Worldsheet Theory Research Articles

Integration over families of Lagrangian submanifolds in BV formalism

Gauge fixing is interpreted in BV formalism as a choice of Lagrangian submanifold in an odd symplectic manifold (the BV phase space). A natural construction defines an integration procedure on families of Lagrangian submanifolds. In string perturbation theory, the moduli space integrals of higher genus amplitudes can be interpreted in this way. We discuss the role of gauge symmetries in this construction. We derive the conditions which should be imposed on gauge symmetries for the consistency of our integration procedure. We explain how these conditions behave under the deformations of the worldsheet theory. In particular, we show that integrated vertex operator is actually an inhomogeneous differential form on the space of Lagrangian submanifolds.

Non-Abelian strings in N = 1 supersymmetric QCD

Non-Abelian flux tubes (strings) are well studied in N = 2 supersymmetric QCD in (3+1) dimensions. In addition to translational zero modes they have also orientational moduli associated with rotations of their fluxes inside a non- Abelian group. The dynamics of the orientational moduli is described by the two dimensional CP(N - 1) model living on the world sheet of the non-Abelian string. In this paper we consider a deformation of N = 2 supersymmetric QCD with the U(N) gauge group and Nf = N quark flavors with a mass term μ of the adjoint matter. In the limit of large μ the theory flows to an N = 1 supersymmetric QCD. We study the solution for the non-Abelian string in this limit and derive an effective theory on the string world sheet. The bosonic sector of this theory is still given by the CP(N -1) model. We study also the fermionic sector of the world sheet theory. Upon the deformation the non-Abelian string is no longer BPS and we show that the fermionic superorientational zero modes are all lifted. This leaves us with the pure bosonic CP(N-1) model on the string world sheet in the limit of N = 1 QCD. We discuss what happens to confined monopoles at large μ.

Mobility edge and Black Hole Horizon

We conjecture that the mobility edge in the 4D Euclidean Dirac operator spectrum in QCD in the deconfined phase found in the lattice studies corresponds to the near black hole (BH) horizon region in the holographic dual. We present some evidences both from the field theory side and from the worldsheet theory of long open string.

Amplitudes on plane waves from ambitwistor strings

In marked contrast to conventional string theory, ambitwistor strings remain solvable worldsheet theories when coupled to curved background fields. We use this fact to consider the quantization of ambitwistor strings on plane wave metric and plane wave gauge field backgrounds. In each case, the worldsheet model is anomaly free as a consequence of the background satisfying the field equations. We derive vertex operators (in both fixed and descended picture numbers) for gravitons and gluons on these backgrounds from the worldsheet CFT, and study the 3-point functions of these vertex operators on the Riemann sphere. These worldsheet correlation functions reproduce the known results for 3-point scattering amplitudes of gravitons and gluons in gravitational and gauge theoretic plane wave backgrounds, respectively.

A superstring field theory for supergravity

A covariant closed superstring field theory, equivalent to classical tendimensional Type II supergravity, is presented. The defining conformal field theory is the ambitwistor string worldsheet theory of Mason and Skinner. This theory is known to reproduce the scattering amplitudes of Cachazo, He and Yuan in which the scattering equations play an important role and the string field theory naturally incorporates these results. We investigate the operator formalism description of the ambitwsitor string and propose an action for the string field theory of the bosonic and supersymmetric theories. The correct linearised gauge symmetries and spacetime actions are explicitly reproduced and evidence is given that the action is correct to all orders. The focus is on the NeveuSchwarz sector and the explicit description of tree level perturbation theory about flat spacetime. Application of the string field theory to general supergravity backgrounds and the inclusion of the Ramond sector are briefly discussed.

Asymptotic fragility, near AdS2 holography and Toverline{T}

We present the exact solution for the scattering problem in the flat space Jackiw-Teitelboim (JT) gravity coupled to an arbitrary quantum field theory. JT gravity results in a gravitational dressing of field theoretical scattering amplitudes. The exact expression for the dressed S-matrix was previously known as a solvable example of a novel UV asymptotic behavior, dubbed asymptotic fragility. This dressing is equivalent to the Toverline{T} deformation of the initial quantum field theory. JT gravity coupled to a single mass-less boson provides a promising action formulation for an integrable approximation to the worldsheet theory of confining strings in 3D gluodynamics. We also derive the dressed S-matrix as a flat space limit of the near AdS2 holography. We show that in order to preserve the flat space unitarity the conventional Schwarzian dressing of boundary correlators needs to be slightly extended. Finally, we propose a new simple expression for flat space amplitudes of massive particles in terms of correlators of holographic CFT’s.

On exceptional instanton strings

According to a recent classification of 6d (1, 0) theories within F-theory there are only six “pure” 6d gauge theories which have a UV superconformal fixed point. The corresponding gauge groups are SU(3), SO(8), F4, E6, E7, and E8. These exceptional models have BPS strings which are also instantons for the corresponding gauge groups. For G simply-laced, we determine the 2d mathcal{N}=left(0,4right) worldsheet theories of such BPS instanton strings by a simple geometric engineering argument. These are given by a twisted S2 compactification of the 4d mathcal{N}=2 theories of type H2, D4, E6, E7 and E8 (and their higher rank generalizations), where the 6d instanton number is mapped to the rank of the corresponding 4d SCFT. This determines their anomaly polynomials and, via topological strings, establishes an interesting relation among the corresponding T2× S2 partition functions and the Hilbert series for moduli spaces of G instantons. Such relations allow to bootstrap the corresponding elliptic genera by modularity. As an example of such procedure, the elliptic genera for a single instanton string are determined. The same method also fixes the elliptic genus for case of one F4 instanton. These results unveil a rather surprising relation with the Schur index of the corresponding 4d mathcal{N}=2 models.

A holographic dual for string theory on AdS3\xd7S3\xd7S3\xd7S1

The CFT dual of string theory on AdS3 × S3 × S3 × S1 is conjectured to be the symmetric orbifold of the {mathcal{S}}_{kappa } theory, provided that one of the two Q5± quantum numbers is a multiple of the other. We determine the BPS spectrum of the symmetric orbifold in detail, and show that it reproduces precisely the BPS spectrum that was recently calculated in supergravity. We also determine the BPS spectrum of the world-sheet theory that is formulated in terms of WZW models, and show that, apart from some gaps (which are reminiscent of those that appear in the corresponding {mathbb{T}}^4 calculation), it also reproduces this BPS spectrum. In fact, the matching seems to work as well as for the familiar {mathbb{T}}^4 case, and thus our results give strong support for this proposal.

Giant magnons of string theory in the lambda background

The analogues of giant magnon configurations are studied on the string world sheet in the lambda background. This is a discrete deformation of the AdS5×S5 background that preserves the integrability of the world sheet theory. Giant magnon solutions are generated using the dressing method and their dispersion relation is found. This reduces to the usual dyonic giant magnon dispersion relation in the appropriate limit and becomes relativistic in another limit where the lambda model becomes the generalized sine-Gordon theory of the Pohlmeyer reduction. The scattering of giant magnons is then shown in the semi-classical limit to be described by the quantum S-matrix that is a quantum group deformation of the conventional giant magnon S-matrix. It is further shown that in the small g limit, a sector of the S-matrix is related to the XXZ spin chain whose spectrum matches the spectrum of magnon bound states.

Quantum inverse scattering and the lambda deformed principal chiral model

The lambda model is a one parameter deformation of the principal chiral model that arises when regularizing the non-compactness of a non-abelian T dual in string theory. It is a current–current deformation of a WZW model that is known to be integrable at the classical and quantum level. The standard techniques of the quantum inverse scattering method cannot be applied because the Poisson bracket is non ultra-local. Inspired by an approach of Faddeev and Reshetikhin, we show that in this class of models, there is a way to deform the symplectic structure of the theory leading to a much simpler theory that is ultra-local and can be quantized on the lattice whilst preserving integrability. This lattice theory takes the form of a generalized spin chain that can be solved by standard algebraic Bethe Ansatz techniques. We then argue that the IR limit of the lattice theory lies in the universality class of the lambda model implying that the spin chain provides a way to apply the quantum inverse scattering method to this non ultra-local theory. This points to a way of applying the same ideas to other lambda models and potentially the string world-sheet theory in the gauge-gravity correspondence.

Non-Abelian strings in N=1 supersymmetric QCD

Non-Abelian flux tubes (strings) are well studied in N=2 supersymmetric QCD in (3+1) dimensions. In addition to translational zero modes they have also orientational moduli associated with rotations of their fluxes inside a non-Abelian group. The dynamics of the orientational moduli is described by the two dimensional CP(N-1) model living on the world sheet of the non-Abelian string. In this paper we consider a deformation of N=2 supersymmetric QCD with the U(N) gauge group and N_f=N quark flavors with a mass term $\mu$ of the adjoint matter. In the limit of large $\mu$ the theory flows to an N=1 supersymmetric QCD. We study the solution for the non-Abelian string in this limit and derive an effective theory on the string world sheet. The bosonic sector of this theory is still given by the CP(N-1) model but its scale is exponentially small as compared to the scale of the four dimensional bulk theory in contrast to the N=2 case where these scales are equal. We study also the fermionic sector of the world sheet theory. Upon the deformation the non-Abelian string is no longer BPS and we show that the fermionic superorientational zero modes are all lifted. This leaves us with the pure bosonic CP(N-1) model on the string world sheet in the limit of N=1 QCD. We also discuss what happens to confined monopoles at large $\mu$.

On the mass of the world-sheet ‘axion’ in SU(N) gauge theories in 3 + 1 dimensions

There is numerical evidence that the world sheet action of the confining flux tube in D=3+1SU(N) gauge theories contains a massive excitation with 0− quantum numbers whose mass shows some decrease as one goes from SU(3) to SU(5). Moreover it has been shown that the natural coupling of this pseudoscalar has a topological interpretation making it natural to call it the world-sheet ‘axion’. Recently it has been pointed out that if the mass of this ‘axion’ vanishes as N→∞ then it becomes possible for the world sheet theory to be integrable in the planar limit. In this paper we perform lattice calculations of this ‘axion’ mass from SU(2) to SU(12), which allows us to make a controlled extrapolation to N=∞ and so test this interesting possibility. We find that the ‘axion’ does not in fact become massless as N→∞. So if the theory is to possess planar integrability then it must be some other world sheet excitation that becomes massless in the planar limit.

Low-energy effective worldsheet theory of a non-Abelian vortex in high-density QCD revisited: A regular gauge construction

Color symmetry is spontaneously broken in quark matter at high density as a consequence of di-quark condensations with exhibiting color superconductivity. Non-Abelian vortices or color magnetic flux tubes stably exist in the color-flavor locked phase at asymptotically high-density. The effective worldsheet theory of a single non-Abelian vortex was previously calculated in the singular gauge to obtain the ${\mathbb C}P^2$ model [ arXiv:0908.4470 , arXiv:0912.1352 ]. Here, we reconstruct the effective theory in a regular gauge without taking a singular gauge, confirming the previous results in the singular gauge. As a byproduct of our analysis, we find that non-Abelian vortices in high-density QCD do not suffer from any obstruction for the global definition of a symmetry breaking.

Protected string spectrum in AdS3/CFT2 from worldsheet integrability

We derive the protected closed-string spectra of AdS3/CFT2 dual pairs with 16 supercharges at arbitrary values of the string tension and of the three-form fluxes. These follow immediately from the all-loop Bethe equations for the spectra of the integrable worldsheet theories. Further, representing the underlying integrable systems as spin chains, we find that their dynamics involves length-changing interactions and that protected states correspond to gapless excitations above the Berenstein-Maldacena-Nastase vacuum. In the case of AdS3 × S3 × T4 the degeneracies of such operators precisely match those of the dual CFT2 and the supergravity spectrum. On the other hand, we find that for AdS3 × S3 × S3 × S1 there are fewer protected states than previous supergravity calculations had suggested. In particular, protected states have the same su(2) charge with respect to the two three-spheres.

Lie algebra lattices and strings on T-folds

We study the world-sheet conformal field theories for T-folds systematically based on the Lie algebra lattices representing the momenta of strings. The fixed point condition required for the T-duality twist restricts the possible Lie algebras. When the T-duality acts as a simple chiral reflection, one is left with the four cases, A1, D2r, E7, E8, among the simple simply-laced algebras. From the corresponding Englert-Neveu lattices, we construct the modular invariant partition functions for the T-fold CFTs in bosonic string theory. Similar construction is possible also by using Euclidean even self-dual lattices. We then apply our formulation to the T-folds in the E8 × E8 heterotic string theory. Incorporating non-trivial phases for the T-duality twist, we obtain, as simple examples, a class of modular invariant partition functions parametrized by three integers. Our construction includes the cases which are not reduced to the free fermion construction.

Multiloop amplitudes of light-cone gauge NSR string field theory in noncritical dimensions

Feynman amplitudes of light-cone gauge superstring field theory are ill-defined because of various divergences. In a previous paper, one of the authors showed that taking the worldsheet theory to be the one in a linear dilaton background Φ = −iQX1 with Feynman iε (ε > 0) and Q2> 10 yields finite amplitudes. In this paper, we apply this worldsheet theory to dimensional regularization of the light-cone gauge NSR superstring field theory. We concentrate on the amplitudes for even spin structure with external lines in the (NS,NS) sector. We show that the multiloop amplitudes are indeed regularized in our scheme and that they coincide with the results in the first-quantized formalism through the analytic continuation Q → 0.

Resurgence of the dressing phase for AdS5 \xd7 S5

We discuss the resummation of the strong coupling asymptotic expansion of the dressing phase of the AdS5 × S5 superstring. The dressing phase proposed by Beisert, Eden and Staudacher can be recovered from a modified Borel-Ecalle resummation of this asymptotic expansion only by completing it with new, non-perturbative and exponentially suppressed terms that can be organized into different sectors labelled by an instanton-like number. We compute the contribution to the dressing phase coming from the sum over all the instanton sectors and show that it satisfies the homogeneous crossing symmetry equation. We comment on the semiclassical origin of the non-perturbative terms from the world-sheet theory point of view even though their precise explanation remains still quite mysterious.

S-matrices and quantum group symmetry of k-deformed sigma models

Recently, two kinds of integrable deformations of the string world sheet theory in the gauge/gravity correspondence have been constructed (Delduc et al 2014 Phys. Rev. Lett. 112 051601; Hollowood et al 2014 J. Phys. A: Math. Theor. 47 495402). One class of these, the k deformations associated to the more general q deformations but with a root of unity, has been shown to be related to a particular discrete deformation of the principal chiral models and (semi-)symmetric space sigma models involving a gauged WZW model. We conjecture a form for the exact S-matrices of the bosonic integrable field theories of this type. The S-matrices imply that the theories have a hidden infinite dimensional affine quantum group symmetry. We provide some evidence, via quantum inverse scattering techniques, that the theories do indeed possess the finite-dimensional part of this quantum group symmetry.

Double genus expansion for general Ω background

We will show how the refined holomorphic anomaly equation obeyed by the Nekrasov partition function at generic [Formula: see text], [Formula: see text] values becomes compatible, in a certain two-parameter expansion, with the assumption that both parameters are associated to genus counting. The underlying worldsheet theory will be analyzed and constrained in various ways, and we will provide both physical interpretation and some alternative evidence for this model. Finally, we will use the Gopakumar–Vafa formulation for the refined topological string in order to give a more quantitative description.

Heterotic non-Abelian string of a finite length

We consider non-Abelian strings in N=2 supersymmetric QCD with the U$(N)$ gauge group and $N_f=N$ quark flavors deformed by a mass term for the adjoint matter. This deformation breaks N=2 supersymmetry down to N=1. Dynamics of orientational zero modes on the string world sheet are described then by CP$(N-1)$ model with N=(0,2) supersymmetry. We study the string of a finite length $L$ assuming compactification on a cylinder (periodic boundary conditions). The world-sheet theory is solved in the large-$N$ approximation. We find a rich phase structure in the $(L, \,u)$ plane where $u$ is a deformation parameter. At large $L$ and intermediate $u$ we find a phase with broken $Z_{2N}$ symmetry, $N$ vacua and a mass gap. At large values of $L$ and $u$ still larger we have the $Z_{2N}$-symmetric phase with a single vacuum and massless fermions. In both phases N=(0,2) supersymmetry is spontaneously broken. We also observe a phase with broken SU$(N)\;$ symmetry at small $L$. In the latter phase the mass gap vanishes and the vacuum energy is zero in the leading $1/N$ approximation. However, we expect that $1/N$ corrections will break N=(0,2) supersymmetry. We also discuss how this rich phase structure matches the N=(2,2) limit in which the world-sheet theory has a single phase with the mass gap independent of $L$.