Showing papers on "Topological string theory published in 2014"

Coupling a QFT to a TQFT and duality

Abstract: We consider coupling an ordinary quantum field theory with an infinite number of degrees of freedom to a topological field theory. On ℝ^d the new theory differs from the original one by the spectrum of operators. Sometimes the local operators are the same but there are different line operators, surface operators, etc. The effects of the added topological degrees of freedom are more dramatic when we compactify ℝ^d, and they are crucial in the context of electric-magnetic duality. We explore several examples including Dijkgraaf-Witten theories and their generalizations both in the continuum and on the lattice. When we couple them to ordinary quantum field theories the topological degrees of freedom allow us to express certain characteristic classes of gauge fields as integrals of local densities, thus simplifying the analysis of their physical consequences.

Non-perturbative effects and the refined topological string

Abstract: The partition function of ABJM theory on the three-sphere has nonperturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi-Yau manifold known as local ℙ1 × ℙ1, in the NekrasovShatashvili limit. Our result can be interpreted as a first-principles derivation of the full series of non-perturbative effects for the closed topological string on this Calabi-Yau background. Based on this, we make a proposal for the non-perturbative free energy of topological strings on general, local Calabi-Yau manifolds.

M-strings, Elliptic Genera and N=4 String Amplitudes

Abstract: We study mass-deformed N = 2 gauge theories from various points of view. Their partition functions can be computed via three dual approaches: firstly, (p,q)-brane webs in type II string theory using Nekrasov's instanton calculus, secondly, the (refined) topological string using the topological vertex formalism and thirdly, M theory via the elliptic genus of certain M-strings configurations. We argue for a large class of theories that these approaches yield the same gauge theory partition function which we study in detail. To make their modular properties more tangible, we consider a fourth approach by connecting the partition function to the equivariant elliptic genus of ℂ2 through a (singular) theta-transform. This form appears naturally as a specific class of one-loop scattering amplitudes in type II string theory on T2, which we calculate explicitly.

Topological Strings on Elliptic Fibrations

Abstract: We study topological string theory on elliptically fibered Calabi-Yau manifolds using mirror symmetry. We compute higher genus topological string amplitudes and express these in terms of polynomials of functions constructed from the special geometry of the deformation spaces. The polynomials are fixed by the holomorphic anomaly equations supplemented by the expected behavior at special loci in moduli space. We further expand the amplitudes in the base moduli of the elliptic fibration and find that the fiber moduli dependence is captured by a finer polynomial structure in terms of the modular forms of the modular group of the elliptic curve. We further find a recursive equation which captures this finer structure and which can be related to the anomaly equations for correlation functions. ∗alim@physics.harvard.edu †emanuel.scheidegger@math.uni-freiburg.de 1 ar X iv :1 20 5. 17 84 v3 [ he pth ] 2 1 Ju n 20 13

Resumming the string perturbation series

Abstract: We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local $\mathbb P^1 \times \mathbb P^1$, which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.

Topological Model for Domain Walls in (Super-)Yang-Mills Theories

Abstract: We derive a topological action that describes the confining phase of (Super-)Yang-Mills theories with gauge group $SU(N)$, similar to the work recently carried out by Seiberg and collaborators. It encodes all the Aharonov-Bohm phases of the possible non-local operators and phases generated by the intersection of flux tubes. Within this topological framework we show that the worldvolume theory of domain walls contains a Chern-Simons term at level $N$ also seen in string theory constructions. The discussion can also illuminate dynamical differences of domain walls in the supersymmetric and non-supersymmetric framework. Two further analogies, to string theory and the fractional quantum Hall effect might lead to additional possibilities to investigate the dynamics.

Light Z′ in heterotic string standardlike models

Abstract: The discovery of the Higgs boson at the LHC supports the hypothesis that the Standard Model provides an effective parametrization of all subatomic experimental data up to the Planck scale. String theory, which provides a viable perturbative approach to quantum gravity, requires for its consistency the existence of additional gauge symmetries beyond the Standard Model. The construction of heterotic string models with a viable light ${Z}^{\ensuremath{'}}$ is, however, highly constrained. We outline the construction of standardlike heterotic string models that allow for an additional Abelian gauge symmetry that may remain unbroken down to low scales. We present a string inspired model, consistent with the string constraints.

Deformations of special geometry: in search of the topological string

Abstract: The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus g � 3 contributions are constructed explicitly for a general class of effective ac.

The spectral problem of the ABJ Fermi gas

Abstract: The partition function on the three-sphere of ABJ theory can be rewritten into a partition function of a non-interacting Fermi gas, with an accompanying one-particle Hamiltonian. We study the spectral problem defined by this Hamiltonian. We determine the exact WKB quantization condition, which involves quantities from refined topological string theory, and test it successfully against numerical calculations of the spectrum.

On the B-twisted topological sigma model and Calabi-Yau geometry

Abstract: We provide a rigorous perturbative quantization of the B-twisted topological sigma model via a first order quantum field theory on derived mapping space in the formal neighborhood of constant maps. We prove that the first Chern class of the target manifold is the obstruction to the quantization via Batalin-Vilkovisky formalism. When the first Chern class vanishes, i.e. on Calabi-Yau manifolds, the factorization algebra of observables gives rise to the expected topological correlation functions in the B-model. We explain a twisting procedure to generalize to the Landau-Ginzburg case, and show that the resulting topological correlations coincide with Vafa's residue formula.

A twistor string for the ABJ(M) theory

Abstract: We construct an open string theory whose single-trace part of the tree-level S-matrix reproduces the S-matrix of the ABJ(M) theory with a unitary gauge group. We also demonstrate that the multi-trace part of the string theory tree-level S-matrix — which has no counterpart in the pure $ \mathcal{N} $ = 6 super-Chern-Simons theory — is due to conformal supergravity interactions and identify certain Lagrangian interaction terms. Our construction suggests that there exists a higher dimensional theory which can be dimensionally-reduced, in a certain sense, to the ABJ(M) theory. It also suggests a generalization of this theory to product gauge groups with more than two factors.

Secondary invariants for string bordism and topological modular forms

Abstract: Using spectral invariants of Dirac operators we construct a secondary version of the Witten genus, namely a bordism invariant of string manifolds in dimensions 4 m − 1 . We prove a secondary index theorem which relates this global-analytic construction with its homotopy-theoretic analog. The latter will be calculated through its factorization over topological modular forms.

Strings of Minimal 6d SCFTs

Abstract: 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 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 whose refined BPS partition function captures the elliptic genus of the SCFT strings. We use the topological string theory to gain insight into the elliptic genus for other values of n.

Exact Instanton Expansion of Superconformal Chern-Simons Theories from Topological Strings

Abstract: It was known that the ABJM matrix model is dual to the topological string theory on a Calabi-Yau manifold. Using this relation it was possible to write down the exact instanton expansion of the partition function of the ABJM matrix model. The expression consists of a universal function constructed from the free energy of the refined topological string theory with an overall topological invariant characterizing the Calabi-Yau manifold. In this paper we explore two other superconformal Chern-Simons theories of the circular quiver type. We find that the partition function of one theory enjoys the same expression from the refined topological string theory as the ABJM matrix model with different topological invariants while that of the other is more general. We also observe an unexpected relation between these two theories.

Extensive Quantum Theory of DNA and Biological String

Abstract: Assume that quantum elements of DNA are A-T and G-C. According to basic thinking of NeuroQuantology, the extensive quantum theory of DNA is researched, and corresponding quantum theory and its many mathematical methods are applied to DNA and molecular biology. From this we discuss symmetry and supersymmetry of DNA, and the quantum theory and equations of DNA, in particular, the SU(2) gauge theory and some solutions of equation. Further, we propose the string theory of DNA and general biological string. Some solutions and functions of these theories may describe probably DNA, biological things and their motions. Finally, we propose quantitatively a universal entropy theory on evolution of any natural and social systems.

Generalized Ablowitz–Ladik hierarchy in topological string theory

Abstract: This paper addresses the issue of integrable structures in topological string theory on generalized conifolds. Open string amplitudes of this theory can be expressed as the matrix elements of an operator on the Fock space of 2D charged free fermion fields. The generating function of these amplitudes with respect to the product of two independent Schur functions becomes a tau function of the 2D Toda hierarchy. The associated Lax operators turn out to have a particular factorized form. This factorized form of the Lax operators characterizes a generalization of the Ablowitz–Ladik hierarchy embedded in the 2D Toda hierarchy. The generalized Ablowitz–Ladik hierarchy is thus identified as a fundamental integrable structure of topological string theory on the generalized conifolds.

Resurgence in topological string theory

Abstract: The study of physical theories in the nonperturbative regime is an interesting but difficult problem. In high energy theoretical physics the use of dualities inspired by the original AdS/CFT correspondence has become the main technique for calculating nonperturbative data. String theories, and in particular, topological string theory, lack nonperturbative definitions. The perturbative free energy, as series in the string coupling constant, is asymptotic, with zero radius of convergence. This is a general feature of many physical systems and it is the main concern of the mathematical theory of resurgence. One of the main results of this theory describes, in a quantitative way, the relation between the perturbative and nonperturbative information of a system. Encoded in the asymptotic growth of the series coefficients of perturbation theory is the information necessary to reconstruct nonperturbative sectors. All these sectors can be put together in a formal object called the transseries, whose different coefficients are related to each other by resurgence relations. The resurgent approach has been applied succesfully to problems in mathematics, on differential and difference equations, and in physics, on quantum mechanics and even quantum field theory. It is currently a very active area of research merging the efforts of both physicists and mathematicians. This thesis performs a resurgent analysis of the perturbative topological string theory. Using the holomorphic anomaly equations it is possible to compute coefficients of the perturbative free energy to very high order and analyze their asymptotic growth. In agreement with resurgence, it is found that nonperturbative sectors coming from a transseries control this growth. It is shown that this transseries can be computed as a solution of a natural extension of the holomorphic anomaly equations. The first half of this thesis is concerned with the main properties of the theory of resurgence and with the computation of the perturbative topological string free energy. These results are then applied to a concrete topological string example. A careful study of the asymptotic growth of the perturbative free energies is performed and various resurgence relations are uncovered. These relations involve elements of the transseries describing the full nonperturbative free energy. General properties of the transseries satisfying the holomorphic anomaly equations are described, including the role of the instanton actions, the presence of holomorphic ambiguities and the possibility of resonance. The numerical results are found to match, to high precision, the elements of the computed transseries. The asymptotic nature of the higher instanton sectors is also studied and a complicated net of resurgence relations is found.

Deformations of special geometry: in search of the topological string

Abstract: The topological string captures certain superstring amplitudes which are also encoded in the underlying string effective action. However, unlike the topological string free energy, the effective action that comprises higher-order derivative couplings is not defined in terms of duality covariant variables. This puzzle is resolved in the context of real special geometry by introducing the so-called Hesse potential, which is defined in terms of duality covariant variables and is related by a Legendre transformation to the function that encodes the effective action. It is demonstrated that the Hesse potential contains a unique subsector that possesses all the characteristic properties of a topological string free energy. Genus $g\leq3$ contributions are constructed explicitly for a general class of effective actions associated with a special-K\"ahler target space and are shown to satisfy the holomorphic anomaly equation of perturbative type-II topological string theory. This identification of a topological string free energy from an effective action is primarily based on conceptual arguments and does not involve any of its more specific properties. It is fully consistent with known results. A general theorem is presented that captures some characteristic features of the equivalence, which demonstrates at the same time that non-holomorphic deformations of special geometry can be dealt with consistently.

Supersymmetric closed string tachyon cosmology: a first approach

Abstract: We give a worldline supersymmetric formulation for the effective action of closed string tachyon in a FRW background. This is done considering that, as shown by Vafa, the effective theory of closed string tachyons can have worldsheet supersymmetry. The Hamiltonian is constructed by means of the Dirac procedure and written in a quantum version. By using the supersymmetry algebra we are able to find solutions to the Wheeler-DeWitt equation via a more simple set of first order differential equations.

Note on intersecting branes in topological strings

Abstract: The intersection of branes is an important object in string theory, in order to study non-perturbative aspects of branes, and also its applicatioins to quantum field theory. In this report we investigate some aspects of the intersecting branes in topological string theory, especially through its matrix model description. We consider the topological B-model on the CalabiYau threefold uv − H(p, x) = 0 with H(p, x) = p − W ′(x)2 − f(x). This geometry realizes at the large N limit of the matrix model with the potential function W (x). There are seemingly two kinds of non-compact branes in the topological B-model, which correspond to the characteristic polynomial and the external source in the matrix model. They play a role of the creation operator of branes for x and p coordinates, respectively. By considering both kinds of the branes simultaneously, we can discuss intersection of branes in the B-model. The corresponding matrix model partition function ΨN,M ({aj}; {λα}) is given by ∫

Field Theories with a Twist

Abstract: This thesis deals with topological field theories derived from maximally supersymmetric gauge theories. The main part of this work is the three appended research papers, each of which investigates some different aspect of these theories. Leading up to these is the first part of the thesis which aims at introducing the above-mentioned concepts in such a fashion that the papers become legible to even those readers who are less familiar with notions such as topological field theory and N=4 Yang-Mills theory from the start. This introduction begins with the concept of symmetry, and both gauge- and supersymmetry are explained. This leads on to the introduction of maximally supersymmetric gauge theories, and then finally to outlining the possible ways to create topological field theories from these. The two last chapters of this thesis are dedicated to the theories featured in the appended papers. Papers I and II both deals with the geometric Langlands-twist of N=4 Yang-Mills theory and the five-dimensional analog of this. In paper I, spherically symmetric solutions to the localisation equations of both the four- and five-dimensional topological field theories are investigated, whereas paper II presents an explicit expression for the action of the five-dimensional theory. The final paper deals with a topological twisting of (2,0) theory, which was conjectured to be relevant in the understanding of the AGT-correspondence. However, this suggested twist is shown to not result in a topological field theory on a general background by showing that the stress-tensor of the theory cannot in general be both $Q$-exact and conserved.

Twisting and Turning in Six Dimensions

Abstract: This thesis investigates certain aspects of a six-dimensional quantum theory known as (2,0) theory. This theory is maximally supersymmetric and conformal, making it the most symmetric higher dimensional quantum theory known. It has resisted an explicit construction as a quantum field theory yet its existence can be inferred from string theory. These properties suggests that an understanding of the theory will create a deeper understanding of the foundations of both. In the first part of the thesis an explicit formulation of the non-interacting ver- sion of the theory is investigated on space-time manifolds that are circle fibrations. The circle fibration geometry enables a compactification to a five dimensional su- persymmetric Yang-Mills theory. A unique extension to an interacting theory is found and conjectured to be the compactification of the interacting theory in six dimensions. The second part of the thesis concerns the topological twisting of the free theory in six dimensions. A space-time manifold which is a product of a four-dimensional and a two-dimensional part is considered. This setup has recently been proposed as an explaination for the conjectured correspondence between four dimensional gauge theory and two-dimensional conformal field theory known as the AGT corre- spondence. We perform the twisting and subsequent compactification on the two- dimensional manifold of the free tensor multiplet in Minkowski signature to avoid the problems associated with the definition of (2,0) theory on Euclidean manifolds. With the same choice of supercharge as in the usually preferred Euclidean scenario we conclude that there is no stress tensor which exhibits the topological properties previously found in similar theories.

Quantum theories and geometric topology: A chapter in physical mathematics

Abstract: In recent years, the interaction between geometric topology and classical and quantum field theories has attracted a great deal of attention from both the mathematicians and physicists. We discuss some topics from low-dimensional topology where this has led to new viewpoints as well as new results. They include categorification of knot polynomials and a special case of the gauge theory to string theory correspondence in the Euclidean version of the theories, where exact results are available. We show how the Witten–Reshetikhin–Turaev invariant in SU(n) Chern–Simons theory on S3 is related via conifold transition to the all-genus generating function of the topological string amplitudes on a Calabi–Yau manifold. This result can be thought of as an interpretation of TQFT as topological quantum gravity (TQG). After a brief discussion of Perelman's work on the geometrization conjecture and its relation to gravity, we comment on some recent work on black hole radiation and its relation to mock moonshine.

M-theory interpretation of the real topological string

Abstract: We describe the type IIA physical realization of the unoriented topological string introduced by Walcher, describe its M-theory lift, and show that it allows to compute the open and unoriented topological amplitude in terms of one-loop diagram of BPS M2-brane states. This confirms and allows to generalize the conjectured BPS integer expansion of the topological amplitude. The M-theory lift of the orientifold is freely acting on the M-theory circle, so that integer multiplicities are a weighted version of the (equivariant subsector of the) original closed oriented Gopakumar-Vafa invariants. The M-theory lift also provides new perspective on the topological tadpole cancellation conditions. We finally comment on the M-theory version of other unoriented topological strings, and clarify certain misidentifications in earlier discussions in the literature.