Homotopical Renormalization and Effective-Theory Maps

Integrating out fields can be expressed as homotopy transfer when the discarded sector is contractible relative to the retained complex. A contraction transfers not only the differential but every higher bracket, producing tree sums with the contracting homotopy as the internal propagator. The resulting map is a quasi-isomorphism of formal classical deformation problems. It is not automatically a Wilsonian quantum renormalization-group flow, and it does not justify a truncation unless the projection, inclusion, and homotopy identities actually hold.

Required background. Obstruction–Deformation Complexes and Quantization Classes supplies the complexes whose deformations are being compared. Quantum Observables, Renormalization, and Factorization supplies the distinction between classical transfer and quantum effective observables. Helpful background. Wilsonian Coarse Graining and Theory Space gives the scale-dependent interpretation. Local Field Redefinitions and the Equivalence Theorem explains when a change of variables preserves scattering data. Renormalization and Coarse Graining as Encoding supplies the limits of information-theoretic analogies.

Contractions and transferred brackets

Let $(C,d)$ be a cochain complex and $(H,d_H)$ a retained complex. A strong deformation retract consists of chain maps

$$H\mathrel{\mathop{\longrightarrow}^{i}}C \mathrel{\mathop{\longrightarrow}^{p}}H,$$

and a degree $-1$ homotopy $h:C\to C[-1]$ such that

$$pi=\operatorname{id}_H, \qquad \operatorname{id}_C-ip=dh+hd.$$

One may impose the side conditions $ph=0$, $hi=0$, and $h^2=0$. These identities imply that $p$ is a quasi-isomorphism. More importantly, they supply the algebra needed to transfer an $L_\infty$ structure $\{\ell_n\}$ on $C$ to one on $H$.

For a differential graded Lie algebra, the first terms are

$$\ell_1^H=d_H, \qquad \ell_2^H(x,y)=p[,ix,iy,],$$

and, up to the Koszul signs fixed by the grading convention,

$$\ell_3^H(x,y,z) =-p\bigl([h[ix,iy],iz]\bigr)+\text{graded cyclic terms}.$$

Higher brackets are sums over rooted trees: original vertices label the vertices, $h$ labels internal edges, $i$ labels inputs, and $p$ labels the output. The square-zero identities for the transferred coderivation follow from the contraction equation; they are not obtained by simply projecting each original bracket. Arvanitakis, Hohm, Hull, and Lekeu give an explicit series for the transferred coderivation and prove its nilpotence 2022, §2.3, pp. 15–20.

A useful finite check is a contractible pair $u,v$ with $du=Mv$, $M\neq0$, and no cohomology. Let $p$ kill $u,v$, let $i$ include the retained variables, and set $h(v)=u/M$, $h(u)=0$. On the heavy pair, $dh+hd=\operatorname{id}$; on the retained subcomplex it vanishes. Thus the contraction equation holds exactly. If $M=0$, the pair is no longer contractible and this transfer is unavailable.

First application: eliminating a heavy field at tree level

The algebraic statement sharpens Integrating Out Heavy Fields. Consider a finite BV model with retained bosonic coordinates $\phi^i$ and heavy coordinates $\chi^a$. Assume the quadratic form is block diagonal and the heavy Hessian $A_{ab}$ is invertible. Keep a cubic coupling and write, suppressing purely light interactions,

$$S(\phi,\chi) =\frac12A_{ab}\chi^a\chi^b +\frac{\lambda}{2}C_{aij}\chi^a\phi^i\phi^j +S_{\mathrm{light}}(\phi).$$

The heavy equation is algebraic:

$$A_{ab}\chi^b+\frac{\lambda}{2}C_{aij}\phi^i\phi^j=0,$$

so

$$\chi_*^a(\phi) =-\frac{\lambda}{2}(A^{-1})^{ab}C_{bij}\phi^i\phi^j.$$

Substitution gives the tree-level effective action

$$S_{\mathrm{eff}}(\phi) =S_{\mathrm{light}}(\phi) -\frac{\lambda^2}{8} C_{aij}(A^{-1})^{ab}C_{bkl} \phi^i\phi^j\phi^k\phi^l.$$

The same term arises from homotopy transfer. The heavy field–antifield sector is the contractible pair, and $h$ is minus the heavy propagator $A^{-1}$ in the convention of the action. Two cubic brackets joined by $h$ produce the transferred ternary bracket, which is dual to the quartic effective vertex. The explicit comparison, including the coefficient and the three tree channels, appears in Arvanitakis–Hohm–Hull–Lekeu 2022, §5.2, pp. 45–47.

The associated $L_\infty$ inclusion is nonlinear:

$$i_\infty(\phi) =\bigl(\phi,\chi_*(\phi)\bigr)+O(\lambda^2).$$

Pulling an observable $F(\phi,\chi)$ back along $i_\infty$ gives $F(\phi,\chi_*(\phi))$ at tree level. Thus the effective observable map agrees with the familiar classical field redefinition, while its higher Taylor coefficients remember the homotopies required for compatibility with brackets.

An independent check is the quartic coefficient. Completing the square in $\chi$ gives

$$\frac12(\chi-\chi_*)^TA(\chi-\chi_*) -\frac{\lambda^2}{8}(C\phi\phi)^TA^{-1}(C\phi\phi),$$

which reproduces both the sign and the factor $1/8$ above. The tree-transfer diagram gives the same inverse Hessian on its internal edge.

Failure test: an invalid projection

Now choose a linear projection $p:C\to H$ that is not a chain map, so $pd\neq d_Hp$. Projecting the action may still reproduce a few low-point amplitudes at a chosen kinematic point, but $p$ cannot extend to an $L_\infty$ morphism with the stated unary term: the first morphism identity already fails. Alternatively, let the proposed heavy sector contain a zero mode. Then no $h$ can satisfy $\operatorname{id}-ip=dh+hd$ on that class, because the right-hand side acts trivially in cohomology while the left does not.

These are concrete adversarial failures. Agreement of selected amplitudes is not a converse to homotopy equivalence. Even a valid classical transfer is a formal tree-level result unless analytic estimates make the tree sums converge. Loops require determinants, measure terms, and quantum master-equation corrections; sharp momentum projections can also jeopardize locality. Filtered $L_\infty$ quasi-isomorphisms preserve Maurer–Cartan infinity-groupoids only under completeness and filtration hypotheses Dolgushev–Rogers 2015, Theorem 2.2, p. 8, so a scale-by-scale physical equivalence needs those domains stated separately.

In particular, matching a finite effective action does not establish equivalence of quantum states or correlation functions.

Exercises

Verify the contraction identity for $du=Mv$ and $h(v)=u/M$.

Solution

On $u$, $(dh+hd)u=h(Mv)=u$. On $v$, $(dh+hd)v=d(u/M)=v$. The projection $ip$ vanishes on the heavy pair, so $\operatorname{id}-ip=dh+hd$ there. On retained variables $ip=\operatorname{id}$ and $h=0$, so both sides vanish.

Derive the quartic term in $S_{\mathrm{eff}}$ by substitution.

Solution

Let $J_a=C_{aij}\phi^i\phi^j$. Substituting $\chi_*=-\lambda A^{-1}J/2$ gives $\frac12\chi_*^TA\chi_*=\lambda^2J^TA^{-1}J/8$ and $(\lambda/2)J^T\chi_*=-\lambda^2J^TA^{-1}J/4$. Their sum is $-\lambda^2J^TA^{-1}J/8$.

References

• Arvanitakis, Hohm, Hull, and Lekeu 2022, Homotopy Transfer and Effective Field Theory I: Tree-Level, Fortschritte der Physik 70, 2200003. Open PDF
• Dolgushev and Rogers 2015, A Version of the Goldman–Millson Theorem for Filtered L∞-Algebras, Journal of Algebra 430, 260–302. Open PDF