Macdonald topological vertices and brane condensates

We show, in a number of simple examples, that Macdonald-type $qt$-deformations of topological string partition functions are equivalent to topological string partition functions that are without $qt$-deformations but with brane condensates, and that these brane condensates lead to geometric transitions.


Introduction
We recall the topological vertex, its refinements and deformations, and ask what the physical interpretation of a specific Macdonald-type deformation is.

A hierarchy of topological vertices.
1.1.1. Abbreviations. To simplify the presentation, we use 1. string, string partition function, vertex, etc. for topological string, topological string partition function, topological vertex, etc., which should cause no confusion, as we only consider the latter, and use topological only for emphasis when that is needed, 2. qt-string partition function, qt-quantum curve, etc. for qt-deformed string partition function, qt-deformed quantum curve, etc. 3. refined as in refined partition functions, etc., when discussing objects that are refined in the sense of [9,10,35]; otherwise, no refinement should be inferred, and unrefined is used only for emphasis when that is needed, 4. the qt-version of · · · for the version of an object that is deformed in the sense of [58,18], and 5. a brane condensate, or simply a condensate is a set of infinitely-many brane insertions.
1.1.2. The original vertex as a normalized 1-parameter generating function of plane partitions with fixed asymptotic boundaries. In [31], Iqbal introduced a systematic way to compute A-model string partition functions in terms of gluing copies of a trivalent topological vertex, and constructed a special case of that vertex where one of the three legs is trivial. In [2], Aganagic, Klemm, Mariño and Vafa constructed the full topological vertex legs are non-trivial, that we refer to in the present work as the original vertex 1 . It depends on a single parameter x, and a set of three Young diagrams, Y 1 , Y 2 and Y 3 , and has a combinatorial interpretation as a normalized partition function of 3D plane partitions [49], where each box in each plane partition is assigned a weight x. All plane partitions generated by C Y 1 Y 2 Y 3   x   satisfy fixed asymptotic boundary conditions specified by Y 1 , Y 2 and Y 3 .
can be glued to form string partition functions. Using geometric engineering [38,39], these string partition functions are identified with instanton partition functions in 5D supersymmetric gauge theories on Ê 4 × S 1 , in a self-dual Ω-background with Nekrasov parameters ǫ 1 + ǫ 2 = 0 [44,45]. Using the AGT/W correspondence [5,59], the Key words and phrases. Topological vertex. Brane condensation. Geometric transition. Topological string partition function. Quantum spectral curve. 1 To streamline the presentation, we make a number of departures from conventional notation. We state these changes as we introduce them, and list them in section 2.1.1. In particular, we use x, instead of q, for the weight of a box in 4D limit of these 5D instanton partition functions are identified with conformal blocks in 2D conformal field theories with an integral central charge c.

The refined vertex as a normalized 2-parameter generating function of plane partitions with fixed asymptotic boundaries.
In [9,10], Awata and Kanno introduced a refined version of C Y 1 Y 2 Y 3   x   , and in [35], Iqbal, Kozcaz and Vafa introduced yet another refined version of the same object. In [7], Awata, Feigin and Shiraishi proved that these two refinements are equivalent. In the present work, we focus on the refined vertex R Y 1 Y 2 Y 3   x, y   of [35]. 2 It depends on two parameters   x, y   , and a set of three Young diagrams, Y 1 , Y 2 and Y 3 , and has a combinatorial interpretation as a normalized partition function of 3D plane partitions. Each box in each plane partition is assigned a weight x or y as follows. One splits each plane partition diagonally into vertical Young diagrams. Scanning the vertical Young diagrams from one end to the other, a box in a plane partition is assigned a weight x if it belongs to a vertical Young diagram that protrude with respect to the preceding Young diagram, and a weight y if it belongs to a vertical Young diagram that does not. All plane partitions can be glued to form refined string partition functions.
Using geometric engineering [38,39], these refined string partition functions are identified with instanton partition functions in 5D supersymmetric gauge theories on Ê 4 × S 1 , in a generic Ω-background, with Nekrasov parameters ǫ 1 + ǫ 2 0 [44,45]. Using the AGT/W correspondence [5,59], the 4D limits of these 5D instanton partition functions are identified with conformal blocks in 2D conformal field theories with a non-integral central charge c.

The Macdonald vertex as a qt-deformation of the refined vertex.
In [58], Vuletić introduced a deformation of MacMahon's generating function of plane partitions, in terms of two Macdonald-type parameters   q, t   . This deformation is independent of the refinement introduced in [9,10] and [35], as one can check by considering R ′ which is a refinement of MacMahon's generating function, but is different from that of [58]. In [18], R Y 1 Y 2 Y 3   x, y   was deformed using the same Macdonaldtype parameters   q, t   that were used in [58], to obtain the Macdonald vertex M [a m , a n ] = n δ m+n, 0 , play a central role [49,35]. Similarly, in constructing the Macdonald vertex, qt-free bosons that satisfy the qt-Heisenberg algebra, 2 We use [35]. We reserve the parameters   q, t   for the Macdonald-type deformation parameters of [58,18] introduced in section 1.1.4. 3 We call the ratio x/y a refinement, and in the limit x → y, the refined vertex reduces to the original one, and we call the ratio q/t a deformation, and in the limit q → t, the Macdonald vertex reduces to the original vertex, for x = y, or to the refined vertex, for x y.
play a central role. In the limit x, y   , and in the limit 1.2. The physical interpretation of the qt-deformation. It is clear by inspection of explicit computations that the Macdonald parameter ratio q/t is a different object from either the M-theory circle radius R or the refinement parameter ratio x/y. 4 The purpose of the present work is to shed light on the geometric and/or physical interpretation of the qt-deformation. To do this, we consider simple string partition functions, and show that in M-theory terms, the deformation q/t 1 describes a condensation of M5-branes that lead to geometric transitions that change the topology of the original Calabi-Yau 3-fold [23]. In conformal field theory terms, we expect that it describes a condensation of vertex operators that push the conformal field theory off criticality [60].

Outline of contents.
In section 2, we include comments on notation used in the text, and definitions of combinatorial objects, including MacMahon's generating function of plane partitions, its refinement and qt-deformation, and in 3, include basic facts related to the original topological vertex, the refined topological vertex, and their qt-deformations. In section 4, we give our first example of the equivalence of qt-deformation and brane condensation, which shows that the refined qt-string partition function on 3 is equivalent to a refined string partition function on 3 with no qt-deformation but in the presence of condensates, and in 5, we give our second example, which shows that a refined qt-deformed partition function on 3 with a single-brane insertion is equivalent to its counterpart (also with a single-brane insertion) with no qt-deformation but in the presence of condensates. In section 6, we discuss the relation of the condensates and geometric transitions in the context of unrefined objects, and in 7, we discuss the qt-quantum curves associated with qt-partition function. Finally, in section 8, we collect a number of remarks, and discuss the various parameters that can appear in topological vertices and the relation with conformal field theory, and in appendix A, we collect useful skew Schur function identities that are used freely in the text.

Notation and definitions
We collect comments on notation, definitions of combinatorial objects, including variations on MacMahon's generating function of plane partitions that appear in the sequel.

Notation.
2.1.1. Deviations from standard notation. We use the variables   x, y   as box weights/refinement parameters, instead of the variables   q, t   used in [9,10,35]. We use [35]. 5 We reserve the variables   q, t   for the Macdonald-type deformation parameters that appear in the Macdonald vertex The first equation in (2.7) is the definition of the MacMahon generating function. The second is obtained by direct expansion of the logarithms of both sides. All refined and qt-versions of this equation, in the sequel, are proven similarly.

2.5.3
. M x y . The refined MacMahon's generating function of plane partitions is [35], The qt-version MacMahon's generating function of plane partitions is, which is the qt-MacMahon generating function introduced by Vuletić in [58]. In the limit These two vertices break the cyclic symmetry and have the preferred leg.
x y . The refined qt-version MacMahon's generating function of plane partitions is [18], x x , and so on.

Topological vertices
We recall basic facts related to the topological vertices introduced in section 1.1.

3.1.
The original vertex of [2]. With reference to the figure on the left in Fig. 3.1, the normalized version of the original vertex 7 of [2] is, where g s is the string coupling constant, and and a set of possibly infinitely-many variables In the second equality, we have used the notation f Y   x   for the framing factor (2.3), and the identities in appendix A.
In the present work, we use x for the weight of a box in a plane partition, instead of q in [31,2]. For a review of the original vertex, see [42].
the open topological A-model partition function on 3 with three special Lagrangian submanifolds. M x x is the closed topological A-model partition function on 3 . The figure on the left in Fig. 3.1 is the toric web diagram of 3 .
3.2. The refined vertex of [35]. With reference to the figure on the right in Fig. 3.1, the normalized refined vertex of [35] is,  is the refined framing factor (2.4). In the limit y → x,  in the present work, and the refined vertex The Macdonald vertex of [18]. With reference to the figure on the right in Fig. 3.1, the normalized Macdonald vertex of [18] is,  are the skew Macdonald and dual Macdonald functions defined for a pair of Young diagrams   Y 1 , Y 2   and a set of possibly infinitely-many variables [18], and none will be needed in the present work.

A qt-partition function from brane condensates
We give an example of a refined qt-deformed partition function that is obtained from its undeformed counterpart via brane condensation.

From M5-branes to surface operators.
Consider M-theory on, where S 1 is the M-theory circle, and X is a local toric Calabi-Yau 3-fold such that the topological A-model on X geometrically engineers a 5D SU(N) supersymmetric gauge theory on -equivariant parameters x, y acting on Ê 4 (Ω-background) [38,39]. We introduce M5-branes on the submanifold, where L S 1 × is a Lagrangian submanifold in X [29] such that an end-point of L is on an edge of the toric web diagram [3]. The M5-branes geometrically engineer simple-type half-BPS surface operators that reduce the gauge group to SU(N − 1) × U(1) on the surface Ê 2 [27,6,13,37].

From surface operators to primary-field vertex operators.
The AGT/W correspondence [5,59] relates a class of 4D N = 2 supersymmetric gauge theories on Ê 4 to 2D Toda conformal field theories. Each of these Toda conformal field theories is defined on a punctured Riemann surface that is related to the Seiberg-Witten curve of the gauge theory and to the mirror curve of the Calabi-Yau 3-fold X. The simple-type surface operators on the gauge theory side correspond to vertex operators that, in turn, correspond to the highest-weight states in irreducible fully-degenerate highest-weight representations on the conformal field theory side [4,17]. In other words, the M5-branes in (4.2) correspond to primary-field vertex operators of fully-degenerate representations in Toda conformal field theory [40,17,57,8].
From that it follows that a condensation of the M5-branes corresponds to a condensation of vertex operators. We expect that such a condensation leads to an off-critical deformation of the chiral blocks in the conformal field theory of the type that leads to correlation functions in off-critical integrable models. We will say more about this in section 8. In the following we show that for X = 3 , M5-brane condensates lead to the refined qt-MacMahon generating function (2.10).

4.3.
A qt-partition function from two brane condensates.
where N branes is a normalization factor, due to the introduction of the branes to be determined in the sequel. Choosing Using the Cauchy identities in appendix A we obtain, The partition function (4.5) includes the contribution of the brane-brane interactions across the brane-stacks. To remove this contribution, we take the normalization factor N branes to be, and obtain the partition function without the brane-brane interactions, 8 We take the holonomies along the un-preferred legs to be proportional to Schur functions [41] (see also [40,17,34]). In the absence of the condensates, we have a closed string partition function on 3 get the unnormalized partition function with a single-brane insertion and two condensates, Splitting the index I → (i, j), as in section 2.4, and setting, We conclude that the refined open-string partition function on 3 with two condensates, with moduli as in (4.10), agrees with the refined qt-MacMahon generating function M q t x y in (2.10) which gives the refined qt-deformed closed string partition function on 3 . By taking the unrefined limit y → x in (4.11), we obtain, where the right hand side is the qt-MacMahon generating function (2.9) of Vuletić [58], and the left hand side can be derived using the original vertex

A qt-partition function with a single-brane insertion from brane condensates
We give an example of a refined qt-deformed partition function with a single-brane insertion that is obtained from its undeformed counterpart via brane condensation.

A partition function with two brane condensates and a single-brane insertion.
Consider the same partition function as in section 4.3, but now with an additional single brane (on the preferred leg of the refined vertex that has no brane-stacks) 9 with an open-string modulus U and a framing factor f 3 , and two brane stacks, as represented in Fig. 5.1, Here N branes is the normalization factor introduced in (4.3) and determined in (4.7), and the Schur function s Y 3   U   with a single variable U is non-zero only for Young diagrams with a single row y 1 = d. Choosing the framing factors as Using the Cauchy identities in appendix A, we obtain, where L   a, x   is the quantum dilogarithm in (4.6).

Normalization.
Dividing the partition function with two condensates and a singlebrane insertion by its counterpart that has no single-brane insertion (4.5), we obtain the normalized partition function, By the inverse arrow in the preferred leg we assign the transpose Y ′ 3 of the Young diagram Y 3 , where we note the relation We now show that for a suitable choice of the moduli   a 1 , a 2 , . . .
Remark. Using the specialization of the one-row Schur function, the partition function (5.5) is expressed in terms of Schur functions as, Using the Cauchy identities in appendix A, the special cases of (5.5) that correspond to f = 0, 1, satisfy, The qt-partition function on 3 with a singlebrane insertion with an open-string modulus U, can be computed using the Macdonald vertex as, this qt-partition function can be considered as the qt-deformation of the undeformed partition function (5.5).

Identification.
To identify the partition function in (5.4) with that in (5.9), we make the choice of moduli, instead of that in (4.10). In other words, in this case, the moduli of the condensates now depend on the length of the single-row Young diagram that labels the Schur function that characterizes the single-brane insertion, d. For this modified choice of moduli, the normalized partition function (5.4) with a single-brane insertion and two condensates becomes, and we find, We conclude that the refined qt-partition function with a single-brane insertion (and no condensates) coincides with its undeformed counterpart (with condensates) for a suitable choice of the framing factors, and of the open-string moduli of the condensates. Note that this refined qt-partition function does not depend on y, and coincides with the result computed by the original vertex C Y 1 Y 2 Y 3   x   in a similar way.

5.3.1.
Remark. We interpret the change in the choice of the moduli of the condensates from that in (4.10) to that in (5.10) as a back-reaction of the condensates to the single-brane insertion.

Remark.
We have shown that the qt-deformed partition functions (2.10) and (5.9) are obtained, in the absence of a qt-deformation, from the partition functions (4.9) and (5.4), respectively. These results depend on the chosen specializations (4.10) and (5.10) that were made to obtain results that can be clearly interpreted. A study of the special significance (if any) of the choices that were made and the consequences of more general choices is beyond the scope of the present work.

qt-Deformations as geometric transitions
We discuss the relation of the brane condensates and geometric transitions in the context of unrefined objects.
6.1. Brane condensates and geometric transitions. Following Gomis and Okuda [21,22], brane insertions change the topology of a Calabi-Yau 3-fold via a geometric transition [23], and a Calabi-Yau 3-fold with brane insertions is equivalent to a bubbling Calabi-Yau 3fold of a more complicated topology, but without brane insertions. Correspondingly, an interpretation of the result in section 4.3 is that a condensate (which is a set of infinitelymany brane insertions) changes the topology of 3 via a geometric transition, and 3 with condensates is equivalent to another Calabi-Yau 3-fold of a more complicated geometry, but without condensates. To test this interpretation, we consider the qt-MacMahon generating function M q t x x in (2.9), which, as we showed in section 4.3, is equal to the open-string partition on 3 with two condensates, and interpret it as an undeformed (no condensates) closed string partition function on a Calabi-Yau 3-fold with more complicated topology than 3 .

Gopakumar-Vafa invariants. The partition function Z X
  x, Q   of the string on a Calabi-Yau 3-fold X with (exponentiated) Kähler moduli Q, is the generating function of Gopakumar-Vafa invariants n β, g ∈ [24], where we have followed the notation used in [42].
the second Betti number of X, S i is a basis of the second homology group H 2   X,   , and Q i are (exponentiated) Kähler parameters, then for any x x in (2.9) normalized by M x x in (2.7) and the expansion in (6.1), we find that n β, 0 = ± 1, n β, g = 0, for g = 1, 2, . . ., which are the Gopakumar-Vafa invariants of a genus-0 manifold with infinitely-many homology 2-cycles β. From (4.10), the infinitely-many branes (in the unrefined case) have holonomies, where g s = − log x, g s N q = log q, g s N t = log t, and according to [21,22], after large N q and N t limit, this yields a Calabi-Yau 3-fold via the bubbling. This agrees with our interpretation of the qt-deformation in terms of a geometric transition driven by a condensate, that is, the insertion of infinitely-many branes. In section 7, we identify this geometry with that of an infinite strip, but before we do that, we consider a simple, but important example.

A simple example of a geometric transition.
In the special case of q = 0, t 0, the qt-MacMahon generating function (2.9) is, This coincides with the undeformed closed string partition function on the resolved conifold, which is the total space of O(−1) ⊕ O(−1) → P 1 with a single (exponentiated) Kähler modulus t, in agreement with the interpretation of the t-deformation of the MacMahon's generating function proposed in [55]. 10 From the perspective of this section, what we have is the simple geometric transition in Fig. 6.1.

qt-Quantum curves
We discuss the qt-quantum curves associated with the unrefined limit of the refined qt-deformed partition function with a single-brane insertion in section 5.

The quantum curve for
7.1.1. Two operators. In the following, we need the operators U and V, where U acts as multiplication by a variable U, and V acts as, and satisfy the x-Weyl relation, The quantum curve. The operators U and V act on Z q t   U   , the unrefined limit of the refined qt-partition function with a single-brane insertion (5.9), as, which is the quantum curve related to Z q t   U   . As discussed below, (7.4) is a qt-version of the quantum curve of 3 in string theory [1,16,15,26].
7.1.3. The classical limit of the quantum curve. Assuming that the asymptotic expansion of Z q t   U   in the classical limit, g s = − log x → 0, has the WKB-form, which is the classical curve related to Z q t   U   . This curve can be identified with the mirror curve related to the infinite-strip geometry that consists of an infinite chain of curves, see the figure on the left in Fig. 7.1 [33] (see also [20]). This infinite-strip geometry agrees with the picture of condensates in sections 4, 5 and 6. In the remainder of this section, we consider a number of spacial cases of quantum curves.
7.2. Case 1. Choosing q = t, the partition function with a single-brane insertion (5.9) reduces to the undeformed partition function Z  with a single-brane insertion on 3 in (5.5), and we find the quantum curve, The classical limit, g s → 0, of the quantum curve (7.8) gives a mirror curve of 3 [3], In other words, the qt-quantum curve (7.4) is a qt-version of the quantum curve (7.8), and (7.6) is a qt-version of the mirror curve (7.9). 7.3. Case 2. Choosing q = 0 and t 0, the partition function with a single-brane insertion (5.9) reduces to, and we find the t-version of the quantum curve of 3 , Note that Z 0 t   U   agrees with the undeformed partition function with a single-brane insertion, up to framing ambiguities, on the resolved conifold with the Kähler modulus t [55], and the classical limit, g s → 0, of the quantum curve (7.11) is the mirror curve of the resolved conifold [3], In other words, the t-deformation of 3 is the resolved conifold as discussed in section 6.3.

Case 3.
Choosing q 0 and t = 0, the partition function with a single-brane insertion (5.9) reduces to, and the q-version of the quantum curve of 3 is, (7.14)  agrees with the undeformed partition function with a single-brane insertion, up to framing ambiguities and a slight modification of the Kähler moduli, for the infinite chain of   −2, 0   -curves, with the same Kähler modulus q for all P 1 , see the figure on the right in Fig. 7.1 [33] (see also [20]). This infinite-strip geometry can be obtained from that in the figure on the left in Fig. 7.1 by suitable blow-downs. 11 The classical limit, g s → 0, of the quantum curve (7.14) is the mirror curve of this strip geometry, We conclude that the q-deformation of 3 is identified with the infinite-strip geometry in the figure on the right in Fig. 7.1, and that this infinite-strip geometry is the result of a geometric transition caused by the condensates.

Remarks
We collect a number of remarks, with particular attention to the interpretation of the various parameters that can appear in topological vertices, and to the relation with conformal field theory.
8.1. The AGT counterpart of brane condensates. We showed that the Macdonald-type qt-deformation introduced in [58], when applied to topological string partition functions [18], leads to qt-partition functions that are equivalent to partition functions without a qtdeformation but with condensates. These condensates are surface operator condensates, and their counterparts on the conformal field theory side of the AGT correspondence are vertex operator condensates in 2D chiral conformal blocks. While this has not been studied in any detail, we expect that these vertex operator condensates play, at the level of conformal blocks, the same role that switching-on off-critical perturbations plays, at the level of the correlation functions [60], and that results in correlation functions in 2D off-critical integrable models. This expectation coincides with the results in [11,12,47,48,51]. 12 8.2. Four parameters. If we start from a 4D instanton partition function in the absence of an Ω-background, or an AGT-equivalent conformal block in a Gaussian 2D conformal field theory with an integral central charge, there are four known ways to modify such a partition function, or conformal block, and each of these ways is characterized by a parameter. circle. For small R, one can think of the 5D instanton partition functions as R-deformations of their 4D limits, in the sense that switching on R gradually is equivalent to including the lighter Kaluza-Klein modes that are infinitely-massive in the R → 0, and that acquire finite masses as R increases [32]. In 2D conformal field theory terms, switching R on is equivalent to deforming the chiral conformal blocks away from criticality to obtain expectation values of type-I vertex operators [14], in some off-critical integrable statistical mechanical models [11,12,47,48,51].

8.2.2.
The refinement parameter x/y. Starting with 4D instanton partition functions in the absence of an Ω-background, one can switch on Nekrasov's Ω-deformation parameters, that is ǫ 1 + ǫ 2 0. In the presence of a finite M-theory circle of radius R, setting x = e − R ǫ 1 , and y = R R ǫ 2 , this refinement is equivalent to setting x/y 1. In 2D conformal field theory terms, we modify the central charge of the conformal field theory while preserving conformal invariance, and the underlying statistical mechanical model remains critical. 8.2.3. The Macdonald deformation parameter q/t. The q/t-deformation of [58,18] is yet another perturbation but, so far, no interpretation of this deformation is known. The purpose of this work is to offer one such interpretation. 8.2.4. The elliptic nome p. In [61,19], two versions were proposed of a topological vertex based on Saito's elliptic deformation of the quantum toroidal algebra U q      gl 1      [52][53][54]. In addition to the refinement parameters   x, y   , and the Macdonald-type deformation parameters   q, t   , this vertex depends on an elliptic nome parameter p and copies of the   q = t   -limit of this vertex can be glued to obtain elliptic conformal blocks. The latter are equal to the elliptic conformal blocks that were computed in [36,46] by gluing copies of the refined vertex of [35], then gluing pairs of external legs.
8.3. Three off-critical deformations. Aside from the refinement parameter x/y, which preserves criticality, it appears that we have three parameters that push the underlying 2D conformal blocks off-criticality, namely the M-theory circle radius R, the Macdonald parameter q/t, and the nome parameter p. One can show by explicit computation that these three parameters coexist and that their effects are different, but it remains unclear how to interpret these effects in statistical mechanics terms.

BPS states in M-theory.
Following [24,50], topological string partition functions on a Calabi-Yau 3-fold encode the degeneracies of the BPS states in M-theory compactified on the Calabi-Yau 3-fold, and the interpretation of the xy-refinement (of the refined topological vertex) was discussed in [30,25]. What is the interpretation of the qt-deformation (of the Macdonald vertex) in the context of M-theory? In section 6, we argued that a topological string partition function on a Calabi-Yau 3-fold with finitely-many homology 2-cycles, in the presence of a qt-deformation is equal, after a geometric transition, to a corresponding topological string partition function in the absence of a qt-deformation, on a Calabi-Yau 3-fold with infinitely-many homology 2-cycles. From this correspondence, we expect that the qtpartition functions encode the degeneracies of BPS states in M-theory compactified on the Calabi-Yau 3-fold with infinitely-many homology 2-cycles. A more direct and perhaps deeper interpretation at the level of the original Calabi-Yau 3-fold with finitely-many homology 2cycles is beyond the scope of the present work.
8.5. Summary. In [9,10,35], a refinement of the original topological vertex was obtained, and the physical meaning of this refinement was clear and related to switching-on a non-self-dual Ω-background. In [58], an independent Macdonald-type qt-deformation of MacMahon's generating function of plane partitions was obtained, and was used in [18] to qt-deformed the refined topological vertex, but no physical meaning of this deformation was proposed.
In the present work, we have presented a number of simple but clear examples of qtdeformed topological string partition functions, and showed in sections 4 and 5 that, in these cases, the qt-deformation is equivalent to switching-on infinitely-many brane insertions, or equivalently brane condensates. In section 6, we showed that a Calabi-Yau 3-fold with a simple topology in the presence of these condensates is equivalent to another Calabi-Yau 3-fold with a more complicated topology without condensates, and argued that the condensates cause the Calabi-Yau 3-fold on which the topological string theory is formulated to undergo a geometric transition that changes its topology. Finally, in section 7, we studied the qt-quantum curves related to the unrefined limit of the qt-partition functions studied in section 5, and showed that their classical limit does indeed correspond to undeformed partition functions on infinite-strip geometry, in agreement with the conclusion that the qt-deformation is equivalent to brane condensates that drive a geometric transition. We expect these conclusions to hold for qt-deformations of more complicated topological string partition functions.