Showing papers on "Topological string theory published in 2007"

String theory and M-theory

Abstract: String theory is one of the most exciting and challenging areas of modern theoretical physics. This book guides the reader from the basics of string theory to very recent developments at the frontier of string theory research. The book begins with the basics of perturbative string theory, world-sheet supersymmetry, space-time supersymmetry, conformal field theory and the heterotic string, and moves on to describe modern developments, including D-branes, string dualities and M-theory. It then covers string geometry (including Calabi–Yau compactifications) and flux compactifications, and applications to cosmology and particle physics. One chapter is dedicated to black holes in string theory and M-theory, and the microscopic origin of black-hole entropy. The book concludes by presenting matrix theory, AdS/CFT duality and its generalizations. This book is ideal for graduate students studyingmodern string theory, and it will make an excellent textbook for a 1-year course on string theory. It will also be useful for researchers interested in learning about developments in modern string theory. The book contains about 120 solved exercises, as well as about 200 homework problems, solutions of which are available for lecturers on a password protected website at www.cambridge.org/9780521860697.

Hurwitz numbers, matrix models and enumerative geometry

Abstract: We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.

Holomorphic anomaly and matrix models

Abstract: The genus g free energies of matrix models can be promoted to modular invariant, non-holomorphic amplitudes which only depend on the geometry of the classical spectral curve. We show that these non-holomorphic amplitudes satisfy the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa. We derive as well holomorphic anomaly equations for the open string sector. These results provide evidence at all genera for the Dijkgraaf-Vafa conjecture relating matrix models to type B topological strings on certain local Calabi-Yau threefolds.

Nonperturbative Effects and the Large-Order Behavior of Matrix Models and Topological Strings

Abstract: This work addresses nonperturbative effects in both matrix models and topological strings, and their relation with the large-order behavior of the 1/N expansion. We study instanton configurations in generic one-cut matrix models, obtaining explicit results for the one-instanton amplitude at both one and two loops. The holographic description of topological strings in terms of matrix models implies that our nonperturbative results also apply to topological strings on toric Calabi-Yau manifolds. This yields very precise predictions for the large-order behavior of the perturbative genus expansion, both in conventional matrix models and in topological string theory. We test these predictions in detail in various examples, including the quartic matrix model, topological strings on the local curve, and Hurwitz theory. In all these cases we provide extensive numerical checks which heavily support our nonperturbative analytical results. Moreover, since all these models have a critical point describing two-dimensional gravity, we also obtain in this way the large-order asymptotics of the relevant solution to the Painleve I equation, including corrections in inverse genus. From a mathematical point of view, our results predict the large-genus asymptotics of simple Hurwitz numbers and of local Gromov-Witten invariants.

Polynomial Structure of the (Open) Topological String Partition Function

Abstract: 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.

Non-Perturbative Corrections and Modularity in N=1 Type IIB Compactifications

Abstract: Non-perturbative corrections and modular properties of four-dimensional type IIB Calabi-Yau orientifolds are discussed. It is shown that certain non-perturbative α' corrections survive in the large volume limit of the orientifold and periodically correct the Kahler potential. These corrections depend on the NS-NS two form and have to be completed by D-instanton contributions to transform covariantely under symmetries of the type IIB orientifold background. It is shown that generically also the D-instanton superpotential depends on the two-form moduli as well as on the complex dilaton. These contributions can arise through theta-functions with the dilaton as modular parameter. An orientifold of the Enriques Calabi-Yau allows to illustrate these general considerations. It is shown that this compactification leads to a controlled four-dimensional = 1 effective theory due to the absence of various quantum corrections. Making contact to the underlying topological string theory the D-instanton superpotential is proposed to be related to a specific modular form counting D3, D1, D(-1) degeneracies on the Enriques Calabi-Yau.

Hurwitz numbers, matrix models and enumerative geometry

Abstract: We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.

D-branes as a bubbling Calabi-Yau

Abstract: We prove that the open topological string partition function on a D-brane configuration in a Calabi-Yau manifold X takes the form of a closed topological string partition function on a different Calabi-Yau manifold Xb. This identification shows that the physics of D-branes in an arbitrary background X of topological string theory can be described either by open+closed string theory in X or by closed string theory in Xb. The physical interpretation of the ``bubbling" Calabi-Yau Xb is as the space obtained by letting the D-branes in X undergo a geometric transition. This implies, in particular, that the partition function of closed topological string theory on certain bubbling Calabi-Yau manifolds are invariants of knots in the three-sphere.

The Wave Function Behavior of the Open Topological String Partition Function on the Conifold

Abstract: We calculate the topological string partition function to all genus on the conifold, in the presence of branes. We demonstrate that the partition functions for different brane backgrounds (smoothly connected along a quantum corrected moduli space) can be interpreted as the same wave function in different polarizations. This behavior has a natural interpretation in the Chern-Simons target space description of the topological theory. Our detailed analysis however indicates that non-perturbatively, a modification of real Chern-Simons theory is required to capture the correct target space theory of the topological string. We perform our calculations in the framework of a free fermion representation of the open topological string, demonstrating that this framework extends beyond the simple 3 geometry. The notion of a fermionic brane creation operator arises in this setting, and we study to what extent the wave function properties of the partition function can be extended to this operator.

Background independence and the open topological string wavefunction

Abstract: The open topological string partition function in the background of a D-brane on a Calabi-Yau threefold specifies a state in the Hilbert space associated with the quantization of the underlying special geometry. This statement is a consequence of the extended holomorphic anomaly equation after an appropriate shift of the closed string variables, and can be viewed as the expression of background independence for the open-closed topological string. We also clarify various other aspects of the structure of the extended holomorphic anomaly equation. We conjecture that the collection of all D-branes furnishes a basis of the Hilbert space, and revisit the BPS interpretation of the open topological string wavefunction in this light.

Evidence for Tadpole Cancellation in the Topological String

Abstract: We study the topological string on compact Calabi-Yau threefolds in the presence of orientifolds and D-branes. In examples, we find that the total topological string amplitude admits a BPS expansion only if the topological charge of the D-brane is equal to that of the orientifold plane. We interpret this as a manifestation of a general tadpole cancellation condition in the topological string that is necessary for decoupling of A- and B-model in loop amplitudes. Our calculations in the A-model involve an adapted version of existing localization techniques, and give predictions for the real enumerative geometry of higher genus curves in Calabi-Yau manifolds. In the B-model, we introduce an extension of the holomorphic anomaly equation to unoriented strings.

D-branes as a Bubbling Calabi-Yau

Abstract: We prove that the open topological string partition function on a D-brane configuration in a Calabi-Yau manifold X takes the form of a closed topological string partition function on a different Calabi-Yau manifold X_b. This identification shows that the physics of D-branes in an arbitrary background X of topological string theory can be described either by open+closed string theory in X or by closed string theory in X_b. The physical interpretation of the ''bubbling'' Calabi-Yau X_b is as the space obtained by letting the D-branes in X undergo a geometric transition. This implies, in particular, that the partition function of closed topological string theory on certain bubbling Calabi-Yau manifolds are invariants of knots in the three-sphere.

Topological conformal field theories and gauge theories

Abstract: This paper gives a construction, using heat kernels, of differential forms on the moduli space of metrised ribbon graphs, or equivalently on the moduli space of Riemann surfaces with boundary. The construction depends on a manifold with a bundle of Frobenius algebras, satisfying various conditions. These forms satisfy gluing conditions which mean they form an open topological conformal field theory, that is, a kind of open string theory. If the integral of these forms converged, it would yield the purely quantum part of the partition function of a Chern-Simons type gauge theory. Yang-Mills theory on a four manifold arises as one of these Chern-Simons type gauge theories.

The Geometric Transition Revisited

Abstract: Our intention in this article is to review known facts and to summarise recent advances in the understanding of geometric transitions and the underlying open/closed duality in string theory We aim to present a pedagogical discussion of the gauge theory underlying the Klebanov--Strassler model and review the Gopakumar--Vafa conjecture based on topological string theory These models are also compared in the T-dual brane constructions We then summarise a series of papers verifying both models on the supergravity level An appendix provides extensive background material about conifold geometries We pay special attention to their complex structures and re-evaluate the supersymmetry conditions on the background flux in constructions with fractional D3-branes on the singular (Klebanov--Strassler) and resolved (Pando Zayas--Tseytlin) conifolds We agree with earlier results that only the singular solution allows a supersymmetric flux, but point out the importance of using the correct complex structure to reach this conclusion

The space-time operator product expansion in string theory duals of field theories

Abstract: We study the operator product expansion (OPE) limit of correlation functions in field theories which possess string theory duals, from the point of view of the string worldsheet. We show how the interesting ("single-trace") terms in the OPE of the field theory arise in this limit from the OPE of the worldsheet theory of the string dual, using a dominant saddle point which appears in computations of worldsheet correlation functions in the space-time OPE limit. The worldsheet OPE generically contains only non-physical operators, but all the non-physical contributions are resummed by the saddle point to a contribution similar to that of a physical operator, which exactly matches the field theory expectations. We verify that the OPE limit of the worldsheet theory does not have any other contributions to the OPE limit of space-time correlation functions. Our discussion is completely general and applies to any local field theory (conformal at high energies) that has a weakly coupled string theory dual (with arbitrary curvature). As a first application, we compare our results to a proposal of R. Gopakumar for the string theory dual of free gauge theories.

Non-Perturbative Corrections and Modularity in N = 1 Type IIB Compactifications

Abstract: Non-perturbative corrections and modular properties of four-dimensional type IIB Calabi-Yau orientifolds are discussed. It is shown that certain non-perturbative alpha' corrections survive in the large volume limit of the orientifold and periodically correct the Kahler potential. These corrections depend on the NS-NS two form and have to be completed by D-instanton contributions to transform covariantely under symmetries of the type IIB orientifold background. It is shown that generically also the D-instanton superpotential depends on the two-form moduli as well as on the complex dilaton. These contributions can arise through theta-functions with the dilaton as modular parameter. An orientifold of the Enriques Calabi-Yau allows to illustrate these general considerations. It is shown that this compactification leads to a controlled four-dimensional N=1 effective theory due to the absence of various quantum corrections. Making contact to the underlying topological string theory the D-instanton superpotential is proposed to be related to a specific modular form counting D3, D1, D(-1) degeneracies on the Enriques Calabi-Yau.

D-Branes in Topological String Theory

Abstract: This thesis is concerned with D-branes in topological string theory, focusing on the description of B-type D-branes in topological Landau-Ginzburg models Such D-branes are characterized by matrix factorizations of the Landau-Ginzburg superpotential After reviewing some aspects of topological string theory we give a detailed discussion of matrix factorizations We discuss methods to calculate the effective superpotential in minimal models, putting emphasis on an algorithm to calculate the effective superpotential via deformations of matrix factorizations We furthermore perform a non-trivial check of homological mirror symmetry for the torus

Generalised hyperKähler manifolds in string theory

Abstract: We discuss the notion of generalised hyperKahler structure in the context of string theory and discuss examples of this geometry.

Quantization and holomorphic anomaly

Abstract: We study wave functions of B-model on a Calabi-Yau threefold in various polarizations.

Twistor strings with flavour

Abstract: We explore the tree-level description of a class of N = 2 UV-finite SYM the- ories with fundamental flavour within a topological B-model twistor string framework. In particular, we identify the twistor dual of the Sp(N) gauge theory with one antisymmetric and four fundamental hypermultiplets, as well as that of the SU(N) theory with 2N hy- permultiplets. This is achieved by suitably orientifolding/orbifolding the original N = 4 setup of Witten and adding a certain number of new topological 'flavour'-branes at the orientifold/orbifold fixed planes to provide the fundamental matter. We further comment on the appearance of these objects in the B-model on | CP 3|4 . An interesting aspect of our construction is that, unlike the IIB description of these theories in terms of D3 and D7-branes, on the twistor side part of the global flavour symmetry is realised geometrically. We provide evidence for this correspondence by calculating and matching amplitudes on both sides.

On the Correspondence of Open and Closed Strings

Abstract: The topic of this thesis is the correspondence of open and closed strings. Already for a long time it has been evident that those both sectors of string theory do not only couple to each other, but it is also possible to identify excitations of the closed string in the open string sector. This correspondence has been shown in a multitude of examples, which indicates a deep connection. This thesis tries to understand this from the viewpoint of coupled open-closed moduli spaces and finally from a string field theoretic point of view. Implications of this conjectures correspondence have gained great importance, among them gauge-gravity correspondence, AdS/CFT correspondence as well as non-perturbative e A new approach to bosonic boundary string field theory on curved target spaces is developed, which allows to demonstrate techniques to identify closed string excitations in the open string sector. Certain factorisation properties of path integrals over WZW-models are derived, which lead to a adequate re-formulation of boundary string field theory. It is shown for the first time that this setting can reproduce curved D-branes by tachyon condensation starting from flat space as soon as non-local interactions terms are permitted. The results coincide with expectations from conformal field theory. Additionally first steps are taken to study more complex supersymmetric string theories on Calabi-Yau manifolds. This includes an exemplary investigation of coupled open-closed moduli spaces. These results are derived in the framework of topological string theory, which constitutes a projection to a finite subspace of the full theory. The recent formalism of matrix factorisations for describing B-type branes is used in order to show how open string moduli spaces can be constructed exactly. Moreover the influence of closed string perturbations appears in form of an e

Confining String and Its Widening in HP 1 Embedding Approach

Abstract: Structure of confining string in terms of the topological charge density and the action density is studied in SU(2) Yang-Mills theory on the lattice using HP 1 �-model embedding approach. We find that the confining flux tube noticeably suppresses both the topological charge and the action densities. Beyond the string formation length the string cross section in terms of these quantities is well described by a Gaussian profile. In both cases the squared string width is found to be a logarithmic function of the string length confirming the Luscher widening of the chromoelectric string. Characteristic string scales in terms of the topological and action densities are estimated as well. PACS numbers: 11.15.-q, 11.15.Ha, 12.38.Aw, 12.38.Gc and the topological charge density was discovered in nu- merical simulation of Ref. (13) in the so called "cooled" SU(3) vacuum. The investigation was developed further in Ref. (14) with the help of the same cooling method which is a gradual transformation of the quantum ("hot") fields towards the classical ("cold") vacuum. The cooling is aimed to reduce the topologically trivial effects of di- vergent zero-point fluctuations. This important result - obtained with the simplest lattice definitions of the topo- logical charge - is a first numerical indication that the QCD string suppresses topological fluctuations. In this paper we investigate, in particular, the structure of con- fining string in terms of the topological charge density. The problem is conceptually and technically challenging and therefore it calls for a few physically motivated sim-

Little string theory from double-scaling limits of field theories

Abstract: We show that little string theory on S5 can be obtained as double-scaling limits of the maximally supersymmetric Yang-Mills theories on R × S2 and R × S3/Zk. By matching the gauge theory parameters with those in the dual supergravity solutions 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 R × S2, the 't Hooft coupling must be scaled like ln 3N, and on R × S3/Zk, like ln 2N. Accordingly, taking these limits in these field theories gives Lagrangian definitions of little string theory on S5.