The S and the T transformations

1.2 The S and the T transformations

The Maxwell equation is given by

\partial_{[μ} F_{ν ρ]} = 0, \partial_{μ} F_{μ ν} = 0

(1.2.1)

or equivalently in the diﬀerential form notation by

d F = d ⋆ F = 0 .

(1.2.2)

This set of equations is invariant under the exchange

F \leftrightarrow ⋆ F .

(1.2.3)

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

F = (\vec{E}, \vec{B}) \to ⋆ F = (\vec{B}, - \vec{E}) \to ⋆^{2} F = (- \vec{E}, - \vec{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 $F_{D}$ and the dual coupling $e_{D}$ need to be deﬁned so that

F_{D} = \frac{4 π}{e^{2}} ⋆ F, \frac{4 π}{e^{2}} \frac{4 π}{e_{D}^{2}} = 1 .

(1.2.5)

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

\begin{matrix} particle 1 & (n, m) & \overset{S}{\to} & (- m, n), \\ particle 2 & (n^{'}, m^{'}) & \overset{S}{\to} & (- m^{'}, n^{'}), \\ Dirac pairing & n m^{'} - n^{'} m & = & - m n^{'} - (- m^{'}) n . \end{matrix}

(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

\frac{1}{2 e {(ϕ)}^{2}} F_{μ ν} F_{μ ν} + \frac{𝜃 (ϕ)}{16 π^{2}} F_{μ ν} {\tilde{F}}_{μ ν} .

(1.2.7)

The Maxwell equation is now

\begin{align} \partial_{[μ} F_{ν ρ]} & = 0, & (1.2.8) \\ \partial_{μ} [\frac{4 π}{e {(ϕ)}^{2}} F_{μ ν} + \frac{𝜃 (ϕ)}{2 π} {\tilde{F}}_{μ ν}] & = 0 . & (1.2.9) \end{align}

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

{\vec{E}}_{conserved} = \frac{4 π}{e {(ϕ)}^{2}} \vec{E} + \frac{𝜃 (ϕ)}{2 π} \vec{B} .

(1.2.10)

Therefore , we have

\int_{S^{2}} \vec{B} \cdot d \vec{n} = 2 π m, \int_{S^{2}} {\vec{E}}_{conserved} \cdot d \vec{n} = 2 π n

(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, $\vec{E}$ gets a contribution proportional to $𝜃 (ϕ) \vec{B}$ to keep ${\vec{E}}_{conserved}$ ﬁxed. This is called the Witten eﬀect [23].

The S transformation, then, exchanges ${(\vec{E})}_{conserved}$ and $\vec{B}$ . The dual gauge ﬁeld strength $F_{D}$ is

F_{D} = \frac{4 π}{e {(ϕ)}^{2}} ⋆ F + \frac{𝜃}{2 π} F,

(1.2.12)

and we have

\frac{1}{2 e {(ϕ)}^{2}} F_{μ ν} F_{μ ν} + \frac{𝜃 (ϕ)}{16 π^{2}} F_{μ ν} {\tilde{F}}_{μ ν} = \frac{1}{2 e_{D} {(ϕ)}^{2}} F_{D, μ ν} F_{D, μ ν} + \frac{𝜃_{D} (ϕ)}{16 π^{2}} F_{D, μ ν} {\tilde{F}}_{D, μ ν}

(1.2.13)

where $e_{D} (ϕ)$ , $𝜃_{D} (ϕ)$ are given by

τ_{D} (ϕ) = - \frac{1}{τ (ϕ)}

(1.2.14)

where

τ (ϕ) = \frac{4 π i}{e {(ϕ)}^{2}} + \frac{𝜃 (ϕ)}{2 π}, τ_{D} (ϕ) = \frac{4 π i}{e_{D} {(ϕ)}^{2}} + \frac{𝜃_{D} (ϕ)}{2 π} .

(1.2.15)

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

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

exp [i \int d^{4} x \frac{1}{8 π} F_{μ ν} {\tilde{F}}_{μ ν}]

(1.2.16)

which is always one² . We call it the $T$ transformation. This does change ${\vec{E}}_{conserved}$ by adding $\vec{B}$ , however. Equivalently, it changes the set of charges $(n, m)$ as follows:

\begin{matrix} particle 1 & (n, m) & \overset{T}{\to} & (n + m, m), \\ particle 2 & (n^{'}, m^{'}) & \overset{T}{\to} & (n^{'} + m^{'}, m^{'}), \\ Dirac pairing & n m^{'} - n^{'} m & = & (n + m) m^{'} - (n^{'} + m^{'}) m . \end{matrix}

(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)}_{old} \overset{T}{\to} τ {(ϕ)}_{new} = τ {(ϕ)}_{old} + 1 .

(1.2.18)

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

\begin{align} S & = (\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}), & S (\begin{matrix} n \\ m \end{matrix}) & = (\begin{matrix} - m \\ n \end{matrix}), & S τ & = - \frac{1}{τ} & (1.2.19) \\ T & = (\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}), & T (\begin{matrix} n \\ m \end{matrix}) & = (\begin{matrix} n + m \\ m \end{matrix}), & T τ & = τ + 1 . & (1.2.20) \end{align}

In general the action on $τ$ is the fractional linear transformation

(\begin{matrix} a & b \\ c & d \end{matrix}) \in S L (2, ℤ) : τ \to \frac{d τ + b}{c τ + a} .

(1.2.21)

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