Current-algebra cocycles and conditional chiral anomaly cancellation

Let $\mathfrak{g}$ be a finite-dimensional real Lie algebra. Let $\kappa :\mathfrak{g}\times \mathfrak{g}\to \mathbb{R}$ be a symmetric bilinear form that is invariant, meaning $\kappa ([U,V],W)=\kappa (U,[V,W])(U,V,W\in \mathfrak{g})\mathrm{.}$ No nondegeneracy, positivity, simplicity, or compactness assumption is imposed. Put $S^1=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ and $\mathcal{L}\mathfrak{g}=C^{\mathrm{\infty }}(S^1,\mathfrak{g})$, with pointwise bracket $[X,Y](\theta )=[X(\theta ),Y(\theta ){]}_{\mathfrak{g}}$. For $Y\in \mathcal{L}\mathfrak{g}$, write $Y^{\mathrm{\prime }}={\mathrm{\partial }}_{\theta }Y$ in the indicated periodic coordinate.

A Lie-algebra $2$-cocycle for the trivial scalar action is an alternating bilinear map $\omega :\mathcal{L}\mathfrak{g}\times \mathcal{L}\mathfrak{g}\to \mathbb{R}$ satisfying $\omega ([X,Y],Z)+\omega ([Y,Z],X)+\omega ([Z,X],Y)=0.$ Define the candidate current cocycle and its candidate central-term bracket by $\omega (X,Y)=\frac{1}{2\pi }{\int }_0^{2\pi }\kappa (X(\theta ),Y^{\mathrm{\prime }}(\theta ))d\theta ,\widehat{\mathcal{L}\mathfrak{g}}=\mathcal{L}\mathfrak{g}\oplus \mathbb{R},$ $[(X,a),(Y,b)]=([X,Y],\omega (X,Y))\mathrm{.}$ The real coordinates $a,b$ are central coordinates and do not enter the right-hand side. A one-dimensional central extension means a short exact sequence of Lie algebras $0\to \mathbb{R}\to \widehat{\mathcal{L}\mathfrak{g}}\to \mathcal{L}\mathfrak{g}\to 0$ whose kernel is central. The proposition on this page verifies that the displayed operation has this Lie-algebra structure. The normalization is fixed as written; no integer level, integration to a loop group, or representation-theoretic existence or unitarity follows from this definition.

For the finite-dimensional real Lie algebra $\mathfrak{g}$, invariant symmetric form $\kappa$, smooth loop algebra $\mathcal{L}\mathfrak{g}$, and bilinear form $\omega$ defined in Loop current Lie algebra and central term, the form $\omega$ is alternating and satisfies $\omega ([X,Y],Z)+\omega ([Y,Z],X)+\omega ([Z,X],Y)=0(X,Y,Z\in \mathcal{L}\mathfrak{g})\mathrm{.}$ Consequently the candidate bracket on $\widehat{\mathcal{L}\mathfrak{g}}$ is a Lie bracket and gives a one-dimensional central extension of $\mathcal{L}\mathfrak{g}$ by $\mathbb{R}$.

Loop current Lie algebra and central term

The arrows are the linear inclusion $i(a)=(0,a)$ and projection $p(X,a)=X$. The proof verifies that these maps are Lie-algebra homomorphisms, that $\ker p=\mathrm{im}i$, and that $\mathrm{im}i$ is central. The diagram records the resulting exact sequence; the proof does not rely on the diagram.

Fix a finite-dimensional real Lie algebra $\mathfrak{g}$, a finite index set $A$ with real basis $\{t^a{\}}_{a\in A}$, and a finite species index set $I$. Let ${\mathfrak{g}}_{\mathbb{C}}:=\mathfrak{g}{\otimes }_{\mathbb{R}}\mathbb{C}$ be its complexification, with complex scalar multiplication on the second tensor factor and the complex-bilinear bracket specified on pure tensors by $[X\otimes z,Y\otimes w]=[X,Y]\otimes zw$ for $X,Y\in \mathfrak{g}$ and $z,w\in \mathbb{C}$. Identify $X\in \mathfrak{g}$ with its canonical real embedding $X\otimes 1$; in particular $t^a$ in a representation argument means $t^a\otimes 1$. For each $i\in I$, let $V_i$ be a finite-dimensional complex vector space and let ${\rho }_i:{\mathfrak{g}}_{\mathbb{C}}\to {\mathrm{End}}_{\mathbb{C}}(V_i)$ be a complex-linear Lie-algebra representation, so ${\rho }_i([U,V])={\rho }_i(U){\rho }_i(V)-{\rho }_i(V){\rho }_i(U)$. Attach the bookkeeping label ${\chi }_i=L$ to each species. In this interface, a “left-handed Weyl species” means exactly the data $(V_i,{\rho }_i,{\chi }_i=L)$; the label is not a spinor field or a chirality operator. No spacetime manifold, spinor bundle or field, kinetic operator or action, integration of ${\rho }_i$ to a gauge group, regulator, effective action, or gauge-variation map is constructed or assumed.

Set $T_{R_i}^a:={\rho }_i(t^a)$ and define $\{A,B\}:=AB+BA$ for endomorphisms of $V_i$. The algebraic cubic representation coefficient is $d^{abc}={\sum }_{i\in I}{\mathrm{tr}}_{V_i}\text{\hspace{0.17em}⁣}(T_{R_i}^a\{T_{R_i}^b,T_{R_i}^c\})\in \mathbb{C}\mathrm{.}$ The basis and generator normalization are fixed as part of the data. No factor $1\mathrm{/}2$ is included in the anticommutator. If $T^{(b}{T}^{c)}=(T^b{T}^c+T^c{T}^b)\mathrm{/}2$, then $d^{abc}$ is twice ${\sum }_i{\mathrm{tr}}_{V_i}(T_{R_i}^a{T}_{R_i}^{(b}{T}_{R_i}^{c)})$. Cancellation of this coefficient means $d^{abc}=0$ for every basis triple in this convention. These are representation traces only; calling them anomaly coefficients in a physical model requires an additional theorem.

Define the separate linear representation coefficient by ${\mathrm{\ell }}^a={\sum }_{i\in I}{\mathrm{tr}}_{V_i}(T_{R_i}^a)\in \mathbb{C}\mathrm{.}$ For one-dimensional representation spaces of a one-dimensional Abelian Lie algebra with fixed generator $t$ and ${\rho }_i(t)=q_i{\mathrm{id}}_{\mathbb{C}}$, this gives $\mathrm{\ell }={\sum }_i{q}_i$. The coefficient ${\mathrm{\ell }}^a$ is recorded separately because a later mixed gauge-gravitational analysis needs its own representation data. Such an analysis must specify its spacetime and gauge connections, curvature conventions, and index polynomial; no mixed-anomaly formula is asserted by this interface.

The only local BRST obstruction statement recorded here is a separate formal result for the specific Abelian jet complex used by the mathematical supplier, not a BRST construction for the general Lie algebra $\mathfrak{g}$ above. In the convention of def-qftx-brst-relative-local-cohomology, a compactly supported local horizontal $p$-form is a finite sum of $f_{i_1\dots i_p}(x,z)d{x}^{i_1}\wedge \cdots \wedge d{x}^{i_p}$, with each coefficient smooth and compactly supported in $x\in {\mathbb{R}}^4$ and polynomial in finitely many field-jet coordinates $z_{\alpha }^i$; $i$ indexes the field and ghost generators and $\alpha$ their derivative multi-indices. Its total derivative is $D_{\mu }=\mathrm{\partial }\mathrm{/}\mathrm{\partial }{x}^{\mu }+{\sum }_{i,\alpha }{z}_{\alpha +e_{\mu }}^i{\mathrm{\partial }}_L\mathrm{/}\mathrm{\partial }{z}_{\alpha }^i$, where $e_{\mu }$ is the unit multi-index and ${\mathrm{\partial }}_L$ is the left graded derivative; $d_H=d{x}^{\mu }\wedge D_{\mu }$. The differential $s$ is the odd ghost-number-one Abelian derivation fixed by def-qftx-abelian-local-brst, with $s{a}_{\mu }={\mathrm{\partial }}_{\mu }{c}_{\mathrm{g}\mathrm{h}}$, $s{c}_{\mathrm{g}\mathrm{h}}=0$, $s\bar{c}=B$, $sB=0$, and $s\psi =ig{c}_{\mathrm{g}\mathrm{h}}\psi$, extended by the graded Leibniz rule and commutation with spacetime derivatives; $a_{\mu },\psi ,B$ are even, $c_{\mathrm{g}\mathrm{h}},\bar{c}$ are odd, and their ghost numbers are $0,0,0,1,-1$, respectively. On a horizontal $p$-form $\omega$, use $s\omega =(-1{)}^p{s}_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}\omega$, where $s_{\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}}$ is this coefficient derivation. Thus $s^2=d_H^2=0$ and $s{d}_H+d_{H}s=0$, with nilpotence of $s$ established by lem-qftx-brst-nilpotence-doublets. Write ${\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}$ for the space of these forms homogeneous of ghost number $r\in \mathbb{Z}$ and form degree $0\le p\le 4$, and set it to zero outside that form-degree range. For each order $n\ge 1$, let ${\mathcal{K}}_n={⨁}_{r,p}{\mathcal{K}}_n^{r,p}$ be a bigraded subspace with ${\mathcal{K}}_n^{r,p}\subseteq {\mathrm{\Omega }}_{\mathrm{l}\mathrm{o}\mathrm{c},c}^{r,p}$ that is stable in the sense that $s({\mathcal{K}}_n^{r,p})\subseteq {\mathcal{K}}_n^{r+1,p}$ and $d_H({\mathcal{K}}_n^{r,p})\subseteq {\mathcal{K}}_n^{r,p+1}$. The relative class is the quotient $H^{1,4}(s\mathrm{\mid }{d}_H;{\mathcal{K}}_n)=\frac{\{a_4\in {\mathcal{K}}_n^{1,4}:s{a}_4+d_H{a}_3=0\text{ for some }{a}_3\in {\mathcal{K}}_n^{2,3}\}}{\{s{b}_4+d_H{c}_3:b_4\in {\mathcal{K}}_n^{0,4},c_3\in {\mathcal{K}}_n^{1,3}\}}\mathrm{.}$ The direct supplier prop-qftx-anomaly-obstruction is a draft proposition with a supplied conditional proof. Its scalar construction premise is the draft theorem thm-qftx-scalar-eg-induction: under the Axiom of Choice (every family of nonempty sets has a choice function) and Countable Choice (every sequence of nonempty sets has a choice function), and for admissible flat scalar free data and a local scalar polynomial class as defined by def-qftx-local-time-ordered-system, that theorem asserts existence of symmetric multilinear time-ordered maps satisfying its contract at every finite order, with finite graph scaling bounds and local extension ambiguities. It asserts no convergence, adiabatic limit, positivity, or Ward identity. The exact statement of prop-qftx-anomaly-obstruction assumes those data and a time-ordering scheme satisfying that theorem. For each ghost number $r$, define the local-functional quotient ${\mathcal{F}}_n^r:={\mathcal{K}}_n^{r,4}\mathrm{/}{d}_H{\mathcal{K}}_n^{r,3}$ and put ${\mathcal{F}}_n^{\bullet }={⨁}_r{\mathcal{F}}_n^r$. The formal series module is ${\mathcal{F}}_n^{\bullet }\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$, where $\mathrm{\hslash }$ is a formal indeterminate. For some $n\ge 1$, it further assumes an odd formal ghost-number-one operator ${\mathcal{W}}_{\mathrm{\hslash }}=s+{\sum }_{j\ge 1}{\mathrm{\hslash }}^j{\mathcal{W}}_j$ on this module, ${\mathcal{W}}_{\mathrm{\hslash }}^2=0$, and an element $X\in {\mathcal{F}}_n^0\mathrm{⟦}\mathrm{\hslash }\mathrm{⟧}$ with ${\mathcal{W}}_{\mathrm{\hslash }}X={\mathrm{\hslash }}^n\mathcal{A}+O({\mathrm{\hslash }}^{n+1})$, where $O({\mathrm{\hslash }}^{n+1})$ denotes terms of formal degree at least $n+1$, all lower coefficients vanish, and $\mathcal{A}=[a_4{]}_{{\mathcal{F}}_n}$ for compactly supported $a_4\in {\mathcal{K}}_n^{1,4}$. It requires ${\mathcal{K}}_n^{0,4}$ to be exactly the admissible local counterterm densities. Under these hypotheses, that proposition states that $s{a}_4$ is $d_H$-exact and that there exists an admissible $b_4\in {\mathcal{K}}_n^{0,4}$ such that shifting $X$ by $-{\mathrm{\hslash }}^n[b_4{]}_{{\mathcal{F}}_n}$ cancels the order-$n$ breaking if and only if $[a_4{]}_{s\mathrm{\mid }{d}_H}=0$. This is used only in the stated conditional form: it does not establish that the hypotheses hold for the representation data above or construct a physical effective action, and it does not prove a quantum action principle.

Accordingly, this interface asserts no physical anomaly-cancellation criterion from $d^{abc}$. A later physical theorem must specify its spacetime, field space, action, regulator and renormalized effective-action map, and the map that assigns a local gauge-variation density to that data. It must then prove the relation of that density to an explicitly specified local BRST cohomology class and invariant six-form polynomial, with mixed gauge-gravitational coefficients and loop-order hypotheses stated separately. No conclusion about transformations outside the identity component of a separately specified full gauge-transformation group follows from this infinitesimal representation tensor. Vanishing $d^{abc}$ alone proves neither a nonperturbative gauge theory nor the absence of failures under such transformations.

Wess–Zumino consistency is an obstruction class

Complexification of a real Lie algebra