Analytic Torsion, 3d Mirror Symmetry,
And Supergroup Chern-Simons Theories
Department of Physics, Princeton University, Princeton, NJ 08544
We consider topological field theories that compute the Reidemeister-Milnor-Turaev torsion in three dimensions. These are the psl(1|1) and the U(1|1) Chern-Simons theories, coupled to a background complex flat gauge field. We use the 3d mirror symmetry to derive the Meng-Taubes theorem, which relates the torsion and the Seiberg-Witten invariants, for a three-manifold with arbitrary first Betti number. We also present the Hamiltonian quantization of our theories, find the modular transformations of states, and various properties of loop operators. Our results for the U(1|1) theory are in general consistent with the results, found for the GL(1|1) WZW model. We also make some comments on more general supergroup Chern-Simons theories.
May 12, 2015
In this paper, we study the topological quantum field theory that computes the Reidemeister-Milnor-Turaev torsion ,  in three dimensions. This is a Gaussian theory of a number of bosonic and fermionic fields in a background flat complex GL gauge field. It can be obtained by topological twisting from a free hypermultiplet with supersymmetry. This theory is very simple and can be given different names – the one-loop Chern-Simons path-integral , or the Rozansky-Witten model  with target space , or the U supergroup Chern-Simons theory  at level equal to one, but we prefer to call it supergroup Chern-Simons theory.
Let us give a brief summary of the paper. In section 2, we describe the definition of the theory. We explain that its functional integral computes a ratio of determinants, which depends holomorphically on a background flat GL bundle . We also define various line operators, the most important of which lead to the Alexander polynomial for knots and links.
In section 3, we use mirror symmetry in three dimensions to represent the theory as the endpoint of an RG flow, that starts from the twisted version of the QED with one fundamental flavor. The computation of the partition function of the QED can be localized on the set of solutions to the three-dimensional version of the Seiberg-Witten equations . This provides a physicist’s derivation of the relation between the Reidemeister-Turaev torsion and the Seiberg-Witten invariants, which is known as the Meng-Taubes theorem , . We consider, in particular, the subtle case of three-manifolds with first Betti number and show, how the quantum field theory manages to reproduce the details of the Meng-Taubes theorem in this case. Previously, the same RG flow has been used in  to derive a special case of the Meng-Taubes theorem for the trivial background bundle, when the torsion degenerates to the Casson-Walker invariant. (We elaborate a little more on this in the end of section 3.) In comparison to , the new ingredient in our paper is the coupling of the QED to the background flat bundle , so let us explain, how this works. In flat space and before twisting, the QED has a triplet of FI terms , which transform as a vector under the SU-subgroup of the R-symmetry. (In our notations, the scalars of the vector multiplet of the QED transform in the vector representation of SU.) These FI terms can be thought of as a vev of the scalars of a background twisted vector multiplet. The vector field of the same multiplet can be coupled in a supersymmetric way to the current of the topological U-symmetry of the QED. Upon twisting the theory by SU, the scalar and the vector fields of the twisted vector multiplet combine into a complex gauge field . Invariance under the topological supercharge requires this background field to be flat. One can easily see that the partition function depends on it holomorphically. In the theory, which emerges in the IR, the field gives rise to the complex flat connection that is used in the definition of the Reidemeister-Turaev torsion.
In section 4, we consider the U supergroup Chern-Simons theory. It is obtained from the theory by coupling it to U Chern-Simons gauge fields. It has been argued previously ,  that this theory computes the torsion that we study. We show that, in fact, the U theory for the compact form of the gauge group is a -orbifold of the theory, and thus, indeed, computes essentially the same invariant. To be more precise, there exist different versions of the U theory, which differ by the global form of the gauge group, but they all are related to the theory. Mirror symmetry maps the U Chern-Simons theory at level to an orbifold of the same twisted QED, or equivalently, to an QED with one electron of charge .
In section 5, we present the Hamiltonian quantization of the theory. This section does not depend on the results of section 3, and can be read separately. By considering braiding transformations of the states on a punctured sphere, we recover the skein relations for the multivariable Alexander polynomial. We consider in some detail the Hilbert space of the theory on a torus, and the correspondence between the states and the loop operators. We find the OPEs of line operators and the action of the modular group. In fact, as long as the background bundle has non-trivial holonomies along the cycles of the Riemann surface, on which the theory is quantized, the Hilbert space is one-dimensional, and our analysis is very straightforward. We also discuss the canonical quantization of the U Chern-Simons theory. We consider modular transformations of the states on the torus, and find results very similar to those obtained from the GL WZW model . To our knowledge, this is the first example of the canonical quantization of a supergroup Chern-Simons theory, that does not assume an a priori relation to the WZW model.
In section 6, we discuss possible generalizations to other supergroup Chern-Simons theories. We make a summary of properties of such theories. (Some of these have previously appeared in .) We also present some brane constructions, and consider possible dualities.
Besides the papers that we have already mentioned, previous work on the topological field theory interpretation of the Meng-Taubes theorem includes , where the subject was approached from the four-dimensional Donaldson theory, and , where a mathematically rigorous proof of the Meng-Taubes theorem using TQFT was presented. All the mathematical facts about the Reidemeister-Turaev torsion, the Seiberg-Witten invariants and the Meng-Taubes theory, that we touch upon in this paper, can be found in a comprehensive review .
Finally, let us mention that there exists yet another approach  to the Reidemeister-Turaev torsion, which presumably can be given a physical interpretation, – in this case, in terms of the first-quantized theory of Seiberg-Witten monopoles. Unfortunately, this will not be considered in the present paper.
2 Electric Theory
In this section, we describe the theory, which computes an analytic analog of the Reidemeister-Turaev torsion. Up to some details, it is simply the theory of the degenerate quadratic functional . One important difference, however, is that we introduce a coupling to a complex background flat bundle, and consider the torsion as a holomorphic function of it. Our definition is similar but not quite identical to the definition of the analytic torsion, known in the mathematical literature . The discussion will be phrased in the language of supergroup Chern-Simons theory. Though this might seem like an unnecessary over-complication, it will make our formulas a little more compact, and will also help, when we discuss generalizations in later sections. Throughout the paper, the theory of this section will be called “electric”, while its mirror, considered in section 3, will be called “magnetic”.
2.1 The Simplest Supergroup Chern-Simons Theory
In this section we introduce the Chern-Simons theory. We work on a closed oriented three-manifold .
The superalgebra is simply the supercommutative Grassmann algebra . The Chern-Simons gauge field will be a -valued fermionic one-form , where and are the superalgebra generators. To make the theory interesting, we want to couple it to a background flat bundle. It could possibly be a GL-bundle, where GL acts on in the obvious way. However, the definition of the Chern-Simons action requires a choice of an invariant bilinear form. This reduces the symmetry to SL, so we couple the theory to a flat SL-bundle . The Chern-Simons action can be written as111Throughout the paper we use Euclidean conventions, in which the functional under the path-integral is .
where the supertrace denotes an invariant two-form, Str, and is the covariant differential, acting on the forms valued in . One could eliminate the flat gauge field from by a suitable choice of trivialization of , but we prefer not to do so.
The supergroup gauge transformations act by . To fix the gauge, we introduce a -valued ghost field . Since our gauge symmetry is fermionic, this field has to be bosonic: its two components are complex scalars and . We also introduce a bosonic -valued antighost field and a -valued fermionic Lagrange multiplier . The BRST generator is defined to act as
Next we have to choose an appropriate gauge-fixing action. It will contain in particular the kinetic term for the bosonic fields and , and we want to make sure that this term is positive-definite. To that end, we pick a hermitian structure on our flat bundle and restrict to unitary gauges. We impose a reality condition . The complex flat connection in can be decomposed as , where is a hermitian connection and is a section of the adjoint bundle. We introduce a covariant derivative , and also introduce notations for the covariant derivative in the flat bundle and for the covariant derivative with the conjugate gauge field. We pick a metric on and take the gauge-fixing action to be
The bosonic part of this action is manifestly positive-definite. The gauge-fixing condition is . The action has a ghost number symmetry U, under which the ghost and the antighost fields have charges . If the background field satisfies , or equivalently , this symmetry is enhanced to SU, which rotates and as a doublet and which we will call SU. If we turn off the background gauge field completely, we also recover the “flavor” SU symmetry, which is the unitary subgroup of the SL automorphism group of the superalgebra. The groups SU and SU commute. Together they generate an action of SO on the real four-dimensional space parameterized by and .
In this paper, we will not consider the general SL analytic torsion222The reason is that the Meng-Taubes theorem, which will be the subject of section 3, does not seem to generalize to SL torsion, since only the abelian part of the symmetry is visible in the UV. However, what could be generalized to the SL torsion (and, in fact, to Sp torsion) is the Hamiltonian quantization that we consider in section 5. This generalization will be discussed elsewhere.. From now on, we restrict our attention to the case that the background flat bundle is abelian, , where333Throughout the paper, the coefficients in homology and cohomology are assumed to be , unless explicitly specified otherwise. . By abuse of notation, we will denote the connection in by the same letters , where now is understood to be a connection in a flat unitary line bundle, and is a closed one-form, whose cohomology class determines the absolute values of the holonomies in .
The abelian background field preserves a U-subgroup of the flavor symmetry group SU. We will furthermore assume that is chosen to be the harmonic representative in its cohomology class, so that the -symmetry is present.
2.2 Relation To A Free Hypermultiplet
Our theory can be obtained by making a topological twist of the theory of a free hypermultiplet. This is a trivial special case of the general relation between supergroup Chern-Simons and Chern-Simons-matter theories, found in . For completeness, we provide some details.
The R-symmetry group of supersymmetry in three dimensions is SU. The supercharges transform in the -representation of . A supersymmetric theory can be twisted by taking the Lorentz spin-connection to act by elements of the diagonal subgroup of SU. This leaves an SU doublet of invariant supercharges. We pick one of them, to be called , and use it to define a cohomological topological theory. The ghost number symmetry U is the subgroup of SU, for which is an eigenvector.
The scalars of the free hyper give rise to the ghost fields and . They parameterize a copy of the quaternionic line , which has a natural action of two commuting SU groups. One of them is identified with the R-symmetry group SU, and the other is the flavor symmetry SU. The hypermultiplet fermions, which transform in the representation of the Lorentz and R-symmetry groups, upon twisting give rise to a vector field and a scalar, which we identify with the fermionic gauge field and the Lagrange multiplier field .
Finally, the imaginary part of the flat connection originates from the SU-triplet of hypermultiplet masses. While they are constant parameters in the untwisted theory, they are promoted to a closed one-form in the topological theory, still preserving the -invariance. Different terms in the action (2.1), (2.3) can be easily seen to originate from the kinetic and the mass terms for the hypermultiplet scalars and fermions.
2.3 A Closer Look At The Analytic Torsion
Here we would like to take a closer look at the invariant that our theory computes. We discuss its properties and relation to other known definitions of the torsion. For simplicity, the manifold is assumed to be closed, unless indicated otherwise.
2.3.1 Definition And Properties
The partition function of the theory can be written as a ratio of determinants,
Here the operator is acting in , where is the space of -forms valued in . The twisted Laplacian is acting in . Note that the operator is hermitian, while is hermitian only when .
The ratio , by construction, is a holomorphic function of the flat bundle, even though the determinants in (2.4) are not. We can understand the analytic properties of rather explicitly. The absolute value of the torsion can be written in the usual Ray-Singer form as
where is the twisted Laplacian, acting on one-forms. The numerator in this formula vanishes, whenever the twisted cohomology is non-empty. This subspace, possibly with the exception of the trivial flat bundle, is the locus of zeros of . The denominator vanishes, when the twisted cohomology is non-empty, which is precisely when the flat bundle is trivial. At this point the function can potentially have a singularity. In fact, if the first Betti number of is greater than one, the singularity would be of codimension at least two, which is not possible for a holomorphic function. For , let the holonomies of around the torsion444A cycle is called “torsion” if it lies in the torsion part of , that is, if some multiple of it is trivial. This use of the word “torsion” is totally unrelated to “torsion” as an invariant of the manifold. Hopefully, this will not cause confusion. one-cycles be trivial, and let be the holonomy around the non-torsion one-cycle. At , the operators and have one zero mode each. At small , these eigenfunctions become quasi-zero modes with eigenvalues of order , according to the non-degenerate perturbation theory. Plugging this into (2.5), we see that the ratio near the trivial flat bundle is proportional to , that is, has a second-order pole. Finally, for the torsion is a function on the discrete set of flat bundles. For the trivial flat bundle and , it is natural to set to be equal to infinity555One could say that for the trivial bundle the path-integral is undefined, since it has both bosonic and fermionic zero modes. But it is natural to set it equal to infinity for , because, thinking in terms of gauge-fixing, the path-integral has a factor of inverse volume of the gauge supergroup, which is infinity, since this volume is zero. Taking also makes the factorization formulas of the ordinary Chern-Simons valid in the supergroup case..
Another important property of the torsion is the relation
which follows from the charge conjugation symmetry that maps the superalgebra generators as , and the line bundle to its dual .
2.3.2 Details Of The Definition
We would like to make a more precise statement about what we mean by the formal definition (2.4). Let us assume for now that the flat bundle is unitary. If we eliminate the ambiguities in the definition of for such bundles, the definition for complex flat bundles will also be unambiguous, by analyticity.
The absolute value (2.5) of is (the inverse of) the Ray-Singer torsion, which is a well-defined and metric-independent object. However, as is well-known in the context of Chern-Simons theory , the definition of the phase of requires more care. With our assumption that is unitary, the operator is hermitian and has real eigenvalues. Since the determinant of comes from a fermionic path-integral, it is natural to choose a regularization, in which it is real. The only possible ambiguity then is in its sign. Note that this is mainly interesting in the case when there is torsion in , so that the space of flat bundles is not connected, and signs can potentially be changed for different connected components.
Let us suggest a way to define the sign of . What we are about to say might not seem particularly natural at first sight, but, as we show later, matches well with known definitions of the analytic and combinatorial torsion. Let us pick a spin structure on the three-manifold , and take some oriented spin four-manifold , of which with a given spin structure is a boundary. The line bundle can be extended onto , though the extension might not be flat. On we consider the Donaldson operator that arises from the linearization of the self-duality equations, twisted by the line bundle . Here is the bundle of anti-selfdual two-forms. We define the sign of the determinant of , and therefore of the torsion , using the index of the elliptic operator ,
where is the Donaldson operator coupled to the trivial line bundle. The motivation behind this definition is that, if we were to compute the change of sign of under a continuous change of , we could naturally do it by using the formula (2.7) with the four-manifold taken to be the cylinder , since the index of on such a cylinder computes the spectral flow of .
We started with a choice of a spin structure, but so far it did not explicitly enter the discussion. Its role is the following. For two different choices of the four-manifold, the change in the sign of is governed by the index of on a closed four-manifold , which, according to the index theorem, is
However, since the spin structure on can be extended to , the four-manifold is spin, and therefore its intersection form is even, and so is the right hand side of (2.8). We conclude that the sign of depends on a spin structure on , but not on the choice of the four-manifold. (This is equivalent to the well-known fact  that a choice of a spin-structure allows to define a half-integral Chern-Simons term for an abelian gauge field.)
It is not hard to calculate the dependence on the spin structure explicitly. Let and be two spin structures on , which differ by some . Let and be four-manifolds with boundary , onto which and extend. Now the closed four-manifold , glued from and along their boundary , need not be spin, and its Stiefel-Whitney class can be non-zero. The intersection form is not even, but its odd part is governed by the Wu’s formula, which tells us that , where is the mod 2 reduction of . (This is true for any class, of course.) The Stiefel-Whitney class of is determined by . For a given good covering of , the two spin structures and define a lift of the transition functions of the tangent bundle of from SO to Spin, and this lift is consistent everywhere, except for a codimension-two chain, lying in . This chain defines the Stiefel-Whitney class of , but it is also the Poincaré dual of the class in . These arguments allow us to write
where stands for Poincaré dual. We conclude that under a change of the spin structure by , the sign of changes by the factor
It will be useful to rearrange this formula a little. For that we need to recall a couple of topological facts. The topology of a flat line bundle is completely defined by its holonomies around the torsion one-cycles. This is formalized by the following exact sequence,
which is associated to the short exact sequence of coefficients . By Pontryagin duality, is the abelian group of (unitary) flat line bundles on . The morphism gives a flat bundle with trivial holonomy around the torsion cycles and given holonomy around the non-torsion cycles666What one means by non-torsion cycles is not canonically defined, but this does not matter, when the holonomies around the torsion cycles are trivial.. The morphism maps a given flat bundle to its first Chern class, which depends only on the holonomies around the torsion cycles, by exactness of the sequence. Pick a pair of classes and from . Let be some flat bundle with Chern class . Its holonomies around the torsion cycles are completely defined by . We can take a holonomy of around the one-cycle, Poincaré-dual to . The logarithm of this number gives a pairing , which is known as the linking form. An important fact is that it is bilinear and symmetric. (Actually, this pairing is just the U Chern-Simons term for flat bundles.)
Returning to the formula (2.10), we note that defines a -bundle, and (2.10) is the holonomy of this bundle around the one-cycle, Poincaré dual to . Since the linking form is symmetric, this holonomy is equal to the holonomy of around the one-cycle, Poincaré dual to , where, to construct , we think of the -bundle defined by as of a U-bundle. This holonomy will be denoted by . We conclude that it defines the change of the sign of , when the spin structure on is changed by . To indicate the dependence on the spin structure explicitly, we will sometimes write the torsion as , so that
It is noteworthy that if the line bundle has trivial holonomies around 2-torsion cycles, the definition of is independent of any choices at all.
In fact, even for a generic flat bundle, depends on something less than a spin structure. There is a natural map from the set of spin structures to the set of spin- structures with trivial determinant, which is given by tensoring with a trivial line bundle,. This map is not an isomorphism, because in general two different spin structures can map to the same spin- structure. Since the change of the sign of under a change of by an element of depends only on the first Chern class of the line bundle obtained from , the sign of really depends only on a spin- structure with trivial determinant, and not on the spin structure itself.
One could consider some trivial generalizations of our definition of the torsion. For example, can be naturally defined for an arbitrary spin- structure , not necessarily with trivial determinant. Let be some arbitrary spin- structure with trivial determinant, be an arbitrary spin- structure, and let be such that . We can set . Clearly, (2.12) implies that depends only on , and not on the choice of . In quantum field theory terms, this modification amounts to adding to the action a local topologically-invariant functional of the background gauge field – the Wilson loop of around the cycle, Poincaré-dual to . Another possible modification of the definition would be to add a Chern-Simons term for the background field . Note that, if we choose the coefficient of this term to be a half-integer, this would eliminate the dependence of on the spin structure. In what follows, we will mostly restrict to our most basic definition of , unless indicated otherwise.
2.3.3 Comparison To Known Definitions
Let us comment on the relation of our torsion to some known definitions from the mathematical literature. A rigorous definition of the complex analytic torsion was given in . The authors consider essentially777There are some differences. The discussion in  is more general: the authors consider a manifold of arbitrary odd dimension, and not necessarily one-dimensional flat vector bundles. Another difference from our approach, if phrased in path-integral language, is that in  the gauge-fixing term in the analog of (2.3) is defined using the derivative , rather than its conjugate. This eliminates the need to pick a hermitian structure on the flat bundle, but makes the functional integral representation of the determinant more formal. Finally, the ratio of determinants in  is actually the inverse of ours. the same ratio of determinants (2.4) and use the -function regularization to define it as a holomorphic function of the flat bundle . An important difference, however, is that for a unitary flat bundle their torsion is not real, but has a phase, proportional to the eta-invariant of . In the language of functional integral, such definition is perhaps more natural , when the determinant of comes from a bosonic, rather than a fermionic functional integral. The relation to our definition is given by the APS index theorem: to transform the eta-invariant into the index, one needs to subtract what might be called a half-integral Chern-Simons term of the flat connection in the line bundle . This is why the dependence on a spin structure appeares in our story, but not in .
In fact, there is a combinatorial definition of torsion, which, as we conjecture, is precisely equal to our . This is the Turaev’s refinement of Reidemeister torsion888Note that sometimes Reidemeister torsion is defined to be the inverse of what we consider here. With the definition that we use, the absolute value of the combinatorial torsion is equal to the inverse of Ray-Singer torsion, defined in the usual way.. We briefly summarize some facts about it. For a detailed review, as well as references, the reader can consult .
Let be a compact three-manifold, which is closed or is a complement of a link neighborhood in a closed three-manifold, so that it has a boundary consisting of a number of tori. (In our language, non-empty boundary will correspond to adding line operators, to which we turn in the next section.) In either case, the Euler characteristic of is zero. Reidemeister torsion of is defined as the determinant of a particular acyclic complex, twisted by a vector representation of the fundamental group of the manifold. The determinant of this combinatorially defined complex can be viewed as a discretisation of the functional integral, which computes the analytic torsion. We will assume that the representation of the fundamental group is given by the flat line bundle . Reidemeister torsion is defined only up to a sign and up to multiplication by a holonomy of around an arbitrary cycle in . This happens because the determinant depends on the basis in the complex, of which there is no canonical choice. Turaev has shown  that this ambiguity can be eliminated, once one makes a choice of what he called an Euler structure999More precisely, the choice of an Euler structure eliminates the freedom to multiply the torsion by a holonomy of , while the overall sign can be fixed by choosing an orientation in the homology . At least for a closed three-manifold, there exists a canonical homology orientation, defined by the Poincaré duality, and we assume that our theory automatically picks this orientation.. In analytical terms, it is a choice of a nowhere vanishing vector field, up to homotopy and up to an arbitrary modification inside a three-ball. Such vector fields always exist on , since . In three dimensions, it is not hard to see that Euler structures are in a canonical one-to-one correspondence with spin- structures. For a spin- structure , let us denote the Reidemeister-Turaev combinatorial torsion by . Under a change of the spin- structure by an element , the torsion changes as
where, as usual in our notations, is the holonomy of around the cycle Poincaré dual to . The combinatorial torsion also has a charge conjugation symmetry
where is the conjugate of the spin- structure , and is the number of connected components of the boundary of . The second equality here follows from (2.13).
If the three-manifold is closed and the spin- structure has trivial determinant, we claim that coincides with our analytic torsion . (Modulo signs, that is, ignoring the dependence on the spin structure, this statement would follow from the results of  and .) For a spin- structure with trivial determinant, the properties (2.13) and (2.14) reduce to our formulas (2.12) and (2.6), respectively. When the three-manifold is not closed but is a complement of a link, the relation between and our should still hold, with an appropriate definition of the analytic torsion in presence of line operators. This will be discussed in the next section.
An important special case is when the flat bundle has trivial holonomies around the torsion one-cycles. Then is a holomorphic function of variables . Let us also ignore the dependence on , so that we consider modulo sign and modulo multiplication by powers of . This variant of the combinatorial torsion is known as the Milnor torsion. A theorem due to Milnor  and Turaev  describes its relation to the Alexander polynomial of , which is a function of the same variables . If , then . If , then , if is a closed three-manifold, and , if is a complement of a knot in a closed three-manifold. For a closed , these statements are in agreement with the analytical properties of our , described in section 2.3.1.
2.4 Line Operators
We would like to define some line operators in our theory, in order to study knot invariants. First thing that comes to mind is to use Wilson lines. For these to be invariant under the transformations (2.2), they should be labeled by representations of . This superalgebra contains as well as one bosonic generator, which acts on the fermionic generators with charges . The Wilson lines should be defined with the connection . In fact, the only irreducible representations of are one-dimensional representations, to be denoted , in which the bosonic generator acts with some charge , and the fermionic generators act trivially. Inserting a Wilson loop in representation along a knot is equivalent to multiplying the path-integral by the -th power of the holonomy of the background bundle around the cycle . Though this operator is of a rather trivial sort, it will be convenient to consider it as a line operator. It will be denoted by , . According to the remarks at the end of section 2.3.2, inserting operators around various cycles is equivalent to changing the spin- structure, with which the torsion is defined.
All the other representations of are reducible, but, in general, can be indecomposable101010That is, they have invariant subspaces, but need not split into direct sums.. Some examples are shown on fig. 1. (There are more such representations – they are listed e.g. in , – but we will not need them.) In this paper, we are mostly interested in closed loop operators. Naively, due to the presence of the supertrace, a closed Wilson loop labeled by a reducible indecomposable representation splits into a sum of Wilson loops for the invariant subspaces and quotients by them. If this were true, the indecomposable representations would not need to be considered separately. We will later find that, due to regularization issues, at least for some indecomposable representations the Wilson loops do not actually reduce to sums of Wilson loops . This will be discussed in section 5, but till then we will not consider indecomposable representations.
In the case that the holonomy of the background field along some loop is trivial, one can construct a line operator by inserting an integral into the path-integral. Note that these operators transform as a doublet under the SU flavor symmetry. These will play the role of creation/annihilation operators in the Hamiltonian picture in section 5, but, again, will not be important till there.
The most useful line operator can be obtained by cutting a knot (or a link) out of , and requiring the background gauge field to have a singularity near with some prescribed holonomy around the meridian of the knot complement111111The meridian is the cycle that can be represented by a small circle, linking around the knot. A longitude is a cycle that goes parallel to the knot. The longitude, unlike the meridian, is not canonically defined. Its choice is equivalent to choosing a framing of the knot.. It is this type of line operators that will give rise to the Alexander knot polynomial.
One has to be careful in defining the determinants (2.4) in presence of such a singularity. In this paper, our understanding of the determinants in this case will be much less complete than in the case of closed three-manifolds. We will not attempt to give a rigorous definition, but will simply state some results that are consistent with other approaches to line operators, which are discussed later in the paper, and with known properties of the Alexander polynomial. Let be the holonomy around the meridian of the knot , and be the holonomy around the longitude. While is a part of the definition of the line operator along , depends on the flat connection and, in particular, on other line operators, linked with . The problem with the determinants (2.4) in presence of line operators is that in general they can be anomalous, that is, they can change sign under large gauge transformations of the background gauge field. Equivalently, one will in general encounter half-integral powers of and in the expectation values. One possible resolution is to choose a square root of the holonomies, or, equivalently, to take , and to consider the knot polynomial as a function of the holonomies of . One expects this to produce a version of the Alexander polynomial known as the Conway function. (See section 4 of  for a review.) Alternative approach, which we will assume in most of the paper, is to add along the longitude of the knot a Wilson line for the background gauge field. So, we will in general consider combined line operators, labeled by two parameters and , with being the charge for the Wilson line for the background field . It will be clear from the discussion of the U theory in section 4 that for gauge invariance, the charge should be taken valued in . It is more convenient to work with an integer parameter , and we will accordingly label our line operators as . Note that, since the longitude cycle is not canonically defined, the definition of these line operators depends on the knot framing. Under a unit change of framing, the Wilson line for the background gauge field will produce a factor of . With suitable choices of framing, half-integral powers of will not appear in the expectation values.
The operators will sometimes be called typical, while and Wilson lines for the indecomposable representations will be called atypical. This terminology originates from the classification of superalgebra representations, as we briefly recall in section 4.1.
3 Magnetic Theory And The Meng-Taubes Theorem
As was explained in section 2.2, our Chern-Simons theory can be obtained from the theory of a free hypermultiplet by twisting. An alternative description of the same topological theory can be obtained, if we recall that the free hypermultiplet describes the infrared limit of the QED with one electron. This is the basic example of mirror symmetry  in three-dimensional abelian theories, which was understood in  as a functional Fourier transform. By metric independence of the topological observables, they can be equally well computed in the UV or in the IR description. We now consider the topologically-twisted version of the UV gauge theory, which we will call the “magnetic” description.
(On a compact manifold, the claim that the RG flow from the UV theory leads to a free hypermultiplet depends on the presence of the non-trivial background flat bundle, which forces the theory to sit near its conformally-invariant vacuum. When the background gauge field is turned off, e.g. as is necessarily the case for a manifold with trivial , the situation is more subtle. This and some other details will be discussed in part 3.3 of the present section.)
3.1 The QED With One Electron
We now describe the bosonic fields of the theory. The fermionic fields, as well as the details on the action, are discussed in the Appendix A. Bosonic fields of the vector multiplet are a gauge field and an SU-triplet of scalars . (Bosonic gauge field here is completely unrelated to the fermionic gauge field of the electric gauge theory. In fact, the fields of the electric description emerge from the monopole operators of the UV theory.) In the twisting construction we use the SU-subgroup of the R-symmetry, so the scalars of the vector multiplet will remain scalars. It is convenient to introduce a combination , which has charge under the ghost number symmetry U. The remaining component has ghost number zero. The hypermultiplet contains an SU-doublet of complex scalars, which upon twisting become a spinor . They have charge one under the gauge group. The imaginary part of the background flat connection originated from the masses in the electric description. Under the mirror symmetry, it is mapped to a Fayet-Iliopoulos parameter.
The flavor symmetry SU is emergent in the infrared limit. In the UV, only its Cartan part is visible – it is identified with the shift symmetry of the dual photon. The current for this symmetry is , where . The real part of the background gauge field couples to this symmetry, so, it should enter the action in the interaction . In fact, the whole action of the topological theory has the form
(More details are given in Appendix A.) This can be more accurately written as
where is a line bundle, in which is the connection. The fields take values in a spin- bundle, and correspondingly, the path-integral includes a sum over spin- structures . We view this spin- bundle as a spin bundle for some fixed spin structure , tensored with the line bundle . We identify the reference spin structure with the spin structure, which was used in the definition of torsion on the electric side. A change of the spin structure by an element is equivalent to twisting the bundle by the -bundle, corresponding to . The formula (3.3) then changes in the same way (2.12) as the torsion , in agreement with the mirror symmetry121212Again, should be more appropriately viewed as a spin- structure with trivial determinant. Of course, we could equally well take an arbitrary reference spin- structure. That would give the trivial generalization of to arbitrary spin- structures, as described at the end of section 2.3.2.. The theory also has a charge conjugation symmetry, which, as on the electric side, implies that the invariants for and are the same.
Note that, instead of (3.2), we could try to use
Here is the determinant line bundle of the spin- bundle, in which the fields live. However, the factor of inside the brackets means that one has to take a square root of the holonomy of , and therefore the sign of this quantity is not well defined. This is the same ambiguity that we encountered in section 2.3.2, and it is resolved, again, by picking a reference spin structure .
The functional integral of the magnetic theory can be localized on the solutions of BPS equations , where is any fermion of the theory. One group of these equations actually tells us that the solution should be invariant under the gauge transformation with parameter, equal to the field . We will only consider irreducible solutions, and therefore must be zero. We also only consider the case that the background field satisfies , so that the twisted theory has the full SU-symmetry. (We have seen on the electric side that is the condition for this symmetry to be present. On the magnetic side, one can also explicitly check this, as shown in the Appendix A.) This symmetry, together with vanishing of , implies that is also zero. With this vanishing assumed, the remaining BPS equations take the form of the three-dimensional Seiberg-Witten equations,
where is the moment map, with being the Pauli matrices contracted with the vielbein, is the gauge coupling, and is the Dirac operator, acting on the sections of . Generically, the localization equations have a discrete set of solutions, and the partition function of the theory can be written as
where the sum goes over the set of solutions of the Seiberg-Witten equations, is a line bundle, corresponding to the given solution, and is the sign of the fermionic determinant.
The relation between the Reidemeister-Turaev torsion and the Seiberg-Witten invariant in three-dimensions is the content of the Meng-Taubes theorem  and its refinement due to Turaev . We have presented a physicist’s derivation of this theorem. Some subtleties that arise for three-manifolds with are discussed later in this section.
3.2 Adding Line Operators
Let us describe the magnetic duals of line operators, which were introduced in section 2.4. The first type of line operators were the integrals of the fermionic gauge field . On the magnetic side, their duals will be the integrals of monopole operators, which we will not discuss. The second type were the Wilson lines for the background gauge field. Obviously, their definition will be the same on the magnetic side.
Non-trivial and interesting line operators were defined by singularities of the background flat connection. We denoted them by in section 2.4. Since the one-form enters the BPS equations (3.5) on the magnetic side, the singularity of implies that those equations will have solutions with a singularity along the knot . The line operator is then defined by requiring the fields to diverge near as in a particular singular model solution. We use notation for the closed three-manifold, and for the manifold, obtained from by cutting out a small toric neighborhood of the singular line operator. Let and be the polar coordinates in the plane, perpendicular to . Near the knot, the singularity of the imaginary part of the background gauge field has the form
(We follow the notations of .) Note that whenever the parameter is non-zero, the closed two-form has a non-vanishing integral around the boundary of the toric neighborhood of the link. This might be forbidden for topological reasons – e.g., if is a one-component link in . In such cases, cannot be turned on. Even when the parameter can be non-zero, we expect the invariants to be independent of it.
To find the model solution, consider the case that is the flat space, and is a straight line. Let and be the two components of , and be the complex coordinate in the plane, perpendicular to . We are looking for a time-independent, scale-invariant solution of the Seiberg-Witten equations. The gauge field in such a solution can be set to zero, so that the remaining equations give
and the scale-invariant solution is simply , , where and . Note that the field here is antiperiodic around . Since we view as a spinor field on the closed three-manifold , it should rather be periodic, so, we make a gauge transformation to bring the model solution to the form
To complete the definition of the line operator, we also need to explain, how the topological action (3.3) is defined in presence of the singularity. The flat bundle is naturally an element of . By Poincaré duality, it can be paired with an element of the relative cohomology , and this pairing will define the action. If we forget for a moment about possible torsion, the relative cohomology class that we need is naturally the cohomology class of the gauge field strength for a given solution. However, here we encounter the mirror of the problem that we had on the electric side: this class in general is not integral. The reason, roughly speaking, is the antiperiodicity of the field around , or equivalently, the half-integral term in the gauge field (3.9) near the line operator. (Depending on the topology, there can also appear a similar term with replaced by the angle along .) This will in general cause half-integral powers of the holonomies and to appear in the torsion invariant. To remove them, just as in section 2.4, one introduces along a Wilson line for the background gauge field with a half integral charge . With a suitable choice of framing, this is enough to remove the half-integral powers of holonomies.
Here we viewed the field as a section of the spin bundle on , tensored with a line bundle with connection . A more systematic way to define these line operators is to allow spin (or spin-) structures on that do not necessarily extend to . The antiperiodicity of in the model solution (3.9) can then be absorbed into the definition of this spin structure. The field then provides an honest cross-section of the line bundle in the neighborhood of the link, and this allows to canonically define an integer-valued relative Chern class . The charge of the background field Wilson line and the choice of the framing are then absorbed into the choice of the spin- structure. This is the approach taken in the mathematical literature131313There is also another difference of our treatment of line operators from mathematical literature. There, the analogs of line operators are typically introduced by gluing in an infinite cylindrical end to the manifold , rather than by considering solutions on with singularities., see e.g. . This point of view is consistent with the picture that inserting line operators of type , or changing the parameter for operators , is equivalent to changing the spin- structure.
We only considered the case that the holonomy of the background flat connection around the meridian of the knot is not unimodular. In the opposite case, we have in eq. (3.7), and, assuming that the parameter is also zero, the singular model solution seems to disappear. This makes it unclear, how to define the magnetic duals of line operators with unimodular holonomy, except by the analytic continuation from . This problem looks analogous to the one described in the end of section 4.4.5 of .
3.3 More Details On The Invariant
In our derivation of the relation between the Seiberg-Witten invariant and the Reidemeister-Turaev torsion we ignored some subtleties , , which occur for three-manifolds with . Here we would like to close this gap. First we look at the UV theory, and then we describe the RG flow to the IR theory in more detail. We will see that the claim that the IR theory is the Chern-Simons model sometimes has to be corrected.
3.3.1 Seiberg-Witten Equations For
Let us look closer at the Seiberg-Witten counting problem. Our goal here is not to derive something new, but merely to understand, how gauge theory takes care of some subtleties in the formulation of the Meng-Taubes theorem.
Note that in the analogous problem in the context of Donaldson theory in four dimensions, the gauge theory gives the first of the Seiberg-Witten equations roughly in the form . To avoid dealing with reducible solutions with , one introduces by hand a deformation two-form in the equation . In our case, the situation is different: the deformation two-form is already there. In nice situations, the counting of solutions does not depend on the choice of this deformation, so any two-form could be taken. But sometimes it is not true, and then it will be important, what deformation two-form is chosen for us by the gauge theory.
The properties of the counting problem depend on the first Betti number , whose role here is analogous to in four dimensions. For , a reducible solution has and . For such a solution to occur, the cohomology class of has to be integral. When in the parameter space we pass through such a point, so that reducible solutions are possible, the counting of solutions can in principle jump. This makes the Seiberg-Witten invariant dependent on the deformation two-form, or the metric and , if we prefer to keep the deformation two-form equal to with fixed . Actually, for no jumping is possible, since in the space of deformation two-forms we can always bypass the point, where reducible solutions can occur. But for , non-trivial wall-crossing phenomena do happen. As we change the two-form , and its cohomology class passes through an integer point, the number of solutions with first Chern class equal to this integer does change in a known way . (For the particular case of , the Seiberg-Witten counting problem is worked out in detail in the Example 4.1 in .)
There is another related issue. As we explained in section 2.3.1, the torsion, to which the Seiberg-Witten invariant is supposed to be equal to, for has a second order pole. Just for concreteness, consider the manifold , for which the torsion is141414The function should have a second order pole at . Also, it cannot have any zeros for , since the twisted cohomology for such is empty. Imposing also invariance under the charge conjugation , we recover the stated result, up to a constant numerical factor. , where is the holonomy around the non-trivial cycle. If we expand this, say, near , we get a semi-infinite Laurent series . However, it is known that for any given deformation two-form the Seiberg-Witten equations have only a finite number of solutions.
The resolution of these puzzles is that we need to take the infrared limit of the theory, e.g. by taking to infinity. This means that we have to take the deformation two-form to be . That is, it should be proportional to with a positive coefficient, and, to count solutions with a given Chern class , we should use a deformation two-form with Chern class much larger than in absolute value. This is equivalent to the prescription of Meng and Taubes. Depending on the sign of , the two expansions that we get in this way for would be and . One can check that the sign of is such that in the first case and in the second, so that in either case the expansion is absolutely convergent.
Just like for closed three-manifolds, for manifolds with links in them, the Seiberg-Witten counting problem for is special. (This case arises e.g. when one cuts out a one-component knot from a simply-connected manifold.) As we reviewed in the end of section 2.3.3, the Reidemeister torsion for such a manifold has a first order pole. Therefore, it has two different Laurent expansions near and . The Seiberg-Witten equations in this case have an infinite number of solutions, with Chern class unbounded from above or from below, depending on the sign of the deformation two-form . The sign of is such that these expansions are absolutely convergent. Unlike the case of a compact three-manifold, here we do not need to explicitly take to infinity, since the deformation two-form already diverges near the knot.
When is a rational homology sphere, that is , there is no way to avoid reducible solutions in working with the Seiberg-Witten equations. Because of this, a simple signed count of solutions is no longer a topological invariant. Still, one can define a topological invariant by adding an appropriate correction term . We will not attempt to derive it from the quantum field theory, but will in what follows use the fact that the definition of the invariant for does exist.
3.3.2 Massive RW Model And The Casson-Walker Invariant
Let us now turn to the IR theory, which is a valid description, when the size of the three-manifold is scaled to be large. The topological theory reduces in this case to the Rozansky-Witten (RW) sigma-model  with the target space being the Coulomb branch manifold, which for the QED is  the smooth Taub-NUT space . The U graviphoton translation symmetry is generated on by a Killing vector field . The coupling of the UV theory to the flat gauge field translates into a coupling of the RW model to the same flat gauge field via the isometry . In the untwisted language, the imaginary part of the gauge field would be a hyperkäler triplet of mass terms. For this reason, we call our IR theory the massive Rozansky-Witten model. An explicit Lagrangian and more detailed treatment of this theory will appear elsewhere. The coupling of the Rozansky-Witten model to a dynamical Chern-Simons gauge field has been previously considered in . We will now see, how and when the massive RW model reduces to the Gaussian theory.
First, let us turn off the background flat gauge field. What we get then is the ordinary RW model for . The path-integral of that theory has the following structure . The kinetic terms have both bosonic and fermionic zero modes. The bosonic ones correspond to constant maps to the target space. The integral over the bosonic zero modes thus is an integral over . The one-loop path-integral produces a simple measure factor, while most of the higher-loop diagrams vanish. The reason is that all the interactions (which involve the curvature of ) are irrelevant in the RG sense, and can be dropped, when the worldsheet metric is scaled to infinity. However, some diagrams do survive due to the presence of the zero modes. Overall, the path-integral for each is given by a simple Feynman diagram, which captures the topological information about , times the integral of the Euler density of . It is important that the path-integral depends on the target space only by this curvature integral. The Euler numbers happen to be the same for and for the Atiyah-Hitchin manifold . This was used in  to derive a special case of the Meng-Taubes theorem by the following argument. The RW model for can be obtained from the IR limit of the topologically-twisted SU Yang-Mills theory , which computes the Casson-Walker invariant , , . Since the Rozansky-Witten invariants computed using and are the same, the Casson-Walker invariant is equal to the Seiberg-Witten invariant, when the background bundle is trivial.
Now let us turn on the background bundle back again. In its presence, the kinetic terms of the RW model have no zero modes. The classical solution, around which one expands, is the map to the fixed point of the vector field , that is, to the conformally-invariant vacuum. In the absence of the zero modes, all the irrelevant curvature couplings can be thrown away. In this way, the RW path-integral reduces to the Gaussian integral of the model. It is natural to expect the path-integral to be continuous in . To the extent that this is true, the torsion evaluated for trivial should thus coincide with the Casson-Walker invariant. Note that on the level of Feynman diagrams this is not completely trivial, since for the interaction vertices come from the curvature terms, while for they come from expanding the Gaussian path-integral in powers of the background gauge field. Still, the actual Feynman integrals should coincide. We will not explicitly analyze the diagrams here (most of them were analyzed in ), but will just use the known relation between and the Casson-Walker invariant to check the continuity of the massive RW path-integral in .
Let denote the torsion evaluated for the trivial background flat bundle151515Note that for the dependence on the spin- structure drops out., and CW be the Casson-Walker invariant. For , it is indeed true that . For , the torsion has the form
where is the Alexander polynomial. Setting and expanding this in , we get
Dropping the term, we define the regularized torsion . This combination, again, is equal to the Casson-Walker invariant for . However, the presence of the extra divergent piece means that the path-integral of the massive RW model in this case is not continuous in its dependence on the background gauge field: for approaching the trivial flat bundle, the torsion tends to infinity, while for taken to be exactly the trivial flat bundle, the invariant is finite. One can trace the origin of this discontinuity to the wall-crossing in the UV theory. Indeed, for non-zero , the Seiberg-Witten invariant is evaluated using the deformation two-form , which in the infrared limit lands us in the infinite wall-crossing chamber. The singularity of the torsion for arises from the infinite number of solutions of the Seiberg-Witten equations in this chamber. On the other hand, for trivial we have , and the deformation two-form vanishes for all . To evaluate the invariant, one should properly deal with reducible solutions. Instead, we will simply assume that the deformation two-form is non-zero, but infinitesimally small. It is known  that in such chamber the Seiberg-Witten invariant is equal to , which, again, is the Casson-Walker invariant, up to a correction , which, presumably, would be recovered with an appropriate treatment of the reducible solutions. Thus, one can say that the discontinuity at trivial in the massive RW model for is a “squeezed version” of the wall-crossing in the UV theory161616It is a “squeezed version”, because the wall-crossing condition is not conformally-invariant, and thus we cannot see all the walls in the IR theory, but only see a discontinuity at . This can be contrasted with the situation in the Donaldson theory in four dimensions, where the wall-crossing condition is conformally-invariant, and the walls can be seen both in the UV and in the IR descriptions .
Finally, for , assuming that the torsion subgroup is non-empty, the Reidemeister-Turaev torsion is a function on the discrete set of flat bundles. For non-trivial , the Seiberg-Witten counting problem computes the torsion, while for trivial it computes the Casson-Walker invariant, which now is not related to the torsion, since there is no way to continuously interpolate to the trivial , starting from a non-trivial . In fact, for the Casson invariant is computed by a two-loop Feynman integral , and it is clearly not possible to obtain it from the one-loop torsion.
Let us summarize. The UV topological theory, and thus the Seiberg-Witten counting problem, is equivalent to the massive RW theory. For non-trivial , this theory reduces to the Chern-Simons theory and computes the Reidemeister-Turaev torsion. For trivial , it computes the Casson-Walker invariant, which for can be obtained from a limit of the invariant, while for is not related to it. Our results agree with the mathematical literature , .
4 U Chern-Simons Theory
In a series of papers [10, 11, 5], it has been shown that the Alexander polynomial and the Milnor torsion can be computed from the U Chern-Simons theory. We would like to revisit this subject and to show, how it fits together with our discussion in previous sections. We point out that for the compact form of the bosonic gauge group, the U Chern-Simons theory is simply an orbifold of the theory. (A direct analog of this statement is well-known in the ABJM context.) In particular, it contains no new information compared to the Chern-Simons with a coupling to a general background flat bundle , and computes, indeed, essentially the same invariant.
4.1 Lie Superalgebra
We start with a brief review of the superalgebra . A more complete discussion can be found e.g. in . Let and be the fermionic generators, and and the generators of the left and right bosonic factors. It will also be convenient to use a different basis in the bosonic subalgebra, which is and . The element acts on the fermionic subalgebra by the U transformations, and the element is central. Explicitly, the non-trivial commutation relations are
The group of even automorphisms of is generated by the charge conjugation , , , rescalings , with , and shifts , .
As for any Lie superalgebra, the representations of can be usefully divided into two classes – the typical and the atypical ones. (For a very brief review of superalgebra representations, with applications to Chern-Simons theory, see section 3 of .) The typicals are precisely the ones, in which the central generator acts non-trivially. They are two-dimensional, and the generators, in some basis, act by matrices
with . These will be called representations of type . To be precise, one has to make a choice, whether to assign a bosonic or a fermionic parity to the highest weight vector. This effectively doubles the number of representations. In our applications, the representations will be labeling closed Wilson loops, which come with a supertrace. Therefore, different parity assignments will be just a matter of overall sign, and we will mostly ignore this.
In the atypical representations the generator acts trivially, and therefore they can be equivalently thought of as representations of . These have already been described in section 2.4. Note that the indecomposable representation of fig. 1 can be obtained as a degeneration of the typical representation for . With a suitable rescaling of the generators before taking the limit, one can similarly obtain the representation of fig. 1. The representations and are known as the atypical Kac module and anti-Kac module.
Let us also write out some tensor products. Tensoring any representation with the one-dimensional atypical simply shifts the -charges. The other tensor products are
where the indecomposables were defined on fig. 1. The prime on the second representation in the r.h.s. of (4.3) means that the highest weight vector in it has reversed parity. The set of representations , , is closed under tensor products.
The superalgebra possesses a two-dimensional family of non-degenerate invariant bilinear two-forms, which can be obtained by taking a supertrace over a representation with . Note that all the representations for different values of and , and therefore also the corresponding invariant forms, are related by the superalgebra automorphisms.
4.2 Global Forms
There exist different versions of Chern-Simons theory based on the superalgebra , and here we would like to classify them. To define such a theory, one needs to pick a global form of the gauge group, and also to choose an invariant bilinear form, with which to define the action. These data should be consistent, in the sense that the action should be invariant under the large gauge transformations. Theories related by the superalgebra automorphisms are equivalent. We can use this symmetry to bring either the invariant bilinear form, or the lattice, which defines the global form of the group, to some simple canonical form. To classify the theories, it is convenient to take the first approach.
Let be the bosonic subalgebra of . The gauge field, in components, is . For the bosonic part of the gauge field, we will also use expansion in a different basis, . The action of the theory can be written as
where is the Chern-Simons term for the bosonic gauge field, is the action (2.1), coupled to the line bundle with connection , and to some background flat bundle . Finally, is the gauge-fixing action (2.3) for the fermionic part of the gauge symmetry.
By using the superalgebra automorphisms, we bring the bosonic Chern-Simons term to the form