Conifold Singularity Research Articles

Conifolds from D-branes

In this note we study the resolution of conifold singularity by D-branes by considering compactification of D-branes on C 3/( Z 2× Z 2) . The resulting vacuum moduli space of D-branes is a toric variety which turns out to be a resolved conifold, that is a nodal variety in C 4 . This has the implication that all the corresponding phases of Type–II string theory are geometrical and are accessible to the D-branes, since they are related by flops.

Enhanced Gauge Symmetry in Three-Modulus Models of Type II String and Hypergeometric Series

The conifold singularities in the type II string are considered as the points of phase transition. In some cases, these singularities can be understood, in the framework of the conventional field theories, as the points of enhanced gauge symmetry. We consider a class of three-modulus type II strings. It is shown the periods can be written in the form of hypergeometric series around the singular points in these models. The leading expansion around the conifold locus turns out to be described by Appell functions. In one singular point, we observe the enhanced gauge symmetry of SU(2) × SU(2) independent of the models. Around another conifold locus, however, the resulting expression of the Appell functions depends on the models. We examine the result by considering a relation between these Appell functions and underlying Riemann surfaces.

Higher derivative F-terms in N = 2 strings

We study a class of higher derivative F-terms of the form Fg,n W2g (Π f)n where W is the N = 2 gravitational superfield and Π is the chiral projector applied to a non-holomorphic function f of the heterotic dilaton vector superfield. We analyze these couplings in the heterotic theory on K3 × T2, where it is found they satisfy an anomaly equation which generalizes the previous found for Fg,0 terms in N = 2 type II vacua and some heterotic dual models. We recognize that, near a point of SU(2) enhancement, a given generating function of the leading singularity of the Fg,n reproduces the free energy of a c = 1 string at an arbitrary radius R. According to the N = 2 heterotic-type II duality in 4D, we then study these couplings near a conifold singularity, using its local description in terms of intersecting D-5-branes. In this context, it turns out that there exists, among the other states involved, a vector gauge field reproducing the heterotic leading singularity structure.

Higher derivative F-terms in N = 2 strings

We study a class of higher derivative F-terms of the form F g, n W 2 g ( Π f) n where W is the N = 2 gravitational superfield and Π is the chiral projector applied to a non-holomorphic function f of the heterotic dilaton vector superfield. We analyze these couplings in the heterotic theory on K3 × T 2, where it is found they satisfy an anomaly equation which generalizes the previous found for F g,0 terms in N = 2 type II vacua and some heterotic dual models. We recognize that, near a point of SU(2) enhancement, a given generating function of the leading singularity of the F g, n reproduces the free energy of a c = 1 string at an arbitrary radius R. According to the N = 2 heterotic-type II duality in 4D, we then study these couplings near a conifold singularity, using its local description in terms of intersecting D-5-branes. In this context, it turns out that there exists, among the other states involved, a vector gauge field reproducing the heterotic leading singularity structure.

D-branes and the conifold singularity

We analyze in detail the description of type IIB theory on a Calabi-Yau three-fold near a conifold singularity in terms of intersecting D-branes. In particular, we study the singularity structure of higher derivative F-terms of the form F g W 2 g where W is the gravitational superfield. This singularity is expected to be due to a one-loop contribution from a charged soliton hypermultiplet becoming massless at the conifold point. In the intersecting D-brane description this soliton is described by an open string stretched between the two D-branes. After identifying the graviphoton vertex as a closed string operator we show that F g 's have the expected singularity structure in the limit of vanishing soliton mass.

Summing up Dirichlet Instantons.

We investigate quantum corrections to the moduli space for hypermultiplets for type IIA near a conifold singularity. We find a unique quantum deformation based on symmetry arguments which is consistent with a recent conjecture. The correction can be interpreted as an infinite sum coming from multiple wrappings of the Euclidean D-branes around the vanishing cycle.

CALABI-YAU COMPACTIFICATION OF TYPE-IIB STRING AND A MASS FORMULA OF THE EXTREME BLACK HOLES

Recently proposed mechanism of the black hole condensation at conifold singularity in type-II string is an interesting idea from which we can interpret the phase of the universal moduli space of the string vacua. It might also be expected that the true physics is on the conifold singularity after supersymmetry breaking. We derive a mass formula for the extreme black holes caused by the self-dual five-form field strength, which is stable and supersymmetric. It is shown that the formula can be written by the moduli parameters of Calabi-Yau manifold and can be calculated explicitly.

D-strings on D-manifolds

We study the mechanism for the appearance of massless solitons in type II string compactifications. We find that by combining T-duality with strong/weak duality of type IIB in ten dimensions enhanced gauge symmetries and massless solitonic hypermultiplets encountered in Calabi-Yau compactifications can be studied perturbatively using D-strings (the strong/weak dual to type IIB string) compactified on “ D-manifolds”. In particular the nearly massless solitonic states of the type IIB compactifications correspond to elementary states of D-strings. As examples we consider the D-string description of enhanced gauge symmetries for type IIA string compactification on ALE spaces with A n singularities and type IIB on a class of singular Calabi-Yau 3-folds. The class we study includes as a special case the conifold singularity in which case the perturbative spectrum of the D-string includes the expected massless hypermultiplet with degeneracy one.

N = 2 heterotic superstring and its dual theory in five dimensions

We study quantum effects in five dimensions in heterotic superstring theory compactified on K 3 × S 1 and analyze the conjecture that its dual effective theory is eleven-dimensional supergravity compactified on a Calabi-Yau threefold. This theory is also equivalent to type II superstring theory compactified on the same Calabi-Yau manifold, in an appropriate large volume limit. In this limit the conifold singularity disappears and is replaced by a singularity associated to enhanced gauge symmetries, as naïvely expected from the heterotic description. Furthermore, we exhibit the existence of additional massless states which appear in the strong coupling regime of the heterotic theory and are related to a different type of singular points on Calabi-Yau threefolds.

N = 2 type II-heterotic duality and higher-derivative F-terms

We test the recently conjectured duality between N = 2 supersymmetric type II and heterotic string models by analysing a class of higher-dimensional interactions in the respective low-energy Lagrangians. These are F-terms of the form F g W 2 g where W is the gravitational superfield. On the type II side these terms are generated at the g-loop level and in fact are given by topological partition functions of the twisted Calabi-Yau sigma model. We show that on the heterotic side these terms arise at the one-loop level. We study in detail a rank-3 example and show that the corresponding couplings F g satisfy the same holomorphic anomaly equations as in the type II case. Moreover we study the leading singularities of F g 's on the heterotic side, near the enhanced symmetry point and show that they are universal poles of order 2 g − 2 with coefficients that are given by the Euler number of the moduli space of genus- g Riemann surfaces. This confirms a recent conjecture that the physics near the conifold singularity is governed by c = 1 string theory at the self-dual point.

String theory near a conifold singularity

We demonstrate that type II string theory compactified on a singular Calabi-Yau manifold is related to c = 1 string theory compactified at the self-dual radius. We establish this result in two ways. First we show that complex structure deformations of the conifold correspond, on the mirror manifold, to the problem of maps from two dimensional surfaces to S 2. Using two dimensional QCD we show that this problem is identical to c = 1 string theory. We then give an alternative derivation of this correspondence by mapping the theory of complex structure deformations of the conifold to Chern-Simons theory on S 3. These results, in conjunction with similar results obtained for the compactification of the heterotic string on K 3 ×T 2 , provide strong evidence in favour of S-duality between type II strings compactified on a Calabi-Yau manifold and the heterotic string on K 3 × T 2.

C = 1 string as the topological theory of the conifold

We show that the non-critical c = 1 string at the self-dual radius is equivalent to topological strings based on the deformation of the conifold singularity of Calabi-Yau threefolds. The Penner sum giving the genus expansion of the free energy of the c = 1 string theory at the self-dual radius therefore gives the universal behaviour of the topological partition function of a Calabi-Yau threefold near a conifold point.

Black hole condensation and the unification of string vacua

It is argued that black hole condensation can occur at conifold singularities in the moduli space of type II Calabi-Yau string vacua. The condensate signals a smooth transition to a new Calabi-Yau space with different Euler characteristic and Hodge numbers. In this manner string theory unifies the moduli spaces of many or possibly all Calabi-Yau vacua. Elementary string states and black holes are smoothly interchanged under the transitions, and therefore cannot be invariantly distinguished. Furthermore, the transitions establish the existence of mirror symmetry for many many or possibly all Calabi-Yau manifolds.

Massless black holes and conifolds in string theory

Low-energy effective field theories arising from Calabi-Yau string compactifications are generically inconsistent or ill-defined at the classical level because of conifold singularities in the moduli space. It is shown, given a plausible assumption on the degenaracies of black hole states, that for type II theories this inconsistency can be cured by nonperturbative quantum effects: the singularities are resolved by the appearance of massless Ramond-Ramond black holes. The Wilsonian effective action including these light black holes is smooth near the conifold, and the singularity is reproduced when they are integrated out. In order for a quantum effect to cure a classical inconsistency, it can not be suppressed by the usual string coupling g s . It is shown how the required g s dependence arises as a result of the peculiar couplings of Ramond-Ramond gauge fields to the dilaton.

A stringy test of the fate of the conifold

By studying string loop corrections to the superpotential of Type II strings compactified on Calabi-Yau threefolds we find a quantum stringy test and a confirmation of a recent proposal of Strominger on the fate of the conifold singularity. We also propose a connection between the spectrum of Bogomolnyi saturated solitons and the one-loop string partition function of N = 2 topological strings.

Tautological systems under the conifold transition on $G(2,4)$

Via a natural degeneration of Grassmannian manifolds $G(k,n)$ to Gorenstein toric Fano varieties $P(k,n)$ with conifold singularities, we suggest an approach to study the relation between the tautological system on $G(k,n)$ and the extended GKZ system on the small resolution $\hat{P}(k,n)$ of $P(k,n)$. We carry out the simplest case $(k,n)=(2,4)$ to ensure its validity and show that the extended GKZ system can be regarded as a tautological system on $\hat{P}(2,4)$.

Conifold transitions in $M$-theory on Calabi-Yau fourfolds with background fluxes

We consider topology changing transitions for M-theory compactifications on Calabi-Yau fourfolds with background G-flux. The local geometry of the transition is generically a genus g curve of conifold singularities, which engineers a 3d gauge theory with four supercharges, near the intersection of Coulomb and Higgs branches. We identify a set of canonical, minimal flux quanta which solve the local quantization condition on G for a given geometry, including new solutions in which the flux is neither of horizontal nor vertical type. A local analysis of the flux superpotential shows that the potential has flat directions for a subset of these fluxes and the topologically different phases can be dynamically connected. For special geometries and background configurations, the local transitions extend to extremal transitions between global fourfold compactifications with flux. By a circle decompactification the M-theory analysis identifies consistent flux configurations in four-dimensional F-theory compactifications and flat directions in the deformation space of branes with bundles.