We first derive the boundary theory from the U(1) Chern-Simons theory. The boundary action on an $n$-sheet manifold appears from its back-reaction of the Wilson line. The reason is that the U(1) Chern-Simons theory can provide an exact effective action when introducing the Wilson line. The Wilson line in the pure AdS$_3$ Einstein gravity is equivalent to entanglement entropy in the boundary theory up to classical gravity. The U(1) Chern-Simons theory deviates by a self-interaction term from the gauge formulation on the boundary. We also compare the Hayward term in the SL(2) Chern-Simons formulation to the Wilson line approach. Introducing two wedges can reproduce the entanglement entropy for a single interval at the classical level. We propose quantum generalization by combining the bulk and Hayward terms. The quantum correction of the partition function vanishes. In the end, we calculate the entanglement entropy for a single interval. The pure AdS$_3$ Einstein gravity theory shows a shift of central charge by 26 at the one-loop level. The U(1) Chern-Simons theory does not show a shift from the quantum effect. The result is the same in the weak gravitational constant limit. The non-vanishing quantum correction shows that the Hayward term is incorrect.
A cutoff regularization for a pure Yang-Mills theory is implemented within the background field method keeping explicit the gauge invariance of the effective action. The method has been applied to compute the beta function at one loop order.
We discuss a resummed perturbation theory based on the Wilson renormalization group. In this formulation the Wilsonian flowing couplings, which generalize the running coupling, enter directly into the loop expansion. In the case of an asymptotically free theory the flowing coupling is well defined since the infrared Landau pole is absent. We show this property in the case of the $ϕ^3_6$ theory. We also extend this formulation to the QED theory and we prove that it is consistent with gauge invariance.
We study the supersymmetric Wilson loops in the four-dimensional N=4 super Yang-Mills theory in the context of AdS/CFT correspondence. In the gauge theory side, it is known that the expectation value of the Wilson loops of circular shape with winding number k is calculable by using a Gaussian matrix model. On the other hand, in the gravity side, it has been conjectured that the expectation value of the Wilson loop is given by the classical value of the action for a probe D3-brane with k electric fluxes. Given such correspondence, we pursue the interpretation of the matrix model eigenvalue dens…
We study dual conformal transformations of minimal area surfaces in $AdS_5 \times S^5$ corresponding to holographic smooth Wilson loops and some other related observables. To act with dual conformal transformations we map the string solutions to the dual space by means of T-duality, then we apply a conformal transformation and finally T-dualize back to the original space. The transformation maps between string solutions with different boundary contours. The boundary contours of the minimal surfaces are not mapped back to the AdS boundary, and the regularized area of the surface changes.
We calculate the expectation value of the Wilson line in the pure AdS$_3$ Einstein gravity theory and also the entanglement entropy in the boundary theory. Our one-loop calculation of entanglement entropy shows a shift of the central charge 26. Finally, we show that the Wilson line provides the equivalent description to the boundary entanglement entropy. This equivalence leads to a concrete example of the building of "minimum surface=entanglement entropy".
The construction of gravitational Wilson lines in the Chern-Simons formulation of $AdS_3$ gravity in terms of composite operators in the dual boundary conformal field theory is reviewed. New evidence is presented that the Wilson line, dimensionally regularized and suitably renormalized, behaves as a bi-local operator of two conformal primaries whose dimension is predicted by SL(2,R) current algebra.
We study the Wilson loops and defects in the three dimensional QFT from the D-branes in the AdS(4) x CP**3 geometry. We find out explicit D-brane configurations in the bulk which correspond to both straight and circular Wilson lines extended to the boundary of AdS(4). We analyze critically the role of boundary contributions to the D2-branes with various topology and to the fundamental string actions.
We give a wilsonian formulation of non-abelian gauge theories explicitly consistent with axial gauge Ward identitities. The issues of unitarity and dependence on the quantization direction are carefully investigated. A wilsonian computation of the one-loop QCD beta function is performed.
We describe a non-perturbative quantization of classical Wilson loops in the WZW model. The quantized Wilson loop is an operator acting on the Hilbert space of closed strings and commuting either with the full Kac-Moody chiral algebra or with one of its subalgebras. We prove that under open/closed string duality, it is dual to a boundary perturbation of the open string theory. As an application, we show that such operators are useful tools for identifying fixed points of the boundary renormalization group flow.
We present integral equations for the area of minimal surfaces in AdS_3 ending on generic smooth boundary contours. The equations are derived from the continuum limit of the AMSV result for null polygonal boundary contours. Remarkably these continuum equations admit exact solutions in some special cases. In particular we describe a novel exact solution which interpolates between the circle and 4-cusp solutions.
We use the Wilson renormalization group (RG) formulation to solve the fine-tuning procedure needed in renormalization schemes breaking the gauge symmetry. To illustrate this method we systematically compute the non-invariant couplings of the ultraviolet action of the SU(2) pure Yang-Mills theory at one-loop order.
We write down the Yang-Mills partition function and the average Wilson loop in terms of local gauge-invariant variables being the six components of the metric tensor of dual space. The Wilson loop becomes the trace of the parallel transporter in curved space, else called the gravitational holonomy. We show that the external coordinates mapping the 3d curved space into a flat 6d space play the role of glueball fields, and there is a natural mechanism for the mass gap generation.
Using covariant phase space formulations for the natural topological invariants associated with the world-surface in closed string theory, we find that certain Wilson loops defined on the world-surface and that preserve topological invariance, correspond to wave functionals for the vacuum state with zero energy. The differences and similarities with the 2-dimensional QED proposed by Schwinger early are discussed.
In this paper, we introduce the overlap Dirac operator, which satisfies the Ginsparg-Wilson relation, to the matter sector of two-dimensional N=(2,2) lattice supersymmetric QCD (SQCD) with preserving one of the supercharges. It realizes the exact chiral flavor symmetry on the lattice, to make possible to define the lattice action for general number of the flavors of fundamental and anti-fundamental matter multiplets and for general twisted masses. Furthermore, superpotential terms can be introduced with exact holomorphic or anti-holomorphic structure on the lattice. We also consider the lattic…
Boundary operators are gauge invariant operators whose form factors correspond to boundary contributions of BCFW shifts. In gauge theory, the boundary operators contain infinite series, which are constrained by gauge symmetry. We compute the boundary operators of all possible BCFW shifts in Yang-Mills theory and QCD, and show that the infinite series can be elegantly organized into Wilson lines, which are natural building blocks for non-local gauge invariant operators. We comment on their connection to jet functions and gauge invariant off-shell amplitudes. We also verify our results by studyi…
There is substantial evidence that string theory on AdS_5 x S_5 is a holographic theory in which the number of degrees of freedom scales as the area of the boundary in Planck units. Precisely how the theory can describe bulk physics using only surface degrees of freedom is not well understood. A particularly paradoxical situation involves an event deep in the interior of the bulk space. The event must be recorded in the (Schroedinger Picture) state vector of the boundary theory long before a signal, such as a gravitational wave, can propagate from the event to the boundary. In a previous paper…
We consider the Chern-Simons theory with Wilson lines in 3D and in 1D in the BV-BFV formalism of Cattaneo-Mnev-Reshetikhin. In particular, we allow for Wilson lines to end on the boundary of the space-time manifold. In the toy model of 1D Chern-Simons theory, the quantized BFV boundary action coincides with the Kostant cubic Dirac operator which plays an important role in representation theory. In the case of 3D Chern-Simons theory, the boundary action turns out to be the odd (degree 1) version of the BF model with source terms for the B field at the points where the Wilson lines meet the boun…
We study solutions of Type IIB supergravity, which describe the geometries dual to supersymmetric Wilson lines in N=4 super-Yang-Mills. We show that the solutions are uniquely specified by one function which satisfies a Laplace equation in two dimensions. We show that if this function obeys a certain Dirichlet boundary condition, the corresponding geometry is regular, and we find a simple interpretation of this boundary condition in terms of D3 and D5 branes which are dissolved in the geometry. While all our metrics have AdS_5 x S^5 asymptotics, they generically have nontrivial topologies, whi…
We apply Renormalization Group techniques to the Real Time formulation of thermal field theory. Due to the separation between the $T=0$ and the $T\neq 0$ parts of the propagator in this formalism, one can derive exact evolution equations for the Green functions describing the effect of integrating out thermal fluctuations of increasing wavelengths, the initial conditions being the renormalized Green functions of the $T=0$ theory. As a first application, we study the phase transition for the real scalar theory, computing the order of the transition, the critical temperature, and critical expone…
We discuss formulation of Wilson polygon - MHV amplitude duality at the perturbative level in various regularizations. For four gluons it is shown that at one loop one can formulate diagrammatic correspondence interpolating between the dimensional regularization and the off-shell one. We suggest new interpretation of all types of box diagrams in terms of the dual simplex in dimensional regularization and describe its degeneration to the Wilson polygon. The interesting nullification phenomena for the low-energy amplitudes in the Higgsed phase has been found.
We investigate the effect of finite fermion density on symmetry breaking by Wilson loops in $(1+1)$ dimensions. We find the breaking and restoration of symmetry at finite density in the models with $SU(2)$ and $SU(3)$ gauge symmetries, in the presence of the adjoint fermions. The transition can occur at a finite density of fermions, regardless of the periodic or antiperiodic boundary condition of the fermion field; this is in contrast to the finite-temperature ease examined by Ho and Hosotani (Nucl. Phys. B345 (1990) 445) where the boundary condition of fractional twist is essential to the occ…
We study supersymmetric Wilson loops from a geometrical perspective. To this end, we propose a new formulation of these operators in terms of an integral form associated to the immersion of the loop into a supermanifold. This approach provides a unifying description of Wilson loops preserving different sets of supercharges, and clarifies the flow between them. Moreover, it allows to exploit the powerful techniques of super-differential calculus for investigating their symmetries. As remarkable examples, we discuss supersymmetry and kappa-symmetry invariance.
We consider general supersymmetric Wilson loops in ABJM model, i.e. Chern-Simons-matter theory in 2+1 dimensions with N=6 supersymmetry. They are so-called Zarembo-type: the Wilson loops of our interest have generic contours in spacetime, but the scalar field coupling is arranged accordingly so that there are unbroken supersymmetries. Based on the supermatrix formulation of Wilson loops by Drukker and Trancanelli, we construct explicitly 1/6-BPS Wilson loops and check that their expectation value is protected using perturbation up to two loops. We also study the dual string configuration in Ad…
We study Euclidean Wilson loops at strong coupling using the AdS/CFT correspondence, where the problem is mapped to finding the area of minimal surfaces in Hyperbolic space. We use a formalism introduced recently by Kruczenski to perturbatively compute the area corresponding to boundary contours which are deformations of the circle. Our perturbative expansion is carried to high orders compared with the wavy approximation and yields new analytic results. The regularized area is invariant under a one parameter family of continuous deformations of the boundary contour which are not related to the…
We extend the Wilson renormalization group (RG) formulation to chiral gauge theories and show that local gauge symmetry can be implemented by a suitable choice of the RG flow boundary conditions. Since the space-time dimension is four, there is no ambiguity in handling the matrix $\g_5$ and left and right fermions are not coupled. As a result the ultraviolet action contains all possible globally chiral invariant interactions. Nevertheless, the correct chiral anomaly is reproduced.
Vector spaces of (framed) BPS states of Lagrangian four-dimensional N=2 field theories can be defined in semiclassical chambers in terms of the $L^2$-cohomology of Dirac-like operators on monopole moduli spaces. This was spelled out previously for theories with only vectormultiplets, taking into account only a subset of the possible half-supersymmetric 't Hooft-Wilson line defects. This note completes the discussion by describing the modifications needed when including matter hypermultiplets together with arbitrary 't Hooft-Wilson line defects. Two applications of this extended discussion are …
Within the quantum affine algebra representation theory we construct linear covariant operators that generate the Askey-Wilson algebra. It has the property of a coideal subalgebra, which can be interpreted as the boundary symmetry algebra of a model with quantum affine symmetry in the bulk. The generators of the Askey-Wilson algebra are implemented to construct an operator valued $K$- matrix, a solution of a spectral dependent reflection equation. We consider the open driven diffusive system where the Askey-Wilson algebra arises as a boundary symmetry and can be used for an exact solution of t…
We consider an open Wilson line as a momentum representation of a boundary state which describes a D-brane in a constant B-field background. Using this picture, we study the Seiberg-Witten map which relates the commutative and noncommutative gauge fields, and determine the products of fields appearing in the general terms in the expansion of this map.
A five-dimensional Chern-Simons gravity theory based on the anti-de Sitter group SO(4,2) is argued to be a useful model in which to understand the details of holography and the relationship between generally covariant and dual local quantum field theories. Defined on a manifold with boundary, conformal geometry arises naturally as a gauge invariance preserving boundary condition. By matching thermodynamic quantities for a particular background geometry, the dimensionless coupling constant of the Chern-Simons theory is directly related to the number of fields in a putative dual theory at high t…
It is shown that the anomaly inflow mechanism can be implemented using Wilson line in odd dimensional gauge theories. An action of Wess-Zumino-Witten (WZW) type can be constructed using Wilson line. The action is understood in the odd dimensional bulk space-time rather than in the even dimensional boundary. This action is not gauge invariant. It gives anomalous gauge variations of the consistent form on boundary space-times. So it can be used to cancel the quantum anomalies localized on boundary space-times. This offers a new way to cancel the gauge anomaly and construct anomaly-free gauge the…
We consider the expectation value of Wilson lines in two defect versions of N = 4 SYM, both with supersymmetry completely broken, where one is described in terms of an integrable boundary state, the other one not. For both cases, imposing a certain double scaling limit, we find agreement to two leading orders between the expectation values calculated from respectively the field theory and the string theory side of the AdS/dCFT correspondence.
The critical exponent $η$ is not well accounted for in the Polchinski exact formulation of the renormalization group (RG). With a particular emphasis laid on the introduction of the critical exponent $η$, I re-establish (after Golner, hep-th/9801124) the explicit relation between the early Wilson exact RG equation, constructed with the incomplete integration as cutoff procedure, and the formulation with an arbitrary cutoff function proposed later on by Polchinski. I (re)-do the analysis of the Wilson-Polchinski equation expanded up to the next to leading order of the derivative expansion. I fi…
We formulate a Ginsparg-Wilson relation on a fuzzy 2-sphere for matter in the adjoint representation of the gauge group. Because of the Ginsparg-Wilson relation, an index theorem is satisfied. Our formulation is applicable to topologically nontrivial configurations as monopoles. It gives a solid basis for obtaining chiral fermions, which are an important ingredient of the standard model, from matrix model formulations of the superstring theory, such as the IIB matrix model, by considering topological configurations in the extra dimensions. We finally discuss whether this mechanism really works…
We study the dual gravity description of supersymmetric Wilson loops whose expectation value is unity. They are described by calibrated surfaces that end on the boundary of anti de-Sitter space and are pseudo-holomorphic with respect to an almost complex structure on an eight-dimensional slice of AdS_5 x S^5. The regularized area of these surfaces vanishes, in agreement with field theory non-renormalization theorems for the corresponding operators.
We consider rigid rotating closed strings with spikes in AdS5 dual to certain higher twist operators in N=4 SYM theory. In the limit of large spin when the spikes reach the boundary of AdS5, the solutions with different numbers of spikes are related by conformal transformations, implying that their energy is determined by the same function of the `t Hooft coupling f(lambda) that appears in the anomalous dimension of twist 2 operators or in the cusp anomaly. In the limit when the number of spikes goes to infinity, we find an equivalent description in terms of a string moving in an AdS pp-wave b…
The AdS/CFT correspondence relates the expectation value of Wilson loops in N=4 SYM to the area of minimal surfaces in AdS_5 In this paper we consider minimal area surfaces in generic Euclidean AdS_{n+1} using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS_3. As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be ex…
We consider different supersymmetric mixed boundary conditions for scalar and fermionic fields in $AdS_2$, searching for the dual description of a family of interpolating Wilson Loops in ABJM theory. The family, which interpolates between the bosonic 1/6 BPS loop and the 1/2 BPS loop, can be thought of as an exact marginal deformation in a defect CFT$_1$. Confronting this property against holographic correlators and vacuum energy corrections singles out a particular boundary condition which we propose as dual to the interpolating family of Wilson loops.
We provide a refined interpretation of a gravitational Wilson line in AdS$_3$ in terms of Ishibashi states in the dual CFT$_2$. Our strategy is to give a method to evaluate the Wilson line that accounts for all the information contained in the representation, and clarify the role of boundary conditions at the endpoints of the line operator. This gives a novel way to explore and reconstruct the local bulk dynamics which we discuss. We also compare our findings with other interpretations of Ishibashi states in AdS$_3$/CFT$_2$.
The purpose of the present thesis is the implementation of symmetries in the Wilsonian Exact Renormalization Group (ERG) approach. After recalling how the ERG can be introduced in a general theory (i.e. containing both bosons and fermions, scalars and vectors) and having applied it to the massless scalar theory as an example of how the method works, we discuss the formulation of the Quantum Action Principle (QAP) in the ERG and show that the Slavnov-Taylor identities can be directly derived for the cutoff effective action at any momentum scale. Firstly the QAP is exploited to analyse the break…
We provide a holographic perspective on correlation functions in Schwarzian quantum mechanics, as boundary-anchored Wilson line correlators in Jackiw-Teitelboim gravity. We first study compact groups and identify the diagrammatic representation of bilocal correlators of the particle-on-a-group model as Wilson line correlators in its 2d holographic BF description. We generalize to the Hamiltonian reduction of SL(2,R) and derive the Schwarzian correlation functions. Out-of-time ordered correlators are determined by crossing Wilson lines, giving a 6j-symbol, in agreement with 2d CFT results.
Recent progresses in the computation of quantum string corrections to holographic Wilson loops are extended to the case of strings in $AdS_{4}\times CP^{3}$. For this, the ratio of 1/2-BPS circular and 1/6-BPS latitude fermionic Wilson loops in ABJM is considered at strong coupling by studying the corresponding semiclassical string partition functions. Explicit evaluation of fluctuation determinants using phaseshifts and diffeomorphism invariant regulators leads to exact matching with the recent field theory proposal. Key to this computation is the choice of boundary conditions for massless fe…
We use entanglement entropy to define a central charge associated to a two-dimensional defect or boundary in a conformal field theory (CFT). We present holographic calculations of this central charge for several maximally supersymmetric CFTs dual to eleven-dimensional supergravity in Anti-de Sitter space, namely the M5-brane theory with a Wilson surface defect and three-dimensional CFTs related to the M2-brane theory with a boundary. Our results for the central charge depend on a partition of the number of M2-branes, $N$, ending on the number of M5-branes, $M$. For the Wilson surface, the part…
Using the AdS/CFT correspondence we study UV behavior of Wilson loops in various noncommutative gauge theories. We get an area law in most cases and try to identify its origin. In D3 case, we may identify the the origin as the D1 dominance over the D3: as we go to the boundary of the AdS space, the effect of the flux of the D3 charge is highly suppressed, while the flux due to the D1 charge is enhenced. So near the boundary the theory is more like a theory on D1 brane than that on D3 brane. This phenomena is closely related to the dimensional reduction due to the strong magnetic field in the c…
JT gravity has a first-order formulation as a two-dimensional BF theory, which can be viewed as the dimensional reduction of the Chern-Simons description of $3d$ gravity. We consider $T\bar{T}$-type deformations of the $(0+1)$-dimensional dual to this $2d$ BF theory and interpret the deformation as a modification of the BF theory boundary conditions. The fundamental observables in this deformed BF theory, and in its $3d$ Chern-Simons lift, are Wilson lines and loops. In the $3d$ Chern-Simons setting, we study modifications to correlators involving boundary-anchored Wilson lines which are induc…
Correlators of local operators inserted on a straight Wilson loop in a conformal gauge theory have the structure of a one-dimensional "defect" CFT. As was shown in arXiv:1706.00756, in the case of supersymmetric Wilson-Maldacena loop in $\mathcal{N}=4$ SYM one can compute the strong-coupling contributions to 4-point correlators of operator insertions by starting with the AdS$_5 \times S^5$ string action expanded near the AdS$_2$ minimal surface and evaluating the corresponding AdS$_2$ Witten diagrams. We perform the analogous computations in the non-supersymmetric case of the standard Wilson l…
We study N=1 superconformal theory in the context of AdS/CFT correspondence in the large central charge limit using Chern-Simons formulation of $3d$ gravity. In this limit conformal dimensions of a subclass of so-called light primary superfields remain finite and are governed by $\mathfrak{osp}(1|2)$ subalgebra of N=1 super-Virasoro algebra. We describe the construction of $\mathfrak{osp}(1|2)$ conformal blocks in terms of Wilson lines of the Chern-Simons $3d$ gravity. We consider examples of two and three-point blocks on the sphere and one-point torus blocks of light superprimary fields, whic…
A major problem with periodic boundary condition on the gauge fields used in current lattice gauge theory simulations is the trapping of topological charge in a particular sector as the continuum limit is approached. To overcome this problem open boundary condition in the temporal direction has been proposed recently. One may ask whether open boundary condition can reproduce the observables calculated with periodic boundary condition. In this work we find that the extracted lowest glueball mass using open and periodic boundary conditions at the same lattice volume and lattice spacing agree for…
One of the methods to calculate tree-level multi-gluon scattering amplitudes is to use the Berends-Giele recursion relation involving off-shell currents or off-shell amplitudes, if working in the light cone gauge. As shown in recent works using the light-front perturbation theory, solutions to these recursions naturally collapse into gauge invariant and gauge-dependent components, at least for some helicity configurations. In this work, we show that such structure is helicity independent and emerges from analytic properties of matrix elements of Wilson line operators, where the slope of the st…
A Path to Quantum Simulations of Topological Phases: (2+1)D Quantum Electrodynamics with Wilson Fermions
Mani L. Bhaumik Institute for Theoretical Physics, Department of Physics and Astronomy, University of California, Los Angeles（加州大学洛杉矶分校物理学与天文学系马尼·L·巴乌米克理论物理研究所）；University of Bonn（波恩大学）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器中心）；Massachusetts Institute of Technology（麻省理工学院）；The Cyprus Institute（塞浦路斯研究所）；Tsinghua University（清华大学）
By explicit construction, we show that one can in a simple way introduce and measure gravitational holonomies and Wilson loops in lattice formulations of nonperturbative quantum gravity based on (Causal) Dynamical Triangulations. We use this set-up to investigate a class of Wilson line observables associated with the world line of a point particle coupled to quantum gravity, and deduce from their expectation values that the underlying holonomies cover the group manifold of SO(4) uniformly
In the Randall-Sundrum compactification of AdS$_5$ with detuned brane tensions, supersymmetry can be spontaneously broken by a non-trivial Wilson line for the graviphoton. The supersymmetry breaking vanishes in the tuned limit. This effect is equivalent to supersymmetry breaking by Scherk-Schwarz boundary conditions.
The problem of unphysical zero modes in lattice QCD with Wilson fermions can be solved in a clean way by including a mass term proportional to $i \psibar γ_5 τ^3 ψ$ in the standard lattice theory with Nf=2 mass degenerate Wilson quarks. We argue that up to cutoff effects, this lattice theory is equivalent to standard lattice QCD, for suitable choices of the mass parameters and with a natural re-interpretation of observables. On-shell O(a) improvement can be implemented in a straightforward way.
