Derived, Deformation, and Homotopical QFT

Derived methods retain symmetries and obstructions that an ordinary solution set loses. This chapter connects those complexes to formal moduli, deformation quantization, boundary matching, and effective field theory.

Enter this chapter

Start with one classical action and its gauge complex; identify deformations, stabilizers, and obstructions before introducing shifted geometry. Helpful background. Chain Homotopy, Quasi-Isomorphisms, and Derived Vocabulary supplies the complex-level equivalences. Derived, Higher, and Factorization Frameworks: a Boundary Map distinguishes the frameworks used here. Master Equations and BV Gauge Fixing supplies the field-theoretic master equation.

Topics

1. Derived Critical Loci and Derived Gauge Quotients — Retains equations and stabilizers through homotopy constructions.
2. L∞ Algebras, Formal Moduli, and Field Equations — Encodes nonlinear formal deformation problems in higher brackets.
3. Shifted Symplectic, Poisson, and BV Geometry — States nondegeneracy at the level of tangent and cotangent complexes.
4. Deformation Quantization, Formality, and Star Products — Constructs associative products order by order in a formal parameter.
5. Obstruction–Deformation Complexes and Quantization Classes — Locates quantization obstructions and choices in cohomology.
6. Derived Intersections, Boundary Conditions, and Correspondences — Preserves excess intersection data when matching boundary conditions.
7. Higher Morita Categories and Theories as Objects — Organizes theories, interfaces, and their higher compositions.
8. Homotopical Renormalization and Effective-Theory Maps — Transfers interactions using a controlled contraction of complexes.
9. Strictification, Comparison, and Foundational Limits — Distinguishes formal comparison from global or analytic equivalence.

How the ideas fit together

A derived critical locus records more than the points solving an equation Pantev, Toën, Vaquié, and Vezzosi 2013, Corollary 2.11. In a complete filtered deformation problem, a suitable quasi-isomorphism preserves the Maurer–Cartan homotopy type Dolgushev and Rogers 2015, Theorem 2.2. Formal quantization Kontsevich 2003, Theorem 4.6.2 and tree-level effective transfer Arvanitakis et al. 2022, §§ 2.3 and 5.2 require further structure. Neither implies convergence or a positive Hilbert-space theory.

Hypotheses and licensed conclusions

Derived and homotopical QFT objects, hypotheses, licensed conclusions, and decisive failure tests.

Object and domain	Essential hypotheses	Licensed conclusion	Excluded converse or adversarial check
Derived critical locus or gauge quotient	Specified equation section and gauge complex; compatible grading; chosen smooth, algebraic, or elliptic category	A tangent complex whose cohomology records stabilizers, deformations, equations, and obstructions	An ordinary orbit set at a reducible flat connection deletes both stabilizer and obstruction groups
L∞ formal moduli problem	Nilpotent base or complete descending filtration; filtration-compatible brackets and morphisms; characteristic-zero comparison where invoked	A Maurer–Cartan infinity-groupoid invariant under the stated filtered quasi-isomorphism	A formal series evaluated at a finite unfiltered displacement need not converge or give a global moduli equivalence
Shifted BV or Poisson geometry	Perfect tangent and cotangent complexes; closed form; quasi-isomorphic nondegeneracy map; orientation and support pairing	A shifted bracket on derived functions and, for Lagrangian maps, shifted structure on a homotopy intersection	Unrestricted Maxwell boundary data produce nonzero symplectic flux, so the bulk pairing is not closed
Formal star product on observables	Poisson structure and formal parameter; for the direct free-field formula, regular functional derivatives and a defined propagator contraction	An associative formal deformation with first commutator $i\hbar$ times the Poisson bracket	Local nonlinear functionals put singular kernels on diagonals; associativity on regular functionals does not supply the missing extension
Perturbative BV quantization	Regulator and removal limit; local deformation complex; lower-order master equation; admissible symmetry and scaling conditions	A degree-one obstruction class; when it vanishes, lifts forming a degree-zero cohomological torsor	A Green-operator primitive can be nonlocal; vanishing at one order does not remove higher obstructions or framing dependence
Boundary gluing and Morita composition	Homotopy fiber products; Lagrangian regularity for shifted claims; existing relative tensor products; duals and adjoints for full extension	Zero-mode-preserving gluing and coherent fusion of algebras, bimodules, and intertwiners	An ordinary transverse intersection loses harmonic cokernels; a nondualizable infinite algebra does not define a fully extended theory
Effective transfer or categorical strictification	Chain contraction for transfer; convergence or formal filtration as stated; local equivalence and essential surjectivity for biequivalence; named extra structures	Transferred tree-level brackets or a strict two-category preserving the specified homotopy type	A non-chain projection fails at the unary identity; forgetting topology, domains, adjoints, or observables leaves only an algebraic surrogate

Structured table data (JSON).

Review the chapter

If an anomaly cocycle is exact among all functionals but not among local functionals, does it admit an allowed local counterterm?

Solution

No. The obstruction must vanish in the cohomology of the permitted local complex. A nonlocal primitive changes the problem and does not prove existence of a local quantization satisfying the original conditions.

Where to go next

Continue to gauge orbit geometry for the global quotient problem, then BRST–BV quantization for its cohomological treatment.

References

• Arvanitakis, Alex S., Olaf Hohm, Chris Hull, and Victor Lekeu. “Homotopy Transfer and Effective Field Theory I: Tree-Level.” Fortschritte der Physik 70 (2022): 2200003. DOI. Open PDF.
• Dolgushev, Vasily A., and Christopher L. Rogers. “A Version of the Goldman–Millson Theorem for Filtered L∞-Algebras.” Journal of Algebra 430 (2015): 260–302. DOI. Open PDF.
• Kontsevich, Maxim. “Deformation Quantization of Poisson Manifolds.” Letters in Mathematical Physics 66 (2003): 157–216. DOI. Open PDF.
• Pantev, Tony, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi. “Shifted Symplectic Structures.” Publications Mathématiques de l’IHÉS 117 (2013): 271–328. DOI. Open PDF.