NS-NS Flux Research Articles

The free field realisation of the BVW string

The symmetric orbifold of \U0001d54b4 was recently shown to be exactly dual to string theory on AdS3 × S3 × \U0001d54b4 with minimal (k = 1) NS-NS flux. The worldsheet theory is best formulated in terms of the hybrid formalism of Berkovits, Vafa & Witten (BVW), in terms of which the AdS3× S3 factor is described by a mathfrak{psu} (1, 1|2)k WZW model. At level k = 1, mathfrak{psu} (1, 1 2)1 has a free field realisation that is obtained from that of mathfrak{u} (1, 1 2)1 upon setting a mathfrak{u} (1) field, often called Z, to zero. We show that the free field version of the mathcal{N} = 2 generators of BVW (whose cohomology defines the physical states) does not give rise to an mathcal{N} = 2 algebra, but is rather contaminated by terms proportional to the Z-field. We also show how to overcome this problem by introducing additional ghost fields that implement the quotienting by Z.

Generalised U-dual solutions in supergravity

We discuss the notion of generalised U-duality as a solution generating technique in supergravity. We demonstrate a method to take solutions of type IIA supergravity on a 3-sphere, with NSNS flux, to new solutions of 11-dimensional supergravity, using exceptional geometry techniques. These new solutions are characterised by an underlying 3-algebra structure, and generalise features of solutions obtained by non-abelian T-duality, which involve an underlying ordinary Lie algebra. We focus on an example where we start with the pp-F1-NS5 solution in type IIA supergravity. We discuss the properties of our resulting new solution, including the possibility of viewing it globally as a U-fold, and its M2 and M5 brane charges. In the extremal case, the new solution admits an AdS3 limit, which falls into a recently constructed class of M-theory AdS3 backgrounds — this provides a global completion of our solution with a well-defined holographic dual, similar to the global completions of non-abelian T-dual solutions. Our full solution is a 6-vector deformation of this AdS3 limit. We also explicitly solve the Killing spinor equation in the AdS3 limit, finding a frac{1}{2}hbox{-} mathrm{BPS} solution.

D-branes in AdS3 \xd7 S3 \xd7 \U0001d54b4 at k = 1 and their holographic duals

String theory on AdS3× S3× \U0001d54b4 with minimal k = 1 NS-NS flux can be described in terms of a free field worldsheet theory in the hybrid formalism. We construct various D-branes of this string theory and calculate their associated cylinder amplitudes. We find that these amplitudes match with the cylinder correlators of certain boundary states of the dual symmetric orbifold CFT Sym(\U0001d54b4), thus suggesting a direct correspondence between these boundary conditions. We also show that the disk amplitudes of these D-branes localise to those points in the worldsheet moduli space where the worldsheet disk holomorphically covers the spacetime disk.

The physical states of the Hybrid Formalism

String theory on AdS3 × S3 × mathbbm{T} 4 with one unit (k = 1) of NS-NS flux is considered in the hybrid formalism of Berkovits, Vafa & Witten (BVW). Using the free field realisation of the world-sheet theory at k = 1, we identify explicitly the BRST cohomology classes corresponding to some of the low-lying states of the dual CFT. In particular, we do this for the mathcal{N} = 4 superconformal generators of the symmetric orbifold theory, and we confirm these identifications by showing that the worldsheet correlators reproduce the expected dual CFT answer. Along the way we note that the physical vertex operators on the worldsheet have a simpler form if one works with a different, but equivalent, choice for the BRST operators relative to BVW.

String correlators on AdS3: four-point functions

We propose a closed-form formula for genus 0 four-point functions in AdS3 string theory with pure NS-NS flux including arbitrary amounts of spectral flow. Our formula passes many non-trivial consistency checks and has intriguing connections to Hurwitz theory. This paper is the second in a series with several instalments.

On supersymmetric AdS3 solutions of Type II

We classify supersymmetric warped AdS3×wM7 backgrounds of Type IIA and Type IIB supergravity with non-constant dilaton, generic RR fluxes and magnetic NSNS flux, in terms of a dynamic SU(3)-structure on M7. We illustrate our results by recovering several solutions with various amounts of supersymmetry. The dynamic SU(3)-structure includes a G2-structure as a limiting case, and we show that in Type IIB this is integrable.

Remarks on single trace TT¯ and other current-current deformations

String theory on AdS$_3$ with NS-NS fluxes admits a solvable irrelevant deformation which is close to the $T\bar{T}$ deformation of the dual CFT$_2$. This consists of deforming the worldsheet action, namely the action of the $SL(2,\mathbb{R})$ WZW model, by adding to it the operator $J^-\bar{J}^-$, constructed with two Kac-Moody currents. The geometrical interpretation of the resulting theory is that of strings on a conformally flat background that interpolates between AdS$_3$ in the IR and a flat linear dilaton spacetime with Hagedorn spectrum in the UV, having passed through a transition region of positive curvature. Here, we study the properties of this string background both from the point of view of the low-energy effective theory and of the worldsheet CFT. We first study the geometrical properties of the semiclassical geometry, then we revise the computation of correlation functions and of the spectrum of the $J^-\bar{J}^-$-deformed worldsheet theory, and finally we discuss how to extend this type of current-current deformation to other conformal models.

Summing over geometries in string theory

We examine the question how string theory achieves a sum over bulk geometries with fixed asymptotic boundary conditions. We discuss this problem with the help of the tensionless string on {mathrm{mathcal{M}}}_3times {mathrm{S}}^3times {mathbbm{T}}^4 (with one unit of NS-NS flux) that was recently understood to be dual to the symmetric orbifold SymN ( {mathbbm{T}}^4 ). We strengthen the analysis of [1] and show that the perturbative string partition function around a fixed bulk background already includes a sum over semi-classical geometries and large stringy corrections can be interpreted as various semi-classical geometries. We argue in particular that the string partition function on a Euclidean wormhole geometry factorizes completely into factors associated to the two boundaries of spacetime. Central to this is the remarkable property of the moduli space integral of string theory to localize on covering spaces of the conformal boundary of ℳ3. We also emphasize the fact that string perturbation theory computes the grand canonical partition function of the family of theories ⊕N SymN ( {mathbbm{T}}^4 ). The boundary partition function is naturally expressed as a sum over winding worldsheets, each of which we interpret as a ‘stringy geometry’. We argue that the semi-classical bulk geometry can be understood as a condensate of such stringy geometries. We also briefly discuss the effect of ensemble averaging over the Narain moduli space of {mathbbm{T}}^4 and of deforming away from the orbifold by the marginal deformation.

Dictionary for the type II nongeometric flux compactifications

We study the $T$-dual completion of the four-dimensional $\mathcal{N}=1$ type II effective potentials in the presence of (non)geometric fluxes. First, we invoke a cohomology version of the $T$-dual transformations among the various moduli, axions, and fluxes appearing in the type IIA and type IIB effective supergravities. This leads to some useful observations about a significant mixing of the standard NS-NS fluxes with the (non)geometric fluxes on the mirror side. Further, using our $T$-duality rules, we establish an explicit mapping among the $F$ terms, $D$ terms, tadpole conditions, and Bianchi identities of the two theories. Second, we propose what we call a set of ``axionic flux polynomials,'' which depend on all of the axionic moduli and the fluxes. This subsequently helps to present the two scalar potentials in a concise and manifestly $T$-dual form, which can be directly utilized for various phenomenological purposes, as we illustrate in a couple of examples.

Partition functions of the tensionless string

We consider string theory on AdS3× S3× \U0001d54b4 in the tensionless limit, with one unit of NS-NS flux. This theory is conjectured to describe the symmetric product orbifold CFT. We consider the string on different Euclidean backgrounds such as thermal AdS3, the BTZ black hole, conical defects and wormhole geometries. In simple examples we compute the full string partition function. We find it to be independent of the precise bulk geometry, but only dependent on the geometry of the conformal boundary. For example, the string partition function on thermal AdS3 and the conical defect with a torus boundary is shown to agree, thus giving evidence for the equivalence of the tensionless string on these different background geometries. We also find that thermal AdS3 and the BTZ black hole are dual descriptions and the vacuum of the BTZ black hole is mapped to a single long string winding many times asymptotically around thermal AdS3. Thus the system yields a concrete example of the string-black hole transition. Consequently, reproducing the boundary partition function does not require a sum over bulk geometries, but rather agrees with the string partition function on any bulk geometry with the appropriate boundary. We argue that the same mechanism can lead to a resolution of the factorization problem when geometries with disconnected boundaries are considered, since the connected and disconnected geometries give the same contribution and we do not have to include them separately.

Free field world-sheet correlators for AdS3

We employ the free field realisation of the mathfrak{psu}{left(1,1left|2right.right)}_1 world-sheet theory to constrain the correlators of string theory on AdS3× S3× \U0001d54b4 with unit NS-NS flux. In particular, we directly obtain the unusual delta function localisation of these correlators onto branched covers of the boundary S2 by the (genus zero) world-sheet — this is the key property which makes the equivalence to the dual symmetric orbifold manifest. In our approach, this feature follows from a remarkable ‘incidence relation’ obeyed by the correlators, which is reminiscent of a twistorial string description. We also illustrate our results with explicit computations in various special cases.

N spike D-strings in AdS Space with mixed flux

We use Dirac-Born-Infeld action to study the spinning D-string in AdS_3 background in the presence of both NS-NS and RR fluxes. We compute the scaling relation between the energy (E) and spin (S) in the ‘long string limit’. The energy of these spiky string is found to be a function of spin with the leading logarithmic behaviour and the scaling relation appears to be independent of the amount of flux present. We further discuss folded D-string solutions in AdS_3 background with pure NS-NS and R-R fluxes.

Gaugino mass term for D-branes and Generalized Complex Geometry

We compute the four-dimensional gaugino mass for a Dp-brane extended in spacetime and wrapping a cycle on the internal geometry in a warped compactification with fluxes. Motivated by the backreaction of gaugino bilinear VEVs, we use Generalized Complex Geometry to characterize the internal geometry as well as the cycle wrapped by the brane. We find that the RR fluxes and the non-closure of the generalized complex structures combine in the gaugino mass terms in the same form as they do in the bulk superpotential, while for the NSNS fluxes there is a crucial minus sign in the component normal to the brane. Our expression extends the known result for D3 and D7-branes in Calabi-Yau manifolds, where the gaugino masses are induced respectively by the imaginary anti-self dual and imaginary self-dual components of the complex 3-form flux G3.

Deriving the AdS3/CFT2 correspondence

It was recently argued that string theory on AdS3× S3× \U0001d54b4 with one unit (k = 1) of NS-NS flux is exactly dual to the symmetric orbifold CFT SymN (\U0001d54b4). In this paper we show how to directly relate the n-point correlators of the two sides to one another. In particular, we argue that the correlators of the world-sheet theory are delta-function- localised in string moduli space to those configurations that allow for a holomorphic covering map of the S2-boundary of AdS3 by the world-sheet. This striking feature can be seen both from a careful Ward identity analysis, as well as from semi-classically exact AdS3 solutions that are pinned to the boundary. The world-sheet correlators therefore have exactly the same structure as in the Lunin-Mathur construction of symmetric orbifold CFT correlators in terms of a covering surface — which now gets identified with the world-sheet. Together with the results of [1, 2] this essentially demonstrates how the k = 1 AdS3 string theory becomes equivalent to the spacetime orbifold CFT in the genus expansion.

Nonrelativistic giant magnons from Newton Cartan strings

We show nonrelativistic (NR) giant magnon dispersion relations by probing the torsional Newton Cartan (TNC) geometry with (semi)classical nonrelativistic rigidly rotating strings. We construct NR sigma models over R × S2 and consider two specific limiting cases those are of particular interest. Both of these limiting conditions give rise to what we identify as the small momentum limit of the giant magnon dispersion relation in the dual SMT at strong coupling. We further generalize our results in the presence of background NS-NS fluxes. Our analysis reveals that unlike its relativistic counterpart, the NR string theory lacks of single spike solutions.

Heating up holography for single-trace Joverline{T} deformations

We study thermodynamic aspects of a tractable toy model of holography for extremal Kerr black holes proposed in [1]. On the gravity side, the theory can be described by the worldsheet action of string theory on a warped AdS3 background supported by NS-NS flux. Once we turn on temperature, the deformed background is described by a black string solution of type IIB supergravity that features a locally warped AdS3 factor. The dual field theory is conjectured to be a single-trace version of a Joverline{T} -deformed CFT at finite temperature. As evidence for the correspondence we show that the spectrum of strings winding on the deformed background agrees with the spectrum of Joverline{T} -deformed CFTs. Furthermore, we show that the gravitational charges of the black string match the averaged charges of a thermal ensemble in the dual field theory. Finally, we reproduce the Bekenstein-Hawking entropy of the black string from the microscopic density of states of Joverline{T} -deformed CFTs.

Stringy CFT duals with mathcal{N} = (2, 2) supersymmetry

It was recently shown that string theory on AdS3× S3× mathbbm{T} 4 with minimal NS-NS flux (k = 1) is exactly dual to the symmetric orbifold of mathbbm{T} 4. Here we show that a similar statement also holds for the Dn orbifolds of these backgrounds that have mathcal{N} = (2, 2) supersymmetry. In this case the CFT dual is the symmetric orbifold of mathbbm{T} 4/Dn.

Three-point functions in AdS3/CFT2 holography

Recently, string theory on AdS3× S3× \U0001d54b4 with one unit of NS-NS flux (k = 1) was argued to be exactly dual to the symmetric orbifold of \U0001d54b 4, and in particular, the full (unprotected) spectrum was matched between the two descriptions. This duality was later extended to the case with higher NS-NS background flux for which the long string sector was argued to be described by the symmetric product orbifold of ( mathcal{N} = 4 Liouville) × \U0001d54b 4. In this paper we study correlation functions for the bosonic analogue of this duality, relating bosonic string theory on AdS3× X to the symmetric orbifold of Liouville × X. More specifically, we show that the low-lying null vectors of Liouville theory correspond to BRST exact states from the worldsheet perspective, and we demonstrate that they give rise to the expected BPZ differential equations for the dual CFT correlators. Since the structure constants of Liouville theory are uniquely fixed by these constraints, this shows that the seed theory of the dual CFT contains indeed the Liouville factor.

String theory on AdS3 and the symmetric orbifold of Liouville theory

For string theory on AdS3 with pure NS-NS flux a complete set of DDF operators is constructed, from which one can read off the symmetry algebra of the spacetime CFT. Together with an analysis of the spacetime spectrum, this allows us to show that the long string sector of superstring theory on AdS3×S3×T4 for generic NS-NS flux is described by the symmetric orbifold of (N=4Liouville theory)×T4. For the case of minimal flux (k=1), the Liouville factor disappears, and we just obtain the symmetric orbifold of T4, thereby giving further support to a previous claim. We also show that a similar analysis can be done for bosonic string theory on AdS×3X.

On N-spike strings in conformal gauge with NS-NS fluxes

The AdS3 × S3 string sigma model supported both by NS-NS and R-R fluxes has become a well known integrable model, however a putative dual field theory description remains incomplete. We study the anomalous dimensions of twist operators in this theory via semiclassical string methods. We describe the construction of a multi-cusp closed string in conformal gauge moving in AdS3 with fluxes, which supposedly is dual to a general higher twist operator. After analyzing the string profiles and conserved charges for the string, we find the exact dispersion relation between the charges in the ‘long’ string limit. This dispersion relation in leading order turns out to be similar to the case of pure RR flux, with the coupling being scaled by a factor that depends on the amount of NS-NS flux turned on. We also analyse the case of pure NS flux, where the dispersion relation simplifies considerably. Furthermore, we discuss the implications of these results at length.