Refined Black Hole Ensembles and Topological Strings
We formulate a refined version of the Ooguri-Strominger-Vafa (OSV) conjecture. The OSV conjecture that relates the BPS black hole partition function to the topological string partition function . In the refined conjecture, is the partition function of BPS black holes counted with spin, or more precisely the protected spin character. becomes the partition function of the refined topological string, which is itself an index. Both the original and the refined conjecture are examples of large duality in the ’t Hooft sense. The refined conjecture applies to non-compact Calabi-Yau manifolds only, so the black holes are really BPS particles with large entropy, of order . The refined OSV conjecture states that the refined BPS partition function has a large dual which is captured by the refined topological string. We provide evidence that the conjecture holds by studying local Calabi-Yau threefolds consisting of line bundles over a genus Riemann surface. We show that the refined topological string partition function on these geometries is computed by a two-dimensional TQFT. We also study the refined black hole partition function arising from D4 branes on the Calabi-Yau, and argue that it reduces to a -deformed version of two-dimensional Yang-Mills. Finally, we show that in the large limit this theory factorizes to the square of the refined topological string in accordance with the refined OSV conjecture.
The Ooguri-Strominger-Vafa (OSV) conjecture gives a beautiful relation between the partition function of four-dimensional BPS black holes in a type IIA string theory compactified on a Calabi-Yau and the topological A-model string partition function. Consider the BPS black hole partition function, , in a mixed ensemble given by fixing the magnetic charge, , and summing over the electric charge with electric potential, . The OSV conjecture relates the exact microscopic entropy of the black hole, captured by to the macroscopic entropy, computed in terms of the topological string
where . This reproduces the Bekenstein-Hawking entropy/area law to the leading order, but otherwise it computes quantum gravitational corrections to it. The entropy on the left is a supersymmetric index. The topological string on the right computes F-terms in the four-dimensional low energy effective action of the IIA string theory on the Calabi-Yau. But from Wald’s formula these F-terms are precisely the corrections to the Bekenstein-Hawking black hole entropy. The OSV relation is an example of gauge/gravity duality, where the gauge theory is the theory on the D-branes comprising the black hole. This aspect of the correspondence was emphasized in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11].111The large dual of a black hole with fixed both magnetic and electric charges is naturally the real version of the topological string, recently studied in . Its partition function has both the holomorphic and the anti-holomorphic piece . Note that both sides of (1) depend only on the Kähler moduli and not the complex structure moduli - on the right, this is a well-known property of A-model topological strings, and on the left this is a consequence of computing a well-behaved BPS index.
It is natural to ask if there are meaningful ways to generalize the conjecture (1). The most general possible black hole partition function would include a chemical potential for angular momentum. This partition function, also known as a spin character, has been extensively studied in the context of motivic wall crossing in four-dimensional field theory [13, 14, 15, 16, 17] and supergravity [18, 19].222Rotating single-centered black holes in four dimensions cannot be supersymmetric. The black holes that one is studying here are multicentered configurations that carry intrinsic angular momentum in their electromagnetic fields. Note that despite the fact that these are multi-centered, they still correspond to bound states . For compact Calabi-Yau manifolds, this spin character will depend sensitively on both the Kähler and complex structure of the manifold. On a noncompact Calabi-Yau, however, the situation improves. We can form a protected spin character by utilizing the -symmetry of four-dimensional theory . The protected spin character is a genuine index that only depends on the Kähler moduli and is constant except at real codimension-one walls of marginal stability. Therefore, this protected spin character gives a well-behaved and computable definition for the refined BPS black hole partition function , depending on one extra parameter to keep track of the spin.
There is also a natural candidate for what may replace the topological string partition function. The refined topological string is a one-parameter deformation of topological string theory. Just like the refined black-hole partition function, the refined topological string partition function also makes sense only in the non-compact limit, and also utilizes the symmetry to be defined. The refined topological string is defined as an index in M-theory [21, 22]. There are several equivalent ways to compute the index, either by counting spinning M2-branes , or alternatively, as the refined BPS index for D2 and D0-branes bound to a single D6 brane[23, 24, 15], to name two. Yet another way to compute the index is as the Nekrasov partition function of the five dimensional gauge theory that arises from M-theory at low energies.333Alternatively, it has a B-model formulation, at least in some cases, in terms of -deformed matrix models [25, 26, 27].
Having found a generalization of both and , defined in the same circumstances, and depending on the same set of parameters, it is natural to conjecture that there is in fact a refinement of the OSV conjecture that relates the two:
Despite the fact that the ingredients for the conjecture fit naturally, one would still like to have a rationale for why the conjecture should hold. Because the Calabi-Yau is non-compact, the BPS black holes we are discussing are really BPS particles with large entropy. While one can imagine the system arising by taking a limit where we take the mass of the black-hole to infinity, at the same rate as we take the Planck mass to infinity so that entropy stays finite, the symmetry we are using makes sense only in the non-compact limit. Correspondingly some of the justifications for the OSV conjecture, for example those based on Wald’s formula, may no longer be sound in the refined setting, since the supergravity solution is singular.
However, as we mentioned, the original OSV relation is fundamentally a large duality. For the BPS states that came from a theory on D-branes, we always get an gauge theory describing the particles. In the ’t Hooft large limit of this theory one expects to get a string theory, whether or not there are dynamical gravitons in the theory. Indeed, many famous examples of large duality are of this kind, see for example . In particular, the duality should still hold even for the non-compact Calabi-Yau; it is merely difficult to check beyond the protected quantities. Finally, from the perspective of large duality, once we modify one side of the correspondence, the duality, at least in principle, fixes what the other side has to be. Furthermore, there is a way to understand directly from the large duality refining the black hole ensemble has to correspond to a refinement of the topological string on the other side.
To precisely test the refined OSV conjecture, one would like to compute both sides of the relation and compare explicitly. For the unrefined OSV conjecture, this was done for local, non-compact Calabi-Yau manifolds in [2, 3, 29], and agreement was found. These local Calabi-Yaus can be thought of as the limit of compact Calabi-Yaus in the neighborhood of a shrinking two- (or four-) cycle. In this limit, gravity decouples so we need to be precise about what we mean by the black hole partition function. We must require that the D-branes forming the BPS black hole wrap a cycle that becomes noncompact in this limit so that their entropy remains nonzero. In this limit, the black hole partition function is simply computed by a Witten index on the noncompact brane worldvolume. In the paper, we will run a similar test in the refined case. To this end, we will develop techniques to solve for the refined partition functions on both sides of the conjecture. Remarkably, the refined OSV conjecture passes the tests just as well as the original OSV.444In subsequent work, [2, 5, 30, 31, 32, 33, 34] several aspects of the conjecture were clarified. For one, while the relation holds to all orders in perturbation theory, the exact relation requires summing over nonperturbative corrections to the macroscopic entropy, taking the form of “baby universes” [5, 33]. Second, at the perturbative level the right side of equation 1 should include a summation over so that the periodicities of match on both sides. Additionally, the right side generically contains an additional measure factor of , which is natural from the viewpoint of Kähler quantization since transforms as a wavefunction [35, 4, 34]. Such a measure factor did not appear in the unrefined local curve examples of [2, 3]. One way to understand this is to observe that these geometries do not have a holomorphic anomaly, which implies that actually transforms as a function rather than a wavefunction – therefore the additional measure factors from Kähler quantization are absent. In this paper when we study the refined OSV relation on these local geometries, we will again find that no measure factors appear. This provides strong evidence in support of our conjecture.
The paper is organized as follows. In section 2, we review the ingredients of the original OSV conjecture for both compact and non-compact Calabi-Yaus. In section 3, we motivate the refined OSV conjecture first from the perspective of the correspondence, and second by studying the wall crossing of branes splitting into - bound states. In the noncompact setting these arguments are necessarily heuristic, but we believe they capture the correct physics. In section 4, we explain how the refined topological string on Calabi-Yau manifolds of the form can be completely solved by a two-dimensional topological quantum field theory (TQFT). We then use this TQFT to compute refined topological string amplitudes for the above geometries. We also show that our results agree precisely with the five-dimensional Nekrasov partition function of gauge theories with adjoints that are engineered by these Calabi-Yaus. In section 5, we study the black hole side of the correspondence. Mathematically, the refined partition function for D4/D2/D0-branes computes the genus of the relevant instanton moduli spaces. As originally suggested in , this can be thought of as a categorification of the euler characteristic invariants computed by the Vafa-Witten theory. We then specialize to branes wrapping the geometry and study the refined BPS partition function of bound states with and -branes. We propose that this partition function is computed by a -deformation of two-dimensional Yang-Mills, which is closely related to the refined Chern-Simons theory studied in [37, 38, 39]. Wrapping branes on the geometry , we show that -deformed Yang Mills precisely reproduces a mathematical result of Yoshioka and Nakajima for the genus of instanton moduli space [40, 41]. In section 6 we connect the black hole and topological string perspectives by studying the large N limit of the -deformed Yang Mills theory. We find that the theory factorizes to all orders in into two copies of the refined topological string partition function. This gives a nontrivial check of our refined OSV conjecture. Finally, in section 7 we explain an alternative way to compute the refined black hole partition function on . The refined bound states are counted by using the semi-primitive refined wall-crossing formula , thus giving a refined extension of the techniques used in [42, 43].
2 The OSV Conjecture: Unrefined and Refined
We start by reviewing the remarkable conjecture of Ooguri, Strominger, and Vafa (OSV) connecting four-dimensional BPS black holes with topological strings . Consider IIA string theory compactified on a Calabi-Yau, , with four-dimensional black holes arising from D-branes wrapping holomorphic cycles in . In terms of four-dimensional gauge fields, the and branes are electrically charged while the and branes are magnetically charged.
The object of interest for the OSV conjecture is the mixed black hole partition function given by fixing the magnetic charges and summing over electric charges with chemical potentials,
where we have denoted charges by , charges by , charges by , and charges by . Here and are chemical potentials associated to the electrically charged D-branes. is computed by the Witten index in the corresponding charge sector,
and only receives contributions from BPS black holes.
The OSV conjecture states that this mixed black hole partition function is equal to the square of the A-model topological string partition function on ,
where is the complexified Kähler form on . The projective coordinates on moduli space are given by,
This implies that the string coupling constant and Kähler moduli are determined by the magnetic charges and electric potentials,
Here the real part of is fixed by the attractor mechanism which determines the near-horizon Calabi-Yau moduli in terms of the black hole charge. Both sides of the relation should be considered as expansions in where is the total graviphoton charge of the black hole. From the change of variables, we have a perturbative expansion in if either or is large. We will usually set so it is natural to take both and to infinity is such a way that becomes small and the Kähler form remains constant.
One way to further understand the OSV relation is by inverting it,
so that black hole degeneracies are formally computed by the topological string. It is known that because the topological string partition function obeys the holomorphic anomaly equations , it transforms as a wavefunction under changes of polarization on the Calabi-Yau moduli space . Thus, from quantum mechanics the appearance of is very natural. In fact, this interpretation implies that is the Wigner quasi-probability function on phase space .
The original OSV conjecture focused on the case of compact Calabi-Yau manifolds, so that the wrapped D-branes correspond to black holes in four-dimensional supergravity. However, as explored in [2, 3, 46, 5, 29, 33], it is interesting to study the OSV conjecture for local Calabi-Yau manifolds which can be thought of as the decompactification limit of the compact case. In this limit gravity decouples which means that the four dimensional planck mass goes to infinity. Since the Bekenstein-Hawking entropy of a black hole is proportional to , to obtain a finite entropy in this limit we should also take . This can be accomplished simply by wrapping or -branes on cycles that become non-compact in this limit. The precise OSV relation remains the same in this limit, except that now the black hole partition function is naturally computed by a partition function on the worldvolume of the noncompact or -branes.
The advantage of taking this limit is that both sides of the OSV relation are exactly solvable, leading to a highly nontrivial check of the conjecture. The conjecture has been tested perturbatively to all orders in [2, 3, 29], and non-perturbative corrections in the form of baby universes have been computed in [5, 33].
2.1 Refining the Conjecture
Now that we have reviewed the OSV conjecture, a natural question to ask is whether the black hole degeneracy computed by is the most general index that counts four-dimensional BPS black holes. In fact, we could include information about spin by replacing the Witten index,
by the spin character,
where is the three-dimensional generator of rotations and is the conjugate chemical potential.555Rotating single-centered black holes in four dimensions cannot be supersymmetric. The black holes that we are studying here are multicentered configurations that carry intrinsic angular momentum in their electromagnetic fields. Note that despite the fact that these are multi-centered, they still typically correspond to bound states . These spin-dependent BPS traces and their wall-crossing behavior have been studied extensively in the context of field theory [13, 15, 14, 16, 17] and supergravity [18, 19]. However, this trace has the drawback of not being an index, which means that it will be sensitive to both the complex and Kähler moduli.
If we consider a local Calabi-Yau manifold by taking the gravity decoupling limit, there is a preserved R-symmetry that appears. As explained in , this can be used to form a protected spin character that is a genuine index, and only depends on the Kähler moduli through wall-crossing,
Now we can form the mixed ensemble of black holes counted with spin where, as in the ordinary case, we fix the magnetic charge and sum over the electric charge,
We will refer to this as the refined black hole partition function.
We would like to know whether there exists a generalization of the topological string whose square is equal to . A natural candidate for this one-parameter deformation is well-known, and is given by the refined topological string.
Recall the definition of the refined topological string as the index of M-theory, depending on Kahler moduli and two additional parameters and . The refined topological string partition function [21, 22, 27] on a Calabi-Yau is given by computing the index of M-theory on the geometry,
where denotes the Taub-NUT spacetime, and upon going around the the Taub-NUT is twisted by,
In addition, we must include an R-symmetry twist to preserve supersymmetry. This twist is implemented by a geometric Killing vector on the non-compact Calabi-Yau, . Note that, in our notation, the unrefined limit is , unlike in much of the literature, where one typically defines with a different sign. The partition function of M-theory in this geometry is computing the index of the resulting theory on ,
where and are the spins in the and directions, respectively, and is the R-charge of the state. We have schematically indicated that the partition function depends on the Kahler moduli , via the M2 brane contributions to the index. Note that although the trace is over all states, only BPS states will make a contribution. The index can be computed in several different ways: by counting spinning M2-branes [21, 22] or as the Omega-deformed instanton partition function . In analogy with the unrefined case , the refined topological string can also be written in terms of the refined Donaldson-Thomas invariants that compute the BPS protected spin character for D2 and D0-branes bound to a single D6 brane. Given that the refined topological string also counts BPS particles with spin and only depends on the Kähler moduli of , it is should be related to the refined black hole partition function.
The index is related to the ordinary topological string partition function is we set , and reduce on the thermal to IIA. Then, we get IIA string theory on , whose partition function is the same as the ordinary topological string partition function, where is the topological string coupling constant . In particular, M2 branes wrapping holomorphic curves in and the thermal become the worldsheet instantons of the topological string. For , the theory has no known worldsheet formulation, at the moment.
There is yet another way to view the partition function (16), which will be useful for us. This corresponds to the TST dual formulation, where instead, we go down to IIA string theory on the in the Taub-Nut space . This turns the Taub-Nut space into a single D6 brane wrapping . In this case, the refined topological string partition function has the interpretation as the refined spin character, counting the bound states of the D6 brane on with D0 and D2 branes. In terms of the rotation symmetry of the Taub-Nut space, the D0 brane charge is the component of the spin in M-theory, the rotation symmetry in IIA is identified, under the dimensional reduction, with the symmetry in M-theory, while the R-symmetry is manifestly the same in both IIA and M-theory. In particular, the refinement is associated with the diagonal spin. This allows us to rewrite 16 as
In writing this, we used the fact that , , which is obvious from the way the acts on the coordinates of the Taub-Nut space, and furthermore, as we just reviewed, that and .
This leads us to propose the refined OSV conjecture relating the protected spin character of black holes to the refined topological string666For an alternate proposal relating the Nekrasov partition function to non-supersymmetric extremal black holes, see .,
To complete the conjecture, we propose that the variables are related by,
where we have defined the variable , and we have included an additional constant .777As explained in , an arbitrary constant is needed in the compact case due to the fact that are not functions on the moduli space, but sections of a line bundle. In the non-compact case, which we study, this degree of freedom is fixed. In the unrefined case, it is typically set to . Our choice of the refined value is such that it reduces to when we set to be equal. The most natural choice for , as we explain in section 3 is
so that when we have,
for the D0 brane chemical potential, and moreover
for the spin chemical potential.
In the next section we will motivate the conjecture, and explain the origin of the change of variables. Note that, in the specific example that we study in sections 4-6, we will find that one gets a slightly different effective value of , for the reason we will explain (having to do with a shift in the zero of the spin for the D0 branes).
3 Motivating the Refined Conjecture
As explained in [1, 2, 4, 5, 6, 7, 8, 9, 10, 11], the OSV conjecture is an instance of large duality. In this case, the gauge theory is the gauge theory on the D-branes wrapping cycles of the Calabi-Yau manifold and comprising the black holes. The large dual of this theory is a string theory in the back-reacted geometry. In the full physical string theory, the near-horizon geometry of a BPS black hole in four dimensions takes the form
The OSV conjecture deals with supersymmetric sub-sectors of the theory. On the black-hole side, we consider the Witten index of the theory on D-branes; this is typically a partition function of a topological gauge theory in one dimension less. On the large dual side, the partition function ends up depending only on F-terms in the low energy effective action, which are captured by the topological string partition function. The OSV conjecture can also be thought of as a consequence of large duality in the topological setting alone. The ’t Hooft large duality, relating a gauge theory to a string theory is a very general phenomenon. It should encompass any gauge theory, including topological ones, and requires a string theory on the dual side, though not one containing dynamical gravity. In particular, in [2, 3] it was shown that OSV conjecture holds even for non-compact Calabi-Yau manifolds. In the physical version of the theories studied there, the Planck mass is infinite, and the the black holes horizon will have zero area, making it difficult to study the large duality in the full physical theory.
In the refined context, we have to take the Calabi-Yau to be non-compact, since the R-symmetry which is necessary to compute the protected index exists only in that case. However, the theory on D-branes is still a gauge theory, with large entropy at large . is simply a one parameter deformation of the ordinary black-hole partition function . The dual description of the theory at large has to be a string theory, on general grounds, and moreover, a suitable one-parameter deformation of the topological string theory. Now, we will explain why this one-parameter deformation is the refined topological string.
The statement of the refined OSV conjecture is that we can refine both sides of the OSV duality, by keeping track of the charge. Why this should be true is most tranparent in yet another way to understand the OSV, namely, using wall crossing . In this case, we do not take the near-horizon limit but instead we study general D4/D2/D0-brane bound states. We can then perform -duality on this partition function so that it is dominated by “polar” states. Generically, these polar states can be made to decay by varying the background Kähler parameters. Along a real co-dimension one wall, the state will decay into a D6/D4/D2/D0 state and a /D4/D2/D0-state. By a chain of dualities (lifting to M-theory, then reducing on a different circle), the geometry (see [6, 34] for details) can be related to IIA string theory on , with a pair.
Further, from the primitive wall-crossing formula we know that the degeneracies will factorize,
Now the key observation is that the degeneracies of D6/D4/D2/D0 brane bound states are precisely computed by Donaldson-Thomas invariants, which are further identified with the topological string. On the topological string side, the S-duality used is precisely what relates the D6 brane partition function to the topological string , as we reviewed in the previous section. Therefore, we can identify the D6-brane bound states with and the -brane bound states with . Therefore, the semiprimitive wall-crossing formula gives precisely the factorization expected from OSV,
In the refined setting, the argument goes through in precisely the same way as in the unrefined case; we simply replace, on both sides of the duality, the Witten index of the D4 brane and the D6 branes by the protected spin character. Then we simply use the refined primitive wall-crossing formula which also factorizes. On the black hole side, the protected spin character of the D4 branes is the refined black hole partition function , and on the topological string side, the protected spin character of the D6 branes is the topological string partition function – moreover, we get both and from the D6 branes and the branes, and thus
This argument makes it obvious that the OSV conjecture should hold in the refined setting, as we conjectured.
The one subtlety in this argument is that on a noncompact Calabi-Yau geometry, there is actually no place in moduli space where the -branes can be made to split into constituents, since D6 and branes will always have opposite central charges because of the noncompactness of the Calabi-Yau. However, this issue was already present in the unrefined case, where it did not affect the validity of the conjecture, as was shown in [2, 3]. Thus, there is no reason to think it would affect our refined conjecture either. Thus, we believe that this - decomposition captures the correct physics of D4/D2/D0-brane bound states in both the unrefined and the refined case, and this moreover leads to a refined OSV formula.
In the rest of this section, we will provide further support for the conjecture, and explain the identification of the parameters we gave previously.
3.1 Refined OSV and The Wave Function on the Moduli Space
The refined topological string partition function is a wave function on the moduli space, just like in the ordinary topological string case [51, 52, 53, 27]. The quantum mechanics on the moduli space played a central role in understanding the original conjecture, and the same is true in the refined case. In this respect, there only two differences between the refined and the unrefined topological string thing: for one, the effective value of the Plank’s constant of the theory , becomes (recall that, in the unrefined case, and coincide),
Secondly, the wave function that the topological string partition function computes changes: in the refined case, this wave function depends on the additional parameter .
For this discussion of the quantum mechanics, it is useful to switch to the mirror perspective and study the refined B-model . The refined B-model only depends on the complex structure moduli space, which can be parametrized by the holomorphic three form . We can choose a symplectic basis for such that , and define coordinates,
From special geometry, we know that classically these variables are not independent and that there exists a prepotential, , such that,
But now it is important to recognize that this prepotential is the genus zero contribution of the refined topological string,
Therefore in the full quantum theory, we can represent as the operator and this leads to the commutation relations,
We could have applied this same reasoning to the conjugated theory, which gives,
and finally all of the barred variables commute with all of the unbarred variables. Note that, in this case, because the Calabi-Yau is non-compact, the moduli space is always governed by the rigid special geometry of the field theory, rather than the local special geometry of supergravity.
Now consider formally inverting the refined OSV relation,
We say that this is a formal inversion, since in the non-compact case is always just a parameter, so in particular, it does not really make sense integrating over it. This aside, note that for the relation such as (31) to make sense, it has to be the case that is indeed a wave function on the moduli space. This is because while the left hand side is independent of the choice of polarization, i.e. the choice of basis of - and -cycles, for an arbitrary function on the moduli space, the right hand side would depend on such a choice, and the conjecture would not have a chance to hold. Because is a wave function, while all the terms on the right hand side depend on the choice of polarization, the integral does not depend on such a choice.
More precisely, for this to hold, one has to have the following commutation relations. We follow the reasoning in , and formally introduce magnetic potentials, in addition to the electric potential that we have already used. In the refined black hole partition function, we cannot specify both the electric charge and the electric potential at the same time, so they must have nontrivial commutation relations. Similarly, we require that the new magnetic potentials are conjugate to the magnetic charges. Therefore, we find
where for convenience we have included an extra normalization factor above. For the right hand side of 31 to be invariant under symplectic transformations one needs the black hole commutation relations and the topological string commutation relations, to be consistent. This requires,
for some arbitrary constant, . Note that is in prefect agreement with the refined OSV change of variables in equation 19 upon fixing the constant to . In fact, these relations were our main motivation for the change of variables we proposed in section 3, as a part of our conjecture. Notice that still has the interpretation as a Wigner quasi-probability distribution on phase space, just as it did in , but now it depends on the additional auxillary parameter, .
To understand the identification of and with the D0 brane chemical potential and the spin fugacity , we can use the wall crossing derivation, which forces the identification of parameters. The only subtlety is that, to relate the refined topological string to the black-hole ensemble, we need to perform the TST duality. The TST duality relates this to the chemical potentials before and after as follows:
The derivation of this is presented in appendix D.
As we reviewed in the previous section, the chemical potential for the D0 branes bound to the D6 brane is
and the spin is captured by
For this to be consistent with TST duality, the chemical potentials in the black hole ensemble need to be
for the D0 brane charge, and moreover the spin needs to be captured by
just as we gave in the previous section. In particular, .
The rest of this paper is devoted to testing our conjecture. There is a class of geometries where both sides of duality are computable explicitly. These correspond to Calabi-Yau manifolds that are complex line bundles over a Riemann surface. After developing the necessary tools to precisely compute both sides of the refined OSV formula, we show that the refined OSV conjecture holds true perturbatively to all orders for these geometries.
4 Refined Topological String on
For some simple Calabi-Yau manifolds, the refined topological string partititon function is exactly computable by cutting the Calabi-Yau into simple pieces, and sewing them back together. The open-string version of this index was computed explicitly on simple geometries in  and used to solve the refined Chern-Simons theory completely. We will follow a similar approach here in the closed string case, for Calabi-Yau manifolds of the form
To obtain a Calabi-Yau manifold, the degrees of the line bundles must satisfy the property,
The key idea is to chop up our geometries by introducing stacks of infinitely many M5 brane/anti-brane pairs wrapping Lagrangian three-cycles as in the original topological vertex .888In the refined setting, we must choose whether to wrap these M5 branes on the or plane of the Taub-NUT space. This gives two types of refined A-branes, which can be denoted as -branes and -branes. At each boundary of our geometry we can place either type of brane, leading to different choices of basis for each Hilbert space. In this paper we will not need this rich structure, and we will implicitly place -branes at each puncture. We refer the reader to [55, 56] for details on general /-brane amplitudes. Then the computation of the refined index on these chopped geometries reduces to counting M2 branes ending on these M5 branes. In this paper, we simply explain the structure of the TQFT, and refer the reader to  for the details of computing these amplitudes by counting M2-brane contributions.
4.1 A TQFT for the Refined Topological String
The basic building blocks of the TQFT are given by the annulus (A), cap (C), and pant (P) geometries. Since degrees of bundles and euler characteristics add upon gluing, this gives a way of building up more complicated bundles over Riemann surfaces.
We start by considering the simplest geometry, which is the annulus (shown in Figure 1) with two trivial complex line bundles over it given by .
Here we are summing over all representations , and is the associated Macdonald polynomial which reduces on setting to the simpler . The Macdonald metric, computes the inner product of a Macdonald polynomial with itself and is given by,999Note we are including additional (q/t) factors in our definition of and compared to the standard definitions in . The advantage of these factors is that they restore the symmetry, (see also  for a similar shift). We refer the reader to Appendix A for more details on our Macdonald polynomial conventions.
Since we are working with representations, this metric is the limit of the ordinary Macdonald metric. We refer the reader to Appendix A for our Macdonald polynomial conventions.
Having discussed the simplest geometry, we should now explain how building blocks are glued together. Recall that ordinarily when we want to glue two boundaries together, we should set their holonomies to be equal except that the boundaries should have opposite orientation. This orientation reversal simply flips one of the holonomies from to . Finally, to glue together the boundaries we must integrate over the Hilbert space at the boundaries.
This is also true in the refined setting, except that the integration measure is deformed to the one natural for Macdonald polynomials. If we denote the eigenvalues of by , then the Macdonald measure is given by,
Then gluing two boundaries gives,
where is the Macdonald metric for infinitely many variables introduced above, which is to be contrasted with the finite N Macdonald metric which we will define below in equation 134. Thus, the give an orthogonal but not orthonormal basis for the boundary Hilbert space. Although we could remove the explicit metric factors by choosing a different normalization for , it will actually be more convenient in this paper to keep them.
As a simple consistency check, gluing two annuli with trivial bundles should give back the original annulus amplitude. But this is clearly true, since the two annuli contribute a total factor of while the gluing process contributes a factor of so that the resulting amplitude is equal to the original annulus amplitude.
Now that we have explained gluing, we also want to know how to introduce nontrivial bundles. Note that since , any choice of line bundles over the annulus must satisfy . The simplest nontrivial choice is the geometry . This geometry can be alternatively understood as implementing a change in framing, which has been studied for the refined topological vertex in . The resulting amplitude is given by,
where and .
Next we study the cap geometries (shown in Figure 1), which are given by two complex line bundles over the disc. Since the euler characteristic of the disc is equal to , the degrees of the line bundles must satisfy . In practice, it suffices to determine the amplitudes, since the rest can be obtained by gluing.
It is helpful to notice that this geometry is equivalent to with a stack of branes inserted on one leg of the vertex. Thus the cap amplitude can be computed by the refined topological vertex amplitude, , with branes on the -leg or from refined Chern-Simons . The result is,
where we have defined the -dimension of a representation by,
where is the Weyl vector, while
are the arm- and leg-lengths, respectively, of a box in the Young Tableau of . The -dimension can be understood as a -deformation of the dimension of the symmetric group representation specified by .101010Our notation differs slightly from the notation used for the unrefined case in . In the limit , our -dimension is related to their by, . To obtain the cap with a different choice of line bundles we can simply glue on the or annuli.
Finally, we must specify the three-punctured sphere amplitude (see Figure 1), which we refer to as the “pant.” Since the three-punctured sphere has the euler characteristic , the degree of the line bundles must add to one in this case. To compute this amplitude it is helpful to recall some general properties that our TQFT must satisfy. Since it computes the refined topological A-model, the TQFT must be independent of complex structure moduli. Specifically this means that the amplitude for a Riemann surface should not depend on how it is formed by gluing simpler geometries. For this to be true, the pant amplitude should be symmetric in the three punctures, and thus should be diagonal in the Macdonald basis,
Now it is helpful to recognize that the pant, cap, and annulus are not all independent. We can form the annulus by capping off one of the punctures of the pant. By consistency and using the fact that is diagonal, we can solve for the pant amplitude,
So far we have described the structure of the A-model on these geometries as a TQFT, but it is important to remember that the theory is not purely topological since it depends on the Kähler moduli. For Calabi-Yaus of the form , there is only one Kähler modulus, , that measures the area of the Riemann surface. In fact, as is familiar from the topological vertex , the partition function depends on this modulus only by introducing a term, in the sum over representations.
Altogether, we have given the necessary data to solve the theory completely. As an application of these results, we can study geometries of the form where is a genus Riemann surface. For this to be a Calabi-Yau manifold we must have and . Then the refined amplitude on this geometry is given by,
where we have defined the exponentiated Kähler modulus as . It can be checked that this has the expected symmetry,
which implies that the Gopakumar-Vafa invariants come in complete multiplets of (as was the case in the unrefined limit). However, the amplitude is not symmetric under the exchange . This implies that the Gopakumar-Vafa invariants for these Calabi-Yaus do not come in full representations of , but only carry charge. The BPS states come from quantizing the moduli space of curves in , together with the bundle on them. The spin content comes from cohomologies of the moduli of the curve itself, while the comes from the bundle. In the present case, the curve is itself. Its moduli space is in general non-compact, as typically one of the two line bundles over has positive degree. Correspondingly, the Lefshetz action on the cohomologies of the moduli space does not have to result in complete multiplets – there can be contributions that escape to infinity. (See Appendix B for some sample computations of Gopakumar-Vafa invariants for these geometries.) In fact, the only case when the moduli space is compact is when , and both line bundles are . It is easy to see that in this case the amplitude does in fact have the symmetry as well.
In addition, the amplitude is not symmetric under exchange of the two line bundles, which is equivalent to taking . This tells us that one of the line bundles is distinguished from the other in the refined setting. In fact, this arises because the index in equation 16 includes an -symmetry twist that rotates a specific bundle in the noncompact Calabi-Yau (for more details see ). Equivalently, as will be explained in section 6, these refined topological string amplitudes can be obtained by taking the large N limit of branes wrapping one of the bundles. In the unrefined case, the large N limit does not retain information about which bundle the branes wrapped, but in the refined case this choice has an effect on the closed string amplitude. This is related to the above observation since in both the construction and in the closed string construction we must choose an R-symmetry rotation to preserve supersymmetry. However, these symmetries are not completely lost since the amplitude is symmetric under simultaneously exchanging and exchanging the bundles,
As we will explain in section 4.3, specifies the five-dimensional Chern-Simons coupling of the geometrically engineered gauge theory. In , it was similarly observed that for geometries that engineer five-dimensional gauge theories, the refined topological string is only symmetric under the simultaneous exchange of and inverting the Chern-Simons level, .
So far, we have computed all the non-trivial contributions to the refined topological string. However, we should also include by hand the additional pieces that appear at genus zero and one. In the unrefined case, these arise from constant maps. In the refined case, for geometries that engineer five-dimensional gauge theories, these contributions arise from the classical prepotential and the one-loop determinant of the instanton partition function. These degree zero pieces take the form,
where is the refined MacMahon function,
and is the euler characteristic of the Calabi-Yau, while is related to the triple intersection of the Kähler class and is the second Chern class of the Calabi-Yau. These numbers are ambiguous because of the non-compactness of our geometries, but it was argued in  that for the connection with black holes the natural values are,
Note, that we have split the MacMahon function into two pieces, related by interchanging and . This split naturally appears in section 6, when making the connection with the refined black hole partition function. A similar splitting was recently observed for the motivic Donaldson-Thomas invariants of the conifold in .
In section 4.3, we will give further evidence that this refined amplitude is the correct one by comparing it with the equivariant instanton partition function of the geometrically engineered five-dimensional field theory. Before doing so, however, it will be helpful to discuss one final aspect of the TQFT that arises when D-branes are included in the fiber of the complex line bundles.
4.2 Branes in the Fiber
So far we have solved for the refined string on bundles over closed Riemann surfaces and Riemann surfaces with boundaries. These boundaries naturally end on branes wrapping an in the base and two dimensions in the fiber. However, for understanding the refined OSV conjecture, it will be helpful to also consider introducing branes in the fiber. For unrefined topological strings, this was studied in , and our analysis will follow a similar approach.
We consider a lagrangian brane at a point, , in the base Riemann surface, . Since the brane is local in the base, we only need to study a neighborhood of . Thus it is natural to introduce branes in the base that chop up the geometry into a disc, , containing , and its complement, . The full amplitude is given by,
where is the holonomy around the branes in the fiber. The amplitude on the complement, , can be solved by gluing using the amplitudes in the previous section, but we still need to solve for the disc amplitude with two sets of branes.
This can be accomplished by noticing that has the topology of with the base and fiber branes on two legs of the vertex. Thus, the full cap amplitude,
is simply computed by the refined topological vertex amplitude with two stacks of branes on different legs, as shown in Figure 2.
Alternatively, as will be explained in , this amplitude can be solved by following the refined Chern-Simons theory through a geometric transition. From this perspective, is computed by the large N limit of the refined Chern-Simons S-matrix,
where we have used the symmetrized definition of the infinite-variable Macdonald polynomials in Appendix A. By including the appropriate metric factors, we obtain,
Note that if we set the representation of the fiber brane to be trivial, , then this geometry is the same as the cap (C) that we studied above. This is consistent with the fact that .
As an example, take the geometry, with branes in the fiber over points. Then the refined amplitude is given by,
It is also useful to understand how an anti-brane can be introduced that wraps the fiber. Recall that in the unrefined topological string, converting a brane into an anti-brane corresponds to taking,
where is the Schur function. The analogue of this reversal in the refined setting corresponds to taking,
where is defined by how it acts on power sums, ,
If we want to rewrite this in the basis, this can be done by using a generalized Cauchy identity (see Appendex A),