Topological Sigma Model Research Articles

CALCULATION OF GROMOV-WITTEN INVARIANTS FOR CP3, CP4AND Gr(2, 4)

Using the associativity relations of the topological sigma models with target spaces, CP3, CP4 and Gr (2, 4), we derive recursion relations of their correlation and evaluate them up to a certain order in the expansion over the instantons. The expansion coefficients are regarded as the number of rational curves in CP3, CP4 and Gr (2, 4) which intersect various types of submanifolds corresponding to the choice of BRST-invariant operators in the correlation functions.

The Calabi-Yau moduli and the Toda equations

For a one-parameter family of Calabi-Yau d-fold embedded in C P d+1 we analyze the hermitian metrics on the moduli space of the non-linear sigma model by topological field techniques. The set of equations turns out to be the non-affine A-type Toda equation. It is contrasted with the C P d+1 case where the affine Toda equation appears. We obtain exact solutions of the hermitian metrics and associate their non-perturbative corrections to the holomorphic maps. In particular for the quintic case, the coefficients in the q-expansion are given by the number of rational curves. From these data, we construct the genus one partition function F 1 of the topological sigma model by the method of the holomorphic anomaly. In the asymmetrical limit of the Kähler parameter t → ∞ , the contribution from the A ∗- part is decoupled and we obtain the partition function of A-matter coupled with topological gravity at the stringy one-loop level. The coefficients of the series expansion are related with the integrals of the top Chem class of some vector bundle over the stable maps with definite degrees. We also comment on the behavior of the F 1 near the conifold locus.

On the Mathai-Quillen formalism of topological sigma models

We present a Mathai-Quillen interpretation of topological sigma models. The key to the construction is a natural connection in a suitable infinite-dimensional vector bundle over the space of maps from a Riemann surface (the world sheet) to an almost complex manifold (the target). We show that the covariant derivative of the section defined by the differential that appears in the equation for pseudo-holomorphic curves is precisely the linearization of the operator itself. We also discuss the Mathai-Quillen formalism of gauged topological sigma models.

Non-scale-invariant topological Landau-Ginzburg models

The Landau-Ginzburg formulation of two-dimensional topological sigma models on the target space with positive first Chern class is considered. The effective Landau-Ginzburg superpotential takes the form of logarithmic type which is characteristic of supersymmetric theories with the mass gap. The equations of motion yield the defining relations of the quantum cohomology ring. Topological correlation functions in the CP n−1 and Grassmannian models are explicitly evaluated with the use of the logarithmic superpotential.

Topological massive sigma models

In this paper we construct topological sigma models which include a potential and are related to twisted massive supersymmetric sigma models. Contrary to a previous construction these models have no central charge and do not require the manifold to admit a Killing vector. We use the topological massive sigma model constructed here to simplify the calculation of the observables. Lastly it is noted that this model can be viewed as interpolating between topological massless sigma models and topological Landau-Ginzburg models.

Sigma-models having supermanifolds as target spaces

We study a topological sigma-model ($A$-model) in the case when the target space is an ($m_0|m_1$)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an $A$-model having an ($m_0-m_1$)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of $A$-model is a complete intersection in a toric manifold, this $A$-model is equivalent to an $A$-model having a toric supermanifold as a target space.

N = 4: a unifying framework for 2d topological gravity, cM ≤ 1 string theory and constrained topological sigma model

It is shown that two-dimensional (2d) topological gravity in the conformal gauge has a larger symmetry than has been hitherto recognized; in the formulation of Labastida, Pernici and Witten it contains a twisted “small” N = 4 superconformal symmetry. There are in fact two distinct twisted N = 2 structures within this N = 4, one of which is shown to be isomorphic to the algebra discussed by the Verlindes and the other corresponds, through bosonization, to c M ≤ 1 string theory discussed by Bershadsky et al. As a byproduct, we find a twisted N = 4 structure in c M ≤ 1 string theory. We also study the “mirror” of this twisted N = 4 algebra and find that it corresponds, through another bosonization, to a constrained topological sigma model in complex dimension one.

Summing the instantons: Quantum cohomology and mirror symmetry in toric varieties

We use the gauged linear sigma model introduced by Witten to calculate instanton expansions for correlation functions in topological sigma models with target space a toric variety V or a Calabi-Yau hypersurface M ⊂ V. In the linear model the instanton moduli spaces are relatively simple objects and the correlators are explicitly computable; moreover, the instantons can be summed, leading to explicit solutions for both kinds of models. In the case of smooth V, our results reproduce and clarify an algebraic solution of the V model due to Batyrev. In addition, we find an algebraic relation determining the solution for M in terms of that for V. Finally, we propose a modification of the linear model which computes instanton expansions about any limiting point in the moduli space. In the smooth case this leads to a (second) algebraic solution of the M model. We use this description to prove some conjectures about mirror symmetry, including the previously conjectured “monomial-divisor mirror map” of Aspinwall, Greene and Morrison.

CM < 1 string theory as a constrained topological sigma model

It has been argued by Ishikawa and Kato that by making use of a specific bosonization, c M = 1 string theory can be regarded as a constrained topological sigma model. We generalize their construction for any ( p, q) minimal model coupled to two dimensional (2d) gravity and show that the energy-momentum tensor and the topological charge of a constrained topological sigma model can be mapped to the energy-momentum tensor and the BRST charge of c M < 1 string theory at zero cosmological constant. We systematically study the physical state spectrum of this topological sigma model and recover the spectrum in the absolute cohomology of c M < 1 string theory. This procedure provides us a manifestly topological representation of the continuum Liouville formulation of c M < 1 string theory.

C=1 STRING AS A TOPOLOGICAL MODEL

The discrete states in the c=1 string are shown to be the physical states of a certain topological sigma model. We define a set of new fields directly from c=1 variables, in terms of which the BRST charge and energy-momentum tensor are rewritten as those of the topological sigma model. Remarkably, ground ring generator x turns out to be a coordinate of the sigma model. All of the discrete states realize a graded ring which contains the ground ring as a subset.

The topological B model as a twisted spinning particle

The B-twisted topological sigma model coupled to topological gravity is supposed to be described by an ordinary field theory: a type of holomorphic Chern-Simons theory for the open string, and the Kodaira-Spencer theory for the closed string. We show that the B model can be represented as a particle theory obtained by reducing the sigma model to one dimension, and replacing the coupling to topological gravity by a coupling to a twisted one-dimensional supergravity. The particle can be defined on any Kähler manifold—it does not require the Calabi-Yau condition—so it may provide a more generalized setting for the B model than the topological sigma model. The one-loop partition function of the particle can be written in terms of the Ray-Singer torsion of the manifold, and agrees with that of the original B model. After showing how to deform the Kähler and complex structures in the particle, we prove the independence of this partition function on the Kähler structure, and investigate the origin of the holomorphic anomaly. To define other amplitudes, one needs to introduce interactions into the particle. The particle will then define a field theory, which may or may not be the Chern-Simons or Kodaira-Spencer theories.

Equivariant topological sigma models

We identify and examine a generalization of topological sigma models suitable for coupling to topological open strings. The targets are Kähler manifolds with a real structure, i.e. with an involution acting as a complex conjugation, compatible with the Kähler metric. These models satisfy axioms of what might be called “equivariant topological quantum field theory”, generalizing the axioms of topological field theory as given by Atiyah. Observables of the equivariant topological sigma models correspond to cohomological classes in an equivariant cohomology theory of the targets. Their correlation functions can be computed, leading to intersection theory on instanton moduli spaces with a natural real structure. An equivariant C P 1 × C P 1 model is discussed in detail, and solved explicitly. Finally, we discuss the equivariant formulation of topological gravity on surfaces of unoriented open and closed string theory, and find a Z 2 anomaly explaining some problems with the formulation of topological open string theory.

Large N 2D Yang-Mills Theory and Topological String Theory

We describe a topological string theory which reproduces many aspects of the 1/N expansion of SU(N) Yang-Mills theory in two spacetime dimensions in the zero coupling (A=0) limit. The string theory is a modified version of topological gravity coupled to a topological sigma model with spacetime as target. The derivation of the string theory relies on a new interpretation of Gross and Taylor's ``\Omega^{-1} points.'' We describe how inclusion of the area, coupling of chiral sectors, and Wilson loop expectation values can be incorporated in the topological string approach.

Gl( N, N) current algebras and topological field theories

The conformal field theory for the gl( N, N) affine Lie superalgebra in two space-time dimensions is studied. The energy-momentum tensor of the model, with vanishing Virasoro anomaly, is constructed. This theory has a topological symmetry generated by operators of dimensions 1, 2 and 3, which are represented as normal-ordered products of gl( N, N) currents. The topological algebra they satisfy is linear and differs from the one obtained by twisting the N = 2 superconformal models. It closes with a set of gl( N) bosonic and fermionic currents. The Wess-Zumino-Witten model for the supergroup GL( N, N) provides an explicit realization of this symmetry and can be used to obtain a free-field representation of the different generators. In this free-field representation, the theory decomposes into two uncoupled components with sl( N) and U(1) symmetries. The non-abelian component is responsible for the extended character of the topological algebra, and it is shown to be equivalent to an SL( N)/SL( N) coset model. In the light of these results, the G/G coset models are interpreted as topological sigma models for the group manifold of G.

Topological current algebras in two dimensions

Two-dimensional topological field theories possessing a non-abelian current symmetry are constructed. The topological conformal algebra of these models is analysed. It differs from the one obtained by twisting the N = 2 superconformal models and contains generators of dimensions 1, 2 and 3 that close a linear algebra. Our construction can be carried out with one and two bosonic currents and the resulting theories can be interpreted as topological sigma models for group manifolds.

On effective theories of topological strings

We study the construction of effective target-space theories of topological string theories. The example of the CP1 topological sigma model is analysed in detail. An effective target-space theory whose correlation functions are defined by the sum over connected Riemann surfaces of all genera is found to be itself topological. The values of the couplings of this effective theory are expressed in terms of those of the world-sheet theory for general CP1-like world-sheet model. Any model of this type can be obtained as an effective theory. The definition of the effective theory's expectation values as a sum over disconnected surfaces as well, is shown not to be compatible with those of a topological thoery, at least as long as the connectivity of the target space is kept fixed. Dilaton-type couplings emerge in the full lagrangian realization of the moduli space of topological theories with n observables.En route, we encounter a nonperturbative duality, an equivalence of theories with different world-sheets and discuss the relation between the cosmological constant in these finite theories and the zero-point function.

Topological matter in two dimensions

Topological quantum field theories containing matter fields are constructed by twisting N = 2 supersymmetric quantum field theories. It is shown that N = 2 chiral (antichiral) multiplets lead to topological sigma models while N = 2 twisted chiral (twisted antichiral) multiplets lead to Landau-Ginzburg type topological quantum field theories. In addition, topological gravity in two dimensions is formulated using a gauge principle applied to the topological algebra which results after the twisting of N = 2 supersymmetry.

FUSION RESIDUES

We discuss when and how the Verlinde dimensions of a rational conformal field theory can be expressed as correlation functions in a topological LG theory. It is seen that a necessary condition is that the RCFT fusion rules must exhibit an extra symmetry. We consider two particular perturbations of the Grassmannian superpotentials. The topological LG residues in one perturbation, introduced by Gepner are shown to be twisted version of the SU (N)k Verlinde dimensions. The residues in the other perturbation are the twisted Verlinde dimensions of another RCFT; these topological LG correlation functions are conjectured to be the correlation functions of the corresponding Grassmannian topological sigma model with a coupling in the action to instanton number.

Observable hierarchies from vector supersymmetry in cohomological field theories

We examine the underlying algebraic structure of cohomological field theories. The pattern of observable hierarchies is shown to be universally derived from the vector supersymmetry algebra. As such, vector supersymmetry plays a fundamental role in all topological field theories of this type. Within this algebraic approach, we present observable hierarchies for topological gravity, gauge theory and sigma models, together with their coupling.

Potentials for topological sigma models

>Topological sigma models with potential terms are obtained by twisting N=2 supersymmetric sigma models. These models exist for manifolds with isometries and potential terms in their actions are built out of the corresponding Killing vector fields. It turns out that, as ordinary topological sigma models, the models resulting after twisting N=2 supersymmetry can be generalized to the case of almost hermitian manifolds.