Lagrangian BRST formulation of massive higher-spin fields of the general symmetry type

Abstract

We describe the procedure of dimensional reduction of massless fields in $(d+1)$-dimensional Minkowski space to massive fields in $d$ dimensions in the first-quantized setting. The procedure is compatible with the Lagrangian and in a straightforward way determines the inner product for massive fields. The use of the Howe duality and the BRST technique allows keeping the description concise. We consider both bosonic and fermionic mixed-symmetry fields.

Appendix A. $$\mathfrak{osp}(1|2n)$$ commutation relations

The $\mathfrak{osp}(1|2n)$ basis elements are defined in (3.5) and (3.6). Their nonzero commutation relations in the even sector are

$$\begin{aligned} \, {} &[T_I^J,T_K^L]=\delta_K^JT_I^L-\delta_I^LT_K^J,\qquad [T^{IJ},T_{KL}]=\delta^I_KT_L^J+\delta^I_LT_K^J +\delta^J_KT_L^I+\delta^J_LT_K^I, \\ &[T_K^L,T_{IJ}]=\delta_J^LT_{KI}+\delta^L_IT_{KJ},\qquad [T^{IJ},T_K^L]=\delta^I_KT^{JL}+\delta^J_KT^{IL}, \end{aligned} $$

(A.1)

those in the odd sector are,

$$\{\Upsilon_I,\Upsilon_J\}=2T_{IJ},\qquad \{\Upsilon_I,\Upsilon^J\}=2T_I^J,\qquad \{\Upsilon^I,\Upsilon^J\}=2T^{IJ} $$

(A.2)

and in the cross sector,

$$\begin{alignedat}{3} &[T_{IJ},\Upsilon^K]=-\delta_I^K\Upsilon_J-\delta_J^K\Upsilon_K,&\qquad &[T^{IJ},\Upsilon^K]=0,&\qquad &[T_I^J,\Upsilon^K]=-\delta_I^K\Upsilon_J, \\ &[T^{IJ},\Upsilon_K]=\delta_I^K\Upsilon^J+\delta_K^J\Upsilon^I,&\qquad &[T_{IJ},\Upsilon_K]=0,&\qquad &[T_I^J,\Upsilon_K]=\delta_K^J\Upsilon_I \end{alignedat} $$

(A.3)

Appendix B. Casimir operators

The quadratic and quartic Casimir operators of the $\mathfrak{iso}(p,q)$ algebra are

$$C_2(\mathfrak{iso}(p,q))=P_aP^a\equiv P^2,\qquad C_4(\mathfrak{iso}(p,q))=M_{ab}P^bM^{ac}P_c-\frac{1}{2}M^2P^2, $$

(B.1)

where $P_a$ stands for translation and $M_{ab}$ for rotation generators. In what follows, we express (B.1) in terms of the $\mathfrak{osp}$ basis elements.

Regular spinor–tensor representation. Let $\mathfrak{iso}(d-1,1)$ basis elements $P_a$, $M_{ab}$, $a,b=0,\dots,d-1$ act on spinor–tensor fields as

$$P_a=\partial_a,\qquad M_{ab}=x_a\,\partial_b-x_b\,\partial_a +a_{ia}\bar a^i_b-a_{ib}\bar a^i_a +\biggl\{\frac{1}{4}(\theta_a\theta_b-\theta_b\theta_a)\biggr\}, $$

(B.2)

$i=1,\dots,n$. Expressing the quadratic and quartic Casimir operators in terms of the $\mathfrak{osp}(1|2n+2)$ basis elements, we find

$$\begin{aligned} \, &C_2(\mathfrak{iso}(d-1,1))=\Box, \end{aligned}$$

(B.3)

$$\begin{aligned} \, &C_4(\mathfrak{iso}(d-1,1))=((d-n-2)N_i^i+N_j^iN_i^j-T_{ij}T^{ij})\Box+{} \nonumber \\ &\hphantom{C_4(\mathfrak{iso}(d-1,1))={}}+T_{ij}D^iD^j+(2-d)D^\dagger_iD^i-2D^\dagger_jN_i^jD^i+D^\dagger_iD^\dagger_jT^{ij}+{} \nonumber \\ &\hphantom{C_4(\mathfrak{iso}(d-1,1))={}}+\biggl\{(\Upsilon_iD^i-D^\dagger_i\Upsilon^i)\widehat D +\biggl(N_i^i-\Upsilon_i\Upsilon^i+\frac{(d-1)(d-2)}{8}\biggr)\Box\biggr\}. \end{aligned}$$

(B.4)

Dropping terms in the curly brackets, we obtain the bosonic Casimir operator. We also note that the above expressions for the $\mathfrak{osp}(1|2n+2)$ algebra elements remain valid for all the $\mathfrak{iso}(k,l)$ algebras with $k+l=d$.

Appendix C. Details of the dimensional reduction

Symmetric fields. We start with the case of symmetric fields ($n=1$). Then

$$\mathsf{\Omega}=\alpha(\widehat D+m\Gamma)+c_0(\Box+m^2) +c\biggl(D+m\,\frac{\partial}{\partial z}\biggr) +(D^\dagger+mz)\,\frac{\partial}{\partial b} -\alpha\alpha\,\frac{\partial}{\partial c_0} -c\,\frac{\partial}{\partial b}\,\frac{\partial}{\partial c_0}. $$

(C.1)

Triplet BRST operator (C.1) acts on the subspace singled out by the BRST-extended constraints

$$\widehat N_a\Psi=s\Psi,\qquad \widehat\Upsilon\Psi=0, $$

(C.2)

where

$$\widehat N_a=N_a+z\,\frac{\partial}{\partial z} +b\,\frac{\partial}{\partial b}+c\,\frac{\partial}{\partial c},\qquad \widehat\Upsilon=\Upsilon+\Gamma\,\frac{\partial}{\partial z} -2\alpha\,\frac{\partial}{\partial c} +\frac{\partial}{\partial\alpha}\,\frac{\partial}{\partial b}. $$

(C.3)

We introduce an additional grading

$$\operatorname{deg}z=1,\qquad \operatorname{deg}a^a=\operatorname{deg}b=\operatorname{deg}c=2. $$

(C.4)

BRST operator (C.1) then decomposes as $\mathsf{\Omega}=\mathsf{\Omega}_{-1}+\mathsf{\Omega}_0+\mathsf{\Omega}_1$, where

$$\begin{aligned} \, &\mathsf{\Omega}_{-1}=mz\,\frac{\partial}{\partial b}, \\ &\mathsf{\Omega}_0=\alpha(\widehat D+m\Gamma) +c_0(\Box+m^2)+cD+D^\dagger\,\frac{\partial}{\partial b} -\alpha\alpha\,\frac{\partial}{\partial c_0} -c\,\frac{\partial}{\partial b}\,\frac{\partial}{\partial c_0}, \\ &\mathsf{\Omega}_1=mc\,\frac{\partial}{\partial z}. \end{aligned} $$

(C.5)

The whole space can be decomposed into the direct sum $\mathcal E\oplus\mathcal G\oplus\mathcal F$, where $\mathcal G=\operatorname{Im}\Omega_{-1}$ and ${\mathcal E\oplus\mathcal G=\operatorname{Ker}\Omega_{-1}}$:

$$\Psi=E+G+F=\varphi_0+\sum_{k=1}^\infty z^k\varphi_k +b\sum_{k=0}^\infty z^k\psi_k. $$

(C.6)

The operator

$$\overset{\mathcal G\mathcal F}{\Omega}\Psi\equiv(\Omega F)|_\mathcal G =\sum_{k=1}^\infty z^k\biggl(\biggl(D^\dagger+c\,\frac{\partial}{\partial c_0}\biggr) \psi_k+m\psi_{k-1}\biggr) $$

(C.7)

is algebraically invertible:

$$\overset{\mathcal E\mathcal E}{\Omega}\Psi \equiv(\Omega E)|_{\mathcal E} =\biggl(\alpha(\widehat D+m\Gamma)+c_0(\Box+m^2)+cD -\alpha\alpha\,\frac{\partial}{\partial c_0}\biggr)\varphi_0,$$

(C.8)

$$\overset{\mathcal G\mathcal E}{\Omega}\Psi\equiv(\Omega E)|_{\mathcal G}=0.$$

(C.9)

Hence, the reduced BRST operator $\widetilde\Omega$ acting on $H(\Omega_{-1})\cong\mathcal E$ is given by

$$\widetilde\Omega=\overset{\mathcal E\mathcal E}{\Omega} -\overset{\mathcal E\mathcal F}{\Omega}(\overset{\mathcal G\mathcal F}{\Omega})^{-1} \overset{\mathcal G\mathcal E}{\Omega} =\alpha(\widehat D+m\Gamma)+c_0(\Box+m^2) +cD-\alpha\alpha\,\frac{\partial}{\partial c_0}. $$

(C.10)

It acts on the intersection of $\mathcal E$ and the subspace selected by (C.2):

$$\biggl(N_a+c\,\frac{\partial}{\partial c}\biggr)\varphi=s\varphi,\qquad \biggl(\Upsilon-2\alpha\,\frac{\partial}{\partial c}\biggr)\varphi=0. $$

(C.11)

We note that because the gauge is fixed, this space does not contain elements with negative ghost degree. The zero-ghost-degree component satisfies the relations

$$(\widehat D+m\Gamma)\varphi^{(0)}=0,\qquad N_a\varphi^{(0)}=s\varphi^{(0)},\qquad \Upsilon\varphi^{(0)}=0 $$

(C.12)

(from which $(\Box+m^2)\varphi^{(0)}=D\varphi^{(0)}=0$ follow.)

Mixed-symmetry fields. We can consequently repeat the argument used for symmetric fields. Namely, we introduce the grading

$$\operatorname{deg}z_i=1,\qquad \operatorname{deg}a^a_i =\operatorname{deg}b_i=\operatorname{deg}c_i=2, $$

(C.13)

for $i=1,\dots,n$ and systematically perform homological reduction for each $i$, effectively throwing out pairs of $z_i$ and $b_i$. The result is the reduced BRST operator

$$\widetilde{\mathsf{\Omega}} =\alpha(\widehat D+m\Gamma)+c_0(\Box+m^2) +c_iD^i-\alpha\alpha\,\frac{\partial}{\partial c_0}, $$

(C.14)

which acts on the subspace

$$\widehat N_i\psi=s_i\psi, \qquad \widehat N_i^j\psi=0\quad (i<j),\qquad \widehat\Upsilon^i\psi=0, $$

(C.15)

where

$$\widehat N_i^j=N_i^j+c_i\,\frac{\partial}{\partial c_j},\quad \widehat N_i=\widehat N_i^i\text{ (no sum)},\quad \widehat\Upsilon^i=\Upsilon^i-2\alpha\,\frac{\partial}{\partial c_i}. $$

(C.16)

The physical content of the system (the zero-ghost-degree cohomology) is

$$\begin{aligned} \, &(\widehat D+m\Gamma)\psi^{(0)}=0,\qquad N_i\psi^{(0)}=s_i\psi^{(0)}, \\ &N_i^j\psi^{(0)}=0\quad (i<j),\qquad \Upsilon^i\psi^{(0)}=0 \end{aligned} $$

(C.17)

(whence the relations $(\Box+m^2)\psi^{(0)}=D^i\psi^{(0)}=0$ follow.)