Factorization algebras in quantum field theory. Volume 1 (Q2833168)

This book is the first volume of a two-volume monograph on the theory of factorization algebras, which presents a rigorous mathematical framework that is designed to provide a useful common platform for accommodating various aspects of existing topological and quantum field theories. In this first volume, the authors concentrate on the general properties of (pre)factorization algebras and some examples about free field theories, while in the second volume, they plan to work on factorization algebras for interacting field theories and derive Batalin-Vilkovisky formalism, automatically leading to deformation quantization.NEWLINENEWLINENEWLINENEWLINEBecause the subject of this book touches many advanced leading theories of quantum physics which utilize heavily mathematical machineries from a diverse range of mathematical topics, the background material needed for this book is immense. So it is very helpful and much appreciated that a 103-page four-section appendix is included in this 387-page book, to provide a very well-organized and fairly detailed review of relevant mathematical background topics, including simplicial techniques, colored operads/multicategories and their algebras, differential graded (dg) Lie algebras and their cohomology, sheaves/cosheaves, formal Hodge theory, and `convenient, differentiable, or bornological'\ topological vector spaces facilitating the homological algebra for infinite-dimensional vector spaces. Although the review of a background topic provided in the appendix is often sufficient for a reader to follow the main text of the book, from time to time it is still unavoidable for a non-expert reader to consult other mathematical literature. Overall, in the reviewer's judgment, this book will benefit a non-expert reader who wants to learn this subject while being already mathematically mature and proficient in the general graduate-level mathematics. Experts in quantum physics and quantum mathematics would also find this well-organized monograph, which puts together various theories in the context of factorization algebras, of interest and use. Furthermore this volume provides six pages of useful references. However it does not contain any exercise problems and hence is not quite a textbook in the ordinary sense.NEWLINENEWLINENEWLINENEWLINEThe main text of this book consists of eight chapters (grouped in three parts). Below is a sketch of some major content in each chapter.NEWLINENEWLINENEWLINENEWLINEIn the first chapter, an extensive introduction is given, which starts with a basic setting of classical and quantum mechanics involving vector spaces of local observables to motivate a preliminary definition of a prefactorization algebra on a topological space \(M\), which resembling the structure of a precosheaf, comprises an assignment \(U\mapsto\mathcal{F}\left( U\right) \) of cochain complexes (or for simplicity, vector spaces) \(\mathcal{F}\left( U\right) \) to open sets \(U\subset M\) and cochain maps \(\mathcal{F}\left( U_{1}\right) \otimes\cdots\otimes\mathcal{F}\left( U_{n}\right) \rightarrow\mathcal{F}\left( V\right) \) for any pairwise disjoint open sets \(U_{1},\dots,U_{n}\subset V\) satisfying some natural compatibility conditions on map compositions to be rigorously presented in Chapter 3. Then an interpretation of this structure in the context of quantum field theory is given, and other existing formulations of quantum field theory with similar flavor are discussed for comparison, including Beilinson and Drinfeld's chiral conformal field theory, Segal's field theory, and topological field theory.NEWLINENEWLINENEWLINENEWLINEIn the second chapter, a detailed construction of a concrete prefactorization algebra from the free scalar field theory on a Riemannian manifold \(\left( M,g\right) \) is presented. It is first recalled how classically the expectation function for a Gaussian measure \(\omega\) on \(\mathbb{R}^{k}\) can be identified with the quotient map from the space \(P\left( \mathbb{R} ^{k}\right) \) of polynomials to the cokernel of the divergence operator \(\mathrm{Div}_{\omega}:\mathrm{Vect}\left( \mathbb{R}^{k}\right) \rightarrow P\left( \mathbb{R}^{k}\right) \) for the space \(\mathrm{Vect}\left( \mathbb{R}^{k}\right) \) of polynomial vector fields on \(\mathbb{R}^{k}\). For an open set \(U\subset M\), one can formally replace (1) \(\mathbb{R}^{k}\) by \(C^{\infty}\left( U\right) \), (2) \(P\left( \mathbb{R}^{k}\right) \) by the direct sum \(P\left( C^{\infty}\left( U\right) \right) \) of the spaces \(C_{c}^{\infty}\left( U^{n}\right) _{S_{n}}\) of permutation-invariant compact-supported smooth functions on the Cartesian product \(U^{n}\) of \(n\) copies of \(U\), where \(f\in C_{c}^{\infty}\left( U^{n}\right) _{S_{n}}\) can be viewed as a polynomial function on \(C^{\infty}\left( U\right) \) sending \(\phi\in C^{\infty}\left( U\right) \) to the integral of the function \(f\left( x_{1},..,x_{n}\right) \phi\left( x_{1}\right) \cdots\phi\left( x_{n}\right) \) over \(U^{n}\) equipped with the product Riemannian volume form, and (3) \(\mathrm{Vect}\left( \mathbb{R}^{k}\right) \) by the direct sum \(\mathrm{Vect}_{c}\left( C^{\infty}\left( U\right) \right) \) of the spaces \(C_{c}^{\infty}\left( U^{n+1}\right) _{S_{n}}\) of \(S_{n}\times\left\{ \mathrm{id}_{U}\right\} \)-invariant elements in \(C_{c}^{\infty}\left( U^{n+1}\right) \), which contains a dense subspace \(P\left( C^{\infty}\left( U\right) \right) \otimes C_{c}^{\infty}\left( U\right) \) of ``polynomial'' vector fields on \(C^{\infty }\left( U\right) \) tangent to the subspace \(C_{c}^{\infty}\left( U\right) \) of \(C^{\infty}\left( U\right) \). Then transcribing the standard formula of \(\mathrm{Div}_{\omega}\), one can construct a well-defined field divergence map \(\mathrm{Div}_{U}:\mathrm{Vect}_{c}\left( C^{\infty}\left( U\right) \right) \rightarrow P\left( C^{\infty}\left( U\right) \right) \) for a formal underlying ``Gaussian measure'' on \(C^{\infty}\left( U\right) \) corresponding to the action functional \(S\) sending \(\phi\) to the Riemannian integral of \(\phi\left( \Delta_{g} +m^{2}\right) \phi\) over \(U\), where \(\Delta_{g}\) is the Laplacian (with nonnegative eigenvalues) and \(m\) is a prescribed ``mass''. The cokernel, denoted as \(H^{0}\left( \mathrm{Obs} ^{q}\left( U\right) \right) \), of \(\mathrm{Div}_{U}\) is taken as the space of expectation-equivalent quantum observables for the fields inside \(U\). The canonical commutative function algebra structure of \(P\left( C^{\infty }\left( U\right) \right) \) does not descend to its quotient \(H^{0}\left( \mathrm{Obs}^{q}\left( U\right) \right) \), but the multiplication map \(P\left( C^{\infty}\left( U\right) \right) \otimes P\left( C^{\infty }\left( U\right) \right) \rightarrow P\left( C^{\infty}\left( U\right) \right) \) does descend to a well-defined ``product'' map\ \(H^{0}\left( \mathrm{Obs}^{q}\left( U_{1}\right) \right) \otimes H^{0}\left( \mathrm{Obs}^{q}\left( U_{2}\right) \right) \rightarrow H^{0}\left( \mathrm{Obs}^{q}\left( U\right) \right) \) for any disjoint open subsets \(U_{1},U_{2}\) of \(U\) where \(P\left( C^{\infty}\left( U_{j}\right) \right) \subset P\left( C^{\infty }\left( U\right) \right) \) canonically since \(C_{c}^{\infty}\left( U_{j}^{n}\right) _{S_{n}}\subset C_{c}^{\infty}\left( U^{n}\right) _{S_{n} }\). This assignment \(U\mapsto H^{0}\left( \mathrm{Obs}^{q}\left( U\right) \right) \) together with the above product maps constitutes a concrete example of prefactorization algebra. Inserting a scaling factor \(\hbar\) in this construction, the authors show that as \(\hbar\rightarrow0\), a limit prefactorization algebra \(U\mapsto H^{0}\left( \mathrm{Obs}^{cl}\left( U\right) \right) \) arises with \(H^{0}\left( \mathrm{Obs}^{cl}\left( U\right) \right) \) being the classical observable algebra of polynomials on the space of solutions to the Euler-Lagrange equations, which puts the theory of prefactorization algebras well in the context of deformation quantization. It is also discussed how the physically important concept of correlation functions of observables appears in the context of prefactorization algebras.NEWLINENEWLINENEWLINENEWLINEIn the third chapter, a rigorous definition of prefactorization algebras on a topological space \(M\) with values in some symmetric monoidal category \(\mathcal{C}\) is presented in an elementary way, enumerating all the compatibility conditions on the product/structure maps \(\mathcal{F}\left( U_{1}\right) \otimes\cdots\otimes\mathcal{F}\left( U_{n}\right) \rightarrow\mathcal{F}\left( V\right) \). Then a more elegant and preferred definition in terms of multicategories (or colored operads) is given. Associative algebras are shown to correspond exactly to those prefactorization algebras on \(\mathbb{R}\) with values \(\mathcal{F}\left( U\right) \) in the category of vector spaces, which are locally constant in the sense that the structure map \(\mathcal{F}\left( U\right) \rightarrow\mathcal{F}\left( V\right) \) is an isomorphism for any inclusion of (nonempty) open intervals \(U\subset V\). For a Lie algebra \(\mathfrak{g}\), the prefactorization algebra \(U\mapsto H^{\ast}\left( C_{\ast}\left( \Omega_{c}^{\ast}\left( U\right) \otimes\mathfrak{g}\right) \right) \) on \(\mathbb{R}\) is locally constant, yielding the associative universal enveloping algebra \(U_{\mathfrak{g}}\) of \(\mathfrak{g}\), where \(\Omega_{c}^{\ast}\left( U\right) \otimes\mathfrak{g}\) endowed with \(\left[ \alpha\otimes x,\beta\otimes y\right] :=\alpha \wedge\beta\otimes\left[ x,y\right] \) is a dg Lie algebra and \(C_{\ast}\) is the Chevalley-Eilenberg functor turning a dg Lie algebra to a (co)chain complex for Lie algebra homology. The standard quantum mechanics with a given Hamiltonian operator \(H\) also provides an example of prefactorization algebra on \(\mathbb{R}\), where structure maps are of the form \(\mathcal{O}_{1} \otimes\cdots\otimes\mathcal{O}_{n}\mapsto e^{i\ell_{0}H}\mathcal{O} _{1}e^{i\ell_{1}H}\cdots\mathcal{O}_{n}e^{i\ell_{n}H}\) for observable operators \(\mathcal{O}_{j}\) and lengths \(\ell_{j}\) of closed intervals between disjoint open intervals \(U_{j}\) and \(U_{j+1}\). To deal with differential geometric examples, the analytical concept of differentiable vector spaces is brought in so that one can talk about prefactorization algebra with values in the category of differentiable cochain complexes. An important class of prefactorization algebras called the factorization envelope \(\mathbb{U} \mathcal{L}\) for a fine sheaf \(\mathcal{L}\) of dg Lie algebras on a manifold \(M\) is introduced as \(U\mapsto\mathbb{U}\mathcal{L}\left( U\right) :=C_{\ast}\left( \mathcal{L}_{c}\left( U\right) \right) \), replacing \(\Omega_{c}^{\ast}\left( U\right) \otimes\mathfrak{g}\) in the earlier discussion by the more general \(\mathcal{L}_{c}\left( U\right) \). A variant of factorization envelope called the twisted factorization envelope gives rise to the Kac-Moody vertex algebra as an example in Chapter 5. For topological spaces \(M\) with an action by a group, an equivariant version of prefactorization algebra on \(M\) is formulated and analyzed for both the discrete equivariance and the smooth equivariance.NEWLINENEWLINENEWLINENEWLINEIn the fourth chapter, the prefactorization algebra with vector space values \(H^{0}\left( \mathrm{Obs}^{q}\left( U\right) \right) \) defined directly as the cokernel of divergence operator \(\mathrm{Div}_{U}\) for the free scalar field theory on a Riemannian manifold \(M\) is given a more complete account. A prefactorization algebra \(\mathrm{Obs}^{q}\) with cochain complex values \(\mathrm{Obs}^{q}\left( U\right) \) is constructed according to a general program for free field theories, such that the above space \(H^{0}\left( \mathrm{Obs}^{q}\left( U\right) \right) \) is indeed the zeroth cohomology of the cochain complex \(\mathrm{Obs}^{q}\left( U\right) \) for all open sets \(U\subset M\). It is shown that \(\mathrm{Obs}^{q}\) for the free scalar field on \(\mathbb{R}\) is locally constant, and the associated locally constant cohomology prefactorization algebra \(H^{\ast}\left( \mathrm{Obs}^{q}\right) \) has the Weyl algebra as its corresponding associative algebra. Being a free field theory, the Abelian Chern-Simons theory is also treated and analyzed as an example.NEWLINENEWLINENEWLINENEWLINEIn the fifth chapter, a holomorphic version of prefactorization algebra is introduced. It is shown that a holomorphic prefactorization algebra on \(\mathbb{C}\) leads to a vertex algebra.NEWLINENEWLINENEWLINENEWLINEIn the sixth chapter, a ``descent'' axiom about local data determining global data, in resemblance with the condition for a presheaf to be called a sheaf, is formulated for prefactorization algebras \(\mathcal{F}\) on \(M\). A prefactorization algebra \(\mathcal{F}\) is called a factorization algebra, if it satisfies this ``descent'' condition. Roughly speaking, this axiom says that \(\oplus_{i,j\in I}\mathcal{F}\left( U_{i}\cap U_{j}\right) \rightarrow \oplus_{i}\mathcal{F}\left( U_{i}\right) \rightarrow\mathcal{F}\left( U\right) \rightarrow0\) is exact for any Weiss cover \(\left\{ U_{i}\right\} _{i\in I}\) of an open set \(U\subset M\). Some examples of prefactorization algebras considered earlier in this book are shown to be a factorization algebra, for example, the observables of a free field theory.NEWLINENEWLINENEWLINENEWLINEIn the seventh chapter, the push-forward of a factorization algebra along a continuous map and the reconstruction of a factorization algebra from local data are presented.NEWLINENEWLINENEWLINENEWLINEIn the eighth chapter, the factorization homology \(\mathcal{F}\left( M\right) \) of global sections is computed, for some factorization algebras \(\mathcal{F}\) on \(M\) considered in this book, including those for the free scalar field in dimension one, the Kac-Moody algebra, and the abelian Chern-Simons theory.