ITEP-TH-27/07 LANDAU-07-01

Surface Operators and Knot Homologies

Sergei Gukov

Department of Physics, University of California

Santa Barbara, CA 93106

Topological gauge theories in four dimensions which admit surface operators provide a natural framework for realizing homological knot invariants. Every such theory leads to an action of the braid group on branes on the corresponding moduli space. This action plays a key role in the construction of homological knot invariants. We illustrate the general construction with examples based on surface operators in and twisted gauge theories which lead to a categorification of the Alexander polynomial, the equivariant knot signature, and certain analogs of the Casson invariant.

This paper is based on a lecture delivered at the International Congress on Mathematical Physics 2006, Rio de Janeiro, and at the RTN Workshop 2006, Napoli.

June, 2007

1. Introduction

Topological field theory is a natural framework for “categorification”, an informal procedure that turns integers into vector spaces (abelian groups), vector spaces into abelian or triangulated categories, operators into functors between these categories [1]. The number becomes the dimension of the vector space, while the vector space becomes the Grothendieck group of the category (tensored with a field). This procedure can be illustrated by the following diagram [2]:

Recently, this idea led to a number of remarkable developments in various branches of mathematics, notably in low-dimensional topology, where many polynomial knot invariants were lifted to homological invariants.

Although the list of homological knot invariants is constantly growing, most of the existing knot homologies fit into the “A-series” of homological knot invariants associated with the fundamental representation of (or ). Each such theory is a doubly graded knot homology whose graded Euler characteristic with respect to one of the gradings gives the corresponding knot invariant,

For example, the Jones polynmial can be obtained in this way as the graded Euler characteristic of the Khovanov homology [3]. Similarly, the so-called knot Floer homology [4,5] provides a categorification of the Alexander polynomial . At first, these as well as other homological knot invariants listed in the table below appear to have very different character. Thus, as the name suggests, knot Floer homology is defined as a symplectic Floer homology of two Lagrangian submanifolds in a certain configuration space, while the other theories are defined combinatorially. In addition, the definition of the knot Floer homology admits a generalization to knots in arbitrary 3-manifolds, whereas at present the definition of the other knot homologies (with ) is known only for knots in .

Knot Polynomial Categorification knot Floer homology “” — Lee’s deformed theory Jones Khovanov homology homology

Table 1: A general picture of knot polynomials and knot homologies.

The knot homology [3,6,7] — whose Euler characteristic is the quantum invariant — has a physical interpretation as the space of BPS states, , in string theory [8]. In order to remind the physical setup of [8], let us recall that polynomial knot invariants, such as , can be related to open topological srting amplitudes (“open Gromov-Witten invariants”) by first embedding Chern-Simons gauge theory in topological string theory [9], and then using the so-called large duality [10,11,12], a close cousin of the celebrated AdS/CFT duality [13]. Moreover, open topological string amplitudes and, hence, the corresponding knot invariants can be reformulated in terms of new integer invariants which capture the spectrum of BPS states in the string Hilbert space, . The BPS states in question are membranes ending on Lagrangian five-branes in M-theory on a non-compact Calabi-Yau space . Specifically, the five-branes have world-volume where is a Lagrangian submanifold (which depends on knot ) and .

Fig. 1: A membrane ending on a Lagrangian five-brane.

Surprisingly, the physical interpretation of the knot homology naturally leads to a triply-graded (rather than doubly-graded) knot homology [8] (see also [14,15]). Indeed, the Hilbert space of BPS states, , is graded by three quantum numbers, which are easy to see in the physical setup described in the previous paragraph. The world-volume of the five-brane breaks the rotation symmetry in five dimensions down to a subgroup , where (resp. ) is a rotation symmetry in the dimensions parallel (resp. transverse) to the five-brane. Therefore, BPS states in the effective theory on the five-brane are labeled by three quantum numbers , , and , where is the relative homology class represented by the membrane world-volume. In other words, apart from the -grading by the fermion number, the Hilbert space of BPS states is triply-graded. The properties of this triply-graded theory were studied in [16]; it turns out that this theory unifies all the doubly-graded knot homologies listed in Table 1, including the knot Floer homology. A mathematical definition of the triply-graded knot homology which appears to have many of the expected properties was constructed in [17].

Apart from realization in (topological) string theory, the homological knot invariants are expected to have a physical realization also in topological gauge theory, roughly as polynomial knot invariants have a physical realization in three-dimensional gauge theory (namely, in the Chern-Simons theory [18]) as well as in the topological string theory [9,10,11]. Although these two realizations are not unrelated, different properties of knot polynomials are easier to see in one description or the other. For example, the dependence on the rank is manifest in the string theory description, while the skein operations and transformations under surgeries are easier to see in the Chern-Simons gauge theory.

Similarly, as we explained above, string theory realization is very useful for understanding relation between knot homologies of different rank. On the other hand, the formal properties of knot homologies which are hard to see in string theory (which, however, would be very natural in topological field theory) have to do with the fact that, in most cases, knot homologies can be extended to a functor from the category of 3-manifolds with links and cobordisms to the category of graded vector spaces and homomorphisms

Moreover, on manifolds with corners, it is expected that extends to a 2-functor from the 2-category of oriented and decorated 4-manifolds with corners to the 2-category of triangulated categories [19,20,21]. In particular, it should associate:

a triangulated category to a closed oriented 2-manifold ;

an exact functor to a 3-dimensional oriented cobordism ;

a natural transformation to a 4-dimensional oriented cobordism .

As we explain below, these are precisely the formal properties of a four-dimensional topological field theory with boundaries and corners. Moreover, links and link cobordisms can be incorporated by introducing “surface operators” in the topological gauge theory.

In section 2, we discuss the general aspects of topological gauge theories which admit surface operators. Of particular importance is the fact that every topological gauge theory which admits surface operators gives rise to an action of the braid group on D-branes. Then, in sections 3 and 4 we illustrate how these general structures are realized in simple examples of and twisted gauge theories. Specifically, in section 3 we study surface operators and the corresponding knot homologies in the Donaldson-Witten theory and in the Seiberg-Witten theory, both of which are obtained by twisting supersymmetric gauge theory. In section 4, we explain that a particular twist of the super-Yang-Mills theory — studied recently in connection with the geometric Langlands program [22,23] — with a simple type of surface operators provides a physical framework for categorification of the Casson invariant.

2. Gauge Theory and Categorification

Let us start by describing the general properties of the topological quantum field theory (TQFT) with boundaries, corners, and surface operators. To a closed 4-manifold , a four-dimensional TQFT associates a number, , the partition function of the topological theory on . Similarly, to a closed 3-manifold , it associates a vector space, , the Hilbert space obtained by quantization of the theory on . Finally, to a closed surface it associates a triangulated category, , which can be understood as the category of D-branes in the topological sigma-model obtained via the dimensional reduction of gauge theory on . The objects of the category describe BRST-invariant boundary conditions in the four-dimensional TQFT on 4-manifolds with corners (locally, such manifolds look like ). Summarizing,

gauge theory on number gauge theory on vector space gauge theory on category

where we assume that , , and are closed.
Depending on whether the topological
reduction of the four-dimensional gauge theory on
gives a topological A-model or B-model, the category
is either the derived Fukaya category^{†}^{†} Notice, according
to the Homological Mirror Symmetry conjecture, this category
is equivalent to the derived category of the mirror B-model [24].
In particular, the category , suitably defined,
must be a triangulated category [25]., , or the derived
category of coherent sheaves, ,

topological A-model: topological B-model:

where is the moduli space of classical solutions in gauge theory on , invariant under translations along . Different topological gauge theories lead to different functors . For example, in the context of Donaldson-Witten theory [26], Fukaya suggested [27] that the category associated to a closed surface should be -category of Lagrangian submanifolds in the moduli space of flat -connections on . This is precisely what one finds from the topological reduction [28] of the twisted gauge theory on , in agreement with the general principle discussed here.

The Atiyah-Floer conjecture and its variants

It is easy to see that defined by the topological gauge theory has all the expected properties of a 2-functor. In particular, to a 3-manifold with boundary it associates a “D-brane”, that is an object in the category .

Fig. 2: A 3-manifold can be obtained as a connected sum of 3-manifolds and , joined along their common boundary . In four-dimensional gauge theory, the space is obtained by gluing two 4-manifolds with corners.

The interpretation of 3-manifolds with boundary as D-branes can be used to reproduce the Atiyah-Floer conjecture, which states [29]:

Here is the moduli space of flat connections on , while and are Lagrangian submanifolds in associated with the Heegard splitting of ,

such that the points of , , correspond to flat connections on which can be extended to .

Similarly, in the B-model, and define the corresponding B-branes, which are objects in the derived category of coherent sheaves on . In both cases, the vector space associated with the compact 3-manifold is the space of “ strings”:

In the Donaldson-Witten theory, this leads to the Atiyah-Floer conjecture (2.1).

‘‘Decategorification’’

The operation represented by the arrow going to the left in (1.1) — “decategorification” — also has a natural interpretation in gauge theory. It corresponds to the dimensional reduction, or compactification on a circle. Indeed, the partition function in gauge theory on is the trace (the index) over the Hilbert space :

Similarly, the vector space associated with is the Grothendieck group of the category

In the case of A-model and B-model, respectively, we find

where .

2.1. Incorporating Surface Operators

In a three-dimensional TQFT, knots and links can be incorporated by inrtoducing topological loop observables. The familiar example is the Wilson loop observable in Chern-Simons theory,

Recall, that canonical quantization of the Chern-Simons theory on associates a vector space — the “physical Hilbert space” — to a Riemann surface [18]. In presence of Wilson lines, quantization gives a Hilbert space canonically associated to a Riemann surface together with marked points (points where Wilson lines meet ) decorated by representations . For example, to marked points on the plane colored by the fundamental representation it associates , where is a -dimensional irreducible representation of the quantum group .

Fig. 3: line operators in a three-dimensional TQFT on and topological “surface operators” in four-dimensional gauge theory on .

We wish to lift this to a four-dimensional gauge theory by including the “time” direction, so that the space-time becomes , where the knot is represented by a topological defect (which was called a “surface operator” in [23]) localized on the surface . In the Feynman path integral, a surface operator is defined by requiring the gauge field (and perhaps other fields as well) to have a prescribed singularity. For example, the simplest type of singularity studied in [23] creates a holonomy of the gauge field on a small loop around ,

Quantization of the four-dimensional topological theory on a 4-manifold with a surface operator on gives rise to a functor that associates to this data (namely, a 3-manifold , a knot , and parameters of the surface operator) a vector space, the space of quantum ground states,

Moreover, we will be interested in surface operators which preserve topological invariance for more general 4-manifolds and embedded surfaces . For example, if the four-dimensional topological gauge theory is obtained by a topological twist of a supersymmetric gauge theory, it is natural to consider a special class of surface operators which preserve supersymmetry, in particular, those supercharges which become BRST charges in the twisted theory. Such surface operators can be defined on a more general embedded surface , which might be either closed or end on the boundary of . An example of this situation is a four-dimensional TQFT with corners, shown on fig. 3, which arisies when we consider a lift of a 3-manifold with boundary and line operators with end-points on .

To summarize, including topological surface operators in the four-dimensional gauge theory, we obtain a functor from the category of 3-manifolds with links and their cobordisms to the category of graded vector spaces and homomorphisms:

Here, the knot homology is the space of quantum ground states in the four-dimensional gauge theory with surface operators and boundaries. Similarly, the functor associates a number (the partition function) to a closed 4-manifold with embedded surfaces, and a category to a surface with marked points, , which correspond to the end-points of the topological surface operators.

As in the theory without surface operators, the category is either the category of A-branes or the category of B-branes on , depending on whether the topological reduction of the four-dimensional gauge theory is A-model or B-model. Here, is the moduli space of -invariant solutions in gauge theory on with surface operators supported at .

2.2. Braid Group Actions

As we just explained, surface operators are the key ingredients for
realizing knot homologies in four-dimensional gauge theory.
Our next goal is to explain that every topological gauge theory
which admits surface operators is, in a sense, a factory that
produces examples of braid group actions on branes,
including some of the known examples as well as the new ones.^{†}^{†} It is worth
pointing out that, compared to [23], where the braid group
action is associated with local singularities in the moduli space ,
in the present context the origin of the braid group action is associated
with global singularities.

In general, the mapping class group of the surface acts on branes on . In particular, when is a plane with punctures, the moduli space is fibered over the configuration space of unordered points on ,

and the braid group (= the mapping class group of the -punctured disk) acts on the category . Recall, that the braid group on strands, , has generators, , which satisfy the following relations

where can be reprsented by a braid with only one crossing between the strands and , as shown on the figure below.

Fig. 4: A braid on four strands.

In gauge theory, the action of the braid group on branes is induced by braiding of the surface operators. Namely, a braid, such as the one on fig. 4, corresponds to a non-contactible loop in the configuration space, . As we go around the loop, the fibration (2.11) has a monodromy, which acts on the category of branes as an autoequivalence,

The simplest situation where one finds the action of the braid group on A-branes (resp. B-branes) on is when contans chain of Lagrangian spheres (resp. spherical objects).

We remind that an chain of Lagrangian spheres is a collection of Lagrangian spheres , such that

These configurations occur when can be degenerated into a manifold with singularity of type . Indeed, to any Lagrangian sphere , one can associate a symplectic automorphism of , the so-called generalized Dehn twist along , which acts on as the Picard-Lefschetz monodromy transformation

As shown in [30], Dehn twists along chains of Lagrangian spheres satisfy the braid relations (2.12), and this induces an action of the braid group with strands on the category of A-branes, .

The mirror of this construction gives an example of the braid group action on B-branes [31]. In this case, the braid group is generated by the twist functors along spherical objects (“spherical B-branes”) which are mirror to the Lagrangian spheres. As the name suggests, an object is called -spherical if is isomorphic to for some ,

A spherical B-brane defines a twist functor which, for any , fits into exact triangle

where the first map is evaluation. At the level of D-brane charges, the twist functor acts as, cf. (2.15),

where is the D-brane charge (the Mukai vector) of .

The mirror of an chain of Lagrangian spheres is an chain of spherical objects, that is a collection of spherical objects which satisfy the condition analogous to (2.14),

With some minor technical assumptions [31], the corresponding twist functors generate an action of the braid group on . As we illustrate below, many examples of braid group actions on branes can be found by studying gauge theory with surface operators.

Fig. 5: A particular brane which corresponds to closing a braid on four strands.

In A-model as well as in B-model, the braid group action on branes can be used to write a convenient expression for knot homology, , of a knot represented as a braid closure. Let be a knot obtained by closing a braid on both ends as shown on fig. 5. Then, the space of quantum ground states, , in the four-dimensional gauge theory with a surface operator on can be represented as the space of open string states between branes and . Here, is the basic brane which corresponds to the configuration on fig. 5, while is the brane obtained from it by applying the functor ; it corresponds to the braid closed on one side. These are A-branes (resp. B-branes) in the case of A-model (resp. B-model), and the space open strings is, cf. (2.3),

In particular, when topological reduction of the gauge theory gives A-model, the branes and are represented by Lagrangian submanifolds in . This leads to a construction of link homologies via symplectic geometry, as in [32,33,34].

3. Surface Operators and Knot Homologies in Gauge Theory

Now, let us illustrate the general structures discussed in the previous section in the context of topological gauge theory in four dimensions. For simplicity, we consider examples of gauge theories with gauge groups and known as the Donaldson-Witten theory and the Seiberg-Witten theory, respectively. In fact, these two theories are closely related [35] — the former describes the low-energy physics of the latter — and below we shall use this fact to compare the corresponding knot homologies.

3.1. Donaldson-Witten Theory and the Equivariant Knot Signature

We start with pure super-Yang-Mills theory with gauge group which for simplicity we take to be . After the topological twist, the gauge theory can be formulated on arbitrary 4-manifold and localizes on the anti-self-dual (“instanton”) field configurations [26]:

The space of quantum ground states on is the instanton
Floer homology defined^{†}^{†} As in the original Floer’s definition,
we mainly assume that is a homology sphere when we talk about
in order to avoid difficulties related to reducible connections.
by studying the dradient flow of the Chern-Simons functional,

and the topological reduction [28] on leads to a topological A-model with the target space , the moduli space of flat connections on . As we already mentioned in the previous section these facts, together with the interpretation of boundaries as D-branes, naturally lead to the statement of the Atiyah-Floer conjecture (2.1). The Euler characteristic of is the Casson invariant, , which computes the Euler characteristic of the moduli space of flat -connections on ,

In the special case of that we are mainly considering here, it is the standard Casson-Walker-Lescop invariant which sometimes we write simply as .

We note that, while the homological invariant is difficult to study on 3-manifolds with , its Euler characteristic — which is, at least formally, computed by the partition function of the four-dimensional gauge theory on — is still given by the Casson invariant [36],

Since for , computing (3.4) is much easier in the case . Indeed, in general the Donaldson-Witten partition function can be written as a sum of the contribution of the Coulomb branch (the -plane integral) and two contributions, and , both of which are described by the Seiberg-Witten theory (that we consider in more detail below):

For manifolds with the -plane integral vanishes and we have , which then add up to (3.3). If , the Donaldson-Witten partition function depends on the metric. In particular, it should be compared with the Euler characteristic of in the chamber , where is the radius of . In this case, the -plane integral is non-zero, and instead of (3.4) one finds a similar expression with the “correction” , see [36] for more details.

Surface Operators

Now let us consider surface operators in the Donaldson-Witten theory which correspond to the singularity of the gauge field of the form

Here, are radial coordinates in the normal plane, is the parameter which labels surface operators and takes values in , the Lie algebra of the maximal torus , and the dots in (3.6) stand for less singular terms. More precisely, inequivalent choices of are labeled by elements in since gauge transformations shift by vectors in the cocharacter lattice of . For example, for we have .

In the presence of a surface operator on , supersymmetric field configurations in this theory are described by the instanton equations (3.1):

perturbed by the term , where denotes the self-dual part of the cohomology class that is Poincaré dual to the surface . In the context of gauge theory, such surface operators were extensively used in the work of Kronheimer and Mrowka on minimal genus problems of embedded surfaces in 4-manifolds [37,38].

According to the general rules outlined in the previous section, to a 4-manifold and a surface operator on labeled by the Donaldson-Witten theory associates a vector space, the space of quantum ground states,

Just like the ordinary instanton Floer homology (3.3), it categorifies a Casson-like invariant,

which counts flat connections on a homology sphere with the prescribed singularity (3.6) along .

In order to describe more explicitly, it is convenient to decompose as in (2.2) into a tubular neighborhood of the knot , , and its complement, , glued along the common boundary . As we already mentioned earlier, topological reduction of the Donaldson-Witten theory on yields a topological A-model with the target space , the moduli space of flat connections on :

For this moduli space is the quotient, , of two copies of the maximal torus by the Weyl group of . In particular, for the corresponding moduli space is often called the “pillowcase”. Similarly, each component in the decomposition defines an A-brane supported on a Lagrangian submanifold in . If we denote Lagrangian submanifolds associated to and , respectively, by and , then the invariant (3.9) is given by their intersection number (in the smooth part of ):

Note, the Lagrangian brane supported on does not depend on or , while the Lagrangian brane supported on does not depend on . Indeed, is simply the set of representations taking the meridian of the knot to a matrix of trace . Similarly, the Lagrangian brane supported on corresponds to flat connections on which can be extended to flat connections on . In other words, is the image of under the restriction map

induced by the inclusion of the torus boundary of the knot complement, .

To summarize, surface operators in the Donaldson-Witten theory lead to a variant of the instanton Floer homology, , whose Euler characteristic is a Casson-like invariant (3.11). This is precisely the definition of the knot invariant which was introduced and studied in [39,40,41] (see also [42,43]). This invariant, sometimes called Casson-Lin invariant, is well-defined away from the roots of the Alexander polynomial of and turns out to be equal to the linear combination of more familiar invariants, ,

where is the Casson invariant of and is the equivariant signature function (a.k.a. Levine-Tristram signature) of the knot . Homology theory categorifying was constructed in [44] (see also [45,46]) and, therefore, is expected to be the same as (3.8).

We remind that, for a knot in a homology sphere , the normalized Alexander polynomial is defined as

where is the Seifert matrix of and . Note, that . The equivariant signature is defined as the signature of the Hermitian matrix

The equivariant signature function changes its value only if is a root of the Alexander polynomial. It vanishes for near or ,

and equals the standard knot signature, , for . In particular, for and we get the original Lin’s invariant [39] and the corresponding homology theory categorifying was constructed — as symplectic Floer homology (2.19) of the braid representative of — in [47].

3.2. Seiberg-Witten Theory

Now let us consider twisted gauge theory with abelian gauge group coupled to a single monopole field . This theory localizes on the solutions to the Seiberg-Witten equations for abelian monopoles [48]:

which follow from the topological gauge theory^{†}^{†}
Up to the finite group ,
the set of Spin structures on a 4-manifold is
parameterized by integral cohomology classes
which reduce to mod 2,

To be more precise, the Seiberg-Witten invariants are defined as integrals over , the moduli space of solutions to the Seiberg-Witten equations (3.16),

where is the virtual dimension of , and is a 2-form which represents the first Chern class of the universal line bundle on the moduli space .

The space of quantum ground states in this theory is the Seiberg-Witten monopole homology,

which is conjectured to be isomorphic to the Heegard Floer homology, see e.g. [49]:

In turn, the Heegard Floer homology — as well as its analog for knots, the knot Floer homology , that is closer to our interest — is defined as the symplectic Floer homology of certain Lagrangian submanifolds in the symmetric product space of the form [4,5,50],

The symmetric product space and Lagrangian submanifolds in it
naturally appear in the topological reduction of the Seiberg-Witten theory.
Indeed, on the equations (3.16) reduce to the vortex equations in the abelian Higgs model,
and the moduli space of solutions to these equations,
namely the moduli space of charge vortices,
is the symmetric product space (3.19), see [51].
As in the case of the Donaldson-Witten theory,
the topological reduction of the Seiberg-Witten theory
leads to the topological A-model^{†}^{†} In fact, this is true
for any four-dimensional gauge theory with the same
type of topological twist [28]. with as the target space,
and the corresponding category of branes in this case
is the category of A-branes,

According to the general rules explained in the previous section, the Euler characteristic of the homology theory (3.17) - (3.18) is given by the partition function of the Seiberg-Witten theory on (in the chamber ):

If , then there are no wall-crossing phenomena and can be equivalently viewed as the partition function of the three-dimensional gauge theory on obtained by the dimensional reduction of the Seiberg-Witten theory. For a fairly general class of 3-manifolds , the partition function (3.21) is equal to the Casson invariant of , c.f. (3.4):

For instance, for 3-manifolds with it follows e.g. from the general result of Meng and Taubes [52] that will be discussed in more detail below. On the other hand, for homology spheres the definition of the Seiberg-Witten invariants requires extra care. However, once this is done, one can show that (3.22) still holds for suitably defined ; see [53,54,55] for a mathematical proof and [56] for a physical argument based on the duality with Rozansky-Witten theory [57].

Surface Operators

As in the Donaldson-Witten theory, we can introduce surface operators by requiring the gauge field to have the singularity of the form (3.6). In the Seiberg-Witten theory, such surface operators are labeled by . In the presence of a surface operator on , supersymmetric field configurations are described by the perturbed Seiberg-Witten monopole equations, cf. [48,58]:

As usual, in order to obtain a homological invariant of a knot in a 3-manifold one should consider the Hilbert space of the gauge theory on with a surface operator on . In the context of Seiberg-Witten theory, this gives a vector space . More generally, given a link with components one can introduce surface operators, each with its own parameter , . The corresponding Hilbert space is

where is the link complement, and parameters determine the boundary conditions on . Namely, the holonomy of the gauge connection along the meridian of the -th link component should be equal to . We will be mainly interested in the case where and is the link complement. In this case, (3.24) gives -graded link homology.

We introduce the graded Euler characteristic of the homological invariant (3.24),

which is a formal power series in , where and are the generators of a free abelian group,

In particular, when is a link complement, the group is generated by the meridians of the link components.

In general, is a non-trivial function of and . It is equal to the partition function of the Seiberg-Witten theory on (in the chamber ):

This function is an interesting generalization of the Reidemeister-Milnor torsion, on the one hand, and the equivariant knot signature, on the other. Indeed, since the Seiberg-Witten theory is the low-energy description of the Donaldson-Witten theory, we expect the relation to the equivariant knot signature. On the other hand, if is near or for all , as in (3.15), then the partition function becomes the ordinary partition function of the link complement studied by Meng and Taubes [52] who showed that it is equal to the Reidemeister-Milnor torsion. Hence,

where is the ordinary Reidemeister-Milnor torsion of . In particular, for we have , so that in this limit the homological invariant (3.24) categorifies the multi-variable Alexander polynomial of the link ,

This suggests to identify the -graded homology theory (3.24) with the link Floer homology [59],

In the case of knots, the relation to the Alexander polynomial is slightly more delicate, in part due to metric dependence and wall crossing. It turns out, however, that even though individual Seiberg-Witten invariants are different in the positive and negative chamber, the corresponding generating functions are both equal to the Milnor torsion [52], so that (3.27) still holds. Note, that specializing (3.27) further to , we recover (3.22). It would be interesting to study the invariant further, in particular, its relation to the equivariant knot signature .

4. Surface Operators and Knot Homologies in Gauge Theory

Now, let us consider surface operators and knot homologies in the context of topological super-Yang-Mills theory in four dimensions. Specifically, we shall consider the GL twist of the theory [22], with surface operators labeled by regular semi-simple conjugacy classes [23]. As we shall explain below, this theory provides a natural framework for categorification of the Casson invariant, which counts flat connections of the complexified gauge group .

The topological reduction of this theory leads to a sigma-model [28,60,22], whose target space is a hyper-Kahler manifold , the moduli space of solutions to the Hitchin equations on [61]:

This twist of the super-Yang-Mills theory has a rich spectrum of supersymmetric surface operators. In particular, here we will be interested in the most basic type of surface operators, which correspond to the singular behavior of the gauge field and the Higgs field of the form [23]:

where , and the dots stand for the terms less singular at . For generic values of the parameters , eq. (4.2) defines a surface operator associated with the regular semi-simple conjugacy class .

According to the general rules explained in section 2,
this topological field theory associates a homological invariant
to a closed 3-manifold and, more generally, a knot homology
to a 3-manifold with a knot (link) .
These homologies can be computed as in (2.3) and (2.19) using the Heegard decomposition of as well as the braid group action on branes.
The branes in questions^{†}^{†} e.g. branes and
associated with the Heegard decomposition
are branes of type with respect to the three complex structures
of the hyper-Kahler space .
We can use this fact and analyze the branes in different complex structures
in order to gain a better understanding of the homological invariant
as well as the Casson invariant itself.
For example, in complex structure it corresponds to counting
parabolic Higgs bundles, a fact that has already been used e.g. in [62] for studying the Casson invariant for Seifert fibered homology spheres.

Complex Structure : Counting Flat Connections

The B-model in complex structure is obtained, e.g. by setting the theta angle to zero, , and choosing (where is a complex parameter that labels a family of GL twists of the super-Yang-Mills [22]). In complex structure , the moduli space is the space of complexified flat connections , and the surface operator (4.2) creates a holonomy,

which is conjugate to . Furthermore, at the supersymmetry equations of the four-dimensional gauge theory are equivalent to the flatness equations, , which explains why (from the viewpoint of complex structure ) the partition function of this theory on with a surface operator on computes the Casson invariant,

The space of ground states, , is a categorification of . In general, both and depend on the holonomy , which characterizes surface operators. However, if is regular semi-simple, as we consider here, then and do not depend on a particular choice of .

Complex Structure

Since the four-dimensional topological gauge theory (even with surface operators) does not depend on the parameter that labels different twists, we can take , which leads to the A-model on with symplectic structure . This theory computes the same Casson invariant and its categorification, , but via counting solutions to the following equations on [22]:

rather than flat connections. In particular, given a Heegard decomposition , the space of solutions to the equations (4.3) on (resp. ) defines a Lagrangian A-brane in with respect to . This allows to express as the space of open string states between the corresponding A-branes and , cf. (2.3),

This alternative definition of the Casson invariant and its categorification that follows from the twisted gauge theory can be useful, for instance, for understanding situations when the branes and intersect at singular points in or over higher-dimensional subvarieties.

Categorification of the Casson Invariant

Now, let us return to the complex structure and, for simplicity, take the gauge group to be . Furthermore, we shall consider an important example of the sphere with four punctures:

which in gauge theory corresponds to inserting four surface operators. In complex structure , is the moduli space of flat connections with fixed conjugacy class of the monodromy around each puncture. It can be identified with the space of conjugacy classes of monodromy representations

where the representations are restricted to take the simple loop around the -th puncture into the conjugacy class .

Fig. 6: Sphere with four punctures.

Using the fact that is free on three generators, we can explicitly describe the moduli space by introducing holonomies of the flat connection around each puncture,

where and each is in a fixed conjugacy class. Following [63,64,65,66,67], we introduce the local monodromy data

and