Worldsheet Theory Research Articles

Chern-Simons theory, decomposition, and the A model

In this paper, we discuss how gauging one-form symmetries in Chern-Simons theories is implemented in an A-twisted topological open string theory. For example, the contribution from a fixed H/Z bundle on a three-manifold M, arising in a BZ gauging of H Chern-Simons, for Z a finite subgroup of the center of H, is described by an open string worldsheet theory whose bulk is a sigma model with target a Z-gerbe (a bundle of one-form symmetries) over T∗M, of characteristic class determined by the H/Z bundle. We give a worldsheet picture of the decomposition of one-form-symmetry-gauged Chern-Simons in three dimensions, and we describe how a target-space constraint on bundles arising in the gauged Chern-Simons theory has a natural worldsheet realization. Our proposal provides examples of the expected correspondence between worldsheet global higher-form symmetries, and target-space gauged higher-form symmetries.

Solving AdS3 string theory at minimal tension: tree-level correlators

We revisit the minimal tension (k = 1) string theory on AdS3 × S3 × \U0001d54b4. We propose a new free-field description of the worldsheet theory and show how localization of string amplitudes emerges from the path integral. We exemplify our proposal by reproducing the worldsheet partition function of the psu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{psu} $$\\end{document}(1, 1|2)1 WZW model and providing explicit expressions for spectrally-flowed vertex operators and DDF operators. We compute string correlators in the path integral formalism and obtain a precise tree-level match with correlation functions of the boundary symmetric orbifold.

A supersymmetric extension of w1+∞ algebra in the celestial holography

We determine the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric topological W∞ algebra by using the λ deformed bosons (β, γ) and fermions (b, c) ghost system. By considering the real bosons and the real fermions at λ = 0 (or λ = 12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document}), the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric W∞2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {W}_{\\frac{\\infty }{2}} $$\\end{document} algebra is obtained. At λ = 14\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{4} $$\\end{document}, other N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric W1+∞[λ = 14\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{4} $$\\end{document}] algebra is determined. We also obtain the extension of Lie superalgebra PSU(2, 2|N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4) appearing in the worldsheet theory by using the symplectic bosons and the fermions. We identify the soft current algebra between the graviton, the gravitino, the photon (the gluon), the photino (the gluino) or the scalars, equivalent to N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supersymmetric W1+∞[λ] algebra, in two dimensions with the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supergravity theory in four dimensions discovered by Freedman, van Nieuwenhuizen and Ferrara in 1976 and its matter coupled theories, via celestial holography.

Asymptotic symmetries from the string worldsheet

In IIB string theory on AdS3 background with NS-NS fluxes, we show that Brown-Henneaux asymptotic Killing vectors can be derived by requiring both the worldsheet equations of motion and Virasoro constraints are preserved near the asymptotic boundary of the target spacetime. The charges on the worldsheet that generate the corresponding transformations can be written down in both the Lagrangian formalism and Hamiltonian formalism. This provides a method of studying asymptotic symmetry of the target spacetime directly from worldsheet string theories, without using results from the supergravity limit. As an example, we apply this method to flat spacetime in three dimensions and obtain the BMS3 generators on the worldsheet theory.

Tropological sigma models

With the use of mathematical techniques of tropical geometry, it was shown by Mikhalkin some twenty years ago that certain Gromov-Witten invariants associated with topological quantum field theories of pseudoholomorphic maps can be computed by going to the tropical limit of the geometries in question. Here we examine this phenomenon from the physics perspective of topological quantum field theory in the path integral representation, beginning with the case of the topological sigma model before coupling it to topological gravity. We identify the tropicalization of the localization equations, investigate its geometry and symmetries, and study the theory and its observables using the standard cohomological BRST methods. We find that the worldsheet theory exhibits a nonrelativistic structure, similar to theories of the Lifshitz type. Its path-integral formulation does not require a worldsheet complex structure; instead, it is based on a worldsheet foliation structure.

On the worldsheet S matrix of the AdS3/CFT2 mixed-flux mirror model

String on AdS3 × S3 × T4 backgrounds are known to be classically integrable in the presence of a mixture of Ramond-Ramond and Neveu-Schwarz-Neveu-Schwarz fluxes. It is expected that this results in the existence of a well-defined factorised worldsheet S matrix. In order to use integrability to compute the string spectrum we need such a factorised S matrix to exist also for the “mirror” model, obtained by a double Wick rotation of the original worldsheet theory. In the mixed-flux case the mirror model has a complex Hamiltonian, which raises questions on its well-definedness. In the paper we study the worldsheet tree-level S matrix of the original and mirror model and discuss some necessary conditions for the integrability and reality of the spectrum.

The u(2|2)1 WZW model

WZW models based on super Lie algebras play an important role for the description of string theory on AdS spaces. In particular, for the case of AdS3×S3 with pure NS–NS flux the super Lie algebra of psu(1,1|2)k appears in the hybrid formalism, and higher dimensional AdS spaces can be described in terms of related supergroup cosets. In this paper we study the WZW models based on u(2|2)1 and psu(2|2)1 that may play a role for the worldsheet theory that is dual to free super Yang–Mills in 4D.

The ALE partition functions of M-strings

We compute the equivariant partition function of the six-dimensional M-string SCFTs on a background with the topology of a product of a two-dimensional torus and an ALE singularity. We determine the result by exploiting BPS strings probing the singularity, whose worldvolume theories we determine via a chain of string dualities. A distinguished feature we observe is that for this class of background the BPS strings’ worldsheet theories become relative field theories that are sensitive to finer discrete data generalizing to 6d the familiar choices of flat connections at infinity for instantons on ALE spaces. We test our proposal against a conjectural 6d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = (1, 0) generalization of the Nekrasov master formula, as well as against known results on ALE partition functions in four dimensions.

Exploring T‐Duality for Self‐Dual Fields

AbstractAvatars of T‐duality within Sen's formalism for self‐dual field strengths in various dimensions are studied. This formalism is shown to naturally accommodate the T‐duality relation between Type IIA/IIB theories when compactified on a circle without the need for imposing the self‐duality constraint by hand, as is usually done. The study of this formalism is continued on two‐dimensional target spacetimes and its study as a worldsheet theory is initiated. In particular, it is shown that Sen's action provides a natural worldsheet‐based understanding of twisted and asymmetrically twisted strings. Finally, it is shown that the ‐deformed theory of left‐ and right‐chiral bosons described in Sen's formalism possesses a scaling limit that is related via field‐theoretic T‐duality to a recently studied integrable deformation of quantum mechanics.

The Virasoro minimal string

We introduce a critical string theory in two dimensions and demonstrate that this theory, viewed as two-dimensional quantum gravity on the worldsheet, is equivalent to a double-scaled matrix integral. The worldsheet theory consists of Liouville CFT with central charge c≥≥ 25 coupled to timelike Liouville CFT with central charge 26-c. The double-scaled matrix integral has as its leading density of states the universal Cardy density of primaries in a two-dimensional CFT, thus motivating the name Virasoro minimal string. The duality holds for any value of the continuous parameter cc and reduces to the JT gravity/matrix integral duality in the large central charge limit. It thus provides a precise stringy realization of JT gravity. The main observables of the Virasoro minimal string are quantum analogues of the Weil-Petersson volumes, which are computed as absolutely convergent integrals of worldsheet CFT correlators over the moduli space of Riemann surfaces. By exploiting a relation of the Virasoro minimal string to three-dimensional gravity and intersection theory on the moduli space of Riemann surfaces, we are able to give a direct derivation of the duality. We provide many checks, such as explicit numerical - and in special cases, analytic - integration of string diagrams, the identification of the CFT boundary conditions with asymptotic boundaries of the two-dimensional spacetime, and the matching between the leading non-perturbative corrections of the worldsheet theory and the matrix integral. As a byproduct, we discover natural conformal boundary conditions for timelike Liouville CFT.

Massive ambitwistor-strings; twistorial models

Ambitwistor-strings are chiral strings whose targets are spaces of complex massless particles, and whose correlation functions directly lead to simple, compact formulae for scattering amplitudes and loop integrands for massless gauge and gravity theories. This article extends the framework to massive particles in 4d, obtained via a symmetry reduction of higher dimensional massless models based on twistors. The target space of the resulting models turns out to be the phase space of 4d massive particles in a twistorial representation, and the worldsheet theory agrees with the two-twistor string previously introduced by the authors. The paper has been written so as to be largely self-contained. We discuss two interesting classes of massive theories in detail. For gauge theories, the reduction procedure is explicitly adapted to supersymmetric gauge theories on the Coulomb branch. For supergravity theories, the reduction is adapted to give theories obtained via Cremmer, Scherk & Schwartz (CSS) reduction, with broken supersymmetry and massive multiplets. The reduction procedure gives explicit and systematic rules to obtain amplitudes for all these theories and their amplitudes from two compact master formulae that have their origins in 6d based on the polarized scattering equations; in the CSS case the formulae are new, and in both cases their derivation is systematic. The freedom to include mass allows the definition of a loop insertion operator, thereby extending the formulae to 1-loop. Unlike the massless 4d twistorial models, these all display a perfect double copy structure, here incorporating massive particles in the relationship between gauge theory and CSS supergravity amplitudes.

Longitudinal Galilean and Carrollian limits of non-relativistic strings

It is well known that one can take an infinite speed of light limit that gives rise to non-relativistic strings with a relativistic worldsheet sigma model but with a non-relativistic target space geometry. In this work we systematically explore two further limits in which the worldsheet becomes non-Lorentzian. The first gives rise to a Galilean string with a Galilean structure on the worldsheet, extending previous work on Spin Matrix-related string theory limits. The second is a completely novel limit leading to a worldsheet theory with a Carrollian structure. We find the Nambu-Goto and Polyakov formulations of both limits and explore gauge fixing choices. Furthermore, we study in detail the case of the Galilean string for a class of target space geometries that are related to Spin Matrix target space geometries, for which the Nambu-Goto action (in static gauge) is quadratic in the fields.

Shock waves in holographic EPR pair

We study real-time correlators for N = 4 super Yang Mill fields coupled to a pair of entangled quarks using holography, in the setup that energy quanta sent from one quark perturb the quantum state of the fields and affect the other quark. We make the connection with the ER=EPR conjecture by considering the situation when two quarks are uniformly accelerating opposite to each other. The dynamics of quarks, in the gravity dual, is described by the string worldsheet theory, which in this case has the induced metric describing a two-sided AdS black hole, or a wormhole. Energy quanta sent by one of the quarks produce the shock wave on the worldsheet. We find the effect of shock wave on the boundary field correlators and we discuss the consequence for the ER=EPR conjecture.

AdS3 orbifolds, BTZ black holes, and holography

Conical defects of the form (AdS3 × S3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{S}}^3 $$\\end{document})/ℤk have an exact orbifold description in worldsheet string theory, which we derive from their known presentation as gauged Wess-Zumino-Witten models. The configuration of strings and fivebranes sourcing this geometry is well-understood, as is the correspondence to states/operators in the dual CFT2. One can analytically continue the construction to Euclidean AdS3 (i.e. the hyperbolic ball ℍ3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{H}}_3^{+} $$\\end{document}) and consider the orbifold by any infinite discrete (Kleinian) group generated by a set of elliptic elements γi ∈ SL(2, ℂ), γiki\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\gamma}_i^{{\ extrm{k}}_i} $$\\end{document} = \U0001d7d9, i = 1, . , K. The resulting geometry consists of multiple conical defects traveling along geodesics in ℍ3+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{H}}_3^{+} $$\\end{document}, and provides a semiclassical bulk description of correlation functions in the dual CFT involving the corresponding defect operators, which is nonperturbatively exact in α′. The Lorentzian continuation of these geometries describes a collection of defects colliding to make a BTZ black hole. We comment on a recent proposal to use such correlators to prepare a basis of black hole microstates, and elaborate on a picture of black hole formation and evaporation in terms of the underlying brane dynamics in the bulk.

A 4d non-BPS NS-NS microstate

We construct a two-parameter four-dimensional non-BPS NS-NS smooth microstate solution that asymptotes to flat spacetime with a linear dilaton in type II superstring theory. From the microscopic point of view, the background is made out of a certain number of decoupled (i.e. gs → 0) NS5 branes wrapping T3 × S1 × S1 with fundamental strings wrapping non-contractable cycles of S1 × S1 with integer momentum modes along them. We show that perturbative worldsheet theory in this background is given by a null-gauged WZW model. We also show that the consistency of the worldsheet theory imposes non-trivial constraints on the supergravity background.

Affine characters at negative level and elliptic genera of non-critical strings

We study the elliptic genera of the non-critical strings of six dimensional superconformal field theories from the point of view of the strings’ worldsheet theory. We formulate a general ansatz for these in terms of characters of the affine Lie algebra associated to the 6d gauge group at negative level, and provide ample evidence for the validity of this ansatz for 6d theories obtained via F-theory compactification on elliptically fibered Calabi-Yau manifolds over a Hirzebruch base. We obtain novel closed form results for many elliptic genera in terms of our ansatz, and show that our results specialize consistently when moving along Higgsing trees.

A Note on Integrability Loss in Fuzzball Geometries

AbstractThe dynamics of certain string configurations in a class of fivebrane supertube backgrounds is studied. In the decoupling limit of the fivebranes, these solutions are known to admit an exact description in worldsheet string theory and string propagation is integrable. For the asymptotically flat solutions, by using analytic tools of classical Hamiltonian systems, the non‐integrability of classical string motion is proved. This suggests that string dynamics in circular supertube geometries exhibit a regime of chaotic behavior.

S-duality in Toverline{T} -deformed CFT

Toverline{T} deformed conformal field theories can be reformulated as worldsheet theories of non-critical strings. We use this correspondence to compute and study the Toverline{T} deformed partition sum of a symmetric product CFT. We find that it takes the form of a partition sum of a second quantized string theory with a worldsheet given by the product of the seed CFT and a gaussian sigma model with the two-torus as target space. We show that deformed symmetric product theory admits a natural UV completion that exhibits a strong weak coupling ℤ2 duality that interchanges the momentum and winding numbers and maps the Toverline{T} -coupling λ to its inverse 1/λ. The ℤ2 duality is part of a full O(2, 2, ℤ)-duality group that includes a PSL(2, ℤ) acting on the complexified Toverline{T} coupling. The duality symmetry eliminates the appearance of complex energies at strong coupling for all seed CFTs with central charge c ≤ 6.

Light strings and strong coupling in F-theory

We consider 4d mathcal{N} = 1 theories arising from F-theory compactifications on elliptically-fibered Calabi-Yau four-folds and investigate the non-perturbative structure of their scalar field space beyond the large volume/large complex structure regime. We focus on regimes where the F-theory field space effectively reduces to the deformation space of the worldsheet theory of a critical string obtained from a wrapped D3-brane. In case that this critical string is a heterotic string with a simple GLSM description, we identify new strong coupling singularities in the interior of the F-theory field space. Whereas from the perturbative perspective these singularities manifest themselves through a breakdown of the perturbative α′-expansion, the dual GLSM perspective reveals that at the non-perturbative level these singularities correspond to loci in field space along which the worldsheet theory of the critical D3-brane string breaks down and a 7-brane gauge theory becomes strongly coupled due to quantum effects. Therefore these singularities signal a transition to a strong coupling phase in the F-theory field space which can be shown to arise due to the failure of the F-theory field space to factorize between complex structure and Kähler sector at the quantum level. Such singularities are hence a feature of a genuine mathcal{N} = 1 theory without a direct counterpart in mathcal{N} = 2 theories in 4d. By relating our setup to recent studies of global string solutions associated to axionic strings we argue that the D3-brane string dual to the perturbative heterotic string leaves the spectrum of BPS strings when traversing into the strong coupling phase. The absence of the perturbative, critical heterotic string then provides a physical explanation for the breakdown of the perturbative expansion and the obstruction of certain classical infinite distance limits in accordance with the Emergent String Conjecture.

Bi-η and bi-λ deformations of ℤ4 permutation supercosets

Integrable string sigma models on AdS3 backgrounds with 16 supersymmetries have the distinguishing feature that their superisometry group is a direct product. As a result the deformation theory of these models is particularly rich since the two supergroups in the product can be deformed independently. We construct bi-η and bi-λ deformations of two classes of ℤ4 permutation supercoset sigma models, which describe sectors of the Green-Schwarz and pure-spinor string worldsheet theories on type II AdS3 backgrounds with pure R-R flux. We discuss an important limit of these models when one supergroup is undeformed. The associated deformed supergravity background should preserve 8 supersymmetries and is expected to have better properties than the full bi-deformation. As a step towards investigating the quantum properties of these models, we study the two-loop RG flow of the bosonic truncation of the bi-λ deformation.