We clarify the relation between various approaches to the manifestly T-duality symmetric string. We explain in detail how the PST covariant doubled string arises from an unusual gauge fixing. We pay careful attention to the role of "spectator" fields in this process and also show how the T-duality invariant doubled dilaton emerges naturally. We extend these ideas to non-Abelian T-duality and show they give rise to the duality invariant formalism based on the semi-Abelian Drinfeld Double. We then develop the N=(0,1) supersymmetric duality invariant formalism.
Manifestly T-duality covariant worldsheet string models can be constructed by doubling the coordinate fields. We describe the underlying gauge symmetry of a recently proposed Lorentz invariant doubled worldsheet theory that makes half of the worldsheet degrees of freedom redundant. By shifting the Lagrange multiplier, that enforces the gauge fixing condition, the worldsheet action can be cast into various guises. We investigate the renormalization of this theory using a non-linear background / quantum split by employing a normal coordinate expansion adapted to the gauge-fixed theory. The propa…
We construct a (locally) supersymmetric worldsheet action for a string in a non-relativistic Newton-Cartan background. We do this using a doubled string action, which describes the target space geometry in an $O(D,D)$ covariant manner using a doubled metric and doubled vielbeins. By adopting different parametrisations of these doubled background fields, we can describe both relativistic and non-relativistic geometries. We focus on the torsional Newton-Cartan geometry which can be obtained by null duality/reduction (such null duality is particularly simple for us to implement). The doubled acti…
We propose a Lorentz invariant version of Tseytlin's doubled worldsheet theory that makes T-duality covariance of the string manifest. This theory can be derived as a gauge fixed version of Buscher's gauging procedure, in which the left-over gauge field component acts as a Lagrange multiplier. This description can naturally account for fractional linear O(D,D) transformations of the metric and b-field. It is capable of describing non-geometric backgrounds; geometric and non-geometric fluxes are encoded in the doubled anti-symmetric tensor field strength.
We revisit partition functions of closed strings on toroidal backgrounds, including their $\mathbb{Z}_N$ shift orbifolds in the formalism where the dimension of the target space is doubled to make T-duality manifest. In such a T-duality covariant formalism, the constraint equation imposes a form of chiral factorization. Our computation furnishes a non-trivial consistency check for the quantum worldsheet theory of the doubled sigma model, when strings are placed on general toroidal backgrounds. The topological term that mixes the physical space and its T-dual is crucial in demonstrating that ch…
We show that twisted doubled tori can be used to construct a general class of worldsheet models describing non-geometric string backgrounds. By employing a first order formulation of interacting chiral bosons, we first refine the analysis on the general conditions of worldsheet Lorentz invariance and then prove that twisted doubled tori provide good duality symmetric backgrounds. Subsequently we apply our general analysis to several examples which enable us to gain new insight on the difference between geometric, locally geometric and genuine non-geometric backgrounds.
The Kaluza-Klein monopole has been recognized as a string background with significant non-geometric features: it must appear "localized" to winding strings to match the NS5-brane's localization on the T-dual circle. In this work, we explicitly construct this T-dual system in the doubled geometry formalism, which proves to successfully describe the duality despite a broken isometry on one side of the duality pair. We further suggest an extension of the doubled formalism to the gauged linear sigma models describing this system (both bosonic and supersymmetric) and show that previous calculations…
We revisit the fundamental string (F1) solution in the doubled formalism. We show that the wave-like solution of double field theory (DFT) corresponding to the F1/pp-wave duality pair is more properly a solution of the DFT action coupled to a doubled sigma model action. The doubled string configuration which sources the pp-wave can be thought of as static gauge with the string oriented in a dual direction. We also discuss the DFT solution corresponding to a vibrating string, carrying both winding and momentum. We further show that the solution dual to the F1 in both time and space can be viewe…
The worldsheet theories that describe Poisson-Lie T-dualisable $σ$-models on group manifolds as well as integrable $η$, $λ$ and $β$-deformations provide examples of ${\cal E}$-models. Here we show how such ${\cal E}$-models can be given an elegant target space description within Double Field Theory by specifying explicitly generalised frame fields forming an algebra under the generalised Lie derivative. With this framework we can extract simple criteria for the R/R fields and the dilaton that extend the ${\cal E}$-model conditions to type II backgrounds. In particular this gives conditions for…
We derive and analyze the conditions for quantum conformal and Lorentz invariance of the duality symmetric interacting chiral boson sigma-models, which are conjectured to describe non-geometric string theory backgrounds. The one-loop Weyl and Lorentz anomalies are computed for the general case using the background field method. Subsequently, our results are applied to a class of (on-shell) Lorentz invariant chiral boson models which are based on twisted doubled tori. Our findings are in agreement with those expected from the effective supergravity approach, thereby firmly establishing that the…
Asymmetric heterotic orbifolds are discussed from the worldsheet perspective. Starting from Buscher's gauging of a theory of D compact bosons the duality covariant description of Tseytlin is obtained after a non-Lorentz invariant gauge fixing. A left-over of the gauge symmetry can be used to removed the doubled constant zero modes so that D physical target space coordinate remain. This can be thought of as the worldsheet realization of the strong constraint of double field theory. The extension of this description to the heterotic theory is straightforward as all results are written in terms o…
The doubled formulation of the worldsheet provides a description of string theory in which T-duality is promoted to a manifest symmetry. Here we extend this approach to $\mathcal{N}=(2,2)$ superspace providing a doubled formulation for bi-Hermitian/generalised Kähler target spaces. The theory is described by a single function, a doubled-generalised Kähler potential, supplemented with a manifestly $\mathcal{N}=(2,2)$ constraint. Several examples serve to illustrate this construction, including a discussion of the $\mathcal{N}=(2,2)$ description of T-folds.
We report on some recent progress in understanding heterotic flux compactifications, from a worldsheet perspective mainly. We consider local models consisting in torus fibration over warped Eguchi-Hanson space and non-Kähler resolved conifold geometries. We analyze the supergravity solutions and define a double-scaling limit of the resolved singularities, defined such that the geometry is smooth and weakly coupled. We show that, remarkably, the heterotic solutions admit solvable worldsheet CFT descriptions in this limit. This allows in particular to understand the important role of worldsheet …
We give a covariant realization of the doubled sigma-model formulation of duality-symmetric string theory within the general framework of para-Hermitian geometry. We define a notion of generalized metric on a para-Hermitian manifold and discuss its relation to Born geometry. We show that a Born geometry uniquely defines a worldsheet sigma-model with a para-Hermitian target space, and we describe its Lie algebroid gauging as a means of recovering the conventional sigma-model description of a physical string background as the leaf space of a foliated para-Hermitian manifold. Applying the Kotov-S…
We continue to study doubled aspects of algebroid structures equipped with the C-bracket in double field theory (DFT). We find that a family of algebroids, the Vaisman (metric or pre-DFT), the pre- and the ante-Courant algebroids are constructed by the analogue of the Drinfel'd double of Lie algebroid pairs. We examine geometric implementations of these algebroids in the para-Hermitian manifold, which is a realization of the doubled space-time in DFT. We show that the strong constraint in DFT is necessary to realize the doubled and non-trivial Poisson structures but can be relaxed for some alg…
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, whic…
We study a generalization of the Alexandrov-Kontsevich-Schwarz-Zaboronsky (AKSZ) formulation of the A- and B-models which involves a doubling of coordinates, and can be understood as a complexification of the Poisson $σ$-model underlying these. In the flat space limit the construction contains models obtained by twisting an N=2 supersymmetric $σ$-model on Hull's doubled geometry. The curved space generalization involves a product of two diffeomorphic Calabi-Yau manifolds, and the $O(d,d)$ metric can be understood as a complexification of the CY metric. In addition, we consider solutions that c…
The pure spinor formalism for the superstring, initiated by N. Berkovits, is derived at the fully quantum level starting from a fundamental reparametrization invariant and super-Poincare invariant worldsheet action. It is a simple extension of the Green-Schwarz action with doubled spinor degrees of freedom with a compensating local supersymmetry on top of the conventional kappa-symmetry. Equivalence to the Green-Schwarz formalism is manifest from the outset. The use of free fields in the pure spinor formalism is justified from the first principle. The basic idea works also for the superparticl…
The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd double of Lie algebroids. Based on a geometric realization of doubled space-time as a para-Hermitian manifold, we examine exterior algebras and a para-Dolbeault cohomology on DFT and discuss the structure of the Drinfel'd double behind the DFT gauge symmetry. Similar to the Courant algebroid in the generalized geometry, Lagrangian subbundles $(L,\tilde{L})$ in a…
The integrability-based solution of string theories related to AdS(n)/CFT(n-1) dualities relies on the worldsheet S matrix. Using generalized unitarity we construct the terms with logarithmic dependence on external momenta at one- and two-loop order in the worldsheet S matrix for strings in a general integrable worldsheet theory. We also discuss aspects of calculations at higher orders. The S-matrix elements are expressed as sums of integrals with coefficients given in terms of tree-level worldsheet four-point scattering amplitudes. One-loop rational functions, not determined by two-dimensiona…
We compute the imaginary parts of genus-one string scattering amplitudes. Following Witten's $i\varepsilon$ prescription for the integration contour on the moduli space of worldsheets, we give a general algorithm for computing unitarity cuts of the annulus, Möbius strip, and torus topologies exactly in $α'$. With the help of tropical analysis, we show how the intricate pattern of thresholds (normal and anomalous) opening up arises from the worldsheet computation. The result is a manifestly-convergent representation of the imaginary parts of amplitudes, which has the analytic form expected from…
Compactifications of the heterotic string are a viable route to phenomenologically realistic vacua and interesting new mathematics. While supergravity aspects of heterotic compactifications are largely well-understood their worldsheet description remains largely unexplored. We review recent work in developing linear sigma model techniques aimed at elucidating the underlying worldsheet description.
The integration problem of a C-bracket and a Vaisman (metric, pre-DFT) algebroid which are geometric structures of double field theory (DFT) is analyzed. We introduce a notion of a pre-rackoid as a global group-like object for an infinitesimal algebroid structure. We propose that several realizations of pre-rackoid structures. One realization is that elements of a pre-rackoid are defined by cotangent paths along doubled foliations in a para-Hermitian manifold. Another realization is proposed as a formal exponential map of the algebroid of DFT. We show that the pre-rackoid reduces to a rackoid …
These pedagogical lectures given at the Corfu Summer Institute 2018 review two generalised notions of T-duality, non-Abelian T-duality and Poisson-Lie duality, and their applications. We explain how each of these has seen recent application in the context of holography. Non-Abelian T-duality has been used to construct new holographic dual geometries. Poisson-Lie duality has been used to construct new integrable string sigma-models including the $η$- and $λ$-deformations of the $AdS_5\times S^5$ superstring thought to encode quantum group deformations of holography. We also comment on the doubl…
In this thesis we discuss non-perturbative phenomena emerging in gauge and in string/supergravity theories. We compute the partition function of 5D minimal supersymmetric U(1) gauge theory with extra adjoint matter in general Omega-background. These N=2^* theories are massive deformations of N=4 SYM and can be thought of as a minimal supersymmetric five dimensional theory compactified on a circle. Such partition functions encode very rich topological information. We show that this partition function at some special values of the parameters directly reproduces the generating function for the Po…
We discuss BPS states preserving 1/4 supersymmetries of N=4 supersymmetric Yang-Mills theory as M2-branes holomorphically embedded and ending on M5-branes. We use techniques in electrodynamics to find the M2-brane configurations, and give some explicit examples. In case the M2-brane worldsheet has handles, the worldsheet moduli of the M2-brane is constrained in a discrete manner. Several aspects of multi-pronged strings in type IIB string theory are beautifully reproduced in the M-theory description. We also discuss the relation between the above construction and the D2-brane dynamics in type …
In the first lecture, we discuss basic aspects of worldsheet and penta-brane instantons as well as (unoriented) D-brane instantons, which is our main focus here, and threshold corrections to BPS-saturated couplings. The second lecture is devoted to non-perturbative superpotentials generated by `gauge' and `exotic' instantons living on D3-branes at orientifold singularities. In the third lecture we discuss the interplay between worldsheet and D-string instantons on $T^4/Z_2$. We focus on a 4-fermi amplitude, give Heterotic and perturbative Type I descriptions, and offer a multi D-string instant…
Aspects of strings without spacetime supersymmetry
Scuola Normale Superiore（高等师范学校）；INFN（意大利国家核物理研究所）；Instituto de Física Teórica IFT-UAM/CSIC（理论物理研究所 IFT-UAM/CSIC）；Universidad Autónoma de Madrid（马德里自治大学）
We find supersymmetric solutions of the D4 brane Born-Infeld action describing D2 supertubes ending on an arbitrary curve inside a D4 brane. From the D4 brane point of view, these are dyonic strings. We also consider various higher dimensional extensions of the usual supertubes, involving expanded D4 and D3 brane configurations. Finally, considering the worldsheet theory for open strings on a supertube, we show that this configuration is an exact solution to all orders in alpha'. Further the causal structure of the open-string metric provides new insight into the arbitrary cross-section of the…
Closed string tachyon condensation in spacetime generates potentials on the worldsheet that model two-dimensional inflationary cosmology. These models illustrate and elucidate a variety of aspects of inflation, in particular the generation of quantum fluctuations and their back-reaction on geometry. We exhibit a class of Liouville gravity models coupled to matter that can exhibit, for example: (a) pure de Sitter gravity; (b) slow-roll inflation; (c) topological inflation; and (d) graceful exit into an FRW phase. The models also provide a quantitative testing ground for ideas about the origin o…
We investigate aspects of holographic duals to time-like warped AdS_3 space-times--which include Gödel's universe--in string theory. Using worldsheet techniques similar to those that have been applied to AdS_3 backgrounds, we are able to identify space-time symmetry algebras that act on the dual boundary theory. In particular, we always find at least one Virasoro algebra with computable central charge. Interestingly, there exists a dense set of points in the moduli space of these models in which there is actually a second commuting Virasoro algebra, typically with different central charge than…
Motivated by the recent interest in the different aspects of the string/field theory duality, we describe an approach for obtaining exact string solutions in general backgrounds, based on two types of string embedding, allowing for separation of the worldsheet variables.
In this PhD thesis we review some aspects of integrable models related to string backgrounds or their deformations. In the first part we develop methods to obtain exact results in the AdS3/CFT2 correspondence. We consider the AdS_3 x S^3 x T^4 background with pure Ramond-Ramond flux and we find the all-loop worldsheet S-matrix by exploiting the symmetries of the model in light-cone gauge. As we naturally include the massless modes on the worldsheet, we derive the full set of Bethe-Yang equations. In the massive sector we give also a spin-chain description and we write down solutions compatible…
In this paper we continue the investigation of aspects of integrability of the type IIA AdS(3) x S(3) x S(3) x S(1) and AdS(3) x S(3) x T(4) superstrings. By constructing a one parameter family of flat connections we prove that the Green-Schwarz string is classically integrable, at least to quadratic order in fermions, without fixing the kappa-symmetry. We then compare the quantum dispersion relation, fixed by integrability up to an unknown interpolating function h(lambda), to explicit one-loop calculations on the string worldsheet. For AdS(3) x S(3) x S(3) x S(1) the spectrum contains heavy, …
This course provides a self contained introduction to the general theory of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation. The first part of the course is concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in …
We investigate various aspects of a proposal by Aharony and Kutasov arXiv:0803.3547 [hep-th] for the gravity dual of an open Wilson line in the Sakai-Sugimoto model or its non-compact version. In particular, we use their proposal to determine the effect of finite temperature, as well as background electric and magnetic fields, on the chiral symmetry breaking order parameter. We also generalize their prescription to more complicated worldsheets and identify the operators dual to such worldsheets.
We review recent progress in string cosmology, where string dualities are applied so as to obtain complete cosmological evolutions, free of any essential singularities. Two classes of models are analyzed. The first class consists of string gas cosmologies associated to certain thermal configurations of type II N=(4,0) models. Finite temperature is introduced along with non-trivial "gravito-magnetic" fluxes that lift the Hagedorn instabilities of the canonical ensemble and restore thermal T-duality symmetry. At a critical maximal temperature additional thermal states become massless sourcing st…
AdS_4/CFT_3 duality relating IIA string theory on AdS_4 x CP^3 to N=6 superconformal Chern-Simons theory provides an arena for studying aspects of integrability in a new potentially exactly solvable system. In this paper we explore the tree-level worldsheet scattering for strings on AdS_4 x CP^3. We compute all bosonic four-, five- and six-point amplitudes in the gauge-fixed action and demonstrate the absence of particle production.
The dissertation consists of two parts. The first presents an account of the effective worldvolume description of $N$ coincident M2-branes ending on an M5-brane in M-theory. It reviews Basu and Harvey's recent description of the worldvolume theory of the M2-branes in terms of a Bogomol'nyi equation, and its solution via a fuzzy (three-) funnel. Tests of the consistency of this picture are then performed and many of the issues with it are addressed. This is followed by a discussion of how a refinement of the fuzzy three-sphere algebra used can lead to the correct $N^{3/2}$ scaling of degrees of…
Supermembrane compactified on a $M_9\times T^2$ target space is globally described by the inequivalent classes of torus bundles over torus. These torus bundles have monodromy in $SL(2,Z)$ when they correspond to the nontrivial central charge sector and they are trivial otherwise. The first ones contain eight inequivalent classes of M2-brane bundles which at low energies, are in correspondence with the eight type II gauged supergravities in $9D$. The relation among them is completely determined by the global action of T-duality which interchanges topological invariants of the two tori. The M2-b…
Worldsheet string theory compactified on exceptional holomony manifolds is revisited following arXiv:1809.06376, where aspects of the chiral symmetry were described for the case where the compact space is a 7-dimensional G$_2$-holonomy manifold constructed as a Twisted Connected Sum. We reinterpret this result and extend it to Extra Twisted Connected Sum G$_2$-manifolds, and to 8-dimensional Generalized Connected Sum Spin(7)-manifolds. Automorphisms of the latter construction lead us to conjecture new mirror maps.
In this thesis, two aspects of string theory are discussed, tensionless strings and supersymmetric sigma models. The equivalent to a massless particle in string theory is a tensionless string. Even almost 30 years after it was first mentioned, it is still quite poorly understood. We discuss how tensionless strings give rise to exact solutions to supergravity and solve closed tensionless string theory in the ten dimensional maximally supersymmetric plane wave background, a contraction of AdS(5)xS(5) where tensionless strings are of great interest due to their proposed relation to higher spin ga…
Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Birla Institute of Technology and Science, Pilani Campus（比拉理工学院皮拉尼校区）；Beijing Institute of Mathematical Sciences and Applications（北京数学科学研究所）；Tata Institute of Fundamental Research（塔塔基础研究所）
We review recent developments on nonrelativistic string theory. In flat spacetime, the theory is defined by a two-dimensional relativistic quantum field theory with nonrelativistic global symmetries acting on the worldsheet fields. This theory arises as a self-contained corner of relativistic string theory. It has a string spectrum with a Galilean dispersion relation, and a spacetime S-matrix with nonrelativistic symmetry. This string theory also gives a unitary and ultraviolet complete framework that connects different corners of string theory, including matrix string theory and noncommutativ…
This is the second in a series of papers discussing in the framework of gerbe theory canonical and geometric aspects of the 2d nonlinear sigma model in the presence of conformal defects in the worldsheet. Employing the formal tools worked out in the first paper of the series, 1101.1126 [hep-th], a thorough analysis of rigid symmetries of the sigma model is carried out with emphasis on algebraic structures on generalised tangent bundles over the target space of the theory and over its state space that give rise to a realisation of the symmetry algebra on states. The analysis leads to a proposal…
We review recent developments in duality symmetric string theory. We begin with the world sheet doubled formalism which describes strings in an extended space time with extra coordinates conjugate to winding modes. This formalism is T-duality symmetric and can accommodate non-geometric T-fold backgrounds which are beyond the scope of Riemannian geometry. Vanishing of the conformal anomaly of this theory can be interpreted as a set of spacetime equations for the background fields. These equations follow from an action principle that has been dubbed Double Field Theory (DFT). We review the aspec…
We discuss an aspect of string theory which has been tackled from many different perspectives, but incompletely: the role of nonlocality in the theory and its relation to the geometric shape of the string. In particular, we will describe in quantitative terms how one can zoom out from an extended object such as the string in such a way that, at sufficiently large scales, it appears structureless. Since there are no free parameters in free-string theory, the notion of large scales will be unambiguously determined. In other words, we will be able to answer the question: How and at which scale ca…
