Speaker
Description
The functional renormalization group (FRG) recasts the flow of the effective action as an initial-value problem in RG "time," which for O(N)-type models can be written as a nonlinear fluid-dynamic conservation law for the field-space derivative of the potential. Physical observables - the IR vertex functions $\Gamma^{(2n)} = \partial_\sigma^{2n-1}u|_{\sigma=0}$ - are highly localized functionals of a solution that develops steep restoration fronts and near-singular structure during the flow. This makes the accurate, certified extraction of vertices genuinely hard, and it raises a broader question that is usually answered only by expensive parameter scans: how do the extracted observables depend on the many choices that enter an FRG calculation — the regulator (cutoff) shape function, the UV initial data, the truncation, and the discretization itself ?
We argue that the adjoint method is the natural and unifying tool for these questions. The adjoint field $\lambda(\sigma,t) = \delta J/\delta u(\sigma,t)$ is the exact sensitivity of any chosen observable $J$ to a perturbation of the state at every RG time, obtained from a single backward integration of the transposed flow — at the cost of essentially one extra solve, independent of the number of parameters. Its interpretation is physical, not merely numerical: $\lambda$ is an influence map that reveals which regions of field space and which RG scales actually determine a given IR vertex, quantifies the RG "freeze-out" of each observable, and — crucially — differentiates $J$ with respect to any ingredient of the flow at once.
We demonstrate the framework on the O(N) model. First, as goal-oriented error control and adaptivity: the dual-weighted-residual estimator built from $\lambda$ yields a certified, exact-solution-free error bound with effectivity $\approx 1$, and - by exposing that a residual plateau under mesh refinement signalled a consistency rather than a resolution error- directly diagnosed and motivated the cure for a persistent bias in the higher vertices. Second, and more generally, we show how the same adjoint gives, at negligible additional cost, the gradient of IR observables with respect to the regulator shape function and the UV initial condition — turning regulator-(in)dependence studies and optimized-cutoff searches from brute-force scans into a single sensitivity computation, and opening the door to gradient-based regulator design and inverse/data-assimilation problems in FRG.
| Affiliation | ITP, Goethe University, Frankfurt |
|---|---|
| Career status | Postdoc |