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\begin{document}

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\section*{free field theory}

\begin{quote}%
This entry is about the concept in [[physics]]. For the concept in [[algebra]] see at \emph{[[free field]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{fields_and_quanta}{}\paragraph*{{Fields and quanta}}\label{fields_and_quanta}

[[!include fields and quanta - table]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_covariant_phase_space_geometrymultisymplectic_geometry}{In covariant phase space geometry/multisymplectic geometry}\dotfill \pageref*{in_covariant_phase_space_geometrymultisymplectic_geometry} \linebreak
\noindent\hyperlink{in_bvformalism}{In BV-formalism}\dotfill \pageref*{in_bvformalism} \linebreak
\noindent\hyperlink{kinematics_and_dynamics}{Kinematics and dynamics}\dotfill \pageref*{kinematics_and_dynamics} \linebreak
\noindent\hyperlink{TheClassicalObservables}{The classical observables}\dotfill \pageref*{TheClassicalObservables} \linebreak
\noindent\hyperlink{TheQuantumObservables}{The quantum observables}\dotfill \pageref*{TheQuantumObservables} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization_of_the_quantum_observables}{Characterization of the quantum observables}\dotfill \pageref*{characterization_of_the_quantum_observables} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{0DimensionalFreeFieldTheory}{0-Dimensional free field theory}\dotfill \pageref*{0DimensionalFreeFieldTheory} \linebreak
\noindent\hyperlink{examples_2}{Examples}\dotfill \pageref*{examples_2} \linebreak
\noindent\hyperlink{locally_free_field_theories}{Locally free field theories}\dotfill \pageref*{locally_free_field_theories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A [[field (physics)|field]] [[theory (physics)|theory]] in [[physics]] is called a \emph{free field theory} if it describes standard [[dynamics]] of fields without any [[interaction]]. Otherwise it is called an \emph{[[interacting field theory]]}.

There is some freedom in formalizing precisely what this means. At the very least the [[equations of motion]] of a free field theory should be [[linear differential equations]]. In [[relativistic field theory]] over a [[Lorentzian manifold|Lorentzian]] [[spacetime]] one typically requires that the [[linear differential equation]] [[equation of motion|of motion]] is, after [[gauge fixing]], in fact the [[wave equation]] or [[Klein-Gordon equation]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_covariant_phase_space_geometrymultisymplectic_geometry}{}\subsubsection*{{In covariant phase space geometry/multisymplectic geometry}}\label{in_covariant_phase_space_geometrymultisymplectic_geometry}

We describe free field theory in the language of [[covariant phase spaces]] of [[local Lagrangians]] and their [[multisymplectic geometry]].

Let $\Sigma = (\mathbb{R}^{d-1;1}, \eta)$ be [[Minkowski spacetime]]. Write the canonical [[coordinates]] as

\begin{displaymath}
\sigma^i \;\colon\; \Sigma \longrightarrow \mathbb{R}
  \,.
\end{displaymath}
Let $(X,g)$ be a [[vector space]] $X$ equipped with a [[bilinear form]] $g$ that makes it a [[Riemannian manifold]]. Write its canonical coordinates as

\begin{displaymath}
\phi^a \;\colon\; X \longrightarrow \mathbb{R}
  \,.
\end{displaymath}
Let then $X \times \Sigma \to \Sigma$ be the [[field bundle]]. Its first [[jet bundle]] then has canonical coordinates

\begin{displaymath}
\{ \sigma^i \}, \{\phi^a\}, \{\phi^a_{,i}\} \;\colon\; j_\infty^1(\Sigma \times X) \longrightarrow X
  \,.
\end{displaymath}
\begin{defn}
\label{FreeFieldLocalLagrangian}\hypertarget{FreeFieldLocalLagrangian}{}
The [[local Lagrangian]] for [[free field theory]] with this [[field bundle]] is

\begin{displaymath}
L  
   \coloneqq
  \left(
     \frac{1}{2} g^{i j} \eta_{a b} \phi^a_{,i} \phi^a_{,j}
  \right)
  \wedge
  \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The [[canonical momentum]]-densities for the free field local Lagrangian of def. \ref{FreeFieldLocalLagrangian} are

\begin{displaymath}
\begin{aligned}
    p_a^i \wedge \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d
      & \coloneqq 
    \frac{\partial}{\partial u^a_i} L
    \\
      & =
    \left(
       g^{i j} \eta_{a b} \phi^a_{,j}
    \right)
    \wedge
    \mathbf{d}\sigma^1 \wedge \cdots \wedge \mathbf{d}\sigma^d    
  \end{aligned}
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
So the boundary term $\theta$ in [[variational calculus]], (see \href{covariant+phase+space#CanonicalThetaDensityInLocalCoordinates}{this remark} at \emph{[[covariant phase space]]} ) is

\begin{displaymath}
\begin{aligned}
    \mathbf{d}u^a 
     \wedge
    \iota_{\partial_i}
    \left(
      \frac{\partial}{\partial \phi^a_{,i}}L 
    \right)
     & =  
    p^i_a \wedge (\iota_{\partial_{\sigma^i}} vol) \wedge \mathbf{d}u^a
    \\
    & =  p^i_a \wedge dq_i^a
    \,,
  \end{aligned}
\end{displaymath}
where in the last line we adopted the notation of \hyperlink{multisymplectic+geometry#CanonicalFormInGoodCoordinates}{this remark} at \emph{[[multisymplectic geometry]]}.

This shows that the canonical [[multisymplectic form]] is the ``covariant symplectic potential current density'' which is induced by the free field Lagrangian.

\end{remark}
\hypertarget{in_bvformalism}{}\subsubsection*{{In BV-formalism}}\label{in_bvformalism}

\hypertarget{kinematics_and_dynamics}{}\paragraph*{{Kinematics and dynamics}}\label{kinematics_and_dynamics}

In the formalization of [[perturbation theory]] via [[BV-quantization]] as in (\hyperlink{CostelloGwilliam}{Costello-Gwilliam}), a free field theory is given by a [[BV-complex]] that arises from the following data.

The following appears for instance as (\hyperlink{Gwilliam}{Gwilliam 2.6.2}).

\begin{defn}
\label{FreeFieldTheory}\hypertarget{FreeFieldTheory}{}
A \textbf{free field theory} ([[local quantum field theory|local]], [[Lagrangian]]) is the following data

\textbf{[[kinematics]]}:

\begin{itemize}%
\item A [[smooth manifold]] $X$ (``[[spacetime]]''/``[[worldvolume]]'');


\item a $\mathbb{Z}$-[[graded object|graded]] [[complex numbers|complex]] [[vector bundle]] $E \to X$ (the ``[[field bundle]]'' containing also in general [[antifields]] and [[ghosts]]);


\item equipped with a [[bundle]] [[homomorphism]] (the ``[[antibracket]] [[density]]'')

\begin{displaymath}
\langle -,-\rangle_{loc} \;\colon\; E \times E \to Dens_X
\end{displaymath}
from the fiberwise [[tensor product]] of $E$ with itself to the compex [[density bundle]] which is fiberwise

\begin{itemize}%
\item non-degenerate


\item anti-symmetric


\item of degree -1



\end{itemize}


\end{itemize}
(See also at \emph{[[Verdier duality]]}.)

Write $\mathcal{E}_c \coloneqq \Gamma_{cp}(E)$ for the space of [[sections]] of the [[field bundle]] of [[compact support]]. Write

\begin{displaymath}
\langle -,-\rangle
  \;\colon\;
  \mathcal{E}_c
  \otimes
  \mathcal{E}_c
  \to 
  \mathbb{C}
\end{displaymath}
for the induced pairing on sections

\begin{displaymath}
\langle \phi, \psi\rangle
  =
  \int_{x \in X} \langle \phi(x), \psi(x)\rangle_{loc}
  \,.
\end{displaymath}
The paring being non-degenerate means that we have an [[isomorphism]] $E \stackrel{\simeq}{\to} E^* \otimes Dens_X$ and we write

\begin{displaymath}
E^! \coloneqq E^* \otimes Dens_X
  \,.
\end{displaymath}
\textbf{[[dynamics]]}

\begin{itemize}%
\item A [[differential operator]] on sections of the [[field bundle]]

\begin{displaymath}
Q \;\colon\; \mathcal{E} \to \mathcal{E}
\end{displaymath}
of degree 1 such that

\begin{enumerate}%
\item $(\mathcal{E}, Q)$ is an [[elliptic complex]];


\item $Q$ is [[self-adjoint operator|self-adjoint]] with respect to $\langle -,-\rangle$ in that for all [[field (physics)|fields]] $\phi,\psi \in \mathcal{E}_c$ of homogeneous degree we have $\langle \phi , Q \psi\rangle = (-1)^{{\vert \phi\vert}} \langle Q \phi, \psi\rangle$.



\end{enumerate}


\end{itemize}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
From this data we obtain:

\begin{itemize}%
\item The [[action functional]] $S \colon \mathcal{E}_c \to \mathbb{C}$ of this corresponding free field theory is

\begin{displaymath}
S \;\colon\; \phi \mapsto \int_X \langle \phi, Q \phi\rangle
  \,.
\end{displaymath}

\item The [[classical BV-complex]] is the [[symmetric algebra]] $Sym  \mathcal{E}^!_c$ of [[compact support|compactly suppported]] [[sections]] of $E^!$ equipped with the induced action of the differential $Q$ and the pairing

\begin{displaymath}
\{\alpha,\beta\} \coloneqq  \int_{x \in X} \langle \alpha(x), \beta(x)\rangle
  \,.
\end{displaymath}
See below at \emph{\hyperlink{TheClassicalObservables}{The classical observables}} for more.


\item The [[quantum BV-complex]]

\begin{displaymath}
Obs^q \coloneqq (Sym \mathcal{E}^!_c[ [\hbar] ], Q + \hbar \Delta)
\end{displaymath}
is the deformation of the above to the [[symmetric algebra]] tensored with the [[formal power series]] in $\hbar$ (``[[Planck's constant]]'') $Sym(\mathcal{E}^!)[ [\hbar] ]$ and differential $Q + \hbar \Delta$ with [[BV-Laplacian]] defined to vanish on $Sym^{\leq 1}$, given by

\begin{displaymath}
\Delta (\alpha \cdot \beta) \coloneqq \{\alpha,\beta\}
\end{displaymath}
for $\alpha,\beta \in \mathcal{E}^!$ and extended by the formula

\begin{displaymath}
\Delta(a \cdot b)
  \coloneqq
  (\Delta a) \cdot b
  + 
  (-1)^{deg(a)}
  a \cdot (\Delta b)
  + 
  \{a,b\}  
  \,.
\end{displaymath}
See below at \emph{\hyperlink{TheQuantumObservables}{The quantum observables}} for more.



\end{itemize}
A closed element $\mathcal{O} \in Obs^q$ is an [[observable]] and its formal [[path integral]] [[expectation value]] $\langle \mathcal{O}\rangle$ is its image in the [[cochain cohomology]] $H^\bullet Obs^q$. Via the [[homological perturbation lemma]] this may be computed in [[perturbation theory]] (order by order in $\hbar$) in terms of [[Feynman diagrams]].

In a non-free field theory the differential would have an additional perturbation of the complex by an [[interaction]] term $I$ to

\begin{displaymath}
Q + \{I,-\} + \hbar \Delta
  \,.
\end{displaymath}
\end{remark}
[[!include action (physics) - table]]

\hypertarget{TheClassicalObservables}{}\paragraph*{{The classical observables}}\label{TheClassicalObservables}

For $(E, Q, \langle-, -\rangle_{loc})$ a free field theory, def. \ref{FreeFieldTheory}, write

\begin{displaymath}
E^! \coloneqq E^\ast \otimes Dens_X
\end{displaymath}
and accordingly write $\overline{\mathcal{E}^!}$ for its [[distribution|distributional]] [[sections]]. This is the distributional dual to the smooth sections $\mathcal{E}$ of $E$.

\begin{defn}
\label{ClassicalObservables}\hypertarget{ClassicalObservables}{}
The complex of \textbf{global classical observables} of the free field theory $(E,Q, \langle-,- \rangle_{loc})$ is the [[classical BV-complex]]

\begin{displaymath}
Obs^{cl} \coloneqq (Sym \mathcal{E}^!_c, Q)
\end{displaymath}
given by the [[symmetric algebra]] of dual sections and quipped with the dual of the differential (which we denote by the same letter) defined on generators and then extended as a graded [[derivation]] to the full [[symmetric algebra]].

The [[factorization algebra]] of local classical observables is the [[cosheaf]] of these observables which assigns to $U \subset X$ the complex

\begin{displaymath}
Obs^{cl} \colon U \mapsto (Sym \mathcal{E}^!_c(U), Q)
  \,.
\end{displaymath}
\end{defn}
in (\hyperlink{Gwilliam}{Gwilliam}), this is def. 5.3.6.

\hypertarget{TheQuantumObservables}{}\paragraph*{{The quantum observables}}\label{TheQuantumObservables}

There is a canonical [[BV-quantization]] of the \hyperlink{TheClassicalObservables}{above classical observable} of a free field theory given by defining the [[BV Laplacian]] as follows.

\begin{defn}
\label{TheStandardBVLaplacian}\hypertarget{TheStandardBVLaplacian}{}
For $(E, Q, \langle -,-\rangle_{loc})$ a free field theory, def. \ref{FreeFieldTheory}, the \textbf{standard [[BV Laplacian]]}

\begin{displaymath}
\Delta 
  \colon
  Sym \mathcal{E}^!_c \to Sym \mathcal{E}^!_c
\end{displaymath}
is given on generators $a,b \in Sym^1 \mathcal{E}^!_c$ of homogeneous degree by

\begin{displaymath}
\Delta(a \cdot b) \coloneqq \{a,b\}
\end{displaymath}
and then extended to arbitrary elements by the formula

\begin{displaymath}
\Delta(a \cdot b)
  \coloneqq
  (\Delta a) \cdot b
  + 
  (-1)^{deg(a)}
  a \cdot (\Delta b)
  + 
  \{a,b\}
\end{displaymath}
\end{defn}
In (\hyperlink{Gwilliam}{Gwilliam}) this is construction 2.4.9 (also construction 3.1.6, and section 5.3.3).

\begin{defn}
\label{QuantumObservables}\hypertarget{QuantumObservables}{}
For $(E, Q, \langle -,-\rangle_{loc})$ a free field theory, def. \ref{FreeFieldTheory} its complex of \textbf{[[quantum observables]]} is then the corresponding [[quantum BV-complex]] deformation of the classical BV-complex, def. \ref{ClassicalObservables}, given by the standard BV-Laplacian of def.\ref{TheStandardBVLaplacian}

\begin{displaymath}
Obs^{q} \colon U \mapsto
  \left(
    Sym(\mathcal{E}^!_c(U))[ [ \hbar ] ], Q + \hbar \Delta
  \right)
  \,.
\end{displaymath}
\end{defn}
In (\hyperlink{Gwilliam}{Gwilliam}) this is def. 5.3.9.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{characterization_of_the_quantum_observables}{}\subsubsection*{{Characterization of the quantum observables}}\label{characterization_of_the_quantum_observables}

We characterize the [[cochain complex]] $Obs^q$ of [[quantum observables]] of def. \ref{QuantumObservables} by an equivalent but small complex built from just the [[cochain cohomology]] of the [[elliptic complex]] of [[field (physics)|fields]] $(\mathcal{E}, Q)$.

\begin{defn}
\label{TheQuantumBVComplexOnTheCohomologyOfTheFieldComplex}\hypertarget{TheQuantumBVComplexOnTheCohomologyOfTheFieldComplex}{}
For $(E, Q, \langle -,-\rangle_{loc})$ a free field theory, def. \ref{FreeFieldTheory}, the global pairing constitutes a dg-[[symplectic vector space]] $(\mathcal{E}, Q, \langle -,-\rangle)$, which descends to the [[cochain cohomology]] to a graded symplectic vector space $(H^\bullet(\mathcal{E}, \langle -,-\rangle)$; hence by def. \ref{TheStandardBVLaplacian} there is a standard [[BV-Laplacian]] $\Delta_{H^\bullet\mathcal{E}}$. Write

\begin{displaymath}
\mathcal{BVQ}(H^\bullet \mathcal{E})
  \coloneqq
  ( Sym (H^\bullet(\mathcal{E}))^*, \Delta_{H^\bullet \mathcal{E}} )
\end{displaymath}
for the corresponding [[quantum BV-complex]].

\end{defn}
This is part of (\hyperlink{Gwilliam}{Gwilliam, prop. 2.4.10, prop. 5.5.1}),

\begin{quote}%
Handle the following with care for the moment.


\end{quote}
\begin{prop}
\label{}\hypertarget{}{}
For a free field theory $(E,Q,\langle-,- \rangle_{loc})$, def. \ref{FreeFieldTheory}, the complex of quantum observables $Obs^q$, def. \ref{QuantumObservables} is [[quasi-isomorphism|quasi-isomorphic]] to the BV-quantization of the cohomology of the field complex, given by def. \ref{TheQuantumBVComplexOnTheCohomologyOfTheFieldComplex}

\begin{displaymath}
Obs^q
  \simeq
  \mathcal{BVQ}(H^\bullet(\mathcal{E}))
  \,.
\end{displaymath}
\end{prop}
This is (\hyperlink{Gwilliam}{Gwilliam, prop. 5.5.1}).

The proof is supposedly along the lines of (\hyperlink{Gwilliam}{Gwilliam, section 2.5}), applying the [[homological perturbation lemma]].

\begin{prop}
\label{}\hypertarget{}{}
The bracket $\{-,-\}$ on the complex of quantum observables $Obs^q$ of def. \ref{QuantumObservables} descends to a bracket on [[cochain cohomology]], making $(H^\bullet (Obs^q), \{-,-\})$ into a graded [[symplectic vector space]].

\end{prop}
\begin{proof}
Let $a,b \in Obs^q$ be closed elements of homogeneous degree. Then by the compatibly of $\Delta$ with $\{-,-\}$ also $\{a,b\}$ is closed:

\begin{displaymath}
\Delta \{a,b\}
  = 
  \{\Delta a, b\} \pm \{a , \Delta b\} = 0
  \,.
\end{displaymath}
Let in addition $c \in Obs^q$ be any element. Then

\begin{displaymath}
\begin{aligned}
    \{a, b + \Delta c\}
    &= 
    \{a,b\} + \{a, \Delta c\}
    \\ 
    & = \{a,b\} + \Delta (a \cdot (\Delta b)) - (\Delta a)\cdot b   -a \cdot (\Delta^2 b)
    \\
    & = \{a,b\} + \Delta (a \cdot (\Delta b))
  \end{aligned}
\end{displaymath}
and hence the cohomology class of $\{a,b\}$ is independent of the representative cocycle $b$, and similarly for $a$.

\end{proof}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{0DimensionalFreeFieldTheory}{}\subsubsection*{{0-Dimensional free field theory}}\label{0DimensionalFreeFieldTheory}

A degenerate but instructive class of examples to compare to is the case where $X = *$ is the 0-dimensional connected manifold: the point. (See (\hyperlink{Gwilliam}{Gwilliam 2.3.1})).

In this case

\begin{displaymath}
E = V \oplus V^*[-1]
\end{displaymath}
is the [[direct sum]] of a vector space and its formal dual shifted in degree. The pairing is the canonical pairing between a vector space and its dual.

If $\{x^i \colon V \to \mathbb{R}\}_i$ is a [[basis]] for functional on $V$ and $\{\xi_i\}$ is the corresponding basis of functions on $V^*[-1]$, then the [[antibracket]] in this case is

\begin{displaymath}
\{
    X^i, x^j
  \}
  = 0
  \;\;\;
  \{\xi_i, \xi_j\} = 0
  \;\;\;
  \{x^i, \x_j\} = \delta^i_j
  \,.
\end{displaymath}
The [[BV-Laplacian]] in this basis is

\begin{displaymath}
\Delta = \sum_{i = 1}^n \frac{\partial}{\partial x^i} \frac{\partial}{\partial \xi_i}
  \,.
\end{displaymath}
The [[action functional]] is a [[Gaussian distribution]] over $V$ defined by a matrix $A = (a_{i j})$. The corresponding differential is

\begin{displaymath}
Q = \sum_{i,j =1}^n x^i a_{i j} \frac{\partial}{\partial \xi_i}
  \,.
\end{displaymath}
Hence for a field of the form

\begin{displaymath}
\phi = \sum_{i = 1}^n \phi^i \xi_i
\end{displaymath}
we have the [[action functional]]

\begin{displaymath}
\begin{aligned}
    \exp(S(\phi))
    & =
    \exp(-\langle \phi , Q \phi\rangle)
    \\
    & = \exp( - \phi^i a_{k j} \phi^j \langle \xi_i, x^k   \rangle )
    \\
    & = \exp(- \phi^i a_{i j} \phi^j )
  \end{aligned}
\end{displaymath}
\hypertarget{examples_2}{}\subsection*{{Examples}}\label{examples_2}

\hypertarget{locally_free_field_theories}{}\subsubsection*{{Locally free field theories}}\label{locally_free_field_theories}

Some [[sigma-model]] quantum field theories have the property that they are fee \emph{locally} on their [[target spaces]]. Under good conditions then [[quantization]] of free field theory locally yields a [[sheaf]] of [[quantum observables]] on [[target space]] from which the full quantization of the field theory may be reconstructed.

A famous example of this is the [[topological twist|topologically twisted]][[2d (2,0)-superconformal QFT]] (see there for more, and see (\hyperlink{{#Gwilliam}}{Gwilliam, section 6} for the description in terms of [[factorization algebras]]).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Wick algebra]]


\item [[Gaussian probability distribution]], [[Gaussian integral]]


\item [[Wick's lemma]]


\item [[gauge theory]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

Discussion of free field theories and their [[quantization]] on [[globally hyperbolic Lorentzian manifolds]] is in

\begin{itemize}%
\item [[Christian Bär]], [[Nicolas Ginoux]], [[Frank Pfäffle]], \emph{Wave Equations on Lorentzian Manifolds and Quantization}, ESI Lectures in Mathematics and Physics, European Mathematical Society Publishing House, ISBN 978-3-03719-037-1, March 2007, Softcover (\href{https://arxiv.org/abs/0806.1036}{arXiv:0806.1036})

\end{itemize}
Discussion on Euclidean manifolds and in terms of [[BV-formalism]] is in

\begin{itemize}%
\item [[Kevin Costello]], [[Owen Gwilliam]], \emph{Factorization algebras in perturbative quantum field theory} (\href{http://math.northwestern.edu/~costello/factorization_public.html}{wiki}, \href{http://math.northwestern.edu/~gwilliam/factorization.pdf}{pdf})


\item [[Owen Gwilliam]], \emph{Factorization algebras and free field theories} PhD thesis ([[GwilliamThesis.pdf:file]])


\item [[Owen Gwilliam]], [[Rune Haugseng]], \emph{Linear Batalin-Vilkovisky quantization as a functor of ∞-categories} (\href{https://arxiv.org/abs/1608.01290}{arXiv:1608.01290})



\end{itemize}
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