LECTURE II-9: WILSON LOOPS,’T HOOFT’S LOOPS,

AND ’T HOOFT’S MODEL OF CONFINEMENT

Edward Witten

Notes by Pavel Etingof and David Kazhdan

9.1. ’t Hooft loop operator

Let us recall abelian duality in 4 dimensions, which we discussed two lectures ago. Consider a free U(1) gauge theory with a θ-angle. Thus we have two dimensionless couplings e, θ which combine into a single complex coupling τ = \frac{2πi}{e^{2}} + \frac{θ}{π}, and the Lagrangian is

(9.1)  L = ∫_{M}(\frac{iτ̄}{4π} F_{+}^{2} − \frac{iτ}{4π} F_{−}^{2}),

where F is the curvature of a U(1) connection A and F_{+}, F_{−} are the selfdual and antiselfdual parts of the curvature. We have seen that if the spacetime M is a spin manifold then this theory is “modular invariant” as a function of τ. One modular symmetry τ → τ + 1 is obvious, as it corresponds to shifting the θ-angle by π, which does nothing because c_{1}^{2} for a spin manifold is even. (On a manifold that is not a spin manifold, the symmetry would be only τ → τ + 2.) The symmetry under the second generator of the modular group, τ → −1/τ, is more interesting and corresponds to electromagnetic duality discovered by Maxwell. More precisely, this means that the theory of a connection A with coupling constant τ is identical both classically and quantum mechanically to the same theory with coupling −1/τ and the connection B such that dA = const ∗ dB. Now, like two lectures ago, we want to see what happens to operators under this duality. In particular, we want to know what happens to the Wilson loop operator.

Recall that the Wilson loop operator has the form W_{γ}(C) = e^{iγHol_{C}(A)}, where Hol_{C}(A) denotes the integral of the connection A along a closed oriented curve C in the spacetime M. This operator is gauge-invariant and well-defined if γ is an integer, or for any real γ if the curve C is homotopically trivial in M. (More generally, there could be several components C_{i} with real numbers γ_{i}, and the condition is that ∑_{i} γ_{i}C_{i} should be an integral class in H_{1}(M).) Matrix elements of this operator are computed, as usual, by inserting the above exponential into the path integral. Similarly to what we found in similar problems in two and three dimensions, we should get that the dual description of the Wilson loop is a recipe which says that rather than insert in the path integral an object living on C, we should integrate over connections having a singularity along C.

The precise answer is the following. For any curve C the expectation value ⟨W_{γ}(C)O_{1}...O_{n}⟩ equals to ∫ e^{−L(B)}O_{1}...O_{n}DB, where the integral is taken over

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connections on M \ C such that the integral of the curvature of B over a small 2-sphere S in a normal 3-space to C at any point equals 2πγ.

Let us prove this. We assume that C is a boundary. Let D be a 2-chain whose boundary is C. Recall the calculation from the lecture on Abelian duality: our fields are A – the original connection, G- the 2-form, and B – the dual connection. We have

∫ e^{−L(A)}W_{γ}(C)DA =

∫ DA DG DBe^{\frac{−iτ̄}{4π} ∫ F_{+}^{2}+ \frac{iτ}{4π} ∫ F_{−}^{2}}e^{\frac{i}{2π} ∫ G∧F_{B}}e^{iγ ∫_{D}(F_{A}−G)}, (9.1)

where F = F_{A} − G, and the last factor corresponds to the insertion of the Wilson loop (recall from the abelian duality discussion that the Wilson loop classically is e^{iγ ∫_{D} F_{A}}, and F_{A} is to be replaced with F_{A} − G in the extended theory). Gauging A to 0, we get

∫ e^{−L(A)}W_{γ}(C)DA =
∫ DG DBe^{\frac{−iτ̄}{4π} ∫ G_{+}^{2}+ \frac{iτ}{4π} ∫ G_{−}^{2}}e^{\frac{i}{2π} ∫ G∧F_{B}}e^{−iγ ∫_{D} G} =

∫ DG DBe^{\frac{−iτ̄}{4π} ∫ G_{+}^{2}+ \frac{iτ}{4π} ∫ G_{−}^{2}}e^{i ∫ G∧( \frac{1}{2π} F_{B}−γ[D])}, (9.2)

where [D] is the delta-function of D. Now we define F̃_{B} := F_{B} − 2πγ[D]. Then after integrating out G, (9.2) can be written as

∫ e^{−L(A)}W_{γ}(C)DA =

∫ DBe^{\frac{i}{4πτ̄} ∫ (F̃_{B})_{+}^{2}+ \frac{−i}{4πτ} ∫ (F̃_{B})_{−}^{2}}, (9.3)

Thus, the effect of the insertion W_{γ}(C) is that F_{B} is replaced in the final answer by F̃_{B}. So B is now a connection on a line bundle with singularity along C, as discussed above.

Note that if C is not a boundary then ⟨W_{γ}(C)⟩ = 0 (even with insertion of any number of local operators). Indeed, we have symmetry A → A + dϕ, and ϕ does not have to be globally defined as a map to a circle; in fact, dϕ can be any closed one-form. So if C is not a boundary then we can choose dϕ in such a way that the operator W_{γ}(C) will multiply by e^{iα} for some nonzero α. Hence its expectation value (even with inclusion of local operators, which are invariant under this transformation) vanishes. More generally, the correlator ⟨W_{γ_{1}}(C_{1})...W_{γ_{n}}(C_{n})⟩ (with any local operators) is zero if ∑ γ_{i}C_{i} ≠ 0 in H_{1}(M), where M is the spacetime. As noted before, for the product of operators in question to be well-defined, we only need ∑ γ_{i}C_{i} to be an integral class.

From our construction so far, for any curve C we have two operators:

1) W_{γ}(C) = e^{iγ ∫_{C} A}

2) T_{γ}(C) = e^{iγ ∫_{C} B}.

The second operator, which is dual to the Wilson loop, is called the ’t Hooft loop operator.

9.2. Hilbert space interpretation of the ’t Hooft loop operator

Now let us consider this picture from the Hamiltonian point of view. Then the spacetime M has the form M = M^{3} × ℝ with Minkowski metric. Let C, C′ be two nonintersecting closed simple curves in M^{3}. They define operators W_{γ}(C) and T_{γ′}(C′) on the Hilbert space H. The following commutation relation for these operators is due to ’t Hooft:

(9.4)  W_{γ}(C)T_{γ′}(C′) = e^{2πiγγ′l(C,C′)}T_{γ′}(C′)W_{γ}(C).

Let us prove this formula. Let us work in terms of the original connection A. Then the Hilbert space consists of wave functions Ψ(A). In this realization, the Wilson loop operator W_{γ}(A) is simply the operator of multiplication by e^{iγ ∫ A}.

However, the ’t Hooft loop operator is a bit harder to define. To do this, consider the homomorphism π_{1}(M^{3} \ C′, x_{0}) → ℤ given by the linking number with C′. Let π_{1}^{0} be the kernel of this homomorphism and X be the ℤ-cover of M_{3} \ C′ corresponding to π_{1}^{0}. Let ϕ be a function X → U(1) such that the monodromy corresponding to the generator of ℤ is e^{iγ}. Any two such functions differ by a gauge transformation, but ϕ itself is not an honest gauge transformation. Then it is not difficult to check that the ’t Hooft loop operator T_{γ′}(C′) is just the “illegal” gauge transformation by ϕ:

(9.5)  (T_{γ′}(C′)Ψ)(A) = Ψ(A^{ϕ}).

Note that this is well-defined since any two such ϕ’s differ by an honest gauge transformation.

Now formula (9.4) is clear since T W T^{−1} = e^{iγ_{C} ∫ dϕ}W, because of the way that T transforms the connection in the definition of W.

9.3. The 2+1-dimensional analogue of the 3+1-dimensional picture

Consider the 2+1-dimensional analogue of this picture. As we saw before, in 2+1 dimensions the theory of a scalar field ϕ is dual to a gauge theory of the dual field A. The path integral in ϕ with insertion of e^{iϕ(x)} is the same as path integral in A where A is a connection on M \ x which has ∫ F = 2π, where the integral is over a small sphere around x. Thus, the operator e^{iϕ} corresponds to a magnetic monopole in gauge theory.

Now consider the 3-dimensional cosine theory, defined by the path integral

(9.6)  ∫ Dϕe^{− ∫ (|dϕ|^{2}+ε(e^{iϕ}+e^{−iϕ}))}.

Decomposing this path integral in a power series, and passing to the dual variable A, we get the sum

(9.7)  ∑_{m,n} ε^{m+n} ∫ \frac{dx_{1}...dx_{m}}{m!} ∫ \frac{dy_{1}...dy_{n}}{n!} ∫_{A_{x,y}} e^{− ∫ F_{A}^{2}},

where A_{x,y} is the space of connections with monopoles at x_{i} and antimonopoles at y_{j}. Thus the cosine theory maps to the theory with monopoles. We saw this more computationally when we discussed the Polyakov model two lectures ago.

9.4. The model of confinement

Now we discuss a picture of confinement developed by ’t Hooft. In general we don’t assume that the gauge group is abelian. Recall the definition of confinement. We have a gauge group G and with universal cover Ĝ. We assume that G is the quotient of Ĝ which acts faithfully on all fields in the Lagrangian. We let R be a representation of Ĝ. As we discussed before, if there is a mass gap, there are two usual patterns of decay of the expectation value ⟨W_{R}(C)⟩ of the Wilson line operator corresponding to the representation R as C gets big:

Pattern 1:

(9.8)  ⟨W_{R}(C)⟩ ∼ e^{−λ Length(C)}

Pattern 2:

(9.9)  ⟨W_{R}(C)⟩ ∼ e^{−λ Area(C)}

(here the parameter γ is a fixed nonzero number and the area of C means the minimal area of the spanning surface). The first regime is called the Higgs regime (the length law) and the second one is called the confinement regime (the area law).

As we discussed in the previous lecture on confinement, the first regime is the case when R is a representation of G itself, and to see confinement one needs to consider the case when R is a representation of Ĝ but not of G. Thus interesting W_{R}(C) correspond to elements of π_{1}(G)^{∗}.

Now let us consider the ’t Hooft loop operator T_{γ}(C). It is defined for any G by analogy with the definition in the abelian case. We fix an element γ ∈ π_{1}(G). Recall that G-bundles on a two-sphere S^{2} are classified by a characteristic class that takes values in H^{2}(S^{2}, π_{1}(G)), which is canonically isomorphic to π_{1}(G). The choice of γ therefore canonically determines an isomorphism class of G-bundles on S^{2}. We can now define the ’t Hooft operator: a path integral with insertion of T_{γ}(C) is computed by integrating over connections on M \ C which have the property that when restricted to a small sphere S that links C, the bundle has characteristic class γ.

’t Hooft’s idea was to consider T_{γ}(C) instead of W_{R}(C) and find conditions under which there is an area law for its expectation value. This occurs, as he showed, for certain Higgs theories. Then, ’t Hooft proposed (following earlier ideas of Nambu, Mandelstam, and others) that confinement would be related to the Higgs mechanism by a duality that maps ’t Hooft loop operators into Wilson loop operators. This does not explain confinement, but it reformulates the problem: to reduce the mysterious phenomenon of confinement to the much more easily understood Higgs phenomenon, one must understand the nonlinear duality that exchanges ’t Hooft and Wilson loop operators.

To illustrate the area law for the ’t Hooft loop in Higgs theories, we consider a familiar example: the U(1) gauge theory with a charged complex scalar ϕ (of charge 1). The Lagrangian is

(9.10)  ∫ (\frac{|F |^{2}}{4e^{2}} + |D_{A}ϕ|^{2} + V (ϕϕ̄)),

where V is a (quartic) potential. We will study the ’t Hooft loop T_{γ}(C), where γ ∈ ℤ. Thus A is a connection and ϕ is a section for a hermitian line bundle over M \ C such that it has first Chern class γ when restricted to a small sphere linking C. We will compute ⟨T_{γ}(C)⟩ for two classes of V:

1) V = λ(ϕϕ̄ + a^{2})^{2};

2) V = λ(ϕϕ̄ − a^{2})^{2}.

This theory was considered in Lecture 2. Recall the results of this consideration. Case 1. In the infrared the theory behaves like the product of the theory of a free

massive field with a free gauge theory. In particular, there is no mass gap. Thus, we can calculate ⟨T_{γ}(C)⟩ for large C using the free theory. But in the free theory this expectation value is the same as ⟨W_{γ}(C)⟩ in the dual theory. It is easy to see that the expectation values of both W_{γ} and T_{γ} behave according to the length law, because of Coulomb law of charge interaction. A theory behaving in this way is said to be in the “Coloumb phase.”

Case 2. In the infrared this theory has breaking of gauge symmetry and a Higgs mechanism. In particular, speaking classically, we have a circle of vacua, and at each of these vacua the low energy part of the Hamiltonian spectrum contains a massive vector and a real massive scalar. So there is a mass gap. This theory is not believed to exhibit confinement, i.e. it is believed that it exhibits the length law for the Wilson loop. This is certainly what one computes in perturbation theory.

In case 2, we will show that there is an area law for the ’t Hooft loop operator, because of the Higgs mechanism. This happens for topological reasons, as explained below.

As a warmup consider a closed spacetime M and a line bundle L with a nontrivial c_{1}. Let us consider the path integral for our theory over sections of this bundle. It turns out that the action of all field configurations in this integral has to be very large: it is bounded below by a constant (which is independent of M) times the area of the minimal 2-surface which represents a cycle Poincare dual to c_{1}(L).

Indeed, if ϕ is a section of L then ϕ has to vanish on a 2-cycle Σ which is dual to c_{1}(L). If we fix Σ, we can look at the configuration of minimal action with such a zero. For this, we can (if the metric of M is scaled up) reduce to the case that M = ℝ^{4} = ℝ^{2} × ℝ^{2}, with Σ equal to the first factor. We can assume that ϕ and A are invariant under translations of the first factor in M. In the second factor, we want ϕ to vanish at the origin and to approach the vacuum at infinity (up to gauge transformation), such that the first Chern class of the bundle, relative to the trivialization at infinity given by ϕ, equals 1. The same problem appeared in lecture 2 (in the guise of finding an instanton solution of the twodimensional version of the same model), and we discussed qualitative properties of the solution. Anyway, let I be the action of this solution in the two-dimensional sense (that is, integrated over just the second factor in M). Going back to a global Σ ⊂ M of smallest area representing the first Chern class, the minimum action field looks in the normal directions to Σ like the instanton just described; its action is approximately I · Area(Σ).

Now let us come back to the ’t Hooft loop in ℝ^{4}. In this case the bundle is over M \ C, where M is the spacetime. If D is a 2-chain in M whose boundary is C then D plays the role of the Σ of the previous discussion. Indeed, if L is a line bundle over M \ C with Chern class 1, and ϕ its section then ϕ must vanish on a 2-surface whose boundary is C. Thus, the same argument as above shows that ⟨T_{1}(C)⟩ ∼ e^{−λ area(D)}, where area(D) is the smallest area of a disk spanning C. This is the area law which we wanted to demonstrate.

This behavior is characteristic of what is called the Higgs regime, or phase.

Now let us discuss in more detail the relation of the established behavior of the ’t Hooft loop operator with confinement. It is believed that if π_{1}(G) ≠ 0 there are at least three possible phases:

1) Coulomb: no mass gap, gauge bosons in the infrared, W and T behave like in the free theory and exhibit the length law.

2) Higgs: mass gap, length law for W, area law for T.

3) Confinement: mass gap, area law for W, length law for T.

As already suggested, ’t Hooft’s idea was that there should be a nonabelian analogue of duality which interchanges W with T, the Higgs and the confinement regimes, and maps the Coloumb phase to itself. Thus, the area law for T in a theory implies confinement in the dual theory. This is what happens for some supersymmetric theories, e.g. the theory relevant to Donaldson theory.

In fact, ’t Hooft showed that if R and γ are such that γ|_{R} ≠ 1 then either W_{R} on T_{γ} exhibit the area law. More specifically, he proved an even stronger statement, namely that the set H of all (c, γ) ∈ π_{1}(G)^{∗} × π_{1}(G) such that for some representation R with central character c the operator W_{R}(C)T_{γ}(C) (suitably renormalized) does not exhibit the area law, is an isotropic subgroup of π_{1}(G)^{∗} × π_{1}(G) with respect to the natural symplectic form. The reason for this, very roughly, is the following. If A(C) = W_{R_{1}}T_{γ_{1}}(C) and B(C) = W_{R_{2}}T_{γ_{2}}(C) exhibit the length law then, when acting on the vacuum, they produce only effects that are localized along C (or there would be an area law instead of a length law). So

(9.11)  ⟨A(C)B(C′)⟩ = ⟨A(C)⟩⟨B(C′)⟩(1 + o(1)), d → ∞

where d is the distance between C and C′. This shows that if AB = qBA where q is a constant then q must be equal to 1. By ’t Hooft’s formula (9.4) (which is clearly valid in the nonabelian case as well), this implies that H is isotropic. By further physical arguments, one shows that if there is a mass gap, then H is maximal isotropic. This leads to a more refined classification of massive phases than was stated above: associated to each massive phase is a maximal isotropic subgroup of π_{1}(G)^{∗} × π_{1}(G). All possibilities can arise, in general.

Remark. The argument showing that loop operators A and B must commute fails if one of the operators, say A, exhibits the area law. In this case, acting on the vacuum with this operator produces an effect that is not in any way localized near C; it rather has an effect which is localized near a minimal area disk D whose boundary is C; such a disk will always intersect C′ when the linking number is not zero. Formula (9.11) is now valid only if d is the distance from C′ to D, which is always 0, so the formula does not tell us anything.

Remark. The area law for the ’t Hooft operator in a Higgs phase has many physical and mathematical applications. For example, with some small adjustments, what we said above in analyzing the behavior of the Higgs phase with a bundle of nonzero first Chern class could serve as an explanation of the Meissner effect, the fact that a superconductor (which is described approximately by the abelian Higgs model that we examined) expels magnetic flux. Perhaps the reader has, at a science fair, seen a demonstration of a magnet floating above a superconductor; this effect has the same origin. Mathematically, C. Taubes’s analysis of the Seiberg-Witten invariants of symplectic four-manifolds made use of the same facts: the localization (in a closely analogous system of equations) of the zeroes of ϕ on a surface of smallest area.