Curtright field

In four spacetime dimensions, the field is not dual to the graviton, if massless, but it can be used to describe massive, pure spin 2 quanta.[6] Similar descriptions exist for other massive higher spins, in D≥4.[7]

The simplest example of the linearized theory is given by a rank three Lorentz tensor ${\displaystyle T_{\alpha \beta \mu }}$ whose indices carry the permutation symmetry of the Young diagram corresponding to the integer partition 3=2+1. That is to say, ${\displaystyle T_{\alpha \beta \mu }=T_{[\alpha \beta ]\mu }}$ and ${\displaystyle T_{[\alpha \beta \mu ]}=0}$ where indices in square brackets are totally antisymmetrized. The corresponding field strength for ${\displaystyle T_{\alpha \beta \mu }}$ is ${\displaystyle F_{\alpha \beta \gamma \mu }=3\partial _{[\gamma }T_{\alpha \beta ]\mu }.}$ This has a nontrivial trace ${\displaystyle F_{\alpha \beta }=\eta ^{\gamma \mu }F_{\alpha \beta \gamma \mu }}$ where ${\displaystyle \eta ^{\gamma \mu }}$ is the Minkowski metric with signature (+,−,−,...).

The action for ${\displaystyle T_{\alpha \beta \mu }}$ in D spacetime dimensions is bilinear in the field strength and its trace.

${\displaystyle S={\frac {-1}{6}}\int d^{D}x(F_{\alpha \beta \gamma \mu }F^{\alpha \beta \gamma \mu }-3F_{\alpha \beta }F^{\alpha \beta }).}$

This action is gauge invariant, assuming there is zero net contribution from any boundaries, while the field strength itself is not. The gauge transformation in question is given by

${\displaystyle \delta T_{\alpha \beta \mu }=\partial _{[\alpha }S_{\beta ]\mu }+\partial _{[\alpha }A_{\beta ]\mu }-\partial _{\mu }A_{\alpha \beta }}$

where S and A are arbitrary symmetric and antisymmetric tensors, respectively.

An infinite family of mixed symmetry gauge fields arises, formally, in the zero tension limit of string theory,[8] especially if D>4. Such mixed symmetry fields can also be used to provide alternate local descriptions for massive particles, either in the context of strings with nonzero tension, or else for individual particle quanta without reference to string theory.

• Kalb–Ramond field
• p-form electrodynamics
• dual graviton
• Massive gravity
• Montonen-Olive duality