{'fraction_non_alphanumeric': 0.03972059656409288, 'fraction_numerical': 0.009137247498584104, 'mean_word_length': 4.131357163615228, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Chern-Simons Gauge Theory As A String Theory

Certain two dimensional topological field theories can be interpreted as string theory backgrounds in which the usual decoupling of ghosts and matter does not hold. Like ordinary string models, these can sometimes be given space-time interpretations. For instance, three-dimensional Chern-Simons gauge theory can arise as a string theory. The world-sheet model in this case involves a topological sigma model. Instanton contributions to the sigma model give rise to Wilson line insertions in the space-time Chern-Simons theory. A certain holomorphic analog of Chern-Simons theory can also arise as a string theory.


Introduction
In this paper, I will describe how Chern-Simons gauge theory in three dimensions can be viewed as a string theory. The string theory in question will be constructed using a topological sigma model [1] (related to Floer/Gromov theory) in which the target space is T * M, M being a three-manifold. The perturbation theory of this string theory will coincide with Chern-Simons perturbation theory, in the form that this has been presented by Axelrod and Singer [4] and further studied by Kontsevich [5]. Mathematically, the idea is roughly that there are no instantons with target space T * M and boundary values in M, so in an appropriate topological field theory, the usual counting of instantons is vacuous. However, there are virtual instantons at infinity; their proper counting leads to Chern-Simons perturbation theory. Chern-Simons theory enters in this particular string theory in much the same way that ordinary space-time physics (with general relativity as the long wavelength limit) arises in conventional string theory.
Physically, one might take the following as the starting point. String theorists usually construct two dimensional field theories describing particular classical solutions of string theory by constructing a "matter" system, of total central charge 26, and coupling it to the ghosts, of central charge −26. The vanishing total central charge ensures the existence of a BRST operator Q, obeying Q 2 = 0, and playing a crucial role in world-sheet and space-time gauge invariance. One knows, however [7], that Q can be interpreted as a generator of linearized gauge transformations, mixing ghosts and matter, so the assumption that the matter and ghosts are decoupled cannot be valid as a fundamental principle; it is merely a (partial) gauge condition and very likely cannot be imposed at all in some situations, perhaps time dependent ones.
So in §2, we will ask what is left of the standard structure if one does not assume decoupling of matter from ghosts. From the discussion, it will become obvious that exotic realizations of the same basic structure can be constructed using topological sigma models; we therefore do this in §3. The resulting models can be considered for either open or closed strings, but the open string case is in some ways easier to understand. In §4, we use open-string field theory as a shortcut to determining the space-time interpretation of the open-string version of our models, with the result -alluded to above -that one type of model is equivalent in perturbation theory to three dimensional Chern-Simons gauge theory. This is possibly the first time that the background independent, gauge invariant spacetime interpretation of a string theory has been completely determined. (But see [8] for a previous investigation of the space-time interpretation of some topological string theories of a rather different flavor.) As one application, our result explains certain observations by Kontsevich about Chern-Simons gauge theory, and we will make a small digression on that account. Another version of the theory has for its space-time interpretation a sort of holomorphic version of Chern-Simons theory.
In §5, we attempt to discuss the closed string sector (which among other things should be more closely related to usual manifestations of mirror symmetry). This concluding section is brief since I do not understand it.
To keep this paper within reasonable length, it has not been possible to give a self-contained explanation of all of the relevant background. The relevant material on topological sigma models can be found in [6]; for their coupling to topological gravity see [1]. The relation of Riemann surfaces to gauge theories will be briefly recalled presently. Apart from this, it is helpful to have some familiarity with construction of string models, the relation between world-sheet and space-time physics, string field theory, and Chern-Simons perturbation theory. The main point of the paper is the unexpected relation of those latter topics to each other and to topological sigma models.

Riemann Surfaces And Gauge Theories
Riemann surfaces enter our story in two ways. On the one hand, when we study two dimensional quantum field theory, Riemann surfaces are present at the beginning. Then we wish to show how gauge theory in the target space emerges.  On the other hand, oriented two dimensional manifolds arise in gauge theories with U(N) gauge group in the following fashion, due originally to 't Hooft [9]; see also [10, §4] for a recent explanation. (SO(N) and Sp(N) can be considered similarly and lead to not necessarily orientable Riemann surfaces.) Though the following general comments also go through for ordinary Yang-Mills theory in any dimension, let us to be definite consider the Chern-Simons action for a U(N)-valued connection A on a three manifold M: Consider expanding this theory in perturbation theory in 1/k, say around the trivial connection. The successive terms are all topological invariants. The coefficient of 1/k r comes from Feynman diagrams with r loops, and is a function of N. For given r, by considering the dependence on N, how many invariants can be extracted? This question can be conveniently answered as follows. The gauge field A is essentially a one-form valued in hermitian N × N matrices. Write A as A i j , i, j = 1 . . . N, making the matrix indices explicit while leaving implicit the fact that A is a one-form. The interaction involves sewing the "lower" index of one A field to the "upper" index of the next.
To exhibit this in drawing Feynman diagrams, represent the A propagator not as a line but as a slightly thickened band in which one edge represents the "upper" The genus of Σ is g = (r − h + 1)/2. As r and h vary, g and h vary independently, and Σ varies over all topological types of oriented two dimensional surface with boundary. Thus, U(N) gauge theory gives us a three manifold invariant Γ g,h (M) for every topological type of oriented two dimensional surface with boundary. On the other hand, in §3-4 we will consider a two dimensional topological field theory (closely related to the counting of almost holomorphic curves) in which one can conveniently take the target space to be T * M, the cotangent bundle of a three-manifold M. The partition function Z g,h (M) of this theory formulated on a Riemann surface of genus g with h holes is, again, a three-manifold invariant depending on g and h. Our main conclusion is that Γ g,h (M) = Z g,h (M). In reaching this conclusion, the link between the Riemann surfaces of the two dimensional field theory and those of U(N) gauge theory is provided by the fatgraph [11][12][13] or open string field theory [14,15] description of the moduli space of Riemann surfaces with boundary; in this description, complex Riemann surfaces are built up from pictures similar to that of figure (1(b)), as we will recall in more detail in §4.

Axioms
In this section, we will discuss what remains of the usual structures in string theory if one does not assume that ghosts and matter are decoupled. Σ will be a Riemann surface of genus g with local coordinates x α , α = 1, 2. The symbols J and h will be used to denote a complex structure and a metric on Σ; of course a metric determines a complex structure. The space of all complex structures will be called J ; the group of diffeomorphisms of Σ will be called G. The quotient J /G is the moduli space of complex structures on Σ. The space of metrics on Σ will be called K.
Usually one considers conformally invariant world-sheeet theories consisting of "matter," of central charge c = 26, and "ghosts," of c = −26. As explained in the introduction, we want to drop the assumption of this decoupling; so if conformal invariance is retained, we preserve the fact that the total central charge is c = 0, but no longer build this from a cancellation between different contributions. We will, however, not necessarily retain the assumption of conformal invariance.
One part of the structure that must be kept for all that follows is the existence of a BRST operator Q with Q 2 = 0, and with the further property that the stresstensor T αβ can be written as for some field b αβ -which in the usual case is called the antighost field. This ensures general covariance -or topological invariance -of the two dimensional theory. In the conformally invariant case, b and T are traceless. The stress tensor is conserved, D α T α β = 0, and so we will assume that the b field obeys which when b is traceless is its usual equation of motion. In addition, in the usual constructions, there are no short-distance singularities in operator products of b fields, and there are only the standard short-distance singularities in products of b and T . Concerning the generalization, topological sigma models are an example showing that these assumptions are too strong. In those models, there is a delta function contact term in the b · T product. (This results in the ρρψψ term in the coupling to topological gravity, eqn. (4.18) of [1].) I do not know what sort of singularities should be allowed in b · b and b · T operator products, in general, except to say that they must be such that a coupling of the model to topological gravity must be possible.
Another feature of the usual case that we wish to preserve is that there is a "ghost number" operator G (mathematically it would usually be called the dimension), with Q and b having G = 1 and G = −1, respectively. Moreover, we wish to preserve the usual fact that the "ghost number of the vacuum" is −3χ(Σ) where χ(Σ) is the Euler characteristic of Σ; this means that only an operator product of total ghost number 3χ(Σ) can have a non-zero vacuum expectation value. For instance, for an orientable Riemann surface of genus g, the case that we will consider in this section for definiteness, 3χ(Σ) = −(6g − 6).
A particularly important case (in genus ≥ 2 for simplicity, to avoid ghost zero modes) is the expectation value of a product of 6g−6 b's. Let δ (k) h, k = 1, . . . , 6g−6 be 6g − 6 variations of h (that is, 6g − 6 tangent vectors to the space K of metrics). Let be the corresponding modes of b, and let with the expectation value in the "measure" determined by the genus g world-sheet path integral. By fermi statistics, Θ is skew-symmetric in the δ (k) h's and so can be interpreted as a 6g − 6 form on K. In the usual case, one proves that Θ is closed starting with Then using (2.1) to write {Q, b (j) } as a moment of T , and interpreting the insertion of T as a derivative on K, one can interpret (2.5) as a first order differential equation for Θ which amounts to This proof goes through in general if the b·b and b·T operator products are standard; however, topological sigma models are an example in which the definition of Θ must be modified by addition of contact terms to ensure dΘ = 0. A successful coupling of the model to topological gravity will always ensure the existence of a suitable modification (by addition of contact terms) of the definition of Θ.
Θ is obviously diffeomorphism invariant. Moreover, in the usual case it is induced by an infinitesimal diffeomorphism x α → x α + ǫv α . This is proved by integrating by parts, using (2.2) and the absence of short distance singularities of the b's. In general (because of the relation of the topological gravity multiplet to the equivariant cohomology of the diffeomorphism group), a successful coupling to topological gravity will always ensure that Θ is basic.
The fact that Θ is basic and diffeomorphism invariant means that it can be interpreted as the pullback of a closed 6g − 6 form ν on the space K/G of metrics modulo diffeomorphisms. In the conformally invariant case, Θ is actually a pullback from a form µ on the moduli space M of complex structures; as M is 6g − 6 dimensional, µ is a top form or measure on M. If conformal invariance does not hold, there is no natural way to construct from µ a measure on M, but the fact that ν is closed, dν = 0, is almost as good. It means that if s is a section of the bundle K/G → M, and µ = s * (ν), then µ is independent of the choice of s, modulo an exact form. µ is the desired 6g − 6-form on M.

Vertex Operators And Ghosts
The most traditional string theory vertex operators can be written W = ccV where c and c are the ghosts and V is a spin (1, 1) conformal field constructed from matter fields only. It is well known in the operator formalism [16] that these conditions can be relaxed. The only really essential properties of W in conventional string theory are that it have ghost number 2, spin zero, and be annihilated by These conditions make sense in our abstract setting, and can be borrowed bodily, except that if we do not assume conformal invariance, the definition of b 0 − must be written b 0 − = dx α b α0 . (The integration is over a parametrized circle and "0" is the normal direction to the circle.) Note, therefore, that in including vertex operators, we need make no explicit mention of the ghosts; only the product W = ccV , and not the separate factors in that product, needs to be generalized.
Where, therefore, do the ghosts appear in the discussion? Of course, the antighosts entered in the definition of the measure for g ≥ 2; but what about the ghosts?
We perhaps could "find" the ghosts by trying to generalize the usual definition of the genus one measure -as c and c each has a zero mode on a surface of genus one.
To get a different perspective, I will instead discuss the issue from the standpoint of closed string field theory.
In closed string field theory as we know it today (see [17] for a review), the string field Ψ is a vector in a string Hilbert space obtained by quantization on a circle S. Picking a parametrization of the circle, let L 0 − = L 0 −L 0 be the generator of a rotation of the circle, and let c 0 The quadratic part of the Lagrangian is Ψ, c 0 − QΨ , (2.8) and the linearized gauge invariance is where ǫ has ghost number 1 and This is proved to be a gauge invariance as follows. In the usual theory, (2.10) implies that for some α. Inserting this in (2.8), and using (which is a consequence of (2.1)) along with (2.7), one verifies the invariance of (2.8).
What are the essential points that should be retained if the ghosts and matter fields are not decoupled? In conventional string theory, one has b 0 − 2 = 0, (2.14) a consequence of the absence of singularities in the b · b operator products. This property also holds in topological sigma models, and it seems reasonable to insist This actually is enough to ensure the gauge invariance of the free Lagrangian (2.8).
(If the cohomology of b 0 − is nonzero in ghost number one, one cannot use (2.11); but the fact that Q annihilates the cohomology of b 0 − means that for ǫ annihilated by b 0 − and L 0 − , one can write Qǫ = b 0 − β for some β; this is good enough.) Topological sigma models give an example in which the cohomology of b 0 − is nonzero, but annihilated by Q. We therefore can write down closed string field theory in this case, but as we will see in §5, it seems difficult to understand it.

A Conundrum
The conundrum that I want to state is obvious. The properties of conventional string backgrounds that I have cited are the only ones I know of that make sense in the general case in which ghosts and matter are not decoupled. Yet they are so general as to permit bizarre realizations, like the topological sigma models that we will consider presently. Are they adequate and if not how should they be supplemented?
The following example will perhaps serve to sharpen the puzzle. In Type II superstring theory, it is usually supposed that a classical solution is described by a superconformal world-sheet theory. One therefore might expect and hope to supplement the above discussion with general properties of superconformal symmetry.
Yet it has been pointed out [18] that Ramond-Ramond vertex operators do not commute with the left-or right-moving superconformal currents, and consequently that when such vertex operators are added to the world-sheet Lagrangian (as one would expect in a generic time-dependent situation), the world-sheet theory is not superconformal. Therefore, the structure of Type II backgrounds, at the general level of our above discussion, looks hard to distinguish from that of bosonic string theory. If we do not wish to claim that bosonic strings and Type II superstrings are the same theory (and I would be skeptical of that interpretation), we apparently must find general properties of the world-sheet theory that go well beyond the ones I have cited.

Realizations Via Topological Field Theories
We will now use topological sigma models to make some realizations of the structure just explained. There are two classes of such models [6,19,20], which I will call the A and B models following some of the papers just cited.
These models govern maps from a Riemann surface Σ to a target space X, which must be presented with an almost complex structure, in the case of the A model, or an actual complex structure, for the B model. The A model is concerned with almost holomorphic maps from Σ to X, while the B model is related to periods of differential forms on X. The B model is only well-defined for X obeying the Calabi-Yau condition c 1 (X) = 0. The A model is defined without that condition, but obeys the axioms of §2 (definite ghost number of the vacuum) only if c 1 (X) = 0.
Therefore, we will limit ourselves to Calabi-Yau target spaces. This is, of course, also the case in which the sigma model is conformally invariant.
For either the A or B models, the ghost number of the vacuum is −d·χ(Σ), with d = dim I C (X). This coincides with the desired valued −3χ(Σ) that played such an important role in §2 precisely if d = 3, and therefore we will limit ourselves to this case. It is curious to note that this is the same value of d that arises in the usual "physical" applications of Calabi-Yau manifolds in superstring compactification.
General tangent space indices to X will be written as I, J, K, while indices of type (1, 0) or (0, 1) will be written as i, j, k or i, j, k, respectively. We consider X as a Kahler manifold endowed with a Ricci-flat Kahler metric g ij . The bosonic fields of the A or B models are simply a map Φ : Σ → X; if we pick local coordinates φ I on X, this map can be described by giving functions φ I (x α ). It is also convenient to pick a local complex coordinate z on Σ.
The A Model The detailed construction of the A and B models is explained in [6]. In the A model, the fermi fields are a section χ I of Φ * (T X), and a one-form ψ with values in Φ * (T X). ψ obeys a self-duality condition which says that its (1, 0) part ψ i z has values in Φ * (T 0,1 X), and its (0, 1) form ψ i z has values in Φ * (T 1,0 X). The BRST transformation laws are with α an anticommuting parameter and Γ the affine connnection of X. The BRST operator Q is defined by writing δΛ = −iα{Q, Λ} for any field Λ. The Lagrangian can be written in the form L = i{Q, V } with any V such that L is nondegenerate.
(with d 2 z = |dz ∧ dz|) and gives with t a constant.
The fact that L = i{Q, V } means that the t dependence (and dependence on the target space metric) of (3.3) is of the form {Q, . . .} and so does not affect the BRST invariant physics. As weak coupling means large t, the fact that there is no t dependence means that weak coupling is exact.
The dependence on the metric of Σ is likewise of the form {Q, . . .}, so we can Quantum mechanically, the formula for b requires modification except in the Calabi-Yau case (T and b will no longer be traceless), but an appropriate b still ex-ists, as the theory admits a higher derivative regularization preserving the fermionic symmetry.

The B Model
In the B model, the fermi fields include first of all sections η i , θ i of Φ * (T 0,1 X); actually, it is convenient to write the formulas in terms of not θī but θ j = g ji θ i . The other fermi fields are a one-form ρ i with values in Φ * (T 1,0 X). The transformation laws are and the Lagrangian is again of the form L = i{Q, V }; for a suitable V , one gets The physics is independent of the coupling parameter t for the same reason as in the A model.
In this case, we can write

Boundary Conditions
In our applications, we will want to formulate these theories on Riemann surfaces Σ that may have non-empty boundary. In doing so, we require a boundary condition. ⋆ We want local, elliptic boundary conditions that preserve the fermionic symmetry. I will explain natural boundary conditions for both the A and B model, but there may very well be other natural boundary conditions -perhaps required for mirror symmetry -that should be considered.
The A model is closely related to Floer theory, and the boundary condition that I want to consider for the A model is the one that was studied by Floer. For the B model, we pick instead "free" boundary conditions that do not require anything analogous to the choice of the M i . We require that the normal derivative of Φ vanishes on ∂Σ, that θ vanishes on ∂Σ, and that the pullback to ∂Σ of ⋆ρ vanishes (⋆ being here the Hodge star operator).
⋆ For some generalities about topological field theories on Riemann surfaces with boundary, and some specifics about the A model, see [21]. † Readers to whom these boundary conditions may seem strange at first sight are invited to try to find other local, elliptic, Q-invariant boundary conditions for the A model.

Boundary Terms
Now, we further wish to couple this system to target space gauge fields in the sense of string theory, that is to gauge fields on X. This will be done by introducing Chan-Paton factors, that is by coupling the gauge fields to "charges" that propagate on ∂Σ. These charges are string theory analogs of the labels on the boundaries in figure (1(b)), and will reduce to the latter when we make contact with target space Chern-Simons theory. Just as in the construction leading to The quantum version of either of the models we have introduced is described by a Feynman path integral where Ψ i are the various fields and L is the Lagrangian. Let now A = A I dφ I be a connection with structure group U(N) on a rank N complex vector bundle E over X. Let Σ be an oriented Riemann surface whose boundary is a disjoint union of circles C i . The orientation of Σ induces an orientation of each C i . Given Φ : Σ → X, for each C i we can take the trace of the holonomy of Φ * (A) around C i ; we write this as (The trace is taken in the defining N dimensional representation of U(N).) Then, tentatively, the coupling to gauge fields in the target space is accomplished by replacing (3.8) by (For the B model, this will require some modification.) If A is trivial, this is simply a factor of N for each boundary component, just as in the evaluation of the Feynman amplitude of figure (1(b)).
We must determine whether this modification of the path integral preserves the fermionic symmetry. In general the variation of the trace of the holonomy about a circle C is Here τ is a coordinate on C, and exp C;τ Φ * (A) is the holonomy of Φ * (A) around C, starting and ending at τ ; and F IJ (or more fastidiously Φ * (F IJ )) are the com- But deeper aspects of the theory involve summing over the topology of Σ and the homotopy type of Φ, and then the factors coming from holonomies of a flat connection on X fit together coherently (to give, as we will see, Chern-Simons theory expanded around the given flat connection A).

The B Model
The situation in the B model is similar but more subtle. Plugging the transfor- curvature says that A is a holomorphic connection (locally it can be represented by a holomorphic one-form of type (1, 0)). The latter condition is rather restrictive and we wish to eliminate it. This can be done as follows. Replace Φ * (A) by the following "improved" connection on the bundle Φ * (E) over Σ: Then using the transformation laws of the B model, one readily sees that for any circle C ⊂ Σ, the trace of the holonomy of A is invariant under the fermionic symmetry, provided that the (0, 2) part of the curvature of A vanishes, that is provided A determines a holomorphic structure on E. Thus, the formula (3.10) for coupling external gauge fields to the A model should be modified, in the case of the B model, to (3.14) I will leave it to the interested reader to verify that up to terms of the form {Q, . . .}, the coupling to gauge fields in (3.14) depends only on the holomorphic structure of E (i.e., the (0, 1) part of A), not on the (1, 0) part of A. This is in accord with the fact that, as we will see in §4, the B model has for its classical solutions in space-time the holomorphic vector bundles. As always, the classical solutions are the objects to which the world-sheet theories can be coupled.

Large t Limit
To set up the Hamiltonian version of these theories, for open strings, we take We consider the A and B models with the boundary conditions just described. In the case of the A model, we use the same Lagrangian submanifold M at the two ends of the strip. A quantum Hilbert space H is introduced in the usual way by quantizing on the initial value surface τ = 0. We want to compute the cohomology of Q and certain aspects of the large t behavior that will be essential in the next section.
The A Model If we write ψ = ψ σ dσ + ψ τ dτ , then, as the self-duality condition determines ψ σ in terms of ψ τ , we can regard χ and ψ τ as the independent fermi variables. The canonical commutation relations are The Hamiltonian is Using the canonical commutation relations to write dφ/dτ in terms of δ/δφ, this can be written implies that the Q cohomology can be computed in the subspace of H annihilated by L 0 . Since the Q cohomology (and everything else of essential interest) is independent of t, it is enough to study the kernel of L 0 for large t.
Actually, with other applications in mind, we want to understand not just the kernel of L 0 but all eigenvalues that are of order 1/t for large t. This has an expansion in powers of χ a in which the successive terms can be interpreted as p-forms on M of degrees 0 ≤ p ≤ 3. If Chan-Paton factors are included in the discussion, these differential forms take values in End(E), the endomorphisms of some flat vector bundle E over M.
The commutation relations [Q, Moreover, (3.17) shows that acting on differential forms on M, L 0 reduces to with ∆ = dd * + d * d the usual Laplacian on forms. The underlying relation L 0 = {Q, b 0 } leads one to guess that in the same approximation 22) and this can be verified using the commutation relations.

The B Model
For the B model we can be brief, as the arguments are so similar. In the large t limit, the wave-functional A reduces to a function of the zero modes of φ I and η i (the zero mode of ρ being represented as ∂/∂η). Expanding we see that for the low-lying states, A reduces to a sum of (0, q) forms on X (valued in End(E), E being a holomorphic vector bundle on X), for 0 ≤ q ≤ 3.
The commutation relations [Q, End(E)). So the cohomology of Q is H 0, * (X, End(E)). Likewise, by arguments similar to those for the A model, L 0 and b 0 reduce to

Space-Time Interpretation Via String Field Theory
In this section we will determine the space-time interpretation of the models just constructed in the case of open strings, that is in the case of orientable worldsheets with boundary. The reason for focussing on this case is that a simple and beautiful answer will arise, as we will see. The more perplexing closed string case (Riemann surfaces without boundary) is the subject of §5.  [14], which in turn is intimately tied up [15] with the existence of a simple cell decomposition of the moduli space of Riemann surfaces with boundary [11][12][13]. This cell decomposition involves the construction of Riemann surfaces by gluing together flat strips S i of width π and length T i ; they are glued, as sketched in figure (2), at vertices that generically (unless some T i = 0) are trivalent. The resulting pictures have been called "fatgraphs" to ordinary graphs corresponding to Feynman diagrams of ordinary field theory.

Construction Of Open String Field Theory
We recall now the highlights of the construction of open string field theory. A key part of the structure is a Z Z-graded associative algebra B, with a multiplication law that we denote as ⋆, and a derivation Q of degree 1 with Q 2 = 0. The degree is usually called ghost number. ⋆ There is a linear functional : The string field is a ghost number 1 element A ∈ B. The Lagrangian is This is invariant under gauge transformations generated by Chan-Paton factors are introduced as follows (in the case of a trivial rank N bundle in space-time with flat connection; the generalization is discussed presently).
⋆ The ghost number as I will count it here is the ghost number as defined in [14]

The A Model
We want to understand the physical content of these models. First we consider the A model. It is necessary to pick boundary conditions, and as in §3 we follow So until further notice, in discussing the A model, our target space will be the cotangent bundle T * M of an oriented three-manifold M. ⋆ Like any symplectic manifold, T * M can be given an almost complex structure such that the symplectic structure is positive and of type (1, 1); indeed, this is essential in Floer/Gromov theory of symplectic manifolds. This is good enough for formulating the A model with target space T * M; though the transformation laws and Lagrangian of the A model were written in §3 in a way that assumed the integrability of the almost complex structure, this assumption can be relaxed, as explained in detail in [1].
⋆ The orientation is needed to consistently define the sign of the fermion determinant in the world-sheet theory of §3.

The Vanishing Theorem
Now let us briefly explain the vanishing theorem, which asserts that instantons mapping Σ to T * M and mapping ∂Σ to M are necessarily constant.
Consider temporarily a general symplectic manifold X with symplectic form ω and an almost complex structure J such that ω is of type (1, 1) and positive.
Positivity means that the metric g IK = J S I ω SK is positive definite; the (1, 1) condition means that if i, j, k and i, j, k are indices of types (1, 0) and (0, 1), respectively, then g ij = g ij = 0, and g ij = g ji = −iω ij . If Σ is a Riemann surface, an instanton or almost holomorphic map is a map Φ : Σ → X with∂φ i = 0. Consider the bosonic sigma model action Instantons minimize this action for a given homotopy class since (4.4) The first term on the right hand on the right hand side of (4. To justify (4.4), pick on M local coordinates q a , a = 1 . . . 3. The symplectic structure of T * M can be written as ω = 3 a=1 dp a ∧dq a , with p a linear coordinates in the fibers that vanish on M. This is ω = dρ, where ρ = p a dq a vanishes on

Low Energy Expansion
As explained following equation (3.3), the key simplification of the A model is that the essential physics is independent of the coupling parameter t. As t and the target space metric g IJ appear only in the combination tg IJ , large t is simply the limit in which the target space metric is scaled up; it is the limit of large distances or long wavelengths. This is the limit in which ordinary string theory reduces approximately to field theory. Since ordinary string theory is t-dependent, the large t behavior is only an approximation. The topological string theories that we are studying are t-independent, so one can hope for an exact description by looking at the large t behavior.
We have analyzed the large t behavior of the string states in §3.3. In particular, as we saw there, the low-lying modes can be described as functions A(q a , χ b ) of the center of mass variables. In §3.3, we computed the cohomology for all ghost numbers, but in the string field theory context, we want to insist that that A have ghost number 1. This means that it must be linear in χ, so we write with A a dq a a one-form on M, valued possibly in End(E), the endomorphisms of a flat vector bundle E.
In §3.3, we identified Q, in the same large t limit, with the exterior derivative d. Then the quadratic part of the string field Lagrangian 1 2 A ⋆ QA reduces (for the low-lying modes in the large t limit) to This is simply a matrix element of the cubic part of the usual Chern-Simons action.
So putting the pieces together, the string field action (4.1) reduces to the ordinary that we will discuss later.

Reconsideration From The Hamiltonian Point Of View
Now I want to reconcile what we have said above with previous discussions of the A model such as [6] (or [1, §4.1] where the coupling to topological gravity is considered).
Formally, the A model is a quantum field theory that counts holomorphic curves obeying various conditions. (This is familiar in applications of the A model to mirror symmetry, for instance.) However, in the vanishing theorem that we have in [23].
To identify S, W, and w in the case at hand, let Σ be a compact oriented two dimensional surface, perhaps with boundary, not endowed with any a priori complex structure. Let X be a symplectic manifold, with an almost complex structure obeying the usual conditions, and with a Lagrangian submanifold M. where Ω 0,1 (Σ) is the space of (0, 1)-forms on Σ in the complex structure J, and  In a given homotopy class of maps of the graph Γ to X, there will be a map Φ for which Φ(Γ) is a "geodesic graph" of least total length. Φ can be regarded ⋆ ⋆ When π 1 (M ) has infinitely many conjugacy classes, so that the number of homotopy classes is infinite, it is scarcely plausible that the full answer can be written as a sum over homotopy classes with each class contributing an integer. For convergence, the contributions would have to be almost all zero. The possible noncommutativity of the sum over homotopy classes and the integration over moduli space may also mean that the final answer (which should be obtained by summing over homotopy classes first) cannot be written as a sum of contributions of homotopy classes, though the measure on moduli space can be so written.
In this case, the question of whether the individual contributions are integers does not even make sense. This seems to be the case for the one loop contributions, as one can see by comparing to D. Fried's formulas expressing the logarithm of analytic torsion as a sum over geodesics [24]. π T Fig. 6. A strip S of width π with one real modulus T .
The essential simplification of the A model always arises because for large t, this measure is concentrated near zeros of w. In the case that the only zeros are virtual zeros at infinity, the measure is concentrated for large t on the region near infinity; but this is the limit in which string theory reduces to field theory, and world-sheet path integrals reduce to Feynman graphs. So the non-vanishing contributions to the world-sheet path integral come entirely from the regions in the moduli space (of complex structures on Σ) in which the string theory reduces to the evaluation of Feynman graphs. This is the equivalence of the string theory to a field theory that was claimed earlier. Of course, one must remember that (as in the figure, where there are two possible degenerations) one Riemann surface will in general be capable of degenerating to several possible Feynman graphs.

Perturbation Theory
To make this a little more explicit, we will now discuss how to extract the field theoretic Feynman rules from the string theory.
We want to describe the measure on the moduli space of Riemann surfaces that follows from the description of moduli space via fatgraphs. This can be worked out systematically by quantizing open string field theory. The necessary techniques were given by Thorn and Boccicchio [25,26] and involve an elegant application of the Batalin-Vilkovisky approach to quantization. Instead of following that road, I will merely use the fatgraph description of moduli space and the definition of the measure in equation (2.4).
The basic fact that we will use is therefore the fact that every Riemann surface Σ with boundary has a canonical flat metric (with some conical singularities) built as in figure (2). This metric is obtained by gluing together some standard strips.
The strips are flat rectangles of width π in the σ − τ plane, say 0 ≤ σ ≤ π, 0 ≤ τ ≤ T , with metric ds 2 = dσ 2 + dτ 2 . Such a strip, say S (figure (6)), has one real modulus, namely T . A deformation of this modulus from T to T + δT can be made by changing the metric to be evaluated to give In equations (3.21) and (3.22), we determined the large t limits of L 0 and b 0 to be π∆/2t and πd * /2t. (Thus, b 0 vanishes for large t, but b 0 /L 0 has a non-vanishing limit because of the small eigenvalues of L 0 .) So the large t limit of (4.13) is (4.14) Now, (4.13) is the open string propagator, while (4.14) is the propagator of Chern-Simons field theory.
In the fatgraph approach to construction of moduli space, the strips we have just analyzed are glued together at cubic vertices to make Riemann surfaces. The structure of the interaction vertices is the same as in the open string multiplication law, figure (3(a)). We have already seen in connection with figure (4) how these vertices reduce at large t to wedge products of differential forms, as in Chern-Simons field theory.
While the underlying string field A had ghost number 1, and the underlying Chern-Simons gauge field A is a one-form, in (4.13) and (4.14) states of all ghost number -or differential forms of all degrees -are propagating. In either case, the vertices of the gauge fixed theory have in a natural sense the same structure (gluing of strings or wedge products) as that in the classical theory, though the constraint on the ghost number or degree is absent. In the case of string theory, this structure was explained by Thorn and Boccicchio [25,26] by solving the master equation and gauge fixing of string field theory. In Chern-Simons theory, the analogous formulation of the gauge fixed theory, combining the fields of different ghost number, is due to Axelrod and Singer [4].
(1) for the case that Σ is a disc.
As we know from §3.3, the "physical states" of the open string field theory are elements of H 1 (Σ, End(E)), E being a flat vector bundle that we will keep fixed in the discussion. The exterior derivative with values in that bundle will be denoted as d. Let α (1) , . . . , α (4) be four such states. We consider their coupling on the disc in that cyclic order, as in figure (7). To calculate the four point function in field theory, all that we really need to know is that there are two possible Feynman diagrams, indicated in figure (7), and that the propagator is d * /∆. The amplitude is therefore

(4.15)
To do the same calculation in string theory, we introduce vertex operators V (i) = α (i) a χ a for the external states, and one antighost insertion corresponding to the one real modulus of the disc. We can map V (1) , V (3) , and V (4) to 0, 1, and ∞, and take the modulus to be the position of V (2) . In that case, the antighost insertion can be taken to be a small contour integral around the position of V (2) , as in the The string theory amplitude is therefore Now we wish to evaluate this, as usual, in the limit of t → ∞. If the integral were limited to a range of σ bounded away from 0 and 1, it would vanish as 1/t for t → ∞. This is so because the φ and ψ − χ propagators are both proportional to 1/t. A non-zero result arises only because if the result is blindly expanded in powers of 1/t, the endpoint contributions are infinite. This is because the term of lowest order in 1/t in the operator product V (1) (0) · A a dq a dτ (σ) is proportional to 1/σ, coming from φ − ∂φ contractions. A simple-minded 1/t expansion of (4.17), evaluating those contractions (and discarding a term proportional to {Q, . . .}), leads to a divergent contribution near σ = 0 that looks like with ǫ an arbitrary fixed positive number.
To cure the infinity, we must remember the anomalous dimensions of the various operators, which are given by the eigenvalues of L 0 . To order 1/t, we had In turn, write B = i u i α (i) + σ v σ β (σ) , where α (i) are a basis of zero modes of the Laplacian ∆ and β (σ) are the non-zero modes; and u i , v σ are real coefficients.
The underlying Lagrangian can be regarded as a function L(u i , v σ ). The tree diagrams that we have discussed above are a calculational technique for "integrating out" the v σ to get an "effective potential" V (u i ). It can be written in terms of the n-point functions on the disc discussed above for n = 4 as Since V is obtained from the underlying Lagrangian L by "integrating out the massive modes," its stationary points, i.e., solutions of ∂V /∂u i = 0, are in one-toone correspondence with the stationary points of the underlying Lagrangian.
The process of "integrating out the massive modes" can be described as follows.
Let L be a function with a critical point at the origin, and let n be the dimension of the kernel of the Hessian or matrix of second derivatives of L. Then by a generalization of the Morse lemma, it is possible to pick coordinates u i , i = 1 . . . n and v σ in a neighborhood of the origin so that with non-zero constants a σ ; V (u i ) is then the "effective potential with massive modes integrated out." The u i are of course only uniquely determined up to diffeomorphism. Describing the possible canonical forms of V (u i ) up to a reparametrization of the variables is more or less the problem of singularity theory.
Since the u i can be described in first order as coordinates near the origin on the naturally defined vector space H 1 (M, End(E)), it is natural to think of the effective potential V as a function defined on that space, and as such it is uniquely defined up to a coordinate transformation Under such conditions, the four point coupling can be given the following interpretation. Since α (i) ∧ α (j) and α (j) ∧ α (k) vanish in cohomology, there are End(E)-valued one-forms y, z with In fact, we can set   The element w of that quotient space is known as the Massey triple product of α (i) , α (j) , and α (k) . Under the further hypothesis that α (k) ∧ α (l) and α (l) ∧ α (i) vanish in cohomology, is well-defined and in fact equals coefficient of the quartic term in the effective potential as computed above. The contribution of the quartic coupling to ∂V /∂u l is therefore As a flat connection is a critical point of the Chern-Simons action, this is in keeping with the observation [27] that the Massey triple product and its higher order analogs are the obstructions to deformation of a flat connection.

Comparison To Remarks Of Kontsevich
What do we learn by interpreting Chern-Simons gauge theory as a string theory? Here I will only point out one obvious consequence. Other applications may involve, for instance, a deeper relation of Chern-Simons theory to mirror manifolds.
The fact that Chern-Simons theory can be interpreted as a topological sigma model coupled to topological gravity means that one can introduce the usual observables of topological gravity -the stable cohomology classes on the moduli space The trouble is that we have fixed X to have complex dimension three precisely to ensure that Θ is a top form. It therefore cannot be integrated over a cycle of positive codimension unless the definition is modified somehow. The following way of doing this will enable us to make contact with observations by Kontsevich.
Instead of considering a single target space X with a fixed metric, we will consider a family of X's with a family of metrics. The partition function of the theory will be not a number but a closed differential form on the space of metrics on X. To implement this idea, introduce along with the metric g IJ (φ) of X a fermionic variable ζ IJ (φ) (of ghost number one) which one can think of as (up to a constant) the exterior derivative of g IJ (φ). Take the new transformation laws Recalculating the Lagrangian starting from (3.2), we find that (3.3) is replaced by Since L ′ is of ghost number −1 in the "matter fields" φ, χ, ψ, every insertion of L ′ shifts the ghost number by one unit. If we want to integrate not over moduli space but over a cycle C of codimension n, the non-vanishing contributions will be precisely n th order in L ′ . As L ′ is linear in ζ IJ , these contributions will be n th order in ζ IJ and so will define an n-form Ω on the space R of metrics on X.
Q-invariance means that Ω is closed. Let F be the group of diffeomorphisms of X. We temporarily postpone introducing the antighosts and auxiliary fields that enter the gauge fixing.
Just from the A − c system, it is possible to construct new observables (discussed, for instance, in [28], where more information can be found). Let T be an By solving the "descent equations" is BRST invariant (and is easily seen, by virture of (4.35), to depend only on the homology class of Y ). In particular, setting i = 3 and Y = M, we get new terms that can be added to the Chern-Simons Lagrangian. To be precise, if L is the usual Chern-Simons Lagrangian and T α are the antisymmetric invariants on the Lie algebra, we can take Tα . Writing also A = A (0) + B, where A (0) is a solution of the classical equations about which one wishes to expand, and denoting the covariant derivative with respect to Computing δΓ, one gets the usual gauge fixing Lagrangian Just as in our field theoretic discussion, to modify this, we will consider not a Since ∆L GF is of ghost number −1 in the matter fields, insertions of O (3) 's can be balanced by insertions of ∆L GF . As ∆L GF is linear in ζ, the resulting amplitudes, just as in our string theoretic discussion, will be naturally not numbers but differential forms on the base space of a fibration. We already know that the contribution to (4.47) of instanton number q = 0 is the ordinary Chern-Simons action. We want to determine the contribution for some non-zero value of q.

General Target Spaces
Consider the moduli space of holomorphic maps Φ : Σ → X with Φ(∂Σ) ⊂ M and with two such maps identified if they differ by an SL(2, IR) transformation.
The fact that c 1 (X) = 0 and dim I C (X) = 3 means that in the moduli problem, the dimensions of the appropriate H 0 and H 1 are zero. "Generically" this means that there are only finitely many such instantons for each value of q. For simplicity, we will consider only this case. If Φ is such an instanton, let C = Φ(∂Σ). Generically, C is a knot in the three-manifold M. Let us work out the contribution of Φ to the path integral (4.47). If we are expanding around a background connection A in M, the contribution of the Chan-Paton factors is (4.48) Note that though C bounds a disc in X, it may not do so in M, so (4.48) can be nontrivial even if A is flat. We also get a factor of exp(−θq) from the instanton-counting term in (4.46). The remaining contributions are nearly trivial since (i) they are independent of t; (ii) they reduce in the large t limit to a ratio of determinants; (iii) except for a possible sign, the boson and fermion determinants cancel because of the Q symmetry. The contribution of an instanton is hence where η = ±1 is the ratio of determinants.
The total action is therefore easy to evaluate. If Φ i , i = 1, 2, 3, . . . are the instantons of non-zero instanton number, with instanton numbers q i , boundaries C i , and determinant factors η i , then the action is  the Lagrangian whose solutions are connections of vanishing (0, 2) curvature is Arguments similar to those that we have given for the A model show that in the large t limit, the open string field theory of the B model reduces to (4.52).
The quantum field theory with Lagrangian (4.52) is unrenormalizable by power counting. However, it has the following all-but-unique property: there are no possible counterterms that respect the classical symmetries. The symmetries of (4.52) include complex gauge transformations, ∂ A → g∂ A g −1 , with g an arbitrary gauge transformation of E not respecting any reality or unitarity condition; and local complex changes of coordinates that preserve λ. There is no local density constructed from A that is invariant under these symmetries. (Even L T itself, though possessing these invariances at least if one considers only gauge transformations c u Fig. 8. A cylinder U and an embedded circle C. Any diagram containing this embedded cylinder has two moduli naturally associated with the cylinder; one is the length of the cylinder and the second comes from the possibility of "cutting" the cylinder on C, and twisting about C before regluing. that are connected to the identity, is not the integral of an invariant local density.) Infinities in quantum field theory are ordinarily integrals of invariant local densities, so if (4.52) could be quantized preserving the symmetries, one would expect this theory to be finite, though superficially unrenormalizable.
Relying only on usual field theory arguments, it is not at all clear that (4.52) can be quantized preserving its symmetries. However, the equivalence of (4.52) to a string theory strongly suggests that it in fact is finite. One might worry about whether closed string poles can ruin the finiteness; at the end of §5 we will argue that this does not occur. One might also wonder whether the finite theory given by the string theory is really (4.52) or some more elaborate theory with (4.52) coupled to closed strings. In §5 we argue that the closed strings are decoupled.

The Closed String Sector
We have discovered that -with some reasonable boundary  It is convenient to set b ± 0 = b 0 ± b 0 . The formula analogous to (5.1) is that the closed string propagator is Here L + 0 = L 0 −L 0 is the total string Hamiltonian, and Π is the projection operator on states invariant under rotation of the circle. This formula is fairly well known, and in any case can be derived similarly to (5.1), replacing the strip in figure This propagator, however, does not seem to arise by gauge fixing of any local Lagrangian. It is a pseudodifferential operator of degree zero, so for it to arise as the inverse of a differential operator, that operator would have to be of degree zero, that is a constant.  What sort of Lagrangian do we get by taking the low energy limit of closed string field theory? As we discussed in §2 in a related context, the free part of the closed string Lagrangian is where ideally Such a c 0 − does not exist, since b 0 − , whose field theory limit is ∂ * − ∂ * , has a non-trivial cohomology. However, one can pick a c 0 where T is the projection operator onto a subspace (say the kernel of L 0 ) annihilated by Q; this is good enough to ensure gauge invariance of (5.7). Such a c 0 − is, in the field theory limit, With this choice of c 0 − , and recalling that the field theory limit of Q is ∂ + ∂, the field theory limit of the Lagrangian is not (5.4) but When this is gauge-fixed and inverted to get a propagator, the factor of ∆ −1 migrates to the numerator, canceling a factor of ∆ −1 in (5.5) and reproducing the field theory limit of the closed string propagator (5.3).

A General Puzzle
The conclusion seems to be that closed string field theory of the A model - and of knots [30]. One-loop anomalies of the field theory related to the B model were calculated long ago by Ray and Singer [31]. These anomalies are possible only because of the ultraviolet difficulties of quantum field theory, which of course are greatly ameliorated in string theory. It therefore seems reasonable to suspect that the closed string contributions in the A and B theories cancel the anomalies of these theories, without, in view of the decoupling argued above, having much effect on the open string "physics." In any event, whatever the closed string contributions, they are finite. In usual string theory, possible infinities come from physical closed string poles, but in these topological models, there are no such poles since the closed string propagator annihilates the physical states or harmonic forms; this is because (in contrast to conventional string backgrounds) the b 0 operator annihilates the kernel of L 0 .