Off-shell gauge complexes, BRST doublets and anomaly obstructions

Use the smooth off-shell coordinate model $\mathbb{M}={\mathbb{R}}^4$, with $x^0=c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}t$ in SI coordinates, metric $\eta =\mathrm{diag}(-1,1,1,1)$, increasing $x^0$ as the future direction, and standard orientation. The scalar time-ordering convention sets $c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}=1$, so $x^0$ is the length coordinate in that convention as well. Here off shell means that no field equation is imposed. Take smooth real components $a_{\mu }\in C^{\mathrm{\infty }}(\mathbb{M};\mathbb{R})$ and a smooth commuting matter field $\psi \in C^{\mathrm{\infty }}(\mathbb{M};{\mathbb{C}}^4)$. Use the normalized real potential $a=a_{\mu }d{x}^{\mu }$ and dimensionless coupling $g=q\sqrt{{\mu }_0{c}_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}\mathrm{/}\mathrm{\hslash }}$ of the charged-gauge-data convention; $a_{\mu }$ has units ${\mathrm{m}}^{-1}$, $g$ is dimensionless, and $\psi$ has units ${\mathrm{m}}^{-3\mathrm{/}2}$. The exact supplier convention is $a_{\mu }^{\mathrm{\prime }}=a_{\mu }+{\mathrm{\partial }}_{\mu }\alpha$, ${\psi }^{\mathrm{\prime }}=e^{ig\alpha }\psi$, for a smooth dimensionless real infinitesimal gauge parameter $\alpha$.

Let $\mathcal{P}$ be the free graded-commutative $\mathbb{C}$-algebra over $C^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$ generated by the finite jets of $a_{\mu }$, $\psi$, an odd ghost $c_{\mathrm{g}\mathrm{h}}$, an odd antighost $\bar{c}$, and an even auxiliary variable $B$. Every element of $\mathcal{P}$ is a polynomial in finitely many jet coordinates. The fields $a_{\mu }$ and $\psi$ have Grassmann parity $0$ and ghost number $0$; $c_{\mathrm{g}\mathrm{h}}$ has parity $1$ and ghost number $1$; $\bar{c}$ has parity $1$ and ghost number $-1$; and $B$ has parity $0$ and ghost number $0$. The ghost and nonminimal variables are formal algebraic generators; no state-space or positivity claim is made. The generators $c_{\mathrm{g}\mathrm{h}},\bar{c},B$ carry no physical unit assignment, and the differential below is dimensionless. The subscript distinguishes the ghost from the speed of light $c_{\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}$.

Let $s$ vanish on the coefficient algebra, so $sf=0$ for every $f\in C^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$. Thus $s$ is $C^{\mathrm{\infty }}(\mathbb{M};\mathbb{C})$-linear.

Define the odd derivation $s:\mathcal{P}\to \mathcal{P}$ of ghost number $1$ by

Extend $s$ by the graded Leibniz rule $s(XY)=(sX)Y+(-1{)}^{\mathrm{\mid }X\mathrm{\mid }}X(sY)$ for homogeneous $X$, where $\mathrm{\mid }X\mathrm{\mid }$ is Grassmann parity, and by $s({\mathrm{\partial }}^{\beta }X)={\mathrm{\partial }}^{\beta }(sX)$ for every multi-index $\beta$. This is an off-shell local algebraic differential for the Abelian gauge transformations; it is not a BRST charge on a Hilbert space and it does not assert existence of a gauge-invariant quantum theory.

Charged gauge data and SI-natural conventions

The Abelian jet derivation $s$ defined in the local BRST algebra satisfies $s^2=0$. More generally, take any finite set of jet doublets $u^j,v^j$ with $\mathrm{\mid }{v}^j\mathrm{\mid }=\mathrm{\mid }{u}^j\mathrm{\mid }+1(\mathrm{m}\mathrm{o}\mathrm{d}2)$, $\mathrm{gh}(v^j)=\mathrm{gh}(u^j)+1$, $s{u}^j=v^j$, and $s{v}^j=0$, and suppose the transformations of all other generators are independent of these doublets. Define the doublet-number operator $N={\sum }_j(u^j\frac{{\mathrm{\partial }}_L}{\mathrm{\partial }{u}^j}+v^j\frac{{\mathrm{\partial }}_L}{\mathrm{\partial }{v}^j})$ and the odd homotopy $h={\sum }_j{u}^j\frac{{\mathrm{\partial }}_L}{\mathrm{\partial }{v}^j}$, using left graded derivatives. Then $[N,s]=0$ and $sh+hs=N$. Consequently the inclusion of the doublet-independent polynomial subalgebra induces an isomorphism on polynomial BRST cohomology. In particular, if $s{a}_k=0$ and $N{a}_k=k{a}_k$ for a positive integer $k$, then $a_k=s(h{a}_k\mathrm{/}k)$.

For a ghost-number grading, write this cohomology as $H^r(s,\mathcal{P})=\ker (s:{\mathcal{P}}^r\to {\mathcal{P}}^{r+1})\mathrm{/}\mathrm{im}(s:{\mathcal{P}}^{r-1}\to {\mathcal{P}}^r)\mathrm{.}$

Abelian local jet BRST differential

For one fixed jet index, the arrows record the differential on the pair: $s(u^j)=v^j$ and $s(v^j)=0$. The zero at the right is the zero element. The proof verifies nilpotence and the homotopy identity algebraically; the diagram is only a record of the generator maps.

Use the four-dimensional coordinate and jet algebra and nilpotent differential $s$ fixed by the earlier Abelian local BRST definition and nilpotence lemma. For integers $r$ and $0\le p\le 4$, a compactly supported local horizontal $p$-form of ghost number $r$ is a finite sum of terms $f_{i_1\dots i_p}(x,z)d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_p},$ where each coefficient is smooth and compactly supported in $x\in {\mathbb{R}}^4$, polynomial in finitely many field-jet coordinates $z$, and homogeneous of ghost number $r$. Denote this space by ${\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}$. A left derivative is used for odd jet coordinates.

For each spacetime index $\mu$, define the total derivative on a local coefficient by $D_{\mu }=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^{\mu }}+{\sum }_{i,\alpha }{z}_{\alpha +e_{\mu }}^i\frac{{\mathrm{\partial }}_L}{\mathrm{\partial }{z}_{\alpha }^i},$ where $\alpha$ ranges over multi-indices, $i$ over the field and ghost generators, and $e_{\mu }$ is the multi-index with one derivative in direction $\mu$. The horizontal exterior derivative is $d_H=d{x}^{\mu }\wedge D_{\mu }$. It satisfies $d_H^2=0$. Extend $s$ to a homogeneous $p$-form by $s\omega =(-1{)}^p{s}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}\omega$, where $s_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}$ acts on its coefficient by the earlier local BRST definition; then $s^2=0$ and $s{d}_H+d_{H}s=0$.

Set ${\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}=0$ for $p<0$ or $p>4$. Evaluation of a compactly supported local four-form on any smooth off-shell configuration is a finite integral.

For a graded subcomplex ${\mathcal{K}}^{r,p}\subseteq {\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}$ stable under $s$ and $d_H$, define $Z^{r,p}(s\mathrm{\mid }{d}_H;\mathcal{K})=\{a_p\in {\mathcal{K}}^{r,p}:s{a}_p+d_H{a}_{p-1}=0\text{ for some }{a}_{p-1}\in {\mathcal{K}}^{r+1,p-1}\}$ and $B^{r,p}(s\mathrm{\mid }{d}_H;\mathcal{K})=\{s{b}_p+d_H{c}_{p-1}:b_p\in {\mathcal{K}}^{r-1,p},c_{p-1}\in {\mathcal{K}}^{r,p-1}\}\mathrm{.}$ The relative local BRST cohomology is $H^{r,p}(s\mathrm{\mid }{d}_H;\mathcal{K})=Z^{r,p}\mathrm{/}{B}^{r,p}$. In four dimensions, $H^1(s\mathrm{\mid }{d}_H)$ is shorthand for $H^{1,4}(s\mathrm{\mid }{d}_H)$. For compactly supported coefficients, ${\int }_{{\mathbb{R}}^4}{d}_H{c}_3=0$ by the one-variable fundamental theorem of calculus in each coordinate and Fubini's theorem, so the quotient by $d_H$ identifies densities that define the same integrated local functional.

Abelian local jet BRST differential

Nilpotence and contraction of nonminimal doublets

Assume AC and Countable Choice, fix admissible flat scalar free data and a local scalar polynomial class under a scalar formal time-ordering theorem, and work conditionally in a time-ordering scheme satisfying that theorem's statement. At order $n$, let ${\mathcal{K}}_n$ be an $s,d_H$-stable subcomplex of the compactly supported local-form spaces defined earlier, with ${\mathcal{K}}_n^{0,4}$ exactly the allowed local counterterm densities. For each ghost number $r$, put ${\mathcal{F}}_n^r:={\mathcal{K}}_n^{r,4}\mathrm{/}{d}_H{\mathcal{K}}_n^{r,3},$ the local-functional quotient formed using this same subcomplex, and let ${\mathcal{F}}_n^{\bullet }\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$ denote formal power series. Let ${\mathcal{W}}_{\mathrm{\hslash }}=s+{\sum }_{j\ge 1}{\mathrm{\hslash }}^j{\mathcal{W}}_j$ be an odd formal operator of ghost number one on this module, with ${\mathcal{W}}_{\mathrm{\hslash }}^2=0$ and leading coefficient the induced $s$. Let $X\in {\mathcal{F}}_n^0\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$ and suppose its first possible breaking is ${\mathcal{W}}_{\mathrm{\hslash }}X={\mathrm{\hslash }}^n\mathcal{A}+O({\mathrm{\hslash }}^{n+1}),n\ge 1,\mathcal{A}=[a_4{]}_{{\mathcal{F}}_n},$ where $\mathrm{\hslash }$ is a formal indeterminate, $O({\mathrm{\hslash }}^{n+1})$ means terms of formal degree at least $n+1$, every lower coefficient vanishes, and $a_4\in {\mathcal{K}}_n^{1,4}$ is a compactly supported local four-form of ghost number one. Then $s{a}_4$ is $d_H$-exact within ${\mathcal{K}}_n$. There exists an admissible counterterm $b_4\in {\mathcal{K}}_n^{0,4}$ such that the shift $X\mapsto X-{\mathrm{\hslash }}^n[b_4{]}_{{\mathcal{F}}_n}$ cancels the order-$n$ breaking in ${\mathcal{F}}_n^1$ if and only if $[a_4{]}_{s\mathrm{\mid }{d}_H}=0\text{in}{H}^{1,4}(s\mathrm{\mid }{d}_H;{\mathcal{K}}_n)\mathrm{.}$ This is a conditional obstruction statement. Locality of $a_4$, nilpotence of the formal Ward differential, and admissibility of the counterterm space are assumptions; the result does not establish a quantum action principle.

Relative local BRST cohomology

Nilpotence and contraction of nonminimal doublets

Scalar Epstein–Glaser induction at finite perturbative order

The Axiom of Choice

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)