. Also the relation between the entanglement entropy and central charges in 4D conformal field theories is examined. We check that the logarithmic part of the 4D entanglement entropy computed in the CFT side agrees with the AdS5 result at least under a specific condition. Finally we estimate the entanglement entropy of massive theories in generic dimensions by making use of our proposal.

https://iopscience.iop.org/article/10.1088/1126-6708/2006/08/045/pdf

Aspects of Holographic Entanglement Entropy

Superstring theoryをめぐって(放談室)

We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the ${\mathcal N=4}$ supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure ${\mathcal N=2}$ and the ${\mathcal N=2^*}$ supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional ${\mathcal N=2}$ superconformal gauge theory is treated similarly.

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

We present a uniform treatment of rigid supersymmetric field theories in a curved spacetime $ \mathcal{M} $, focusing on four-dimensional theories with four supercharges. Our discussion is significantly simpler than earlier treatments, because we use classical background values of the auxiliary fields in the supergravity multiplet. We demonstrate our procedure using several examples. For $ \mathcal{M} = Ad{S_4} $ we reproduce the known results in the literature. A supersymmetric Lagrangian for $ \mathcal{M} = {\mathbb{S}^4} $ exists, but unless the field theory is conformal, it is not reflection positive. We derive the Lagrangian for $ \mathcal{M} = {\mathbb{S}^3} \times \mathbb{R} $ and note that the time direction $ \mathbb{R} $ can be rotated to Euclidean signature and be compactified to $ {\mathbb{S}^1} $ only when the theory has a continuous R-symmetry. The partition function on $ \mathcal{M} = {\mathbb{S}^3} \times {\mathbb{S}^1} $ is independent of the parameters of the flat space theory and depends holomorphically on some complex background gauge fields. We also consider R-invariant $ \mathcal{N} = 2 $ theories on $ {\mathbb{S}^3} $ and clarify a few points about them.

Rigid Supersymmetric Theories in Curved Superspace

We explore the Mellin representation of conformal correlation functions recently proposed by Mack. Examples in the AdS/CFT context reinforce the analogy between Mellin amplitudes and scattering amplitudes. We conjecture a simple formula relating the bulk scattering amplitudes to the asymptotic behavior of Mellin amplitudes and show that previous results on the flat space limit of AdS follow from our new formula. We find that the Mellin amplitudes are particularly useful in the case of conformal gauge theories in the planar limit. In this case, the four point Mellin amplitudes are meromorphic functions whose poles and their residues are entirely determined by two and three point functions of single-trace operators. This makes the Mellin amplitudes the ideal objects to attempt the conformal bootstrap program in higher dimensions.

Writing CFT correlation functions as AdS scattering amplitudes

We propose a correspondence between loop operators in a family of four dimensional $$ \mathcal{N} $$
 = 2 gauge theories on S
 4 — including Wilson, ‘t Hooft and dyonic operators — and Liouville theory loop operators on a Riemann surface. This extends the beautiful relation between the partition function of these $$ \mathcal{N} $$
 = 2 gauge theories and Liouville correlators found by Alday, Gaiotto and Tachikawa. We show that the computation of these Liouville correlators with the insertion of a Liouville loop operator reproduces Pestun’s formula capturing the expectation value of a Wilson loop operator in the corresponding gauge theory. We prove that our definition of Liouville loop operators is invariant under modular transformations, which given our correspondence, implies the conjectured action of S-duality on the gauge theory loop operators. Our computations in Liouville theory make an explicit prediction for the exact expectation value of ’t Hooft and dyonic loop operators in these $$ \mathcal{N} $$
 = 2 gauge theories. The Liouville loop operators are also found to admit a simple geometric interpretation within quantum Teichmuller theory as the quantum operators representing the length of geodesics. We study the algebra of Liouville loop operators and show that it gives evidence for our proposal as well as providing definite predictions for the operator product expansion of loop operators in gauge theory.

/pdf/gauge-theory-loop-operators-and-liouville-theory-2p6a5p853y.pdf

Gauge theory loop operators and Liouville theory

We study Wilson-'t Hooft loop operators in a class of = 2 superconformal field theories recently introduced by Gaiotto. In the case that the gauge group is a product of SU(2) groups, we classify all possible loop operators in terms of their electric and magnetic charges subject to the Dirac quantization condition. We then show that this precisely matches Dehn's classification of homotopy classes of non-self-intersecting curves on an associated Riemann surface — the same surface which characterizes the gauge theory. Our analysis provides an explicit prediction for the action of S-duality on loop operators in these theories which we check against the known duality transformation in several examples.

http://iopscience.iop.org/article/10.1088/1126-6708/2009/09/031/pdf

Loop operators and S-duality from curves on Riemann surfaces

The path integral of a general $\mathcal{N} = 2$
 supersymmetric gauge theory on S
 4 is exactly evaluated in the presence of a supersymmetric ’t Hooft loop operator. The result we find — obtained using localization techniques — captures all perturbative quantum corrections as well as non-perturbative effects due to instantons and monopoles, which are supported at the north pole, south pole and equator of S
 4. As a by-product, our gauge theory calculations successfully confirm the predictions made for ’t Hooft loops obtained from the calculation of topological defect correlators in Liouville/Toda conformal field theory.

/pdf/exact-results-for-t-hooft-loops-in-gauge-theories-on-s-4-1ufdu80b7h.pdf

Exact Results for 't Hooft Loops in Gauge Theories on S 4

Coupling of rolling tachyon to closed strings

We show that the physical N=4 super Yang-Mills theory on a four-sphere with an arbitrary gauge group receives no instanton contributions, by clarifying the relation between the hypermultiplet mass and the equivariant parameters of the mass-deformed theory preserving N=2 supersymmetry. The correct relation also implies that N=4 superconformal Yang-Mills theory with gauge group SU(2) corresponds to Liouville theory on a torus with the insertion of a non-trivial operator, rather than the identity as have been claimed in the literature.

Takuya Okuda

Papers

Gauge theory loop operators and Liouville theory

Loop operators and S-duality from curves on Riemann surfaces

Exact Results for 't Hooft Loops in Gauge Theories on S 4

Coupling of rolling tachyon to closed strings

On the instantons and the hypermultiplet mass of N=2* super Yang-Mills on S^4