Orientifolds and the Refined Topological String
We study refined topological string theory in the presence of orientifolds by counting second-quantized BPS states in M-theory. This leads us to propose a new integrality condition for both refined and unrefined topological strings when orientifolds are present. We define the refined Chern-Simons theory which computes refined open string amplitudes for branes wrapping Seifert three-manifolds. We use the refined Chern-Simons theory to compute new invariants of torus knots that generalize the Kauffman polynomials. At large N, the refined Chern-Simons theory on is dual to refined topological strings on an orientifold of the resolved conifold, generalizing the Gopakumar-Sinha-Vafa duality. Finally, we use the theory to define and solve refined Chern-Simons theory for all ADE gauge groups.
Recently in , a refinement of Chern-Simons theory on Seifert manifolds was constructed by studying certain M-theory backgrounds. There it was also shown that in the large N limit, the partition function of refined Chern-Simons theory on is equal to the partition function of the refined topological string on the resolved conifold, thus providing a refinement of the celebrated Gopakumar-Vafa duality .
It was also shown that the refined Chern-Simons theory can be used to explicitly compute new two-variable polynomials associated to torus knots. These invariants generalize the one-variable quantum knot invariants. Further, the authors of  discovered that under appropriate changes of variables, these new polynomials could be used to determine the superpolynomial  of certain torus knots. The superpolynomial is the Poincare polynomial of a knot homology theory categorifying the HOMFLY polynomial. It encodes the large behavior of Khovanov-Rozansky knot invariants.
In analogy with the unrefined case, refined Chern-Simons on a three-manifold is equivalent to open refined string theory on the Calabi-Yau, . Thus refined Chern-Simons theory is part of the broader program of understanding the refinement of topological strings in both the closed and open settings. In the absence of a worldsheet definition, this refinement is best understood in terms of counting BPS states in M-theory.
In this paper, we analyze refined topological string theory when there is an orientifold acting on the spacetime. Again, we find that refinement is most naturally understood by studying an index that counts second-quantized BPS contributions in M-theory. This analysis leads us to propose new integrality structures for orientifolds of both the refined string partition function, and the unrefined string studied previously in [4, 5, 6, 7, 8, 9, 10, 11].
Our analysis also sheds light on the conjecture [12, 13] that the refined topological string at is equal to the ordinary topological string in the presence of an orientifold. From the second-quantized M-theory perspective, it is clear that in the noncompact part of spacetime, the trace that computes the refined topological A-model at is the same as the unrefined trace when an orientifold acts on two of the spacetime directions. However, on the internal Calabi-Yau, , the two computations are different – in the refined case, there is an internal rotation and in the unrefined case, there is an anti-holomorphic involution acting on .
For simple geometries, such as the Dijkgraaf-Vafa models that engineer gauge theory with matter in the symmetric or antisymmetric representation, we expect there will be no significant difference between the presence or absence of the involution and the conjectured correspondence will hold [14, 15, 16, 17, 18]. However, from this line of reasoning, we expect that the correspondence will not hold more generally unless we drop the involution acting on the internal Calabi-Yau.
Having understood the general structure of refined topological strings in the presence of orientifolds, we turn to the refined Chern-Simons theory that arises from wrapping branes on an orientifold plane. We explain how to solve the theory, and study the invariants that come from the expectation value of torus knots. We also study the large N limit of the theory and show that it is consistent with the expected form of refined closed strings propagating on an orientifold of the resoved conifold. This gives a new refinement of the Chern-Simons geometric transition studied in .
The organization of this paper is as follows. In section 2, we review the construction of open and closed refined topological string theory from M-theory. We also review the connection to instanton counting, vortex counting, and knot theory. In section 3, we introduce orientifolds and explain how refinement can be extended to unoriented strings. From this analysis we propose a new integrality condition in Section 4 for both unrefined and refined topological strings on orientifolds.
In section 5, we review the definition of refined Chern-Simons theory and its connection with refined open string theory. We also explain the straightforward generalization of refined Chern-Simons theory to all ADE gauge groups. In section 6 we clarify the connection between refined Chern-Simons theory and knot homology, and use the refined Chern-Simons theory to compute invariants of torus knots that generalize the Kauffman polynomials. We obtain new knot invariants associated to the gauge group in the fundamental and spinor representations. Unlike the case, we find that these polynomials cannot generally be related to the Kauffman superpolynomial  of knot homology by a change of variables. This is not unexpected as, by construction, refined Chern-Simons theory computes an index, and not a Poincare polynomial.
Finally, in section 7 we study the large N limit of the refined Chern-Simons theories on . The result is naturally interpreted as the partition function of refined closed strings on an orientifold of .
In Appendix A we give a detailed description of the topologically twisted theory on Seifert manifolds. Some useful facts about Macdonald polynomials are reviewed in Appendix B, and specific results about and its Macdonald polynomials are given in Appendix C. Finally, in Appendix D we review the refined indices in five and three dimensions that compute the refined topological string.
2 M-Theory and Refined Topological Strings
In this paper we will mainly focus on a one-parameter deformation of topological string theory that is known as the refined topological string. Before discussing refinement, we review some useful facts about unrefined topological string theory. Recall that the (A-model) closed topological string localizes on holomorphic maps from Riemann surfaces into a Calabi-Yau, , and is only sensitive to the Kahler structure of . We can also introduce branes wrapping Lagrangian three-cycles, , so that the open topological string localizes on holomorphic two-chains with boundary on . Although topological string theory was originally defined from this worldsheet point of view, it was later realized that both open and closed topological strings naturally compute certain physical indices in M-theory [20, 21, 22, 23, 24, 25, 26, 27]. In many ways, this modern viewpoint is advantageous since it reveals an integrality structure that is hidden on the worldsheet.
Although there is not yet a worldsheet definition of the refined topological string, it has a very natural definition in M-theory. Refining the topological string simply corresponds to computing a more general trace, which is a five-dimensional analogue of the refined spin character of .
In this section we review the definition of the refined topological string in both the closed and open cases. Further, we explain how this is connected to K-theoretic instanton counting and knot homology, and emphasize the integrality properties that arise from M-theory. This will lay the foundation for introducing Calabi-Yau orientifolds in the next section.
In the pioneering work of [20, 21], the topological A-model was reinterpreted in terms of integer Gopakumar-Vafa invariants that count BPS M2 branes. The connection was made by starting with IIA physical string theory on the geometry, and considering the low-energy theory in the four spacetime dimensions. It is known that the topological A-model computes terms in the low energy effective action of the form,
where is the genus free energy of the topological A-model, the are the vector multiplets whose lowest components parametrize the Kahler moduli space of , and is the Weyl multiplet. Expanding this out in components gives,
where is the self-dual graviphoton field strength and is the self-dual part of the Riemann tensor. The crucial observation of Gopakumar and Vafa was that by turning on a background graviphoton field-strength, , these terms could be reproduced by integrating out massive BPS matter. This charged matter comes precisely from D2-D0 bound states wrapping two-cycles in . It is natural to go to large coupling so that IIA becomes M-theory, and the D2-D0 states lift to M2 branes with momentum around the M-theory circle. Integrating out these states by a Schwinger-type calculation gives a new way of writing the topological string free energy,
where and counts M2 branes wrapping the cycle with intrinsic spin, . It important to note that the Gopakumar-Vafa invariants are in fact integers, since they count BPS states. Thus by integrating out -branes, this gives a new integrality structure underlying topological string theory.
Also note that only depends on the combinations,
so these are usually defined as the Gopakumar-Vafa invariants. The starting point for refining the topological string is the desire to define a more general theory that keeps track of the full spin information, and not just the spin content of branes. This can be done by turning on a non-self-dual graviphoton background, . Then we define the refined topological string to be the result of preforming the same Schwinger calculation in this modified background,
where and . This gives a working definition of the refined topological string, and makes it clear that refinement captures much more information about the spin structure of the BPS spectrum. In the rest of this paper, we will often refer to this picture as the “first-quantized” perspective, since the free-energy is related to counting of single BPS states, rather than the “second-quantized” perspective which we now introduce.
As explained in [22, 23], the partition function, of the unrefined topological A-model on a Calabi-Yau threefold, , can be computed as a trace over the second-quantized Hilbert space of BPS states. We take M-theory on and compute the five-dimensional trace,
where and are rotations in the and planes respectively, and is the radius of the thermal circle. This trace only receives contributions from BPS states (see Appendix D for details), and is equivalent to the geometry,
where denotes the Taub-NUT space. The Taub-NUT is twisted along the “thermal” so that upon going around the we have the rotation,
where as before. Note that the integrality structure of the Gopakumar-Vafa invariants in the graviphoton background computation is precisely what ensures that the result can be interpreted as a second-quantized trace.
This relation between M-Theory and the topological A-model extends to the open topological string sector as follows [24, 25, 26, 27]. Consider the open topological A-model with A-branes wrapping a special lagrangian 3-cycle, , inside a Calabi-Yau threefold, . The corresponding M-theory partition function comes from wrapping M5-branes on,
where the is the cigar-shaped submanifold, , sitting inside the Taub-NUT space. Note that the M5-brane partition function is given explicitly by the same index,
Note however, that now has the interpretation of R-charge from the perspective of the brane while corresponds to a rotation along the brane. For more details on the indices that are relevant for the open and closed, refined and unrefined topological strings, see Appendix D.
Now we would like to introduce a refined index for both the closed and open A-model. Instead of 7, consider the refined M-Theory geometry,
where upon going around the , the Taub-NUT space is twisted by,
As explained in [29, 1], when this configuration will break supersymmetry. However, when is a non-compact Calabi-Yau threefold, M-Theory on geometrically engineers a five-dimensional theory which has a conserved symmetry. Then we can modify the construction to preserve supersymmetry by including an R-symmetry twist as we go around the . When is non-compact, this is actually realized geometrically by a Killing vector in the Calabi-Yau (see  for a recent discussion of this symmetry). This geometry then computes the index,
and gives a definition of the refined closed topological string on .111It is important to emphasize that the refined topological string computes this protected spin character rather than simply counting the refined BPS multiplicities (as it would without the additional twist). The refined BPS multiplicities themselves, , are not invariant under changes of the complex structure , but the protected spin character is invariant. When the Calabi-Yau has no complex structure deformations, the naive and protected indices agree if no “exotic” BPS states (states with nonzero R-charge) are present in the spectrum. The absence of such exotic BPS states in the four-dimensional field theory limit was conjectured in .
It is instructive to consider the two possible dimensional reductions along either the thermal or the of the Taub-NUT space, as studied in the unrefined case in [32, 23]. If we consider to be noncompact, then this geometry engineers a five dimensional gauge theory and by reducing on the thermal we precisely obtain the Nekrasov partition function of the gauge theory at .
If we instead reduce along the Taub-NUT we obtain IIA string theory on the geometry,
with a D6 brane wrapping and sitting at the origin of . Here it is helpful to recall some useful facts about the Taub-NUT geometry. The geometry has a isometry, which we have used above in the definition of the index. Asymptotically, Taub-NUT looks like and the isometry rotates the , while the rotates the base geometry. So upon dimensional reduction, the charge under becomes the charge, while the charge under becomes the spin in the base . Explicitly, becomes the chemical potential, while is the chemical potential for a combination of spin and R-charge, and the index on the D6 brane can be rewritten as,
where , , is the brane charge, and is the generator of the rotation group in . Upon setting , this gives the unrefined topological string, which is known to be equivalent to the topologically twisted theory living on a D6 brane wrapping X. In the refined case, this computation of refined BPS states bound to one brane should be equivalent to the refined topological string.
It is natural to extend this construction to the open string case by inserting a stack of N M5-branes wrapping a special lagrangian inside . We will also require that is fixed by the geometric killing vector in . Then the full geometry wrapped by the branes is given by,
where again denotes the locus inside the Taub-NUT space. Now we can compute the same index as in the closed case,
This gives a definition of the refined open topological string theory on in the presence of N refined A-branes wrapping . Now recall that in the unrefined case, this M-theory partition function was related to ordinary Chern-Simons theory for the choice, . In the refined case, we do not yet have a path integral definition of the refined Chern-Simons theory, so it is natural to take this M-theoretic construction as the definition of refined Chern-Simons theory,
Again we can gain further insight by alternatively reducing the M-theory geometry on either the thermal or the Taub-NUT . Reducing on the thermal we obtain the omega background in the presence of a surface operator. Of course, it is worth noting that in some examples (), the “geometrically engineered” gauge theory in the bulk will be trivial and the only nontrivial dynamics will live on the surface operator. However, more generally we will obtain a surface operator coupled to a gauge theory, with the Omega-deformed theory computing a coupled instanton-vortex partition function . The wall crossing behavior of such coupled 2d-4d systems has recently been studied in .
Alternatively, reducing on the Taub-NUT we are left with a brane wrapping and ending on the D6 brane. This is precisely the geometry considered by Witten in connection with Khovanov homology . The only difference is that here we compute an index by utilizing the additional symmetry, whereas Witten studies Khovanov homology directly, which cannot be computed as an index as it only involves the gradings, and .
3 Orientifolds and M-Theory
Having reformulated refined topological string theory as a trace in M-theory, we now introduce orientifolds. We will restrict to orientifolds that act as an anti-holomorphic involution, , on the Calabi-Yau, including both possibilities of having fixed points or acting freely.
In physical string theory, orientifolds are defined by specifying a simultaneous involution on the spacetime, , and orientation reversal, , on the worldsheet. This definition can be extended to the ordinary topological string , by starting with the covering space, , of an unorientable worldsheet. Then the A-model will localize on holomorphic maps, satsifying the additional constraint that . Mathematically, this means that in the presence of an orientifold, the topological string is counting holomorphic maps which are -equivariant. Note that since orientation reversal on the worldsheet is anti-holomorphic, this constraint only makes sense if acts anti-holomorphically on as we have required. To fully specify an orientifold both in the physical and the topological string, we must make a choice of the sign of the cross-cap amplitude. In this paper we will restrict to the negative sign case (for open strings, this leads to an gauge group), as this choice simplifies the lift to M-theory.
This gives a working worldsheet definition of unoriented topological strings, but it was further conjectured in [4, 6, 7, 8] that the unoriented topological string can be rewritten by counting single-particle BPS states. There it was argued that the orientifolded topological string computes terms in the low energy effective action of IIA string theory on , with the involution acting simultaneously as on and as a reflection on two of the spacetime coordinates, . For future reference, we refer to this combined action as . In the case when has a fixed locus, , this corresponds to wrapping an plane on . As in the ordinary case, these terms in the effective action should also be computable by introducing a self-dual graviphoton background and studying the contribution of wrapped branes222 In order to show this more rigorously, we would need to determine exactly which terms in the orientifolded low energy effective action are computed by the topological string. Then by integrating out wrapped brane contributions to these terms, we could derive this integrality structure. This was done at genus zero in , but it would be interesting to study the higher genus amplitudes in more detail.. This results in the free energy of the closed topological A-model taking the general form,
where the are integers that count branes wrapping the cycle, .333In [8, 9, 10], it was found experimentally that for some orientifold actions with a non-toric fixed locus, the integrality of the BPS states only holds when a D-brane is introduced wrapping the same locus as the orientifold fixed plane. For example, such a D-brane must be introduced in the local geometry when the orientifold fixed plane wraps a real line bundle over the real locus, . In such cases, it was argued that only the combined contribution from open strings and unoriented strings gives the expected BPS integrality structure, and the resulting integers are known as real Gopakumar-Vafa invariants. Although we will not discuss these cases explicitly, we will find that our M-theory integrality conjecture, explained in the following section, applies to these geometries as well. In fact, the equivalent of our strong integrality property was originally stated for real GV invariants by Walcher in . In the unoriented sector, the sums are over either odd or even , depending on the details of the orientifold action . Note that the orientifold action explicitly breaks the spacetime rotational symmetry, down to . Because of this breaking, it is no longer guaranteed that the BPS states will come in full spin multiplets. However, the factors come from the moduli space of flat connections on a wrapped brane, even if the brane is wrapping an unorientable cycle. As explained below, this moduli space generically takes the form of so we expect that the BPS contributions will continue to sit in full multiplets.
To understand this conjecture physically, note that except for the factor of , the first term is the same as the free energy for the topological string on in the absence of an orientifold. This term counts BPS states that wrap oriented cycles in the and are free to move in the four spacetime directions. Since these states can move in , we obtain a factor of in the denominator. Although we do not know of a convincing target space interpretation for the factor of ,  argued that it arises on the worldsheet from dividing by the orientation reversal symmetry, .
Now we turn to the unoriented contributions to the free energy. These come from branes wrapping BPS cycles in . These cycles will lift to surfaces with boundaries in the covering space (if they did lift to closed BPS two-cycles, then this would be an oriented rather than an unoriented contribution). Now recall that the orientifold acts on the full geometry as on and as in two of the spacetime dimensions. When we take into account this full orientifold action, in order to get a genuine closed BPS state with no boundaries, it is necessary that the brane sits at . This means that the unoriented branes effectively propagate in only two noncompact dimensions, so the denominator of the unoriented contributions will give only one power of .
The first unoriented contribution to the free energy comes from D-branes wrapping unorientable surfaces with one crosscap. Recall that a crosscap is inserted into a surface by cutting open a hole and identifying antipodal points on the boundary. The lowest order contribution comes from curves with genus and one crosscap, which have the topology of .
To understand the structure in more detail, it is helpful to recall that the first homology of a genus g surface with one crosscap, , is given by,
This tells us that the continuous part of the moduli space of flat connections on is generically given by as in the original analysis of Gopakumar and Vafa. This accounts for the factor in the numerator of the term. Because of the factor, we also have the option of turning on one unit of discrete flux, which corresponds to dissolving half a unit of brane charge in the wrapped brane. As argued in , this half unit of brane flux will shift , or equivalently, . Depending on how acts, this may change the fermion number assignment and introduce an additional overall minus sign. Adding together both choices of discrete flux, we obtain the cancellation that accounts for summation over only even/odd .
Finally, we can do the same analysis on the terms with two crosscaps. It is helpful to recall that the first homology of a genus , surface is given by,
As in the single crosscap case, this means that a genus surface will have a moduli space of flat connections given by, , which leads to a factor of . Note that for the genus surface with two crosscaps, the Klein bottle, this factor in the numerator cancels the factor in the denominator from moving in two noncompact dimensions. Again, the additional torsion factor accounts for the restriction to even/odd . Note that we have exhausted all possible unorientable surfaces, since in the presence of an additional crosscap, two crosscaps can be traded for a handle. Although we will not discuss it in detail here, by similar arguments we expect the open topological string in the presence of orientifolds to possess nice integrality properties from counting BPS states ending on branes.
Now that we have given the first-quantized BPS interpretation, it is natural to conjecture that, as in the oriented case, the partition function of the topological string, is computed by a second quantized trace in M-theory. As the first step, we must explain how the IIA orientifold action lifts to M-theory.
Of course, since there is no worldsheet in M-theory, the orientifolds must lift to orbifolds in eleven dimensions. The orientifold, with acting on the spacetime and acting on the worldsheet, lifts to the M-theory orbifold by . To fully specify the orbifold, we need to explain how the three-form, is affected by the orbifold action. As explained in [37, 38, 39], acts as . One simple way to see this is to recall that the low-energy action of M-theory contains a Chern-Simons-like term,
All terms in the action should be invariant under the orbifold action, and since is orientation-reversing, the Chern-Simons term will only be invariant if .
In the case where has a fixed locus, the string theoretic plane will lift to the six-dimensional fixed locus of the M-theory orbifold, which we refer to as an -plane. However, when acts freely, the orientifold action lifts to M-theory on the unorientable, smooth geometry, . It is important to remember that the action of multiplies by , so that in this unorientable M-theory geometry, is actually a twisted three-form rather than an ordinary three-form .
Having explained the relevant objects in M-theory, we are now ready to study the topological string. We would like to compute the M-theory trace by taking the geometry,
where as before, indicates that as we go around the , we rotate the by . Note that the orbifold takes , while the the twist takes , so these operations commute with each other, as they must in order to define a sensible geometry. This defines the closed topological string in the presence of orientifolds by representing it as an M-theory trace. Similarly, the open topological string is computed exactly as before by introducing branes wrapping special lagrangain submanifolds in , but now in the presence of orientifolds.
Now we are finally ready to give a definition of the refined topological string in the presence of orientifolds. Although it may be possible to give a definition from the first-quantized graviphoton picture, we have found that refinement is simpler and clearer in the second-quantized picture.
As in the oriented case, to refine we simply need to compute a more general M-theory partition function. This leads us to consider the geometry,
where as in the oriented case, the geometry is rotated as we go around the ,
where we also must include a rotation by the to preserve supersymmetry 444For this to make sense, we must also require that the orientifold action is compatible with the isometry generating the symmetry.. Then we define the refined topological string in the presence of orientifolds to be equal to this M-theory partition function,
Similarly, the refined open topological string is defined by introducing branes wrapping special lagrangians and computing the same trace.
Before moving on to explicit computations, it is interesting to look at the general form of this M-theory partition function. We will focus on the closed case, although the open case is completely analogous. There are two types of BPS states contributing to the partition function. The first contribution comes from BPS M2-branes wrapping closed, orientable two-cycles. Each BPS state gives a field in four dimensions, , and this field has additional excitations which are BPS, provided that depends holomorphically on and . It is natural to decompose into modes as,
In the full M-theory partition function, we must include the contributions of these modes, which carry angular momentum in the and planes. However, here it is important to remember that the M-theory orbifold acts nontrivially on the spacetime as . The field, , should have a well-defined transformation property under this orbifold, so that , where the choice of is related to how the orbifold acts in . This means that we should only keep either the even or odd modes of .
Thus, the contribution to from an brane with intrinsic spin, , wrapping a two-cycle in the class takes the form,
The second type of contribution comes from M2 branes ending on the orbifold (when has fixed points) or M2 branes wrapping unorientable cylces (when does not have fixed points). These branes are frozen at the fixed locus, (since otherwise they would have a boundary). This means that they make a contribution in the form of a quantum dilogarithm,
Putting these contributions together, we find that takes the form,
where the partition function is completely determined by the integer invariants, which counts oriented branes and , which counts the unoriented branes in .
4 A New Integrality Conjecture from M-Theory
It is interesting to take equation 31 and return to the unrefined case, . We then have integrality properties following from both the second-quantized M-theory picture and from the first-quantized Gopakumar-Vafa picture. When no orientifolds are present, these integrality properties are equivalent. More explicitly, the first-quantized integer invariants appearing in the free energy, , guarantee the integrality in the second-quantized partition function, .
However, once we introduce orientifolds, the first-quantized structure (equation 3) includes overall factors of in the free energy. This poses a serious problem since such half-integers will result in terms of the form in the partition function. Such terms have no sensible interpretation as counting second-quantized states in a free Fock space.
Thus, if the topological string partition function can truly be defined as an index in M-theory, we require a stronger integrality constraint on the Gopakumar-Vafa invariants appearing in the free energy. To see precisely which constraints are needed, it is helpful to rewrite the unoriented contributions to the free energy as 555If we instead looked at the case of an even sum over , only the second term would appear, with opposite sign. Since the second term is the crucial one in our analysis, precisely the same conclusions hold regardless of whether the sum is over even or odd .
This form makes it clear that half-integers can appear in both the oriented and unoriented sector. Of course, the simplest way to guarantee integrality is if all the Gopakumar-Vafa invariants, and , are even. However, generically, we do not expect this to be the case. Instead, integrality can be achieved more generally if the half-integer contributions in the oriented and unoriented sector partially cancel each other to give integers.
To see explicitly how this occurs, let us focus on the oriented genus contribution of an M2 brane wrapping , and take to be even. Now we can combine this with the one-crosscap (), genus g amplitude for curves wrapping . Grouping the half-integer pieces of each term gives,
This can be rewritten as,
Now observe that the q-factors in front of and are identical except for signs. When we expand these terms out, we will obtain a polynomial in whose coefficients are multiples of either or . So to cancel the factor of in front of the expression, we require that the Gopakumar-Vafa invariants are both even or both odd.
We can repeat the same analysis by combining the odd genus contributions with the terms. Putting everything together, we find that the second quantized M-theory interpretation of the topological string implies the strong integrality property,
Note that in come cases, there cannot be any BPS state wrapping the class . In this case, the strong integrality property tells us that itself must be even. We have verified that this integrality property is satisfied for all of the geometries studied in [6, 8, 9], including orientifolds of the conifold, the quintic, local , and more general toric geometries.
An equivalent observation was made by Walcher for the case of certain real orientifolds, with a D-brane wrapping the fixed locus . Walcher argued that if we pretend the moduli space of oriented wrapped branes is a collection of points, then acts on this set. The points that are fixed by are precisely the BPS states wrapping the fixed real locus, while the other points must come in pairs that are exchanged by . This implies that the “real” (unoriented) and oriented Gopakumar-Vafa invariants must differ by a multiple of two. Here we have generalized this statement to include arbitrary orientifolds, and have given it a natural M-theory interpretation.
At this point we could go one step further and obtain expressions for the natural M-theoretic integer invariants, and , in terms of the by expanding out the polynomials in q. It is interesting to note that these variables are complimentary, in the sense that the variables make integrality manifest, but obscure the organization of invariants in full spin multiplets. In contrast, the natural free energy variables, make manifest the spin multiplet structure, while hiding the full integrality properties.
We can also consider the integrality structure when we have branes wrapping a special Lagrangian 3-cycle, , inside . When the orientifold action, , has fixed points, we will assume that the new branes do not sit at the fixed locus. As explained in , the open topological string will receive contributions from an oriented and an unoriented sector and the free energy will include half-integer contributions. As discussed above, the partition function, , of the open topological string can be written as a second-quantized trace in M-theory counting branes ending on branes wrapping . For this M-theory interpretation to hold, the half-integers in the free energy must cancel, leading us to a new integrality condition in the open case.
To explain this in more detail, we begin with the first-quantized form of the open string free energy conjectured in ,
The first term comes from oriented open strings and can be computed by going to the covering space. Since the branes are not fixed by , in the covering space there will be branes wrapping both and its image under the involution, . However, since these two stacks of branes are related by , their open string moduli must be equal to each other, giving the trace . The second term comes from unoriented strings, with the sum over restricted to odd or even positive integers depending on the orientifold action, as in the closed case.
It is convenient to rewrite the oriented piece as,
where is the Littlewood-Richardson coefficient for decomposing the tensor product . We can also rewrite the unoriented piece (assuming the sum is over odd ) as,
where is the coefficient of the second Adams Operation defined by,
It is natural to combine the half-integer pieces in the oriented and unoriented amplitudes to give,
We can expand the functions as,
where and are integers counting BPS states with spin , wrapping the relative homology class, , and in representation . Then the absence of half-integers in the free energy imposes the condition,
It is important to note that the integers and that we have used above are not the most fundamental BPS invariants. To exhibit the full BPS structure it is necessary to include the structure of the moduli space of flat connections and geometric deformations of an open brane . In general this structure is more complicated and involves the Clebsch-Gordon coefficients of the symmetric group. For simplicity, we focus on the case of , which leads to a simple integrality constraint on the fundamental BPS invariants, .
We start by expanding out Equation 40 in traces over ,
Now since exchanges the two stacks of branes in the covering space, it follows that . Therefore, the term and the first two terms of do not contribute any half integers, but the last two terms could. Imposing integrality leads to the condition,
Since we are working with the relatively simple functions can be expanded in terms of the fundamental BPS invariants,representation, these
The invariant counts BPS states of genus with crosscaps wrapping the class and in representation . We can group terms exactly as we did in the closed case, and we find the integrality condition,
It was argued in , that when is the resolved conifold and is the special lagrangian corresponding to a knot , then the composite BPS invariants, are related to the HOMFLY polynomial in the composite representation . By making use of this connection, the BPS invariants have been computed for many knots in [11, 42, 43]. Using this data, we have explicitly verified our strong integrality conjecture for the unknot, trefoil, and knots. It would be interesting to perform these checks for more complicated geometries.
In this section we have focused on unrefined amplitudes. In the case of refined amplitudes, we do not yet have a first-quantized definition, but we still demand integrality from the second quantized M-theory index that defines the theory. In Section 7, we will test this integrality by studying the large N limit of refined Chern-Simons theory, and interpreting the result as refined closed strings on the resolved conifold.
5 Open Strings from Refined Chern-Simons Theory
Now that we have given a definition of the refined topological string for oriented geometries and in the presence of orientifolds, we would like to solve the theory. In the case of open, refined topological strings, this was done in  by carefully analyzing the contributions of BPS states to the refined index. As explained above, if we choose our Calabi-Yau to be the cotangent bundle over a three-manifold, , with branes wrapping , then we expect that the open refined string theory should reduce to a three-dimensional field theory living on . This theory is a refined version of Chern-Simons theory, and was defined for oriented strings in . The crucial idea  used to compute the M-theory indices was to cut up (along with ) into pieces on which one can solve the theory explicitly, and glue the pieces back together. In this way, the and matrices of refined Chern-Simons theory could be deduced from M-theory.
In the unoriented case, the natural anti-holomorphic involution of is given by reversing the fibers of the cotangent bundle, so that . This is equivalent to inserting an orientifold plane that wraps . As is standard in both physical and topological string theory , the inclusion of an orientifold plane (with the appropriate choice of crosscap sign) simply changes the gauge group from to . We can now follow the rest of the steps from  in the present context. Taking to be a solid torus, the refined M-theory index can be computed explicitly in the presence of orientifold action. Instead of going through this here, we refer the reader to Appendix A for a detailed discussion of the derivation in the case. We will simply state the answer. We find that the theory is solved, analogously to the case, by replacing characters of with D-type Macdonald polynomials associated to the root system of .
Moreover, in Appendix A, we also extend this to refined Chern-Simons theory with any simply-laced gauge group. Although there is no known brane realization of the gauge groups, the same analysis still works in the corresponding six-dimensional theory. In the appendix, we also give an explicit example of the connection with three dimensional field theory when .
In this section we describe the refined Chern-Simons theory in detail from the perspective of topological field theory. This perspective will be especially helpful when we begin studying knot invariants in Section 6. Although a Lagrangian for refined Chern-Simons theory is not yet known, we can still define the theory by describing its amplitudes on simple geometries.
5.1 Refined Chern-Simons as a Topological Field Theory
Recall that any three-dimensional topological field theory should assign a number to any closed three manifold, , and an element of the appropriate Hilbert space to any three manifold with boundary, , where . Further, diffeomorphisms that act geometrically on the boundary should be represented by unitary operators acting on the Hilbert space, .
The refined Chern-Simons theory can be thought of as a restricted topological field theory in three-dimensions. It is only well-defined on Seifert three-manifolds, M, to which it assigns a number, . This restriction can be understood by recalling the definition of open refined string theory, which requires the existence of a isometry on . The refined Chern-Simons theory also assigns an element of the Hilbert space to any “Seifert manifold with boundary,” possessing a nondegenerate fibration. Necessarily, the boundary of such a manifold will be a disjoint union of two-tori. Finally, the modular group acts geometrically on , so the refined Chern-Simons theory gives a unitary representation of acting on .
To fully specify the refined Chern-Simons theory, we must choose a simply-laced compact gauge group, , an integer level, , and a continuous deformation parameter, . It will be useful in the following formulas to define equivalent variables,
where is the dual Coxeter number of .666The dual Coxeter number is given by for , for , 12 for , 18 for , and 30 for . Note that the unrefined limit corresponds to .
We start by describing the Hilbert space, , associated to a boundary. Recall that in ordinary Chern-Simons theory, with gauge group, , and coupling, , the Hilbert space is given by the space of conformal blocks on of the associated Wess-Zumino-Witten model. This Hilbert space has a natural orthonormal basis, given by integrable representations of at level . In the Chern-Simons theory, this basis can be understood physically by taking the solid torus, and inserting a Wilson line in the representation running along . Then performing the path integral on this geometry gives the state .
Note that to define this basis, we must decide which cycle of the boundary will be filled in to make a solid torus. This gives two natural bases associated with filling in the and cycles respectively. The modular transformation, then will be represented as a unitary operator that transforms the basis into the basis.
The vector space structure of does not change under refinement. This is expected heuristically, since is a continuous parameter that can be adiabatically changed from the unrefined theory to the refined theory . This can be seen more precisely by noting that the metric, , defined below, vanishes for any representations that are not integrable at level .
However, under refinement the inner product on these spaces does change. In ordinary Chern-Simons theory, the basis is an orthonormal one so that,
In the refined theory, this natural basis remains orthogonal but the normalization is nontrivial,
where is the metric defined by taking the Macdonald inner product of the Macdonald polynomial, with itself (see Appendix B for background on Macdonald polynomials). In the special case when , the metric factor is given explicitly by,
where the product is over the positive roots, , and is the Weyl vector, . For more general choices of , there exists a combinatorial formula for , which is given for the case in Appendix C. Note that we could rescale the to obtain a normalized basis, but for subsequent formulas it is actually more convenient to leave the inner product nontrivial.
Now that we have described the Hilbert space associated to each boundary, we should consider arbitrary Seifert three-manifolds, with boundaries. The refined Chern-Simons theory should assign a specific element of the Hilbert space, , to each three-manifold.
We begin by defining the theory on the “pair of paints” geometry given by the three-punctured sphere times a circle, , shown in Figure 1(a). In Figure 1, we have included Wilson lines wrapping the A-cycles of the boundaries. This depiction is useful for two reasons: first, we will use these Wilson lines to keep track of the orientation of the boundaries, since in a TQFT changing the orientation of a boundary takes the Hilbert Space to its dual, .
Second, recall from the discussion above that to specify a basis of , we must choose a cycle on . In all of the following discussion, the element will correspond to taking the boundary and gluing in a solid torus, where the fiber becomes contractible. Then we insert a Wilson loop, in the representation running along one of the boundaries of the base pair of pants, or annulus, , and sitting at a point in the now-contractible fiber. Computing the path integral then gives the state . We have depicted this choice of basis graphically by showing the Wilson loops on .
Then the wavefunction of the refined Chern-Simons theory on is given by,
where is the metric defined above and can be thought of as the -dimension of the representation . This -dimension is a refinement of the quantum-dimension that appears in ordinary Chern-Simons theory, and is given by,
where is the Macdonald polynomial for the gauge group and representation , and is a normalization factor defined below.