# What conserved charge follows from the dual field strength tensor?

The homogenous Maxwell equations

$$\partial_\mu \tilde{F}^{\mu \nu} =0$$

follow "trivially" from the definition of the dual field strength tensor $\tilde{F}^{\mu \nu} = \epsilon^{\mu \nu \sigma \rho} F_{\sigma \rho}.$

They express a conserved current equation for each component $\nu$. For example, for $\nu=1$, we have

$$\partial_\mu \tilde{F}^{\mu 1} =0 \quad \leftrightarrow \quad \partial_\mu j^\mu =0 ,$$

where I have defined $j^\mu \equiv \tilde{F}^{\mu 1}$.

When we have a conserved current, we have a conserved charge thanks to Gauss' theorem:

$$Q \equiv \int_V dx^3 j^0 \quad \text{ with} \quad \partial_0 Q =0 .$$

What is this conserved charge?

It doesn't follow from Noether's theorem and is always conserved irrespectively of if the equations of motion are fulfilled or not, i.e. even "off-shell".

In addition, we get one such conserved charge for each component $\nu$ as mentioned above, so there are, in total, 4 conserved charges here.

In terms of the gauge potential $A_\mu$, the conserved charge reads:

$$Q= \int_V dx^3 j^0 = \int_V dx^3 \tilde{F}^{0 1} = \int_V dx^3 \epsilon^{01 \sigma \rho} F_{\sigma \rho} = \int_V dx^3 \epsilon^{01 \sigma \rho} ( \partial_\sigma A_\rho - \partial_\rho A_\sigma ).$$

The usual electrical charge associated with the electromagnetic field is

$$Q= \int_V dx^3 j^0 \quad \text{ where } \quad j^\mu = \partial_\nu F^{\nu\mu}.$$ $$\rightarrow Q= \int_V dx^3 \partial_\nu F^{\nu0} = \int_V dx^3 \partial_nu ( \partial_\nu A_0 - \partial_0 A_\nu ).$$

• It is a U(1) “1-form” global symmetry (not a set of 4 ordinary symmetries). The charged operators are ‘t Hooft loops, which you can think of as inserting the world line of a probe magnetic monopole. The charge on a 2 sphere linking the line measures the magnetic flux. See this brief answer I wrote the other day, or the paper generalized global symmetries by Seiberg et al from a couple of years ago Mar 14, 2018 at 14:40
• @ElliotSchneider, what is the meaning of such conservation? For example, conservation of electric current have very clear meaning: conservation of charge . What is meaning of such "strange" conservation, without using of e.o.m.? Mar 31, 2020 at 23:34
• @Nikita Take a look at this answer Apr 4, 2020 at 15:00