We study FZZT-branes and open string amplitudes in (p,q) minimal string theory. We focus on the simplest boundary changing operators in two-matrix models, and identify the corresponding operators in worldsheet theory through the comparison of amplitudes. Along the way, we find a novel linear relation among FZZT boundary states in minimal string theory. We also show that the boundary ground ring is realized on physical open string operators in a very simple manner, and discuss its use for perturbative computation of higher open string amplitudes.
Loop amplitudes in (p,q) minimal string theory are studied in terms of the continuum string field theory based on the free fermion realization of the KP hierarchy. We derive the Schwinger-Dyson equations for FZZT disk amplitudes directly from the W_{1+\infty} constraints in the string field formulation and give explicitly the algebraic curves of disk amplitudes for general backgrounds. We further give annulus amplitudes of FZZT-FZZT, FZZT-ZZ and ZZ-ZZ branes, generalizing our previous D-instanton calculus from the minimal unitary series (p,p+1) to general (p,q) series. We also give a detailed …
We revisit the annulus amplitude of FZZT branes with general matter sectors (r,s) using the recent development of matrix model and minimal Liouville gravity. Following the boundary description of the 1-matrix model and bulk resonance transformation between primary operators we find the consistency of the brane decomposition into (1,1)-branes. We also investigate the corresponding results obtained directly from the minimal Liouville gravity and demonstrate the perfect agreement with the matrix results.
We study the annulus amplitudes of (p,q) minimal string theory. Focusing on the ZZ-FZZT annulus amplitude as a target-space probe of the ZZ brane, we use it to confirm that the ZZ branes are localized in the strong-coupling region. Along the way we learn that the ZZ-FZZT open strings are fermions, even though our theory is bosonic! We also provide a geometrical interpretation of the annulus amplitudes in terms of the Riemann surface M_{p,q} that emerges from the FZZT branes. The ZZ-FZZT annulus amplitude measures the deformation of M_{p,q} due to the presence of background ZZ branes; each kind…
In this paper we continue studying the decay of unstable FZZT branes initiated in [1],[2]. The mass of tachyonic mode in this model can be chosen arbitrarily small and we use it as a perturbative parameter. In [2] a time-dependent boundary conformal field theory (BCFT) describing the decay process was studied and it was shown that in a certain sense this BCFT interpolates between two stationary BCFT's corresponding to the UV and IR fixed points of the associated RG flow. In the present work we find in the leading order vertex operators of the time-dependent BCFT. We identify the "in" and "out"…
The exact FZZT brane partition function for topological gravity with matter is computed using the dual two-matrix model. We show how the effective theory of open strings on a stack of FZZT branes is described by the generalized Kontsevich matrix integral, extending the earlier result for pure topological gravity. Using the well-known relation between the Kontsevich integral and a certain shift in the closed-string background, we conclude that these models exhibit open/closed string duality explicitly. Just as in pure topological gravity, the unphysical sheets of the classical FZZT moduli space…
Minimal string theory has a number of FZZT brane boundary states; one for each Cardy state of the minimal model. It was conjectured by Seiberg and Shih that all branes in a minimal string theory could be expressed as a linear combination of the brane associated to the identity operator of the minimal model with complex shifts in the boundary cosmological constant. Subsequently it was found that this identification of FZZT branes does not hold exactly for some cylinder amplitudes but was spoiled by terms that are associated with vanishing worldsheet area and are therefore non-universal. In this…
We study boundary states in (p,q) minimal superstring theory, combining the explicit form of matter wave functions. Within the modular bootstrap framework, Cardy states of (p,q) minimal superconformal field theory are completely determined in both cases of the different supercharge combinations, and the remaining consistency checks in the super-Liouville case are also performed. Using these boundary states, we determine the explicit form of FZZT- and ZZ-brane boundary states both in type 0A and 0B GSO projections. Annulus mplitudes of FZZT branes are evaluated and principal FZZT branes are ide…
We point out that insertions of matrix fields in (connected amputated) amplitudes of (generalized) Kontsevich models are given by covariant derivatives with respect to the Kontsevich moduli. This implies that correlators are sections of symmetric products of the (holomorphic) tangent bundle on the (complexified) moduli space of FZZT Liouville branes. We discuss the relation of Kontsevich parametrization of moduli space with that provided by either the (p,1) or the (1,p) boundary conformal field theories. It turns out that the Kontsevich connection captures the contribution of contact terms to …
We study the annulus amplitudes in the (2,4) minimal superstring theory using the continuum worldsheet approach. Our results reproduce the semiclassical behavior of the wavefunctions of FZZT-branes recently studied in hep-th/0412315 using the dual matrix model. We also study the multi-point functions of neutral FZZT-branes and find the agreement between their semiclassical limit and the worldsheet annulus calculation.
We study both the classical and the quantum target space of (p,q) minimal string theory, using the FZZT brane as a probe. By thinking of the target space as the moduli space of FZZT branes, parametrized by the boundary cosmological constant x, we see that classically it consists of a Riemann surface \CM_{p,q} which is a p-sheeted cover of the complex x plane. However, we show using the dual matrix model that the exact quantum FZZT observables exhibit Stokes' phenomenon and are entire functions of x. Along the way we clarify some points about the semiclassical limit of D-brane correlation funct…
The 2D quantum gravity on a disc, or the non-critical theory of open strings, is known to exhibit an integrable structure, the boundary ground ring, which determines completely the boundary correlation functions. Inspired by the recent progress in boundary Liouville theory, we extend the ground ring relations to the case of non-vanishing boundary Liouville interaction known also as FZZT brane in the context of the 2D string theory. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions. The ring action closes on an infinite arr…
We continue to investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence begun in arxiv:0708.0645. We derive the master matrix of the $(2,1)$ minimal and $(3,1)$ minimal matrix model. We use it's characteristic polynomial to understand why the zeros of the FZZT partition function, which is the Airy function, lie on the real axis. We also introduce an iterative procedure that can describe the Riemann $Ξ$ function as a deformed minimal model whose deformation parameters are related to a Konsevich integr…
We interpret the matrix boundaries of the one matrix model (1MM) recently constructed by two of the authors as an outcome of a relation among FZZT branes. In the double scaling limit, the 1MM is described by the (2,2p+1) minimal Liouville gravity. These matrix operators are shown to create a boundary with matter boundary conditions given by the Cardy states. We also demonstrate a recursion relation among the matrix disc correlator with two different boundaries. This construction is then extended to the two matrix model and the disc correlator with two boundaries is compared with the Liouville …
We study both bosonic and supersymmetric (p,q) minimal models coupled to Liouville theory using the ground ring and the various branes of the theory. From the FZZT brane partition function, there emerges a unified, geometric description of all these theories in terms of an auxiliary Riemann surface M_{p,q} and the corresponding matrix model. In terms of this geometric description, both the FZZT and ZZ branes correspond to line integrals of a certain one-form on M_{p,q}. Moreover, we argue that there are a finite number of distinct (m,n) ZZ branes, and we show that these ZZ branes are located a…
We propose a paradigm for realizing the SYK model within string theory. Using the large $N$ matrix description of $c<1$ string theory, we show that the effective theory on a large number $Q$ of FZZT D-branes in $(p,1)$ minimal string theory takes the form of the disorder averaged SYK model with $J ψ^{p}$ interaction. The SYK fermions represent open strings between the FZZT branes and the ZZ branes that underly the matrix model. The continuum SYK dynamics arises upon taking the large $Q$ limit. We observe several qualitative and quantitative links between the SYK model and $(p,q)$ minimal strin…
This thesis discusses the worldsheet geometries in the (p,q) minimal model coupled to 2D euclidean Quantum Gravity with particular focus on non-compact geometries. We first calculate the FZZT-FZZT cylinder amplitudes for all pairs of Cardy matter states thereby generalizing the result obtained in arXiv:hep-th/0406030 and discuss the decompositions of the cylinder amplitudes in terms of solutions to the homogeneous Wheeler-DeWitt equation. We then show, that the principal ZZ boundary conditions in Liouville theory can be viewed as effective boundary conditions obtained by integrating out the ma…
We demonstrate that a Hermitian matrix model can be derived from level truncated open string field theory with Chan-Paton factors. The Hermitian matrix is coupled with a scalar and a $U(N)$ vector which are responsible for the D-brane at the tachyon vacuum. Effective potential for the scalar is evaluated both in finite and large $N$. Increase of potential height is observed in both examples. The large $N$ matrix integral is identified with a system of $N$ ZZ branes and a ghost FZZT brane.
The FZZT and ZZ branes in (p,p+1) minimal string theory are studied in terms of continuum loop equations. We show that systems in the presence of ZZ branes (D-instantons) can be easily investigated within the framework of the continuum string field theory developed by Yahikozawa and one of the present authors (hep-th/9609210). We explicitly calculate the partition function of a single ZZ brane for arbitrary p. We also show that the annulus amplitudes of ZZ branes are correctly reproduced.
We study the connection between minimal Liouville string theory and generalized open KdV hierarchies. We are interested in generalizing Douglas string equation formalism to the open topology case. We show that combining the results of the closed topology, based on the Frobenius manifold structure and resonance transformations, with the appropriate open case modification, which requires the insertion of macroscopic loop operators, we reproduce the well-known result for the expectation value of a bulk operator for the FZZT brane coupled to the general (q,p) minimal model. The matching of the res…
We clarify some aspects of the map between the c=1 string theory at self-dual radius and the topologically twisted cigar at level one. We map the ZZ and FZZT D-branes in the c=1 string theory at self dual radius to the localized and extended branes in the topological theory on the cigar. We show that the open string spectrum on the branes in the two theories are in correspondence with each other, and their two point correlators are equal. We also find a representation of an extended N=2 algebra on the worldsheet which incorporates higher spin currents in terms of asymptotic variables on the ci…
We study open strings in the noncritical $c=1$ bosonic string theory compactified on a circle at self-dual radius. These strings live on D-branes that are extended along the Liouville direction ({\it FZZT} branes). We present explicit expressions for the disc two- and three-point functions of boundary operators in this theory, as well as the bulk-boundary two-point function. The expressions obtained are divergent because of resonant behaviour at self-dual radius. However, these can be regularised and renormalized in a precise way to get finite results. The boundary correlators are found to dep…
We study the scattering of long strings in c = 1 string theory, both in the worldsheet description and in the non-singlet sector of the dual matrix quantum mechanics. From the worldsheet perspective, the scattering amplitudes of long strings are obtained from a decoupling limit of open strings amplitudes on FZZT branes, which we compute by integrating Virasoro conformal blocks along with structure constants of boundary Liouville theory. In particular, we study the tree level amplitudes of (1) a long string decaying by emitting a closed string, and (2) the scattering of a pair of long strings. …
We explore the non-singlet sector of matrix quantum mechanics dual to $c=1$ Liouville theory. The non-singlets are obtained by adding $N_{f}\times N$ bi-fundamental fields in the gauged matrix quantum mechanics model as well as a one dimensional Chern-Simons term. The present model is associated with a spin-Calogero model in the presence of external magnetic field. In chiral variables, the low energy excitations-currents satisfy an ${SU(2N_{f})}_{\tilde{k}}$ Kǎc-Moody algebra at large $N$. We analyse the canonical partition function as well as two and four point correlation functions, discuss …
The c=1 Liouville theory has received some attention recently as the Euclidean version of an exact rolling tachyon background. In an earlier paper it was shown that the bulk theory can be identified with the interacting c=1 limit of unitary minimal models. Here we extend the analysis of the c=1-limit to the boundary problem. Most importantly, we show that the FZZT branes of Liouville theory give rise to a new 1-parameter family of boundary theories at c=1. These models share many features with the boundary Sine-Gordon theory, in particular they possess an open string spectrum with band-gaps of…
A string field theory of (p,q) minimal superstrings is constructed with the free-fermion realization of 2-component KP (2cKP) hierarchy, starting from 2-cut ansatz of two-matrix models. Differential operators of 2cKP hierarchy are identified with operators in super Liouville theory, and we obtain algebraic curves for the disk amplitudes of η=-1 FZZT-branes and the partition functions of neutral/charged η=-1 ZZ branes, which correctly reproduce those of type 0B (p,q) minimal superstrings in conformal backgrounds. In the course of study, some subtle points are clarified, including a difference o…
We show how a boundary state different from the (1,1) Cardy state may be realised in the (m,m+1) minimal string by the introduction of an auxiliary matrix into the standard two hermitian matrix model. This boundary is a natural generalisation of the free spin boundary state in the Ising model. The resolvent for the auxiliary matrix is computed using an extension of the saddle-point method of Zinn-Justin to the case of non-identical potentials. The structure of the saddle-point equations result in a Seiberg-Shih like relation between the boundary states which is valid away from the continuum li…
We construct the boundary ground ring in c < 1 open string theories with non-zero boundary cosmological constant (FZZT brane), using the Coulomb gas representation. The ring relations yield an over-determined set of functional recurrence equations for the boundary correlation functions, which involve shifts of the target space momenta of the boundary fields as well as the boundary parameters on the different segments of the boundary.
We argue that topological matrix models (matrix models of the Kontsevich type) are examples of exact open/closed duality. The duality works at finite N and for generic `t Hooft couplings. We consider in detail the paradigm of the Kontsevich model for two-dimensional topological gravity. We demonstrate that the Kontsevich model arises by topological localization of cubic open string field theory on N stable branes. Our analysis is based on standard worldsheet methods in the context of non-critical bosonic string theory. The stable branes have Neumann (FZZT) boundary conditions in the Liouville …
We show how non-compact space-time (ZZ branes) emerges as a limit of compact space-time (FZZT branes) for specific ratios between the square of the boundary cosmological constant and the bulk cosmological constant in the (2,2m - 1) minimal model coupled to two-dimensional euclidean quantum gravity. Furthermore, we show that the principal (r,s) ZZ brane can be viewed as the basic (1,1) ZZ boundary state tensored with a (r,s) Cardy boundary state for a general (p,q) minimal model coupled to two-dimensional quantum gravity. In this sense there exists only one ZZ boundary state, the basic (1,1) bo…
We study the boundary N=2 Liouville theory based on the ``modular bootstrap'' approach. As fundamental conformal blocks we introduce the ``extended characters'' that are defined as the proper sums over spectral flows of irreducible characters of the N=2 superconformal algebra (SCA) and clarify their modular transformation properties in models with rational central charges. We then try to classify the Cardy states describing consistent D-branes based on the modular data. We construct the analogues of ZZ-branes (hep-th/0101152), localized at the strong coupling region, and the FZZT-branes (hep-t…
We propose a formalism inspired by matrix models to compute open and closed topological string amplitudes in the B-model on toric Calabi-Yau manifolds. We find closed expressions for various open string amplitudes beyond the disk, and in particular we write down the annulus amplitude in terms of theta functions on a Riemann surface. We test these ideas on local curves and local surfaces, providing in this way generating functionals for open Gromov-Witten invariants in the spirit of mirror symmetry. In the case of local curves, we study the open string sector near the critical point which leads…
Type 0A string theory in the (2,4k) superconformal minimal model backgrounds, with background ZZ D-branes or R-R fluxes can be formulated non-perturbatively. The branes and fluxes have a description as threshold bound states in an associated one-dimensional quantum mechanics which has a supersymmetric structure, familiar from studies of the generalized KdV system. The relevant bound state wavefunctions in this problem have unusual asymptotics (they are not normalizable in general, and break supersymmetry) which are consistent with the underlying description in terms of open and closed string s…
We study the c_L=25 limit, which corresponds to c=1 string theory, of bulk and boundary correlation functions of Liouville theory with FZZT boundary conditions. This limit is singular and requires a renormalization of vertex operators. We formulate a regularization procedure which allows to extract finite physical results. A particular attention is paid to c=1 string theory compactified at the self-dual radius R=1. In this case, the boundary correlation functions diverge even after the multiplicative renormalization. We show that all infinite contributions can be interpreted as contact terms a…
We determine all the correlators of the H3+ model on a disc with AdS2-brane boundary conditions in terms of correlators of Liouville theory on a disc with FZZT-brane boundary conditions. We argue that the Cardy-Lewellen constraints are weaker in the H3+ model than in rational conformal field theories due to extra singularities of the correlators, but strong enough to uniquely determine the bulk two-point function on a disc. We confirm our results by detailed analyses of the bulk-boundary two-point function and of the boundary two-point function. In particular we find that, although the target …
Multi-cut two-matrix models are studied in the Z_k symmetry breaking k-cut (\hat p,\hat q) critical points which should correspond to (\hat p,\hat q) minimal k-fractional superstring theory. FZZT-brane or macroscopic loop amplitudes are obtained in all of these critical points and found to have two kinds of solutions in general. Each of these solutions is expressed by hyperbolic cosine or sine functions with proper phase shifts. The algebraic geometries and ZZ-brane disk amplitudes (instanton actions) of these solutions are also studied. In particular, our results suggest that minimal \infty-f…
In this paper we consider tree-level scattering in the minimal N=4 topological string and show that a large class of N-point functions can be recast in terms of corresponding amplitudes in the (1,k) minimal bosonic string. This suggests a non-trivial relation between the minimal N=4 topological strings, the (1,k) minimal bosonic strings and their corresponding ADE matrix models. This relation has interesting and far-reaching implications for the topological sector of six-dimensional Little String Theories.
We study the scattering amplitudes in the N=2 minimal string or equivalently in the N=4 topological string on ALE spaces. We find an interesting connection between the tree level amplitudes of the N=2 minimal string and those of the (1,n) minimal bosonic string. In particular we show that the four and five-point functions of the N=2 string can be directly rewritten in terms of those of the latter theory. This relation offers a map of physical states between these two string theories. Finally we propose a possible matrix model dual for the N=2 minimal string in the light of this connection.
Some results are presented concerning duality invariant effective string actions and the construction of automorphic functions for general (2,2) string compactifications. These considerations are applied in order to discuss the {\it minimal} unification of gauge coupling constants in orbifold compactifications with special emphasis on string threshold corrections.
We summarize recent progress in the understanding of minimal string theory, focusing on the worldsheet description of physical operators and D-branes. We review how a geometric interpretation of minimal string theory emerges naturally from the study of the D-branes. This simple geometric picture ties together many otherwise unrelated features of minimal string theory, and it leads directly to a worldsheet derivation of the dual matrix model.
We obtain relations among boundary states in bosonic minimal open string theory using the boundary ground ring. We also obtain a difference equation that boundary correlators must satisfy.
We study the non-minimal pure spinor string in a curved background. We find that the minimal BRST invariance implies the existence of a non-trivial stress-energy tensor for the minimal and non-minimal variables in the heterotic curved background. We find constraint equations for the b ghost. We construct the b ghost as a solution of these constraints.
We use insights from string field theory to analyze and cure the divergences in the cylinder diagram in minimal string theory with both boundaries lying on a ZZ brane. We focus on theories with worldsheet matter consisting of the $(2,p)$ minimal model plus Liouville theory, with total central charge 26, together with the usual $bc$-ghosts. The string field theory procedure gives a finite, purely imaginary normalization constant for non-perturbative effects in minimal string theory, or doubly non-perturbative effects in JT gravity. We find precise agreement with the prediction from the dual dou…
Minimal N=1/2 supersymmetric extension of bosonic Polyakov's string is constructed. This model is natural generalization of Di Vecchia-Ravndal superparticle. The classical sector of the model is investigated, Noether currents and Virosoro supercondition are found. Minimal spinning string is more simple, than the standard N=1 spinning string of Neveu-Schwarz-Ramond and has a number of unusial properties such as a chiral symmetry, parabolic type of equations of movement, non-triviality fermionic sectors for closed strings only and e.t.c.
We compute the normalization of the general multi-instanton contribution to the partition function of $(p',p)$ minimal string theory and also to the dual two-matrix integral. We find perfect agreement between the two results.
We study the coupling of the non-minimal ghost fields of the pure spinor superstring in general curved backgrounds. The coupling is found solving the consistent relations from the nilpotency of the non-minimal BRST charge.
We reconsider the well-known issue of string corrections to Supergravity theory. Our treatment is carried out to second order in the string slope parameter. We establish a procedure for solving the Bianchi identities in the non minimal case, and we solve a long standing problem in the perturbative expansion of D=10, N=1 string corrected Supergravity, obtaining the H sector tensors, torsions and curvatures.
We use minimal area metrics to generate all nonorientable string diagrams. The surfaces in unoriented string theory have nontrivial open curves and nontrivial closed curves whose neighborhoods are either annuli or Mobius strips. We define a minimal area problem by imposing length conditions on open curves and on annular closed curves only. We verify that the minimal area conditions are respected by the sewing operations. The natural objects that satisfy recursion relations involving the propagator, which performs both orientable and nonorientable sewing, are classes of moduli spaces grouped by…
We study the spectrum of physical states for higher-spin generalisations of string theory, based on two-dimensional theories with local spin-2 and spin-$s$ symmetries. We explore the relation of the resulting effective Virasoro string theories to certain $W$ minimal models. In particular, we show how the highest-weight states of the $W$ minimal models decompose into Virasoro primaries.
We complete the solution to string corrected (deformed), D=10, N=1 Supergravity as the non-minimal low energy limit of string theory. We reaffirm a previously given solution, and we make important corrections to that solution. We solve what was an apparently intractable Bianchi identity in superspace, and we introduce a new important modification to the known first order results. In so doing we show that this approach to string corrected supergravity is indeed a consistent approach and we pave the way for many applications of the results.
We study strings associated with minimal 6d SCFTs, which by definition have only one string charge and no Higgs branch. These theories are labelled by a number n with 1 <= n <= 8 or n = 12. Quiver theories have previously been proposed which describe strings of SCFTs for n = 1, 2. For n > 2 the strings interact with the bulk gauge symmetry. In this paper we find a quiver description for the n = 4 string using Sen's limit of F-theory and calculate its elliptic genus with localization techniques. This result is checked using the duality of F-theory with M-theory and topological string theory who…
We study the simplest examples of minimal string theory whose worldsheet description is the unitary (p,q) minimal model coupled to two-dimensional gravity (Liouville field theory). In the Liouville sector, we show that four-point correlation functions of `tachyons' exhibit logarithmic singularities, and that the theory turns out to be logarithmic. The relation with Zamolodchikov's logarithmic degenerate fields is also discussed. Our result holds for generic values of (p,q).
We give a complete and consistent solution to string corrected (deformed), D=10, N=1 supergravity as the non-minimal low energy limit of string theory. We solve the Bianchi identities with suitable constraints to second order in the string slope parameter. In so doing we pave the way for continuing the study of the many applications of these results. We also modify, reaffirm and correct a previously given incomplete solution, and we introduce an important adjustment to the known first order results.
We show that Type IIB string theory on AdS_3 X S^3 X M_4 with p units of NS flux contains an integrable subsector, isomorphic to the minimal (p,1) bosonic string. To this end, we construct a topological string theory with target space Euclidean AdS_3 X S^3. We use a variant of Hamiltonian reduction to prove its equivalence to the minimal (p,1) string. The topological theory is then embedded in the physical ten-dimensional IIB string theory. Correlators of tachyons in the minimal string are mapped to correlators of spacetime chiral primaries in the IIB theory, in the presence of background 5-fo…
The target space $M_{p,q}$ of $(p,q)$ minimal strings is embedded into the phase space of an associated integrable classical mechanical model. This map is derived from the matrix model representation of minimal strings. Quantum effects on the target space are obtained from the semiclassical mechanics in phase space as described by the Wigner function. In the classical limit the target space is a fold catastrophe of the Wigner function that is smoothed out by quantum effects. Double scaling limit is obtained by resolving the singularity of the Wigner function. The quantization rules for backgro…
