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Homotopy theory of algebraic quantum field theories. (English) Zbl 1419.81026
The authors claim that algebraic quantum field theory developed to investigate quantum field theory on Lorentzian spacetime has no consensus of the description of gauge theories, which can answer what is a quantum gauge theory and how it differs from a theory without gauge symmetries. The key to address and solve these problems is to develop a generalization of algebraic quantum field theory that takes into account the higher structures which are present in gauge theories. Homotopical quantum field theory is initiated by this argument. It combines the conceptual/physical ideas of algebraic quantum field theory with modern techniques from homotopy theory, in the sense of model theory [W. G. Dwyer and J. Spalinski, in: Handbook of algebraic topology. Amsterdam: North-Holland. 73–126 (1995; Zbl 0869.55018)], or higher category theory [J. Lurie, Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)].
Smooth structure on a higher space of fields can be encoded by using the concept of a stack [the first author et al., Commun. Math. Phys. 359, No. 2, 765–820 (2018; Zbl 1423.14007)]. Its corresponding observable algebra should be e kind of higher algebra there shadows are known as BRST/BV formalism. Therefore the generalization from ordinary algebras in vector spaces to higher algebras in chain complexes is motivated by the necessity to encode and describe the higher structures of quantum gauge theories in the framework of algebraic quantum field theory.
In [the first author et al., “Operads for algebraic quantum field theory”, Preprint, arXiv:1709.08657], the authors show algebraic quantum field theories admit a description in terms of algebras over a suitable colored operad.(cf,Th,3,8 of this paper). Based on this result, the main result of this paper is to show that there exists a canonical model structure on the category of chain complex-valued algebraic quantum field theories and to explain the immense relevance of such structures both from a conceptual and from a more practical point of view.
Basic definitions and results about the model category $$\mathbf{Ch}(k)$$ of chain complexes of $$k$$-modules, $$k\supseteq\mathbb{Q}$$, colored operads and their homotopy theory are reviewed in §2. In [the first author et al., Preprint,loc. cit.], orthogonal category $$\bar{\mathbf{C}}=(\mathbf{C},\perp)$$ is defined, where $$\mathbf{C}$$ is any small category (in this paper, $$\mathbf{C}$$ is the category of (Lorentzian) spacetime). This is reviewed in §3 as Definition 3.1, Then denoting $$\mathbf{dgAlg}(k)$$, $$k\supseteq\mathbb{Q}$$, and define $$\perp$$-commutativity of a covariant functor $$\mathbf{\mathfrak{A}}:\mathbf{C}\to \mathbf{dgAlg}(k)$$ (Definition 3.3. a), the category of $$\mathbf{Ch}(k)$$-valued quantum field theory on $$\bar{\mathbf{C}}$$ is defined as the full subcategory of the functor category; $\mathbf{QFT}(\bar{\mathbf{C}})\subseteq\mathbf{dgAlg}(k)^{\mathbf{C}},$ where objects are all $$\perp$$-commutative functors (Definition 3.2, b)). Then the description of this category in terems of algebra of colored operad is given in §3. 2 (Theorem 3.8; $$\mathbf{Alg}(\mathcal{O}_{\bar{\mathbf{C}}})\cong \mathbf{QFT}(\bar{\mathbf{C}})$$, cf. [the first author et al., Preprint, loc. cit.]). From this result, the following Theorem is obtained.
Theorem 3.10. Define a morphism $$\zeta:\mathbf{\mathfrak{A}}\to \mathbf{\mathfrak{B}}$$ in $$\mathbf{QFT}(\bar{\mathbf{C}}$$ to be
a weak equivalence if the un derlying $$\mathbf{Ch}(k)$$-morphism of each component $$\zeta_c:\mathbf{\mathfrak{A}}(c)\to \mathbf{\mathfrak{B}}(c)$$ is a quasi-isomorphism,
a fibration, if the underlying $$\mathbf{Ch}(k)$$-morphism of each component $$\zeta_c:\mathbf{\mathfrak{A}}(c)\to \mathbf{\mathfrak{B}}(c)$$ is degree-wise fibration.
a cofibration if the left lifting property (cf. Remark 2.2) withrespeect to all acyclicfibrations,
These choices endow $$\mathbf{QFT}(\bar{\mathbf{C}})$$ with thee structure of model categories.
Several examples of this theorem, such as coincidence of weak equivalence and the notions used in BRST/BV formalism (Example 3.12) are follows. Some technical details used in these examples are postponed to the appendix.
In §4, a $$\Sigma$$-cofibrant resolution of a colored operad $$\mathcal{O}$$ is defined as an acyclic fibration $$w:\mathcal{O}_\infty\to \mathcal{O}$$, $$\mathcal{O}_\infty$$ is a $$\Sigma$$-covariant operad (Definition 4.1). The model category of homotopy quantum field theories of $$\bar{\mathbf{C}}$$ corresponding to a $$\Sigma$$-cofibrant resolution $$w$$ is denoted by $$\mathbf{QFT}_w(\bar{\mathbf{C}}$$. Then it is proved for every $$\Sigma$$ cofibrant resolution $$w$$, there exists a Quillen equivalence $w_!;\mathbf{QFT}_w(\bar{\mathbf{C}})\rightleftharpoons\mathbf{QFT}(\bar{\mathbf{C}}):w^\ast,$ between the model categories strict and homotopy quantum field theories on $$\bar{\mathbf{C}}$$ (Theorem 4.3). Then a detailed study on the $$E$$-resolution; a $$\Sigma$$-cofibrant resolution obtained by a component-wise tensor product Barratt-Eccles $$E_\infty$$-operad, and toy models via cochain algebras on stacks, follows.
§5, the last section, provides another class of examples of homotopy quantum field theories over the resolution $$\mathcal{O}_{\bar{\mathbf{C}}}\otimes\mathcal{E}_\infty\to \mathcal{O}_{\bar{\mathbf{C}}}$$. The authors say this is an improvement of construction in [the first and the second author, Commun. Math. Phys. 356, No. 1, 19–64 (2017; Zbl 1378.81080)], supplying homotopy-coherence of the construction. This construction starts with a category a fibered in groupoids $$\pi:\mathbf{D}\to \mathbf{C}$$, where $$\mathbf{D}$$ is a category of spacetimes with background guage fields. Taking a strict $$\mathbf{Ch}(k)$$-valued quantum field theory on the total category $$\mathbf{D}$$, its underlying functor $$\mathbf{\mathfrak{A}}:\mathbf{D}\to \mathbf{Ch}'k)$$ is considered. Over each spacetime $$c\in\mathbf{C}$$, there exists a groupoid $$\pi^{-1}(c)$$ of background fields. Then take homotopy invariants of the corresponding groupoid actions on the quantum field theory $$\mathbf{\mathfrak{A}}$$. This construction is formalized in terms of a homotopy right Kan extension $$\mathbf{hoRan}_\pi$$. It is shown $$\mathbf{hoRan}_\pi\in \mathbf{QFT}_w(\mathbf{C})$$ is a homotopy quantum field theory on the base category $$\bar{\mathbf{C}}$$ corresponding to the resolution $$w_{\bar{\mathbf{C}}}:\mathcal{O}_{\bar{\mathbf{C}}}\otimes\mathcal{E}_\infty\to \mathcal{O}_{\bar{\mathbf{C}}}$$ (Theorem 5.5).
The authors say that the construction in §5 seems to be relevant to perturbative quantum gauge theory (cf. Example 5.1). From a mathematical perspective, the homotopy quantum field theory $$\mathbf{hoRan}_\pi\mathbf{\mathfrak{A}}$$ admits an interpretation in terms of fiber-wise groupoid cohomology on $$\mathbf{D}\to \mathbf{C}$$ with coefficients in the strict quantum field theory $$\mathbf{\mathfrak{A}}$$ (cf. Remark 5.6).

##### MSC:
 81T10 Model quantum field theories 81T05 Axiomatic quantum field theory; operator algebras 81T13 Yang-Mills and other gauge theories in quantum field theory 81T20 Quantum field theory on curved space or space-time backgrounds 18D50 Operads (MSC2010) 18G55 Nonabelian homotopical algebra (MSC2010) 55U35 Abstract and axiomatic homotopy theory in algebraic topology
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