We study the topological string partition function of a class of toric, double elliptically fibered Calabi-Yau threefolds $X_{N,M}$ at a generic point in the Kähler moduli space. These manifolds engineer little string theories in five dimensions or lower and are dual to stacks of M5-branes probing a transverse orbifold singularity. Using the refined topological vertex formalism, we explicitly calculate a generic building block which allows to compute the topological string partition function of $X_{N,M}$ as a series expansion in different Kähler parameters. Using this result we give further explicit proof for a duality found previously in the literature, which relates $X_{N,M}\sim X_{N',M'}$ for $NM=N'M'$ and $\text{gcd}(N,M)=\text{gcd}(N',M')$.
We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as polynomials of five generators. We also compute the explicit polynomial forms of the partition functions for genus 2, 3, and 4. Moreover, some coefficients are written down for all genus.
We study a class of ${\cal N}=(1,0)$ little string theories obtained from orbifolds of M-brane configurations. These are realised in two different ways that are dual to each other: either as $M$ parallel M5-branes probing a transverse $A_{N-1}$ singularity or $N$ M5-branes probing an $A_{M-1}$ singularity. These backgrounds can further be dualised into toric, non-compact Calabi-Yau threefolds $X_{N,M}$ which have double elliptic fibrations and thus give a natural geometric description of T-duality of the little string theories. The little string partition functions are captured by the topologi…
We find a Nekrasov-type expression for the Seiberg-Witten prepotential for the six-dimensional non-critical E_8 string theory toroidally compactified down to four dimensions. The prepotential represents the BPS partition function of the E_8 strings wound around one of the circles of the toroidal compactification with general winding numbers and momenta. We show that our expression exhibits expected modular properties. In particular, we prove that it obeys the modular anomaly equation known to be satisfied by the prepotential.
A particular two-parameter class of little string theories can be described by $M$ parallel M5-branes probing a transverse affine $A_{N-1}$ singularity. We previously discussed the duality between the theories labelled by $(N,M)$ and $(M,N)$. In this work, we propose that these two are in fact only part of a larger web of dual theories. We provide evidence that the theories labelled by $(N,M)$ and $(\tfrac{NM}{k},k)$ are dual to each other, where $k=\text{gcd}(N,M)$. To argue for this duality, we use a geometric realization of these little string theories in terms of F-theory compactifications…
We show that the partition function on a generalised conifold $C_{m,n}$ with ${m+n \choose m}$ crepant resolutions can be equivalently computed on the compound du Val singularity $A_{m+n-1}\times \mathbb C$ with a unique crepant resolution.
We explore two classes of 6d $\mathcal{N}=(1,0)$ little string theories obtained from type IIA/IIB NS5-branes probing $D_n$ singularities. Their tensor branches are described by effective gauge theories whose instanton solitons are macroscopic little strings. We specifically study two families of 2d $\mathcal{N}=(0,4)$ gauge theories which describe at low energy the worldsheet dynamics of the type IIA/IIB little strings. These gauge theories are useful to calculate the supersymmetric partition functions of the little string theories on $\mathbf{R}^4 \times T^2$. We establish the T-duality of t…
We argue that the holographic dual to little string theories at finite temperature suffers from a Gregory-Laflamme like instability, providing an alternative explanation to the results of hep-th/0012258.
We calculate, using holographic duality, the thermal two-point function in finite temperature little string theory. The analysis of those correlators reveals possible instabilities of the thermal ensemble, as in previous discussions of the thermodynamics of little string theory. We comment on the dependence of the instability on the spatial volume of the system.
We consider the string, like point particles and branes, to be an irreducible representation of the semi-direct product of the Cartan involution invariant subalgebra of E11 and its vector representation. We show that the subalgebra that preserves the string charges, the string little algebra, is essentially the Borel subalgebra of E9. We also show that the known string physical states carry a representation of parts of this algebra.
This is a brief review of the current state of knowledge on "little string theories", which are non-gravitational theories having several string-like properties. We focus on the six dimensional maximally supersymmetric "little string theories" and describe their definition, some of their simple properties, the motivations for studying them, the DLCQ and holographic constructions of these theories and their behaviour at finite energy density. (Contribution to the proceedings of Strings '99 in Potsdam, Germany.)
Following the conjectured duality between near-horizon NS5-branes and little string theory, the string-corrected thermodynamics of near-horizon NS5-branes is studied and found to agree with the statistical thermodynamics of a 5+1 dimensional supersymmetric string theory near the Hagedorn temperature. Specifically, tree-level corrections to the temperature are argued to vanish, in accordance with the duality, while the one-loop string correction to the NS5-brane thermodynamics is shown to generate the correct temperature dependence of the entropy.
We study the propagation of a massless minimally coupled scalar in the near horizon geometry of non-extremal NS5-branes. Using the holographic principle for dilatonic backgrounds we compute the two-point function of an operator in Little String Theory at the Hagedorn temperature. We then comment on relations with correlation functions in two dimensional string theory.
We discuss partition functions of N=(4,4) type IIA string theory on the pp-wave background. This theory is shown to be modular invariant. The boundary states are constructed and possible D-brane instantons are classified. Then we calculate cylinder amplitudes in both closed and open string descriptions and check the open/closed string duality. Furthermore we consider general properties of modular invariant partition functions in the case of pp-waves.
We propose to use the identification of topological string partition functions as equivariant indices on framed moduli spaces of instantons to study the Gopakumar-Vafa conjecture for some local Calabi-Yau geometries.
We formulate a refined version of the Ooguri-Strominger-Vafa (OSV) conjecture. The OSV conjecture that Z_{BH} = |Z_{top}|^2 relates the BPS black hole partition function to the topological string partition function Z_{top}. In the refined conjecture, Z_{BH} is the partition function of BPS black holes counted with spin, or more precisely the protected spin character. Z_{top} becomes the partition function of the refined topological string, which is itself an index. Both the original and the refined conjecture are examples of large N duality in the 't Hooft sense. The refined conjecture applies…
We refine the relation between the renormalized partition function of the open bosonic string in background fields and the effective action. In the process, we get some leading derivative corrections to the Born-Infeld action which include all powers of the gauge field.
We examine the Hagedorn behavior of little string theory using its conjectured duality with near-horizon NS5-branes. In particular, by studying the string-corrected NS5-brane supergravity solution, it is shown that tree-level corrections to the temperature vanish, while the leading one-loop string correction generates the correct temperature dependence of the entropy near the Hagedorn temperature. Finally, the Hagedorn behavior of ODp-brane theories, which are deformed versions of little string theory, is considered via their supergravity duals.
We study BPS bound states of little strings in a limit where they realise monopole strings in five dimensional gauge theories. The latter have gauge group $U(M)^N$ and arise from compactification of $(1,0)$ little string theories of type $A_{M-1} \times A_{N-1}$. We find evidence that the partition function of a certain subclass of monopole strings of charge $(k,\ldots,k)$ ($k\geq 1$) is expressible as the partition function of a symmetric orbifold sigma model, whose target space is precisely the symmetric product of the moduli space of monopoles with charge $(1, \ldots, 1)$.
At high energy densities, fivebranes are populated by a Hagedorn phase of so-called "little strings", whose statistical mechanics underlies black fivebrane thermodynamics. A particular limit of this phase yields BTZ black holes in $AdS_3$, leading us to the idea that in this context fuzzballs and highly excited little strings are one and the same. We explore these ideas through an analysis of D-brane probes of fivebrane supertube backgrounds. String theory dynamics on these backgrounds is described by an exactly solvable null-gauged WZW model. We develop the formalism of null gauging on worlds…
In this paper we show that the polynomial structure of the topological string partition function found by Yamaguchi and Yau for the quintic holds for an arbitrary Calabi-Yau manifold with any number of moduli. Furthermore, we generalize these results to the open topological string partition function as discussed recently by Walcher and reproduce his results for the real quintic.
It has been conjectured that Little String Theories in six dimensions are holographic to critical string theory in a linear dilaton background. We test this conjecture for theories arising on the worldvolume of heterotic fivebranes. We compute the spectrum of chiral primaries in these theories and compare with results following from Type I-heterotic duality and the AdS/CFT correspondence. We also construct holographic duals for heterotic fivebranes near orbifold singularities. Finally we find several new Little String Theories which have Spin(32)/Z_2 or E_8 \times E_8 global symmetry but do no…
We study the 2d N = 4 gauge theory descriptions of little strings on type II NS5- branes. The IIB strings on N NS5-branes are described by the N = (4,4) gauge theories, whose Higgs branch CFTs on U(N) instanton moduli spaces are relevant. The IIA strings are described by N = (4,4) circular A_{N-1} quiver theories, whose Coulomb branch CFTs are relevant. We study new N = (0,4) quiver gauge theories for the IIA strings, which make it easier to study some infrared observables. In particular, we show that the elliptic genera of the IIA / IIB strings precisely map to each other by T-duality.
We construct an M-theory dual of a 6 dimensional little string theory with reduced supersymmetry, along the lines of Polchinski and Strassler. We find that upon perturbing the (2,0) theory with an R-current, the M5 branes polarize into a wrapped Kaluza Klein monopole, whose isometry direction is along the R current. We investigate the properties of this theory.
We argue that two four-dimensional strongly coupled superconformal field theories, on the Higgs branch in certain large N limits, become respectively (2,0) theory and (1,1) little string theory in six dimensions. We identify the spectrum of states responsible for the generation of the two extra dimensions and string winding modes. We establish the equivalence using orbifold realizations of the field theories and exploiting string dualities. We also speculate on deconstructions of M-theory.
Certain six-dimensional (1,0) supersymmetric little string theories, when compactified on $T^3$, have moduli spaces of vacua given by smooth K3 surfaces. Using ideas of Gaiotto-Moore-Neitzke, we show that this provides a systematic procedure for determining the Ricci-flat metric on a smooth K3 surface in terms of BPS degeneracies of (compactified) little string theories.
We study little string theory (LST) compactified on $\mathbf{T}^2$, partially breaking supersymmetry by a discrete T-duality twist acting on both the Kähler and the complex structure of the torus. This setup gives raise to 4d $\mathcal{N}=3$ models and it can be performed in both the type IIA and type IIB LSTs. We comment on the relation with other constructions proposed in the literature.
We continue to explore the conjectural expressions of the Gromov-Witten potentials for a class of elliptically and K3 fibered Calabi-Yau 3-folds in the limit where the base P^1 of the K3 fibration becomes infinitely large. At least in this limit we argue that the string partition function (=the exponential generating function of the Gromov-Witten potentials) can be expressed as an infinite product in which the Kahler moduli and the string coupling are treated somewhat on an equal footing. Technically speaking, we use the exponential lifting of a weight zero Jacobi form to reach the infinite pr…
We initiate the study of (2,0) little string theory of ADE type using its definition in terms of IIB string compactified on an ADE singularity. As one application, we derive a 5d ADE quiver gauge theory that describes the little string compactified on a sphere with three full punctures, at low energies. As a second application, we show the partition function of this theory equals the 3-point conformal block of ADE Toda CFT, q-deformed. To establish this, we generalize the A_n triality of \cite{AHS} to all ADE Lie algebras; IIB string perspective is crucial for this as well.
We show that the system of $k$ NS5-branes wrapping $\mathbb{T}^4\times S^1$ has non-trivial vacuum structure. Different vacua have different spectra of 1/4 BPS states that carry momentum and winding around the $S^1$. In one vacuum, such states are described by black holes; in another, they can be thought of as perturbative BPS states in Double Scaled Little String Theory. In general, both kinds of states are present. We compute the degeneracy of perturbative BPS states exactly, and show that it differs from that of the corresponding black holes. We comment on the implication of our results to …
We discuss a class of Little String Theories (LSTs) whose low energy descriptions are supersymmetric gauge theories on the $Ω$-background with gauge group $U(N)$ and matter in the adjoint representation. We show that the instanton partition function of these theories can be written in terms of Kronecker-Eisenstein series, which in a particular limit of the deformation parameters of the $Ω$-background organise themselves into Greens functions of free scalar fields on a torus. We provide a concrete identification between (differences of) such propagators and Nekrasov subfunctions. The latter are…
A double scaling limit can be defined in string theory on a Calabi-Yau (CY) manifold by approaching a point in moduli space where the CY space develops an isolated singularity and at the same time taking the string coupling to zero, while keeping a particular combination of the two parameters fixed. This leads to a decoupled theory without gravity which has a weak coupling expansion, and can be studied using a holographically dual non-critical superstring description. The usual ``Little String Theory'' corresponds to the strong coupling limit of this theory. We use holography to compute two an…
It is emphasized that compactified little string theories have compact moduli spaces of vacua, which globally probe compact string geometry. Compactifying various little string theories on T^3 leads to 3d theories with exact, quantum Coulomb branch given by: an arbitrary T^4 of volume M_s^2, an arbitrary K3 of volume M_s^2, and moduli spaces of G=SU(N), SO(2N), or E_{6,7,8} instantons on an arbitrary T^4 or K3 of fixed volume. Compactifying instead on a T^2 leads to 4d theories with a compact Coulomb branch base which, when combined with the exact photon gauge coupling fiber, is a compact, ell…
We propose that Little String Theories in six dimensions are quasilocal quantum field theories. Such field theories obey a modification of Wightman axioms which allows Wightman functions (i.e. vacuum expectation values of products of fundamental fields) to grow exponentially in momentum space. Wightman functions of quasilocal fields in x-space violate microlocality at short distances. With additional assumptions about the ultraviolet behavior of quasilocal fields, one can define approximately local observables associated to big enough compact regions. The minimum size of such a region can be i…
In this paper we consider the interrelation between compactified string theories on torus and gauge fields on it. We start from open string theories with background gauge fields and derive partition functions by path integral. Since the effects of background fields and compactification correlate only through string zero modes, we investigate these zero modes. From this point of view, we discuss the Wilson loop mechanism at finite temperature. For the closed string, only a few comments are mentioned.
Higher genus partition functions of string world sheets with boundaries are relevant, e.g. for computation of quantum corrections to Wilson loop expectation values. As a preparation for a possible study of strings in curved space like AdS here we consider examples of genus one partition functions of string world-sheets ending on a circle in the bosonic string theory in flat space. We begin with the partition function for annular topology, writing it as an integral over the modulus of the annulus. In the process, we compute the determinant of the Laplacian on the annulus for Dirichlet-Dirichlet…
We compute the partition function of the supersymmetric two-dimensional Euclidean black hole geometry described by the SL(2,R)/U(1) superconformal field theory. We decompose the result in terms of characters of the N=2 superconformal symmetry. We point out puzzling sectors of states besides finding expected discrete and continuous contributions to the partition function. By adding an N=2 minimal model factor of the correct central charge and projecting on integral N=2 charges we compute the partition function of the background dual to little string theory in a double scaling limit. We show the…
Quadratic tachyon profile has been discussed in the boundary string field theory. We here compute the g-function by factorizing the cylinder amplitude. The answer is compared with the disc partition function. The boundary state is constructed. We extend these computations to those of the boundary sine-Gordon model at the free fermion point.
We study the compactification of the $(2,0)$ and type-II little-string theories on $S^1$, $T^2$ and $T^3$ with an R-symmetry twist that preserves half the supersymmetry. We argue that it produces the same moduli spaces of vacua as compactification of the $(1,0)$ theory with $E_8$ Wilson lines given by a maximal embedding of SU(2). In certain limits, this reproduces the moduli space of SU(2) with a massive adjoint hyper-multiplet. In the type-II little-string theory case, we observe a peculiar phase transition where the strings condense. We conjecture a generalization to more than two 5-branes …
We determine the 2-group structure constants for all the six-dimensional little string theories (LSTs) geometrically engineered in F-theory without frozen singularities. We use this result as a consistency check for T-duality: the 2- groups of a pair of T-dual LSTs have to match. When the T-duality involves a discrete symmetry twist the 2-group used in the matching is modified. We demonstrate the matching of the 2-groups in several examples.
We study NSR strings in the Nappi-Witten background, which is the Penrose limit of a certain NS5-brane supergravity solution. We solve the theory in the light-cone gauge, obtaining the spectrum, which is space-time supersymmetric. In light of the LST/NS5-brane duality, this spectrum should be in correspondence with the states of little string theory in the appropriate limit. A semiclassical analysis verifies that the relationship between energy and angular momentum, after a field redefinition, matches the known result for a flat background.
We show that little string theory on S^5 can be obtained as double-scaling limits of the maximally supersymmetric Yang-Mills theories on RxS^2 and RxS^3/Z_k. By matching the gauge theory parameters with those in the gravity duals found by Lin and Maldacena, we determine the limits in the gauge theories that correspond to decoupling of NS5-brane degrees of freedom. We find that for the theory on RxS^2, the 't Hooft coupling must be scaled like ln^3(N), and on RxS^3/Z_k, like ln^2(N). Accordingly, taking these limits in these field theories gives Lagrangian definitions of little string theory on…
This is a short review of the newly discovered ODp-theories that are non-gravitational six-dimensional theories defined as the decoupling limit of NS5-branes in the presence of a near-critical (p+1)-form RR fields. We discuss the motivation for these new theories, their definitions and properties, and their relation to NCOS theory, OM theory and Little String Theory, focusing on the cases p=1,2.
The dominant contribution to the semicanonical partition function of dyonic black holes of N=4 string theory is computed for generic charges, generalizing recent results of Shih and Yin. The result is compared to the black hole free energy obtained from the conjectured relation to topological strings. If certain perturbative corrections are included agreement is found to subleading order. These corrections modify the conjectured relation and implement covariance with respect to electric-magnetic duality transformations.
MacMahon's classic generating function of random plane partitions, which is related to Schur polynomials, was recently extended by Vuletic to a generating function of weighted plane partitions that is related to Hall-Littlewood polynomials, S(t), and further to one related to Macdonald polynomials, S(t,q). Using Jing's 1-parameter deformation of charged free fermions, we obtain a Fock space derivation of the Hall-Littlewood extension. Confining the plane partitions to a finite s-by-s square base, we show that the resulting generating function, S_{s-by-s}(t), is an evaluation of a tau-function …
We study the elliptic genera of 6d strings based on their modular properties. They are weak Jacobi forms of weight 0, whose indices are determined from the 2d chiral anomalies. We propose the ansatz for the elliptic genera which reflects the analytic structure of instanton partition functions. Given a finite amount of initial BPS data, we completely determine the elliptic genera of 6d strings in various 6d SCFTs. We also apply our ansatz to study $\mathcal{N}=(2,0)$ and $(1,1)$ little strings as well as $\mathcal{N}=(1,0)$ heterotic little strings, for which T-duality of little string theories…
We present evidence for a new deconstruction of Little String Theory (LST). The starting point is a four-dimensional conformal field theory on its Higgs branch which provides a lattice regularization of six-dimensional gauge theory. We argue that the corresponding continuum limit is a 't Hooft large-N limit of the same four-dimensional theory on an S-dual confining branch. The AdS/CFT correspondence is then used to study this limit in a controlled way. We find that the limit yields LST compactified to four dimensions on a torus of fixed size. The limiting theory also contains other massive and…
T-duality is one of the essential elements of string theory. Recently, Hull has developed a formalism where the dimension of the target space is doubled so as to make T-duality manifest. This is then supplemented with a constraint equation that allows the connection to the usual string sigma model. This paper analyses the partition function of the doubled formalism by interpreting the constraint equation as that of a chiral scalar and then using holomorphic factorisation techniques to determine the partition function. We find there is quantum equivalence to the ordinary string once the topolog…
We calculate gauge instanton corrections to a class of higher derivative string effective couplings introduced in [1]. We work in Type I string theory compactified on K3xT2 and realise gauge instantons in terms of D5-branes wrapping the internal space. In the field theory limit we reproduce the deformed ADHM action on a general Ω-background from which one can compute the non-perturbative gauge theory partition function using localisation. This is a non-perturbative extension of [1] and provides further evidence for our proposal of a string theory realisation of the Ω-background.
We study the high energy thermodynamics of Little String Theory, using its holographic description. This leads to the entropy-energy relation $S=β_H E+α\log E+O(1/E)$. We compute $α$ and show that it is negative; as a consequence, the high energy thermodynamics is unstable. We exhibit a mode localized near the horizon of the black brane, which has winding number one around Euclidean time and a mass that vanishes at large $E$ (or $β\toβ_H$). We argue that the high temperature phase of the theory involves condensation of this mode.
We study tree level scattering amplitudes of four massless states in the double scaled little string theory, and compare them to perturbative loop amplitudes in six-dimensional super-Yang-Mills theory. The little string amplitudes are computed from correlators in the cigar coset CFT and in N=2 minimal models. The results are expressed in terms of integrals of conformal blocks and evaluated numerically in the alpha' expansion. We find striking agreements with up to 2-loop scattering amplitudes of massless gluons in 6D SU(k) SYM at a Z_k invariant point on the Coulomb branch. We comment on the i…
We study the stringy instanton partition function of four dimensional ${\cal N}=2$ $U(N)$ supersymmetric gauge theory which was obtained by Bonelli et al in 2013. In type IIB string theory on $\mathbb{C}^2\times T^*\mathbb{P}^1\times \mathbb{C}$, the stringy $U(N)$ instantons of charge $k$ are described by $k$ D1-branes wrapping around the $\mathbb{P}^1$ bound to $N$ D5-branes on $\mathbb{C}^2\times \mathbb{P}^1$. The KK corrections induced by compactification of the $\mathbb{P}^1$ give the stringy corrections. We find a relation between the stringy instanton partition function whose quantum s…
We study up to 8-derivative terms in the Coulomb branch effective action of (1,1) little string theory, by collecting results of 4-gluon scattering amplitudes from both perturbative 6D super-Yang-Mills theory up to 4-loop order, and tree-level double scaled little string theory (DSLST). In previous work we have matched the 6-derivative term from the 6D gauge theory to DSLST, indicating that this term is protected on the entire Coulomb branch. The 8-derivative term, on the other hand, is unprotected. In this paper we compute the 8-derivative term by interpolating from the two limits, near the o…
A formula was recently proposed for the perturbative partition function of certain $\mathcal N=1$ gauge theories on the round four-sphere, using an analytic-continuation argument in the number of dimensions. These partition functions are not currently accessible via the usual supersymmetric-localisation technique. We provide a natural refinement of this result to the case of the ellipsoid. We then use it to write down the perturbative partition function of an $\mathcal N=1$ toroidal-quiver theory (a double orbifold of $\mathcal N=4$ super Yang-Mills) and show that, in the deconstruction limit,…
