LECTURE II-8, PART II: ABELIAN DUALITY IN FOUR DIMENSIONS AND Sl(2, ℤ)

EDWARD WITTEN^{∗}

1. Duality and Sl(2, ℤ)

In this second part of lecture II-8, we discuss abelian duality in four dimensions, and give an application to an Sl(2, ℤ) symmetry of the free U(1) theory in four dimensions. We postpone discussion of Sl(2, ℤ) symmetries of non-free theories to a later lecture, since all known examples of that involve supersymmetry.

We work with a U(1) bundle ℒ on a 4-manifold M, and a connection A on ℒ , whose curvature is F =F_{A}. The gauge theory Lagrangian (in Euclidean signature) including the topological term is

\begin{aligned}L(A) & = \frac{1}{4e^{2}} ∫ d^{4}x \sqrt{g}F_{mn}F^{mn} + \frac{iθ}{16π^{2}} ∫ d^{4}x \sqrt{g}ϵ_{mnpq}F^{mn}F^{pq} \\ & = \frac{1}{2e^{2}} ∫ F_{A} ∧ ∗ F_{A} + \frac{iθ}{4π^{2}} ∫ F_{A} ∧ F_{A}. \end{aligned}  (1.1)

We have used the standard normalization on the kinetic term, and have normalized the topological term so that replacing θ by θ+2π does not change the physics. (This property of the topological term derives from the fact that c_{1}(ℒ )^{2} = ∫ (F_{A}/2π)∧(F_{A}/2π) is always an integer. Notice that on a spin manifold, c_{1}(ℒ )^{2} is always an even integer, and we gain an additional equivalence under replacement of θ by θ + π.)

Let τ = \frac{θ}{π} + \frac{2πi}{e^{2}} ∈ 𝔥. As we have just observed, τ ↦ τ + 2 is a symmetry of this theory, and τ ↦ τ + 1 is a symmetry when working on a spin manifold. To extend this to an Sl(2, ℤ) action (in the spin manifold case) we also need a symmetry which maps τ to −1/τ; this will be given by a duality transformation F_{A} ↔ ∗ F_{C} (with C being a new “dual” connection).

The computations for this duality transformation are similar to those in lecture II-7. We begin by defining F_{±} = \frac{1}{2}(F_{A} ± ∗ F_{A}), and rewriting our Lagrangian (1.1) as

\begin{aligned}L(A) & = \frac{i\overline{τ}}{4π} ∫ F_{+} ∧ F_{+} + \frac{iτ}{4π} ∫ F_{−} ∧ F_{−} \\ & = \frac{i\overline{τ}}{4π} ∫ ∥F_{+}∥^{2} − \frac{iτ}{4π} ∫ ∥F_{−}∥^{2}. \end{aligned}  (1.2)

Date: 6 March 1997.

∗Notes by David R. Morrison.

Letting G denote the gauge group associated to A, the partition function for this theory can be written as

Z(τ) = \frac{1}{vol(G)} ∑_{ℒ}∫ DA e^{− \frac{i\overline{τ}}{4π} ∫ ∥F_{+}∥^{2}+ \frac{iτ}{4π} ∫ ∥F_{−}∥^{2}}.  (1.3)

Our earlier examples of duality began with a theory of a scalar field ϕ which entered into the Lagrangian only through its derivative dϕ so that the theory had a symmetry under ϕ ↦ ϕ + c (with c constant); the first step in the duality transformation was to gauge this symmetry, introducing also an appropriate Lagrange multiplier field.

The present theory is already a gauge theory, being a theory of a connection A which enters into the Lagrangian only through its curvature F_{A}, so that there is a symmetry under A ↦ A + B (with B a flat connection). We want to do the analogue of gauging this symmetry, by allowing B to be an arbitary connection on an arbitrary bundle, introducing a kind of “exotic gauge field” G which is a 2-form field, and extending the symmetry to

\begin{aligned}A & → A + B \\ G & → G + F_{B}. \end{aligned}  (1.4)

Then F := F_{A} − G plays the role of the “gauge-invariant” quantity, analogous to the covariant derivative of a scalar field. It is to be stressed that two G fields will be considered gauge-equivalent if they differ by G → G + F_{B} for F_{B} the curvature of any connection on any line bundle. In our analysis, we will assume for simplicity that there is no torsion in H^{2}(M).

We need a “gauge-invariant” extension of our Lagrangian. We might try

L(A, G) = \frac{i\overline{τ}}{4π} ∫ ∥F_{+}∥^{2} − \frac{iτ}{4π} ∫ ∥F_{−}∥^{2},  (1.5)

but this is too simple (because, for example, we could gauge F to zero). To improve this, we introduce a new connection C on a line bundle 𝒩 , with curvature F_{C}, and consider the Lagrangian

L(A, G, C) = \frac{i\overline{τ}}{4π} ∫ ∥F_{+}∥^{2} − \frac{iτ}{4π} ∫ ∥F_{−}∥^{2} − \frac{i}{2π} ∫ F_{C} ∧ G.  (1.6)

The partition function for this new theory can be represented as a path integral, which includes sectors associated to all choices of bundles ℒ and 𝒩 :

\frac{1}{vol(G̃)} \frac{1}{vol(G)} \frac{1}{vol(G_{C})} ∑_{ℒ ,𝒩} ∫ DA DG DC e^{− \frac{i\overline{τ}}{4π} ∫ ∥F_{+}∥^{2}+ \frac{iτ}{4π} ∫ ∥F_{−}∥^{2}+ \frac{i}{2π} ∫ F_{C} ∧G}, (1.7)

where G and G_{C} denote the gauge groups associated to A and C, respectively, and G̃ denotes the “exotic” gauge group.

To see that this new theory is equivalent to the original one, we first do the C-integral in (1.7): write C = C_{0} + C′, for C_{0} a fixed connection on the line bundle 𝒩 . Then the C′ integral is

\frac{1}{vol G_{C}} ∫ DC′ e^{\frac{i}{2π} ∫ C′∧dG} = δ(dG).  (1.8)

Thus, when we sum over 𝒩 we find

\frac{1}{vol G_{C}} ∑_{𝒩}∫ DCe^{− \frac{i}{2π} ∫ F_{C} ∧G} = ∑_{x∈H^{2}(M)} e^{i(x,G)}δ(dG) = δ([\frac{G}{2π} ] ∈ ℤ)δ(dG). (1.9)

The conditions that dG = 0 and that [G/2π] is an integral class precisely mean that G is of the form F_{B} for some connection on some line bundle and hence that G can be gauged to zero. After doing this, it follows that the partition function (1.7) coincides with Z(τ), and we recover the original theory.

Alternatively, we can evaluate the partition function (1.7) by gauging A to 0, using the “exotic” gauge invariance (which has an ordinary gauge invariance as an ambiguity). This leaves the path integral

\frac{1}{vol G_{C}} ∑_{𝒩}∫ DG ∫ DC e^{− \frac{i\overline{τ}}{4π} ∫ ∥G_{+}∥^{2}+ \frac{iτ}{4π} ∫ ∥G_{−}∥^{2}+ \frac{i}{2π} ∫ F_{C} ∧G}.  (1.10)

To evaluate the G integral, we complete the square, bearing in mind that

∫ F_{C} ∧ G = ∫ (F_{C}_{+} ∧ ∗ G_{+} − F_{C}_{−} ∧ ∗ G_{−}).  (1.11)

In fact, if we define G′ = G − \frac{1}{\overline{τ}}F_{C}_{+} + \frac{1}{τ}F_{C}_{−}, then we can write the exponent from eq. (1.10) as

−\frac{i\overline{τ}}{4π} ∫ ∥G′_{+}∥^{2} + \frac{iτ}{4π} ∫ ∥G′_{−}∥^{2} + \frac{i}{4π\overline{τ}} ∫ ∥F_{C}_{+}∥^{2} − \frac{i}{4πτ} ∫ ∥F_{C}_{−}∥^{2}.  (1.12)

When we carry out the G′ integral, the first two terms give a Gaussian integral which contributes to the overall normalization; integrating out G′ leaves the path integral

\frac{1}{vol G_{C}} ∑_{𝒩}∫ DC e^{\frac{i}{4π\overline{τ}} ∫ ∥F_{C}_{+}∥^{2}− \frac{i}{4πτ} ∫ ∥F_{C}_{−}∥^{2}}.  (1.13)

This is the same as the original path integral, but with τ replaced by −1/τ, precisely what we wanted to show.

As we did in the case of two dimensions, it is possible to analyze the τ-dependence of the normalization of the path-integral, and obtain further interesting results. Some hint of the flavor of the results to be obtained this way is seen if we evaluate the Gaussian integral indicated above, which yields

(\frac{2π}{\sqrt{i\overline{τ}}} )^{n_{2}_{+}} (\frac{2π}{\sqrt{−iτ}} )^{n_{2}_{−}} ,  (1.14)

where n_{2}_{±} denote the numbers of self-dual and anti-self-dual 2-forms. Of course, these numbers are infinite, so there must be some cancellation against other normalization factors. When this is worked out in detail,^{1} the result is found to be

Z(τ) = τ^{− \frac{χ+σ}{4}} \overline{τ}^{− \frac{χ−σ}{4}} Z(−1/τ),  (1.15)

where χ and σ are the Euler number and signature of M, respectively. Thus, the partition function Z(τ) is actually a modular form for Sl(2, ℤ) (or for a subgroup, when the manifold is not spin) of weight (\frac{χ+σ}{4}, \frac{χ−σ}{4}).

We can also follow certain operator insertions through the duality transformation, as we did in lower dimensions. An insertion of F_{±} in the original theory can be realized by inserting F_{±} = F_{±} − G in the extended theory, which can be written

F_{+} = F_{+} − G′_{+} − \frac{1}{\overline{τ}}F_{C}_{+}, or F_{−} = F_{−} − G′_{−} + \frac{1}{τ}F_{C}_{−},  (1.16)

respectively, after making the change of variables to G′. Thus, when we gauge A to zero, and integrate out G′, we are left with operator insertions proportional to F_{C}_{±}, namely:

F_{+} ↦ (−1/\overline{τ})F_{C}_{+}, and F_{−} ↦ (1/τ)F_{C}_{−}.  (1.17)

Notice that as a consequence of the τ-dependence of these mappings, a correlation function involving insertions of F_{+} and F_{−} will have a different modular weight than that of the partition function.

2. The Hamiltonian formalism

Returning to the case that the gauge group is U(1), let us briefly discuss abelian fourdimensional duality in a Hamiltonian framework. Take a 4-manifold of the form M_{3} ×ℝ, where ℝ is a timelike direction. Note that this is a spin manifold, so we expect full Sl(2, ℤ) symmetry. For simplicity we suppose that there is no torsion in H_{1}(M_{3}). Each class x ∈ H^{2}(M_{3}) determines a complex line bundle ℒ _{x} on the 3-manifold M_{3} (satisfying c_{1}(ℒ _{x}) = x). The Hilbert space for our theory on the 3-manifold M_{3} can be written in the form

H_{τ}(M_{3}) = ⨁_{x∈H^{2}(M_{3},ℤ)}H_{x},  (2.1)

where H_{x} is the Hilbert space which comes from quantizing connections on ℒ _{x}. (On the left, we have explicitly indicated the dependence on the coupling constant τ.) To construct H_{x}, write an arbitrary connection in the form A = A_{0} + β, where A_{0} is a harmonic connection (a connection whose curvature is a harmonic two-form) and β is a 1-form which is co-closed. Let T_{x} be the space of harmonic connections on the line bundle L_{x}. Then the quantization yields

H_{x} = H_{β} ⊗ L^{2}(T_{x}).  (2.2)

Here H_{β} is a Hilbert space obtained by quantizing the space of β’s, and L^{2}(T_{x}) is just the space of L^{2} functions on T_{x}.

1E. Witten, On S-duality in abelian gauge theory, Selecta Math (N.S.) 1 (1995), 383–410.

Note that the factor H_{β} is independent of x, since the space of co-closed one-forms is defined with no reference to x. Duality maps H_{β} to itself while acting separately on H′ = ⊕_{x}L^{2}(T_{x}). The duality action on H_{β} follows from the operator mapping in (1.17).

The action of duality on H′ can be described as follows. Note that the T_{x}’s are all principal homogeneous spaces acted on by the torus H^{1}(M_{3}, ℝ/ℤ), which parametrizes flat line bundles on M_{3}; the action is defined by tensoring any given line bundle with connection by a flat line bundle determined by a point in H^{1}(M_{3}, ℝ/ℤ). Let y denote a character of the abelian group H^{1}(M_{3}, ℝ/ℤ). There is a decomposition L^{2}(T_{x}) = ⊕_{y}T_{x,y}, where T_{x,y} is the subspace of L^{2}(T_{x}) transforming in the character y. Each T_{x,y} is onedimensional. Hence

H′ = ⊕_{x,y}T_{x,y}  (2.3)

Note that, by Poincaré and Pontryagin duality, the character group of H^{1}(M_{3}, ℝ/ℤ) is H^{2}(M_{3}, ℤ). Thus, x and y take values in the same space. It is hence possible to exchange them, and this is what the τ → −1/τ transformation does (more precisely, it acts by (x, y) → (−y, x)). Thus duality exchanges a classical notion – the decomposition with respect to x – with a quantum notion – the decomposition with respect to y. The claim about how the duality acts will be justified below where we introduce the operators Q_{E} and Q_{M}.

Upon quantization—and suppressing θ for a moment—one writes the four-dimensional curvature as F′_{A} + e^{2}π_{A}dt, where F′_{A} is a two-form on M_{3} and π_{A}—a one-form on M_{3}—is the momentum conjugate to the connection A. The Hamiltonian becomes

H = \frac{1}{2e^{2}} ∫ F_{A_{0}}^{2} + \frac{e^{2}}{2} ∇_{A_{0}} + H(β).  (2.4)

Here H(β) is the part of the Hamiltonian that acts on H_{β}. The other terms act on H′. The first term is the magnetic energy of the harmonic connection A_{0}; it comes from the part of the Lagrangian quadratic in F′_{A} and is a multiple of ∫_{M_{3}} x∧ ∗ x. The second term, which comes from the part of the Lagrangian quadratic in π_{A}, is the electric energy, the Laplacian on T_{x}; it is a multiple of ∫_{M_{3}} y∧ ∗ y.

Including the θ term shifts the quantization. In fact, the canonical momentum F_{A}^{∨} = ∗ π_{A} as determined from the original Lagrangian (1.1) is

F_{A}^{∨} = 2πi \frac{δS}{δF_{A}} = \frac{2πi}{e^{2}} ∗ F_{A} − \frac{θ}{π} F_{A}  (2.5)

At non-zero θ, one has not H′ = ⊕_{x}L^{2}(T_{x}) but H′ = ⊕_{x}Γ_{L^{2}}(T_{x}, S_{θ}), where S_{θ} is a certain flat line bundle over T_{x} and Γ_{L^{2}} is the space of L^{2} sections. I leave it as an exercise to the reader to identify S_{θ}. Of course, S_{θ} is trivial at θ = 0 and only depends on θ modulo 2π.

Rewriting the formula for F_{A}^{∨} in terms of τ, we can determine how this operator transforms under τ → −1/τ. Indeed, under the operator mapping (1.17) one gets

F_{A}^{∨} = −\overline{τ}F_{+} + τF_{−} ↦ F_{C}_{+} + F_{C}_{−} = F_{C}.  (2.6)

We will also need the “dual” version of this computation:

F_{A} = F_{+} + F_{−} ↦ \overline{(−1/τ)}F_{C}_{+} − (−1/τ)F_{C}_{−} = −F_{C}^{∨}.  (2.7)

In the Hamiltonian formalism, for any 2-cycle Σ ⊂ M_{3} we can define associated “electric” and “magnetic” operators on the Hilbert space H_{τ}(M_{3}) for any 2-cycle Σ ⊂ M_{3}, by

\begin{aligned}Q_{E}(Σ) & = ∫_{Σ} \frac{F_{A}^{∨}}{2π} = ∫_{Σ} \frac{F_{C}}{2π} \\ Q_{M}(Σ) & = ∫_{Σ} \frac{F_{A}}{2π} = − ∫_{Σ} \frac{F_{C}^{∨}}{2π} . \end{aligned}  (2.8)

(These operators only depend on the class of Σ in H_{2}(M_{3}, ℤ).) Clearly, under τ → −1/τ, that is under A → C, one has Q_{E} → Q_{M}, Q_{M} → −Q_{E}. Since x and y are the eigenvalues of Q_{M} and Q_{E}, this means that τ → −1/τ acts by (x, y) → (−y, x). Moreover, from the explicit formula (2.5) for F_{A}^{∨}, one sees that when θ is increased by 2π (an operation which leaves the Hilbert space unchanged), the operator Q_{M} is unaltered, but the operator Q_{E} maps to Q_{E} + Q_{M}.

The statements made in the last paragraph can be combined to the following: Sl(2, ℤ) acts on H′ via the natural action of Sl(2, ℤ) on the pair (x, y).