World-sheet Boundary Research Articles

Boundary driven turbulence on string worldsheet

We study the origin of turbulence on the string worldsheet with boundaries laid in anti de Sitter (AdS) spacetime. While the classical motion of a single closed string in AdS is integrable, it has recently been recognized that weak turbulence arises in the case of an open string suspended from the AdS boundary. In the open string case, it is necessary to impose boundary conditions on the worldsheet boundaries. We classify which boundary conditions preserve integrability. Based on this classification, we anticipate that turbulence may occur on the string worldsheet if integrability is not guaranteed by the boundary conditions. Numerical investigations of the classical open-string dynamics support that turbulence occurs when the boundary conditions are not integrable.

Steps towards generalization of tensionless string theory with contact interactions as Wilson loop of non-Abelian Yang–Mills theory

We propose a possible modification to the tensionless string model with contact interactions. The proposed model aims to reproduce the expectation value of a non-Abelian Wilson loop in Yang–Mills theory by integrating out string degrees of freedom with a fixed worldsheet boundary. To reproduce path-ordering along the worldsheet boundary, we introduce Lie algebra-valued fields on the string worldsheet, whose dynamics are determined by the topological BF action. Without bulk contributions, we show that the model describes the non-Abelian Wilson loop, neglecting the effects of self-interactions. Finally, we test the reproduction of the Wilson loop with three-point interaction in the case of SU(2).

Poisson-Lie T-duality defects and target space fusion

Topological defects have long been known to encode symmetries and dualities between physical systems. In the context of string theory, defects have been intensively studied at the level of the worldsheet. Although marked by a number of pioneering milestones, the target space picture of defects is much less understood. In this paper, we show, at the level of the target space, that Poisson-Lie T-duality can be encoded as a topological defect. With this result at hand, we can postulate the kernel capturing the Fourier-Mukai transform associated to the action of Poisson-Lie T-duality on the RR-sector. Topological defects have the remarkable property that they can be fused together or, alternatively, with worldsheet boundary conditions. We study how fusion of the proposed generalised T-duality topological defect consistently leads to the known duality transformations for boundary conditions. Finally, taking a step back from generalised T-duality, we tackle the general problem of understanding the effect of fusion at the level of the target space. We propose to use the framework of Dirac geometry and formulate the fusion of topological defects and D-branes in this language.

Cutkosky rules and unitarity (violation) in D-instanton amplitudes

In perturbative amplitudes in quantum field theory and string field theory, Cutkosky rule expresses the anti-hermitian part of a Feynman diagram in terms of sum over all its cut diagrams, and this in turn is used to prove unitarity of the theory. For D-instanton contribution to a string theory amplitude, the cutting rule needed for the proof of unitarity is somewhat different; we need to sum over only those cut diagrams for which all the world-sheet boundaries ending on some particular D-instanton lie on the same side of the cut. By working with the closed string effective action, obtained after integrating out the open string modes, we prove that the D-instanton amplitudes actually satisfy these cutting rules, provided the effective action is real. The violation of unitarity in the closed string sector of two dimensional string theory can be traced to the failure of this reality condition. In the critical superstring theory, multi-instanton and multi anti-instanton amplitudes satisfy the reality condition. Contribution to the amplitudes from the instanton anti-instanton sector satisfies the reality condition if we make a specific choice of integration cycle over the configuration space of string fields, whereas contribution due to the non-BPS D-instantons will need to either vanish or have an overall real normalization in order for it to give real contribution. We use Picard-Lefschetz theory to argue that these conditions are indeed satisfied in superstring theories.

Fusion for AdS/CFT boundary S-matrices

We propose a fusion formula for AdS/CFT worldsheet boundary S-matrices. We show that, starting from the fundamental Y=0 boundary S-matrix, this formula correctly reproduces the two-particle bound-state boundary S-matrices.

Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group H4

We show that the WZW model on the Heisenberg Lie group H4 has Poisson–Lie symmetry only when the dual Lie group is A2⊕2A1. In this way, we construct the mutual T-dual sigma models on Drinfel'd double generated by the Heisenberg Lie group H4 and its dual pair, A2⊕2A1, as the target space in such a way that the original model is the same as the H4 WZW model. Furthermore, we show that the dual model is conformal up to two-loop order. Finally, we discuss D-branes and the worldsheet boundary conditions defined by a gluing matrix on the H4 WZW model. Using the duality map obtained from the canonical transformation description of the Poisson–Lie T-duality transformations for the gluing matrix which locally defines the properties of the D-brane, we find two different cases of the gluing matrices for the WZW model based on the Heisenberg Lie group H4 and its dual model.

Integrable boundaries in AdS/CFT: revisiting the Z=0 giant graviton and D7-brane

We consider the worldsheet boundary scattering and the corresponding boundary algebras for the Z=0 giant graviton and the Z=0 D7-brane in the AdS/CFT correspondence. We consider two approaches to the boundary scattering, the usual one governed by the (generalized) twisted Yangians and the q-deformed model of these boundaries governed by the quantum affine coideal subalgebras. We show that the q-deformed approach leads to boundary algebras that are of a more compact form than the corresponding twisted Yangians, and thus are favourable to use for explicit calculations. We obtain the q-deformed reflection matrices for both boundaries which in the q->1 limit specialize to the ones obtained using twisted Yangians.

Closed string self-energy on the lightcone worldsheet lattice

We study the one loop correction to the closed bosonic string propagator, including the possible presence of D-branes, by discretizing the lightcone worldsheet on an $M\ifmmode\times\else\texttimes\fi{}N$ rectangular lattice, with $M\ensuremath{\propto}{P}^{+}$ and $N+1\ensuremath{\propto}i{x}^{+}$. The integrals over the moduli then become sums which we evaluate numerically. The main purpose of this study is to assess the reliability of the worldsheet lattice as a regulator of the divergences in string perturbation theory. There are two natural geometrical counterterms for the lightcone worldsheet, one proportional to the area of the worldsheet and the other proportional to the length of worldsheet boundaries, tracing the ends of open strings. We show that the divergences in the closed string self-energy can be cancelled by the area counterterm and a renormalization of the Regge slope parameter. The residual finite part is compatible with Lorentz invariance, provided a novel regularization, natural to the lightcone worldsheet lattice and described in this article, is employed.

Super Poisson–Lie symmetry of the [formula omitted] WZNW model and worldsheet boundary conditions

We show that the WZNW model on the Lie supergroup GL(1|1) has super Poisson–Lie symmetry with the dual Lie supergroup B⊕A⊕A1,1|.i. Then, we discuss about D-branes and worldsheet boundary conditions on supermanifolds, in general, and obtain the algebraic relations on the gluing supermatrix for the Lie supergroup case. Finally, using the supercanonical transformation description of the super Poisson–Lie T-duality transformation, we obtain formulae for the description of the dual gluing supermatrix, then, we find the gluing supermatrix for the WZNW model on GL(1|1) and its dual model. We also discuss about different boundary conditions.

Notes on Euclidean Wilson loops and Riemann theta functions

The AdS/CFT correspondence relates Wilson loops in $\mathcal{N}=4$ super Yang-Mills theory to minimal area surfaces in ${\mathrm{AdS}}_{5}$ space. In this paper we consider the case of Euclidean flat Wilson loops, which are related to minimal area surfaces in Euclidean ${\mathrm{AdS}}_{3}$ space. Using known mathematical results for such minimal area surfaces, we describe an infinite parameter family of analytic solutions for closed Wilson loops. The solutions are given in terms of Riemann theta functions, and the validity of the equations of motion is proven based on the trisecant identity. The world sheet has the topology of a disk, and the renormalized area is written as a finite, one-dimensional contour integral over the world-sheet boundary. An example is discussed in detail with plots of the corresponding surfaces. Further, for each Wilson loops we explicitly construct a one parameter family of deformations that preserve the area. The parameter is the so-called spectral parameter.

Generalized [formula omitted] topological amplitudes and holomorphic anomaly equation

In arXiv:0905.3629 we described a new class of N = 2 topological amplitudes that depends both on vector and hypermultiplet moduli. Here we find that this class is actually a particular case of much more general topological amplitudes which appear at higher loops in heterotic string theory compactified on K 3 × T 2 . We analyze their effective field theory interpretation and derive particular (first order) differential equations as a consequence of supersymmetry Ward identities and the 1/2-BPS nature of the corresponding effective action terms. In string theory the latter get modified due to anomalous world-sheet boundary contributions, generalizing in a non-trivial way the familiar holomorphic and harmonicity anomalies studied in the past. We prove by direct computation that the subclass of topological amplitudes studied in arXiv:0905.3629 forms a closed set under these anomaly equations and that these equations are integrable.

Wilson loop renormalization group flows

The locally BPS Wilson loop and the pure gauge Wilson loop map under AdS/CFT duality to string world-sheet boundaries with standard and alternate quantizations of the world-sheet fields. This implies an RG flow between the two operators, which we verify at weak coupling. Many additional loop operators exist at strong coupling, with a rich pattern of RG flows.

Geometric construction of D-branes in WZW models

The geometric description of D-branes in WZW models is pushed forward. Our starting point is a gluing condition\, $J_{+}=FJ_-$ that matches the model's chiral currents at the worldsheet boundary through a linear map $F$ acting on the WZW Lie algebra. The equivalence of boundary and gluing conditions of this type is studied in detail. The analysis involves a thorough discussion of Frobenius integrability, shows that $F$ must be an isometry, and applies to both metrically degenerate and nondegenerate D-branes. The isometry $F$ need not be a Lie algebra automorphism nor constantly defined over the brane. This approach, when applied to isometries of the form $F=R$ with $R$ a constant Lie algebra automorphism, validates metrically degenerate $R$-twined conjugacy classes as D-branes. It also shows that no D-branes exist in semisimple WZW models for constant\, $F=-R$.

Double field theory for double D-branes

We consider Hull's doubled formalism for open strings on D-branes in flat space and construct the corresponding effective double field theory. We show that the worldsheet boundary conditions of the doubled formalism describe in a unified way a T-dual pair of D-branes, which we call double D-branes. We evaluate the one-loop beta function for the boundary gauge coupling and then obtain the effective field theory for the double D-branes. The effective field theory is described by a DBI action of double fields. The T-duality covariant form of this DBI action is thus a kind of "master" action, which describes all the double D-brane configurations related by T-duality transformations. We discuss a number of aspects of this effective theory.

Noncommutativity in weakly curved background by canonical methods

Using the canonical method, we investigate the Dp-brane world-volume noncommutativity in a weakly curved background. The term ``weakly curved'' means that, in the leading order, the source of nonflatness is an infinitesimally small Kalb-Ramond field ${B}_{\ensuremath{\mu}\ensuremath{\nu}}$, linear in coordinate, while the Ricci tensor does not contribute, being an infinitesimal of the second order. On the solution of boundary conditions, we find a simple expression for the space-time coordinates in terms of the effective coordinates and momenta. This basic relation helped us to prove that noncommutativity appears only on the world sheet boundary. The noncommutativity parameter has a standard form, but with the infinitesimally small and coordinate-dependent antisymmetric tensor ${B}_{\ensuremath{\mu}\ensuremath{\nu}}$. This result coincides with that obtained on the group manifolds in the limit of the large level $n$ of the current algebra. After quantization, the algebra of the functions on the Dp-brane world volume is represented with the Kontsevich star product instead of the Moyal one in the flat background.

On the effective theory of long open strings

We study the general low-energy effective action on long open strings, such as confining strings in pure gauge theories. Using Lorentz invariance, we find that for a string of length R, the leading deviation from the Nambu-Goto energy levels generically occurs at order 1/R^4 (including a correction to the ground state energy), as opposed to 1/R^5 for excited closed strings in four dimensions, and 1/R^7 for closed strings in three dimensions. This is true both for Dirichlet and for Neumann boundary conditions for the transverse directions, though the worldsheet boundary actions are different. The Dirichlet case is relevant (for instance) for the force between external quarks in a confining gauge theory, and the Neumann case for a string stretched between domain walls. In the specific case of confining gauge theories with a weakly curved holographic dual, we compute the coefficient of the leading correction when the open string ends on two D-branes, and find a non-vanishing result.

A Solution for Relevant Deformations in Open String Field Theory

This is my contribution to the proceedings of the SFT2010 conference, held in Kyoto at the YITP in October 2010. We propose a simple classical solution which describes the IR fixed point of a worldsheet boundary relevant deformation within the context of cubic bosonic string field theory. We show that the simplest gauge invariant quantity, the closed string overlap, precisely agrees with the shift in the closed string tadpole between the UV and the IR fixed point of the corresponding worldsheet relevant deformation.

Harmonicity in [formula omitted] supersymmetry and its quantum anomaly

The holomorphicity property of N = 1 superpotentials or of N = 2 F-terms involving vector multiplets is generalized to the case of N = 4 1/2-BPS effective operators defined in harmonic superspace. The resulting harmonicity equations are shown to control the moduli dependence of the couplings of higher-dimensional operators involving powers of the N = 4 Weyl superfield, computed by N = 4 topological amplitudes. These equations can also be derived on the string side, exhibiting an anomaly from world-sheet boundary contributions that leads to recursion relations for the non-analytic part of the couplings.

Classical spinning branes in curved backgrounds

The dynamics of a classical branelike object in a curved background is derived from the covariant stress-energy conservation of the brane matter. The world sheet equations and boundary conditions are obtained in the pole-dipole approximation, where nontrivial brane thickness gives rise to its intrinsic angular momentum. It is shown that intrinsic angular momentum couples to both, the background curvature and the brane orbital degrees of freedom. The whole procedure is manifestly covariant with respect to spacetime diffeomorphisms and world sheet reparametrizations. In addition, two extra gauge symmetries are discovered and utilized. The examples of the point particle and the string in 4 spacetime dimensions are analyzed in more detail. A particular attention is paid to the Nambu-Goto string with massive spinning particles attached to its ends.

Worldsheet boundary conditions in Poisson-Lie T-duality

We apply canonical Poisson-Lie T-duality transformations to bosonic open string worldsheet boundary conditions, showing that the form of these conditions is invariant at the classical level, and therefore they are compatible with Poisson-Lie T-duality. In particular the conditions for conformal invariance are automatically preserved, rendering also the dual model conformal. The boundary conditions are defined in terms of a gluing matrix which encodes the properties of D-branes, and we derive the duality map for this matrix. We demonstrate explicitly the implications of this map for D-branes in two non-Abelian Drinfel'd doubles.