1.2 The S and the T transformations

The Maxwell equation is given by

[μFνρ] = 0,μFμν = 0 (1.2.1)

or equivalently in the differential form notation by

dF = dF = 0. (1.2.2)

This set of equations is invariant under the exchange

F F. (1.2.3)

In terms of the electric field E and the magnetic field B, which we schematically denote by F = (E,B), the transformation does

F = (E,B)F = (B,E)2F = (E,B). (1.2.4)

This operation is often called the S transformation.

To preserve the quantization of the electric and magnetic charges (1.1.8), (1.1.12), the dual field strength FD and the dual coupling eD need to be defined so that

FD = 4π e2F, 4π e2 4π eD2 = 1. (1.2.5)

Under this transformation, the charge (n,m) is transformed as

particle 1 (n,m) S (m,n), particle 2 (n,m) S (m,n), Dirac pairingnm nm = mn (m)n. (1.2.6)

Note that the Dirac pairing is preserved under the operation.

Let us suppose that we have a neutral real scalar field ϕ and the action of the U(1) gauge field is given by

1 2e(ϕ)2FμνFμν + 𝜃(ϕ) 16π2FμνF̃μν. (1.2.7)

The Maxwell equation is now

[μFνρ] = 0, (1.2.8) μ 4π e(ϕ)2Fμν + 𝜃(ϕ) 2π F̃μν = 0. (1.2.9)

Decompose F = (E,B) as before. The equations above show that the magnetic field satisfying the Gauss law is still B, but the electric field satisfying the Gauss law is now the combination

Econserved = 4π e(ϕ)2E + 𝜃(ϕ) 2π B. (1.2.10)

Therefore , we have

S2B dn = 2πm,S2Econserved dn = 2πn (1.2.11)

where m and n are the integers introduced in Sec. 1.1. This shows an interesting fact: let us change ϕ adiabatically to change 𝜃(ϕ). As n is an integer, it cannot change. Therefore, E gets a contribution proportional to 𝜃(ϕ)B to keep Econserved fixed. This is called the Witten effect [23].

The S transformation, then, exchanges (E)conserved and B. The dual gauge field strength FD is

FD = 4π e(ϕ)2F + 𝜃 2πF, (1.2.12)

and we have

1 2e(ϕ)2FμνFμν + 𝜃(ϕ) 16π2FμνF̃μν = 1 2eD(ϕ)2FD,μνFD,μν + 𝜃D(ϕ) 16π2FD,μνF̃D,μν (1.2.13)

where eD(ϕ), 𝜃D(ϕ) are given by

τD(ϕ) = 1 τ(ϕ) (1.2.14)


τ(ϕ) = 4πi e(ϕ)2 + 𝜃(ϕ) 2π ,τD(ϕ) = 4πi eD(ϕ)2 + 𝜃D(ϕ) 2π . (1.2.15)

This combination τ(ϕ) is called the complexified coupling.

We also know that, quantum mechanically, 𝜃(ϕ) and 𝜃(ϕ) + 2π cannot be distinguished, since the change in the integrand of the Euclidean path integral is

exp id4x 1 8πFμνF̃μν (1.2.16)

which is always one2 . We call it the T transformation. This does change Econserved by adding B, however. Equivalently, it changes the set of charges (n,m) as follows:

particle 1 (n,m) T (n + m,m), particle 2 (n,m) T (n + m,m), Dirac pairingnm nm = (n + m)m (n + m)m. (1.2.17)

We see that the Dirac pairing of two particles remain unchanged. On the complexified coupling τ(ϕ), it operates as

τ(ϕ)oldTτ(ϕ)new = τ(ϕ)old + 1. (1.2.18)

The transformations S and T generates the action of SL(2, ) on the set of charge (n,m):

S = 01 1 0 , S nm = m n , Sτ = 1 τ (1.2.19) T = 11 0 1 , T nm = n + m m , Tτ = τ + 1. (1.2.20)

In general the action on τ is the fractional linear transformation

ab c d SL(2, ) : τ dτ + b cτ + a. (1.2.21)