# Topological B-model and ^c=1 String Theory

Seungjoon Hyun1, Kyungho Oh2, Jong-Dae Park3 and Sang-Heon Yi4

Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, Korea
Department of Mathematics and Department of Physics and Astronomy,
University of Missouri-St. Louis, Saint Louis, MO 63121, USA
Korea Institute for Advanced Study, Cheongnyangni 2-dong, Dongdaemun-gu, Seoul, 130-722, Korea
11
22
33
44
###### Abstract:

We study the topological B-model on a deformed orbifolded conifold by investigating variation of complex structures via quantum Kodaira-Spencer theories. The fermionic/brane formulation together with systematic utilization of symmetries of the geometry gives rise to a free fermion realization of the amplitudes. We derive Ward identities which solve the perturbed free energy exactly. We also obtain the corresponding Kontsevich-like matrix model. All these confirm the recent conjecture on the connection of the theory with type 0A string theory compactified at the radius .

Topological B-Model, Deformed Orbifolded Conifold, 0A String Theory
preprint: hep-th/0502075
KIAS-P05018

## 1 Introduction

Topological string theories on Calabi-Yau(CY) threefolds were introduced [1] to probe the topological nature of the space. There are two different types, the A-model and the B-model, of topological string theories, which study the Kähler structure moduli space and the complex structure moduli space of the CY threefolds, respectively. These theories have proven to be very useful to study various aspects of critical and noncritical string theories. Among others, they were used to compute F-terms in superstring compactifications to four dimensions and were shown to be equivalent to non-critical strings. Furthermore, it was shown [2, 3, 4] that the topological B-model on local CY threefolds is large dual to the Dijkgraaf-Vafa(DV) matrix model, from which the computation of nonperturbative superpotentials of gauge theories reduces to perturbative computations in a DV matrix model. For the recent review on the topological string theory, see [5, 6, 7].

Recently, an extremely powerful method to solve exactly the topological B-model on an interesting class of local CY threefolds was developed in [8]. It was shown that the model is governed by the -algebra symmetries of the CY threefolds, namely the holomorphic diffeomorphism. In particular, it was powerful enough to show the full equivalence between the topological B-model on deformed conifold and the noncritical bosonic string theory at the self-dual radius. Both theories admit the same free fermion description and have the symmetries which characterize the theories completely with the emergence of dual Kontsevich-like matrix model.

In somewhat different context, there have been interesting developments [9, 10] in type 0 noncritical string theory. There are two different types, so-called 0A and 0B, of noncritical string theories, both of which have a matrix model description. Since all the matrix models are believed to be equivalent to the topological B-models on some CY threefolds, it is natural to expect that there is a topological B-model on some CY threefold which is equivalent to type 0 string theory. Indeed, it was suggested in an interesting paper [11] that type 0A string theory at the radius is equivalent to the topological B-model on the (deformed) quotient of the conifold. In  [11] the ground ring structure of string theory was identified with certain orbifolded conifold. The partition functions, integrable structures and the associated Ward identities of theory were considered and the corresponding Kontsevich-like matrix model was constructed. From the topological B-model on the orbifolded conifold, several results are obtained leading to the stated conjecture. However, because the orbifolded conifold has non-isolated singularities, topological strings are not well-defined and thus the computation in  [11] is, at best, suggestive.

In this paper, we consider the topological B-model on the deformation of orbifolded conifold, which is well-defined since the deformed geometry is smooth. We adopt the method given in [8] to study the topological B-model and confirm that the model is indeed equivalent to the type 0A string theory compactified at the radius with non-vanishing RR flux background. The key role is played by the -algebra symmetry which comes from the holomorphic diffeomorphism of the CY threefolds. The type 0 string theory also enjoys the same kind of -algebra symmetry. As this symmetry is powerful enough to constrain the whole theory, we can regard this, more or less, as a proof of their equivalence. Indeed we will show that the topological B-model on that space has the free fermion description just like the theory. We use the symmetry to derive the Ward identities from which we determine the perturbed partition function of the topological B-model under complex deformations. It matches exactly with the generating functional of matrix model under the perturbation by tachyon momentum modes. This also guarantees that both theories admit the same Kontsevich-like matrix model description.

As will be clear, the RR flux in 0A string theory corresponds to a deformation parameter of the orbifolded conifold which makes the space non-singular. Therefore the non-vanishing RR flux is crucial in order to have a well-defined topological B-model on the non-singular space. We can consider the limit where the corresponding deformation parameter vanishes. The resultant formulae we obtain in this paper are well-defined under the limit and correspond to the formulae in theory with vanishing RR flux.

The organization of this paper is as follows: In section 2, we explain the prescription to solve the topological B-model on some class of local CY threefolds. In particular, we take an example of the model on the deformed conifold which is equivalent to the bosonic string theory compactified at the self-dual radius. Using this example we explain salient features of those models which show their equivalence most clearly. In section 3, we turn to 0A matrix model and describe the relevant features for our study on the topological B-model. In section 4, firstly we review the deformation and the resolution of the orbifolded conifold, which give rise to non-singular geometries on which string theory can be well-defined. And then we explain the basic set-up to solve the topological B-model on the deformation of orbifolded conifold. Using this, in section 5, we study the integrable structure of the model and show that it is exactly the same as the one of the 0A string theory. We show that both theories have the same Ward identities and thus the same perturbed free energy and, as a result, correspond to the same Kontsevich-like matrix model. In section 6, we draw our conclusions and further comments.

## 2 Topological B-model on the Local Calabi-Yau threefold

In this section we review the topological B-model on non-compact CY geometries and explain general strategies to solve the model on some class of CY geometries following [8]. In particular we focus on the topological B-model on the deformed conifold which was shown to be equivalent to bosonic string theory compactified at the self-dual radius. These models share a lot of characteristic features with those models we consider in this paper.

### 2.1 Basic set-up

The topological B-model describes the quantum theory of the complex structure deformation of Calabi-Yau geometries, which corresponds to the quantum Kodaira-Spencer theory of gravity [12]. In the theory we consider maps from a Riemann surface of genus to the target CY manifold. For each Riemann surface , we compute the corresponding free energy, , where ’s denote the complex structure deformation parameters. Those free energies are summed with weight to give the free energy of the B-model topological closed strings for all genus as

 F(gs,ti)=∞∑g=0g2g−2sFg(ti), (1)

where is the string coupling constant.

Recently, the B-model topological string theory on the local CY threefold of the form

 uv−H(x,y)=0 (2)

was studied by considering -algebra symmetries underlying in this type of CY threefolds [8]. These symmetries correspond to the holomorphic diffeomorphisms of the target CY threefold which preserve the equation of CY threefold (2) and the holomorphic three-form555The normalization of the holomorphic three-form is chosen for later convenience.

 Ω=14π2dx∧dy∧duu . (3)

The CY geometry (2) can be viewed as a fibration over the () plane with one dimensional fibers. in the base manifold is the locus where the fiber degenerates into two components and . By integrating along a contour around , the periods of the three-form over three-cycles become integrals of the two-form

 ∫Didx∧dy (4)

over domains in the -plane whose boundary is one-cycles on the algebraic curve . Hence by Stokes’ theorem, the periods reduce to integrals of the one-form

 ∫ciydx . (5)

Therefore the CY geometry is characterized by the algebraic curve and thus the complex deformations of the CY geometry are captured by the canonical one-form, .

If non-compact B-branes wrap the fiber, the worldvolume theory of the B-branes is given by a dimensional reduction of holomorphic Chern-Simons theory to one complex dimension [13]. It becomes the theory of the Higgs fields and whose zero modes are identified with coordinates of the moduli space of these non-compact B-branes, which is nothing but the base manifold. In the action, the Higgs fields and play roles as canonically conjugated fields and thus their zero modes have a canonical commutation relation

 [x,y]=igs. (6)

All these imply that and are conjugate variables which form a symplectic pair with symplectic structure . After the reduction to the base manifold, () plane, the holomorphic diffeomorphisms that preserve the equation (2) of the CY geometry and the corresponding holomorphic three form descend to the diffeomorphisms of the plane that preserve the symplectic form , and therefore are given by general holomorphic canonical transformations on ().

The complex structure deformation of the curve can appear only at ‘infinity’ i.e. only at the boundary if the ‘compactified’ curve is a Riemann surface of genus zero which does not have any complex deformation moduli. Let us introduce a local coordinate near each boundary such that at the boundary. Near each boundary the canonical one-form is given by which can be identified with the Kodaira-Spencer field as

 ydx=∂ϕ . (7)

The background field can be determined from the relation as . Arbitrary deformations of chiral bosonic scalar field which correspond to complex deformations of the curve near the boundaries are given by the reidentification of or, equivalently, by mode expansion of around of the form

 y(x)=∂xϕ(x)=ycl(x)+t0x−1−gs∞∑n=1ntnxn−1+gs∞∑n=1∂∂tnx−n−1 . (8)

The quantum free energy is a function of the infinite set of couplings , where labels the boundaries. Then one can regard the quantum free energy as a state in the Hilbert space , where is the Hilbert space of a single free boson and is the number of boundaries [14]. One convenient representation of the state is a coherent state representation. In terms of the standard mode expansion of a chiral boson

 ∂xϕ(x)=−i∑n∈Zαnx−n−1 ,[αn,αm]=ng2sδn+m,0 , (9)

the coherent state is defined as

 |t⟩≡exp[igs∞∑n=1tnα−n]|0⟩ . (10)

In this coherent state representation, the partition function can be expressed as

 Z(ti)=expF(ti)=⟨t1|⊗⋯⊗⟨tk|V⟩ , (11)

where is the number of boundaries.

One way to perturb the geometry is to insert B-branes near the boundaries. Non-compact B-branes which wrap the fiber and reside on the locus can be realized in the closed string sector by introducing (anti-)brane creation operator, ():

 ψ(x)= :exp[−igsϕ(x)]: ,ψ∗(x)= :exp[igsϕ(x)]: , (12)

which is the fermionization of Kodaira-Spencer field  [15].

Let us consider two patches with symplectic pairs of coordinates () and (), respectively, and the corresponding fermions. Those two symplectic pairs are related by a canonical transformation preserving the symplectic form with a generating function . The corresponding fermions transform just like the wave function on the geometry, and hence the transformation of fermions between two patches, , is given by [8]

 ψj(xj)=∫dxi eigsS(xi,xj)ψi(xi). (13)

In the next subsections we consider an explicit example in which the quantum free energy of the topological B-model can be computed explicitly in this setup.

### 2.2 The c=1 bosonic string

In this subsection we review some aspects of the matrix model description of bosonic string theory which are relevant for our study. For the review on various aspects of theory, see [16, 17, 18, 19, 20].

The bosonic string theory compactified at the self-dual radius is equivalent [21, 22, 23] to the topological B-model on the deformed conifold which is given by the hypersurface (2) with

 H(x,y)=xy−μ , (14)

where is the complex deformation parameter. The first indication of this equivalence came from the nonchiral ground ring of bosonic string at the self-dual radius [21]. The operator product of the spin zero ghost number zero BRST invariant operators is again BRST invariant and gives a commutative and associative ring structure, modulo BRST exact terms, as

 O(z)O′(0)∼O′′(0)+{Q,...} . (15)

This is called the ground ring. The right- and left-moving sectors of bosonic string give chiral rings and thus, in combination, the corresponding closed string theory has nonchiral ground ring. At the self-dual radius, it is generated by four operators , which correspond to tachyon momentum and winding states, and they obey the relation

 uv−xy=μM , (16)

where is the level of the Fermi sea of the matrix model.

Noncritical bosonic string theory has a free-fermion description with inverted harmonic oscillator potential. The Hamiltonian is given by

 H=12(p2−x2) , (17)

in the usual convention of bosonic string theory. It is convenient to introduce the light-cone variables , in terms of which the Hamiltonian becomes [24, 25, 26, 27]

 H=−12(x+x−+x−x+). (18)

In this formulation it is clear that and are conjugate with commutation relation, and thus the Schrödinger wave function can be represented by either or . Let us denote the wave function in the and representations as and , respectively. The energy eigenfunctions are given by

 ψE±(x±)=1√2πx±iE−1/2±. (19)

The vacuum state of the system corresponds to the Fermi sea in which fermions are filled in the left-hand side of inverted harmonic oscillator. The correlation functions of tachyon operators in string theory correspond to the perturbation of the Fermi sea in the matrix quantum mechanics.

In the -matrix formulation of the matrix model, right-moving modes, , and left-moving modes, , correspond to incoming and outgoing excitations, respectively. These in and out wave functions are related by -matrix as

 ψ+(x+)=(Sψ−)(x+)=∫dx− K(x+,x−)ψ−(x−), (20)

where the kernel, can be taken as . Note that one may choose different kernel such as or . All these kernels give the same results modulo non-perturbative terms. Since we are concerned about the free energy only at the perturbative level, we can choose any one of them as a kernel. It is convenient to use the exponential kernel for matching the corresponding expression from the B-model topological string side. Also note that, since fermions fill in the left-hand side of the inverted harmonic oscillator potential, it is reasonable to take the integration region along the negative real axis666When fermions fill the right-hand side of inverted harmonic oscillator, the positive real axis is taken as the integration region [20, 24].. Reflection coefficient can be introduced as

 R(E)ψE+(x+)=(SψE−)(x+)=1√2π∫0−∞dx− eix+x−ψE−(x−), (21)

from which it can be computed as

 R(E)=−1√2πeiπ2(−iE+12)Γ(−iE+12). (22)

One can euclideanize the time coordinate in the target space and consider the compactification of the theory with compactification radius . The above reflection coefficient can be used to obtain the ‘free energy’ of the model. The perturbative ‘free energy’ of grand canonical ensemble is defined by

 F(μ)=∫∞−∞dE ρ(E)ln[1+e−2πR(E+μ)], (23)

where is the density of states and is a chemical potential. The relation between the density of states and the reflection coefficient is known to be [25, 10, 20]

 ρ(E)=12π[dϕ(E)dE−lnΛ], (24)

where and is a cut-off.

In order to get an expression free from cut-off dependence, it is convenient to consider the third order derivative of the free energy

 ∂3F∂μ3 = −2πR∫∞−∞dE d2ρ(E)dE21e2πR(E+μ)+1. (25)

In the matrix model, we have

 d3ϕc=1(E)dE3=Im[(−i)3ψ(2)(−iE+12)], (26)

where the polygamma function is defined as

 ψ(n)(z)≡dn+1dzn+1lnΓ(z)=(−1)n+1∫∞0dt tne−zt1−e−t. (27)

The computation of is straightforward, given by the contour integral in the upper half-plane, and leads to

 ∂3Fc=1(μM)∂μ3M = −2πR∫∞−∞dE d2ρc=1(E)dE21e2πR(E+μM)+1 (28) = R Im[∫∞0dt e−iμMtt/2sinh(t/2)t/(2R)sinh(t/2R)].

The resultant perturbative free energy of matrix model at the self-dual radius is given by

 Fc=1(μM)=−12μ2MlnμM−112lnμM+∞∑g≥2|B2g|2g(2g−2)μ2−2gM , (29)

where

 B2g=(−1)g−1|B2g|=(−1)g−12(2n)!(2π)2nζ(2n) . (30)

### 2.3 The Topological B-model on the deformed conifold

As explained in sect. 2.1, the Kodaira-Spencer theory on the deformed conifold which corresponds to strings at the self-dual radius is described by the chiral boson on the Riemann surface,

 H(x,y)=xy−μ=0 . (31)

The period integrals over the symplectic basis of the three-cycles in the case of CY geometry of the type (2) are given by

 Xi=∮AiΩ=12πi∮aiydx,Fi=2πi∫BiΩ=∫biydx. (32)

These pairs of periods are related by the following relation

 Fi=∂F0∂Xi , (33)

where denotes the tree level free energy or prepotential.

We need to introduce a cut-off for the integral over noncompact -cycle (-cycle). Then the period, for the curve (31), can be computed as

 X=12πi∮aydx=μ,F=2∫√Λ√μydx=μln(Λμ) , (34)

from which one can easily read off the tree level free energy as

 F0=12XF=−12μ2lnμ+O(Λ) . (35)

The full free energy of the topological B-model, all order in the string coupling, can be obtained via Dijkgraaf-Vafa(DV) conjecture on the equivalence between the topological B-model and the DV matrix model [2, 3, 4]. The DV matrix model dual to the topological B-model on the deformed conifold is known to be the Gaussian matrix model whose partition function is given by

 Z=eF=1Vol(U(N))∫DM  e−12gsTrM2 . (36)

In this correspondence, the ’t Hooft coupling is identified with in the topological B-model. In the ’t Hooft limit, the free energy can be explicitly computed and is given by

 (37)

Therefore we have

 Ftop(μ=gsμM)=FDV(t=igsμM)=Fc=1(μM) , (38)

up to irrelevant regular terms.

The Riemann surface (31) has two boundaries and can be described by two patches whose coordinate is chosen to be and , respectively. Those two asymptotic regions, which correspond to and , may describe the incoming and outgoing states in the matrix model, respectively. As explained in the previous subsection, we introduce Kodaira-Spencer fields in each patch as

 y=∂xϕ(x),x=−∂y~ϕ(y). (39)

Therefore, from the correspondence with matrix model, can be regarded as “incoming” modes and as “outgoing” ones.

The classical part of which describes the original background geometry is given by

 ∂xϕcl(x)=μx,∂y~ϕcl(y)=−μy. (40)

The quantum parts of describe the complex deformations of the geometry and have mode expansions

 ϕqu(x) = −gs∞∑n=1tnxn−gs∞∑n=11n∂∂tnx−n, ~ϕqu(y) = −gs∞∑n=1~tnyn−gs∞∑n=11n∂∂~tny−n, (41)

where, in the classical limit, the couplings are given by periods

 gstn=−1n∮x→∞yx−n ,gs∂F0∂tn=∮x→∞yxn . (42)

Non-compact B-brane creation operator in each patch can be introduced as (12). Suppose we put branes at positions near the boundary , then the gravitational backreaction is given by

 N∏i=1~ψ(yi)=∏i:exp[−igs~ϕ(yi)]:=Δ(y):exp[−igsN∑i=1~ϕ(yi)]: , (43)

where denotes the Vandermonde determinant. The expectation value of in this perturbed background becomes

 ⟨N∏i=1~ψ(yi)∂~ϕ(y)⟩=−igsN∑i=11yi−y=−igsN∑i=1∞∑n=1y−niyn−1 . (44)

This tells us that we can use B-branes to deform the curve with the coupling

 ~tn=inN∑i=1y−ni . (45)

Since and are conjugate each other as shown in (6), they play dual roles as a coordinate in one patch and a momentum in the other, and vice versa. Therefore the canonical transformation between two coordinate patches, which leads to the transformation (13), is nothing but the Fourier transform:

 ~ψ(y)=(Sψ)(y)=1√2π∫dx eiyx/gsψ(x), (46)

which is reminiscent of relation in  (20).

One can compute the quantum free energy of the topological B-model on the deformed conifold, or the corresponding state , using these B-branes and symmetry of the model. Ward identities associated with symmetry are enough to fix the quantum free energy. The curve has two punctures and, in this case, the symmetry generators relate an operation on one puncture to the one on the other. For example, consider an action of the generator given by the Hamiltonian

 f(xi,pj)=xni , (47)

which can be written in the fermionic representation as

 Wn+1m=∮ψ(xi)xniψ∗(xi) . (48)

The corresponding Ward identity is given by [8]

 ∮x→∞dx ψ(x)xnψ∗(x)|V⟩=−∮y→∞dy ~ψ(y)(igs∂y)n~ψ∗(y)|V⟩. (49)

This is identical with the Ward identity [28] of the string amplitude. This ensures that both theories are equivalent, sharing the same integrable structure.

After successive integrations by parts, the Ward identity can have the following alternative form:

 ∮x→∞dx ψ∗(x)xnψ(x)|V⟩=−∮y→∞dy ~ψ∗(y)(−igs∂y)n~ψ(y)|V⟩. (50)

In the following sections, we will use this form of Ward identity to show the equivalence between the topological B-model on the deformed orbifolded conifold and the compactified 0A string theory.

## 3 The ^c=1 type 0A string theory

The string has superconformal symmetry on the worldsheet with one scalar superfield , whose bosonic component, , corresponds to the time coordinate in the target manifold, and one super Liouville field , which comes from the worldsheet supergraviton multiplet. Under the nonchiral GSO projection, it becomes either type 0A or 0B theory [9, 10]. The spectrum of 0A(0B) theory consists of massless tachyon field and R-R vector(scalar) fields. One can euclideanize the time coordinate, , of the target manifold and consider the circle compactification with compactification radius . After the compactification, these two theories, type 0A and type 0B theories are T-dual under .

In this paper we focus on the type 0A theory, especially at the radius , which must be equivalent to type 0B theory at the dual radius . After the compactification, the NS-NS spectrum of type 0A theory consists of tachyon field momentum states with momentum and winding states with where take integer values, while in the R-R sector, the theory has winding modes with , only [10].

The ground ring is generated by four elements, , , and , among which and come from NS-NS momentum modes and and from R-R winding modes. It was argued in [11] that the ground ring of type 0A string theory at the radius is given by

 uv−(xy−μ)2−q24=uv−(xy−μ+i2q)(xy−μ−i2q)=0 , (51)

where is the cosmological constant and is the net D0-brane charge in the background. This is the same form as the equation of the deformed orbifolded conifold which will be studied in detail using the topological B-model.

In this section we review some aspects of the matrix model description of the 0A string theory, which will be relevant in connection with the topological B-model we will consider.

### 3.1 Type 0A matrix quantum mechanics

The matrix model description of the 0A theory is given by the world volume theory of D0-branes and anti-D0-branes, which is matrix quantum mechanics [10]:

 L=Tr[(D0t)†D0t+12α′t†t] , (52)

where denotes the tachyon field in the bifundamental representation under . This model can be described by a non-relativistic free-fermion in two dimensions with upside-down harmonic oscillator potential. The single particle Hamiltonian is given by

 H=12(p2x+p2y)−14α′(x2+y2) . (53)

The conserved charge of the angular momentum is identified with the net D0-brane charge  [10]. Note that in each sector of the angular momentum , the model becomes [10, 29, 30] effectively one-dimensional model which is known as deformed matrix quantum mechanics [31] with the Hamiltonian,

 H′=−12d2dr2−14α′r2+q2−142r2 , (54)

where .

It is again convenient to introduce the light cone variables [32]

 z±=1√2[1√2α′(x+iy)±(px+ipy)] , (55)

and their complex conjugates . The only nontrivial commutators are

 [z+,¯z−]=[¯z+,z−]=−2i√2α′ , (56)

which tells that () and () form conjugate pairs. Therefore in terms of these new variables, the wave function can be expressed either in representation or in representation, denoted by and , respectively. Furthermore the wave functions in representation and those in representation should be related by the Fourier transform:

 ψ+(z+,¯z+)=12π∫0−∞dz−d¯z−ei¯z+z−+iz+¯z−ψ−(z−,¯z−), (57)

where the integration region is taken along the negative real axis, in the same fashion as the case.

From now on we set , which is the standard convention for theory. In the representation, the Hamiltonian and the angular momentum can be expressed as

 H=i2(z+∂∂z++¯z+∂∂¯z++1) , (58)

and

 J=z+∂∂z+−¯z+∂∂¯z+ . (59)

Therefore the wave functions in the sector of angular momentum are of the form

 ψ±(z±,¯z±)=(z±¯z±)q/2f(z±¯z±) . (60)

The energy eigenstates with energy and angular momentum are given by

 ψE,q±(z±,¯z±)=(z±¯z±)q/2(z±¯z±)±iE−12 . (61)

As will be clear, this is the most natural approach when we compare this theory with the topological B-model on the deformed orbifolded conifold. As was the case in the matrix quantum mechanics, all the results from this formalism agree with the exact results, modulo nonperturbative terms. In the next subsection, we use this formalism to derive the reflection coefficient and the free energy of the theory and to describe the general perturbation of the system.

### 3.2 The reflection coefficient and the free energy

In the -matrix formulation of the matrix model, and may be regarded as incoming and outgoing excitations, respectively. These in and out wave functions are related by -matrix:

 ψ+(z+,¯z+)=(Sψ−)(z+,¯z+), (62)

which is nothing but the Fourier transform given in eq. (57).

The reflection coefficient can be introduced in the same way as the one in the matrix model described earlier as

 (SψE,q−)(z+,¯z+)=R(E,q)ψE,q+(z+,¯z+) . (63)

 R(E,q)=12πeπ(E+i/2)Γ(−iE+q2+12)Γ(−iE−q2+12). (64)

Note that the exact expression of the reflection coefficient from the deformed matrix model is given by [10, 33]

 R(E,q)=Γ(−iE+q2+12)Γ(iE+q2+12), (65)

which differs from the expression, (64), in only nonperturbative corrections for .

One can euclideanize the time coordinate in the target space and consider the compactification of the noncritical string theory along that direction. In the context of the matrix quantum mechanics, the Euclidean version of the above reflection coefficient can be obtained by replacing with . Furthermore, since we are dealing with fermions which is anti-periodic in compactified Euclidean time, the momentum modes should be quantized as under the compactification with the radius . As alluded earlier, the topological B-model on the deformed orbifolded conifold is equivalent to the 0A string theory compactified with the radius . Indeed, in the topological B-model we will obtain the expression of the form (64) after the replacement .

One can obtain the ‘perturbative’ free energy of the theory from the ‘perturbative’ reflection coefficient (64) in the similar fashion as in the model outlined in the previous section. Now is given by

 ϕ0A(E)=Im[lnΓ(−iE+q2+12)+lnΓ(−iE−q2+12)] , (66)

modulo irrelevant terms, or

 d3ϕ0A(E)dE3=Im[iψ(2)(−iE+q2+12)+iψ(2)(−iE−q2+12)]. (67)

Therefore we have

 ∂3∂μ3F0A(μ,q) = R Im[∫∞0dt {e−i(μ+iq/2)t+e−i(μ−iq/2)t}t/2sinh(t/2)t/(2R)sinh(t/2R)] (68) = ∂3∂μ3[Fc=1(μ+i2q)+Fc=1(μ−i2q)] .

One may note that the free energy obtained from the deformed matrix model is given by

 ∂3∂μ3F0A(μ,q)=2R Im[∫∞0dt e−i(μ−iq/2)tt/2sinh(t/2)t/(2R)sinh(t/2R)] , (69)

which is the same as the above expression (68) modulo nonperturbative terms.

We can consider the general perturbation by momentum modes of the tachyon field in the 0A string theory. It was suggested in [32] that it can be incorporated in the matrix model by considering the new eigenfunctions

 ΨE,q±=e∓iφ±(z±¯z±;E,q)ψE,q± , (70)

where the phases have Laurent expansion

 φ±(z±¯z±;E,q)=12ϕ(E,q)+R∑k≥1t±k(z±¯z±)k/R−R∑k≥11kv±k(z±¯z±)−k/R . (71)

The matrix model is the theory of two dimensional free fermions with fixed angular momentum , which corresponds to the net D0-brane charge. Since the perturbation by tachyon momentum modes preserves the background net D0-brane charge, it should appear symmetrically in the Hamiltonian under . Therefore the perturbed Hamiltonian in the () representation may be given by

 Htot=H+z+∂φ+∂z++¯z+∂φ+∂¯z+ , (72)

whose eigenfunctions become (70). Later we will show that similar structure appears in the deformation of complex moduli in the topological B-model on the deformed orbifolded conifold.

## 4 The Topological B model on the deformed orbifolded conifold

Now we are ready to study the topological B model on the deformation of orbifolded conifold. The CY space we consider is the hypersurface

 uv−(xy−μ1)(xy−μ2)=0 (73)

with the deformation parameters and . In order to have a non-singular geometry, we should have . Eventually we would like to show that this model is equivalent to the type 0A string theory with the compactification radius in the background of net D0-brane charge . In this correspondence, the deformation parameters are related to the cosmological constant and the net D0-brane charge as

 μ1=gs(μ+i2q) ,μ2=gs(μ−i2q) . (74)

The Riemann surface

 H=(xy−μ1)(xy−μ2)=0 (75)

is given by the union of two sheets and each sheet corresponds to genus zero surface with two boundaries.

In this section we describe the general set-up to solve the model. In the next section we study the integrable structure of the model and show the equivalence of various models.

### 4.1 Orbifolded conifolds

In recent years, topological string theory on conifold has been extensively studied [34, 35, 36]. The conifold is three dimensional singularity in defined by

 uv−xy=0. (76)

The conifold can be realized as a holomorphic quotient of by the action given by [37, 38]

 (A1,A2,B1,B2)↦(λA1,λA2,λ−1B1,λ−1B2) for λ∈C∗. (77)

Thus it is a toric variety with a charge vector and the fan is given by a convex polyhedral cone in generated by where

 v1=(1,0,0),v2=(0,1,0),v3=(0,0,1),v4=(1,1,−1). (78)

The isomorphism between the conifold and the holomorphic quotient is given by

 x=A1B1,y=A2B2,u=A1B2,v=A2B1. (79)

We take a further quotient of the conifold by a discrete group . Here acts on by

 (A1,A2,B1,B2)↦(e−2πi/kA1,A2,e2πi/kB1,B2), (80)

and acts by

 (A1,A2,B1,B2)↦(e−2πi/lA1,A2,B1,e2πi/lB2). (81)

Thus they will act on the conifold by

 (x,y,u,v)↦(x,y,e−2πi/ku,e2πi/kv) (82)

and

 (x,y,u,v)↦(e−2πi/lx,e2πi/ly,u,v). (83)

Its quotient is called the hyper-quotient of the conifold or the orbifolded conifold and denoted by . To put the actions (77), (80) and (81) on an equal footing, consider the over-lattice :

 N=N′+1k(v3−v1)+1l(v4−v1). (84)

Now the lattice points of in is generated by lattice points as a semigroup (These lattice points will be referred as a toric diagram.). The charge matrix will be by . The discrete group will act on the conifold and its quotient will be the symplectic reduction with the moment map associated with the charge matrix . The new toric diagram for will also lie on the plane at a distance from the origin with a normal vector .
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