Flop Invariance of Refined Topological Vertex
and Link Homologies
Department of Physics, Faculty of Science, University of Tokyo,
Bunkyo-ku, Tokyo 113-0033, Japan.
It has been proposed recently that the topological A-model string theory on local toric Calabi-Yau manifolds has a two parameter extension. Amplitudes of the two parameter topological strings can be computed using a diagrammatic method called the refined topological vertex. In this paper we study properties of the refined amplitudes under the flop transition of toric Calabi-Yau three-folds. We also discuss that the slicing invariance and the flop transition imply a simple formula for the homological invariants of the Hopf link. The new expression for the invariants gives a simple refinement of the Hopf link invariant of Chern-Simons theory.
Topological string theory is a special class of two-dimentional topological sigma model coupled to two-dimentional gravity. One of the importances of topological strings is that they provide many insights into physics and mathematics .
There exist two types of topological strings according to the twist procedures in sigma model which are called A-model and B-model respectively. In this articles we study A-model topological strings for local Calabi-Yau manifolds. In general topological strings partition function have the following form
where is the topological string coupling constant and are Kähler parameters of the geometry. It is known that A-model partition functions for the local toric Calabi-Yau threefolds can be solved by the topological vertex formalism . Using these partition functions we can count BPS states of effective theories of Type IIA superstrings on a toric Calabi-Yau manifold in the presence of the self-dual graviphoton background .
Recently the refined topological vertex formalism has been proposed via the melting crystal picture of A-model . From the target-space perspective the refined vertex captures information about the Lefshetz action on the moduli space of the BPS states. Therefore it enable us to count the BPS particles in the non self-dual background , and reproduce the K-theoretic Nekrasov formulas from string theory. Corresponding to the background, the refined vertex has two parameters , .
In the first half of this article we study transformation properties of refined partition functions under the flop transition of toric Calabi-Yau manifolds. The flop invariance of the topological vertex has been studied in . So we generalize their arguments to the refined vertex using the free fermion techniques .
In the latter half of the paper we apply our results to the homological link invariants of Hopf link. By assuming that the slicing invariance  is satisfied we find a simple formula for the superpolynomial of . Using this result, one can save computational costs because there exists no need to sum up the Macdonald functions.
This paper is organized as follows. In section 2, we study the invariance of the refined topological vertex under the flop transition. The application of the flop to the homological link invariant is studied and the formula for the superpolynomial is proposed in section 3. Conclusions are found in section 4. In appendix A, we give some useful formulas for the Schur functions.
2 Flop Transition of Reined Vertex
2.1 Calculation of Subdiagrams
The topological vertex formalism is a cut-and-paste method to compute A-model partition functions for toric Calabi-Yau manifolds. The method consists of some rules. Let us consider A-model on a toric Calabi-Yau. First we divide the web diagram of the Calabi-Yau into trivalent vertices and propagator lines. On the edges of these parts we associate Young diagrams as an analogue of momentums for the Feynman rules. Then we associate the vertex functions and the propagators for these parts. Here is a Kähler parameter of a corresponding . Then inserting some factors111There is no framing factor for the conifold. coming from the framing dependence we glue these parts into one partition function . The gluing procedure is done by summing over the Young diagrams.
where is a specialization of the Macdonald function 
The preferred direction with representation is indicated by short red lines in Fig.1.
In this section we calculate topological string maplitudes for Fig.2 in order to show that the refined patririon functions possess the invariances under the flop transitions. The flop invariance of the topological vertex has been studied in . So we generalize their arguments for the refined topological vertex. By the rules of the topological vertex, the partition function of Fig.2(a) is given by
Then one can calculate the sum over by using (A.8) as follows
Thus this partition function becomes
Let us denote the second line of the above equation by .
A similar calculation gives the partition function for Fig.2(b)
After some algebras we get the following expression
Let us rewrite these partition functions using the free fermions. See the next subsection for the definitions and basic properties of the free fermions. The property (2.19) which will be studied in the next subsection enables us to change the representations in the Schur functions into their transposes. Then the partition function becomes
Using and we can express it as a single correlation function as
This is very similar to . So let us study the relationship between them. The energy operator acts on the vertex operators as and its action commutes with -action. From these facts we obtain
One can show the following identity using (3.11) of 
The coefficient and are such that
We can apply the equation (2.5) for sub-diagrams of refined partition functions. Therefore, we can obtain the flop invariance of the refined topological vertex222The flop invariance of the refined vertex which was proposed in  is studied in  as the result of (2.5).
2.2 Proof of formula
In this section we introduce the free fermion techniques  in order to prove a formula for a special class of the Schur functions. The formula plays an important role in the refined topological vertex calculations of the flop transition.
The free fermions in two dimensions are characterized by the anti-commutation relations
The bosonisation of these fermions gives rise to chiral bosons and their modes are given by
The Young diagrams are in one-to-one correspondence with the elements of the fermionic Fock space such that
where is a diagonal length of the diagram and is the Dirac vacuum which is annihilated by for . The Frobenius coordinates of the Young diagram and are given by
The vertex operators which act on these states are defined by
The property  allows us to find a free fermion representation of the skew Schur functions
Here is an operation act on a power sum333Notice that the Schur functions are related to the Newton polynomial as as
Proof: First we introduce an automorphism of the Fock space
One can show the following properties
Using this automorphism, one can also show that the skew Schur functions (2.13) become
where is a -number. Rewriting the last line of the above equation we obtain the relation between the power summations of
Then using and (2.19), we get the formula
3 Flop, Slicing Invariance, and Homological Link Invariants
3.1 Homological Link Invariants
The latest developments in the homological link invariants provide many insights into a refinement of the polynomial invariants. In the theory of homological link invariants, one regards the link invariants as Euler characteristics of graded homologies associated with links
Then we can introduce the homological link invariants as Poincaré polynomials
These invariants provide a two-parameter refinement of link invariants. The homological link invariants can be embedded into open topological strings  using an idea of the geometric transition and the refinement of the polynomial invariants is interpreted by using the refined topological vertex. Thus a superpolynomial of the Hopf link which gives the homological link invariants for has been proposed in 
where we introduce new variables as
The superpolynomials for many representations were calculated in . For instance
As discussed in  the superpolynomial for provides the Khovanov-Rozansky homological invariant of the Hopf link
3.2 Homological Link Invariants from Flop
In the previous section we have calculated the refined partition functions , () of Fig.2. Now we calculate the following normalized partition function
Let us calculate the normalized partition function for in the first instance
Therefore is the superpolynomial of the unknot up to an overall factor. We can extend this relation between the conifold partition functions (3.2) and superpolynomials for some other representations. For we have
Using we obtain
This is precisely the superpolynomial for (3.6):
In the last instance we calculate the partition function for . Using we have
The Schur function for the single box Young diagram becomes
From these examples it is very natural to expect that there exists the relation between them such that
Then how can we relate the superpolynomials to the partition functions of Fig.2(a)? The slicing invariance proposal  plays a key role to convince that the relationship hold water. The slicing invariance means that a refined partition function is independent of the choices of the preferred direction. With this assumption we can relate the partition function to as shown in Fig.3. The proportional factor can be determined by comparing the leading terms of their expansions in powers of . One can expand (3.2) in powers of