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\begin{document}

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\section*{derived critical locus}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{higher_geometry}{}\paragraph*{{Higher geometry}}\label{higher_geometry}

[[!include higher geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{BasicDgGeometry}{Basic dg-geometry}\dotfill \pageref*{BasicDgGeometry} \linebreak
\noindent\hyperlink{formal_cotangent_bundle}{Formal cotangent bundle}\dotfill \pageref*{formal_cotangent_bundle} \linebreak
\noindent\hyperlink{derived_critical_locus}{Derived critical locus}\dotfill \pageref*{derived_critical_locus} \linebreak
\noindent\hyperlink{ComparisonToBVBRSTExample}{BV-BRST complex}\dotfill \pageref*{ComparisonToBVBRSTExample} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The concept of \emph{derived critical locus} is the refinement of the notion of [[critical locus]] from [[geometry]] to [[derived geometry]].

The formal duals to derived critical loci are described by [[BV-BRST formalism]].

\hypertarget{details}{}\subsection*{{Details}}\label{details}

The following is a basic setup in [[dg-geometry]] aimed at exhibiting the (formal dual of) the [[BV-BRST complex]] as the derived critical locus of (the formal dual of) the [[BRST complex]] (the example \hyperlink{ComparisonToBVBRSTExample}{below})

\hypertarget{BasicDgGeometry}{}\subsubsection*{{Basic dg-geometry}}\label{BasicDgGeometry}

Let $k$ be a [[field]] of [[characteristic zero]].

Write $dgcAlg_k$ for the [[category]] of unbounded cochain [[differential graded-commutative algebras]] (dgc-algebras) over $k$.

An object in the [[opposite category]]

\begin{displaymath}
C \in cdgAlg_k^{op}
\end{displaymath}
we may regard as an affine space in [[dg-geometry]], and hence we write

\begin{displaymath}
\mathcal{O}(C) \in cdgAlg_k
\end{displaymath}
for the corresponding dgc-algebra.

Let $\mathcal{O}(C)$[[Mod]] be the [[category of dg-modules]] over $\mathcal{O}(C)$ equipped with the standard [[model structure on dg-modules]].

\begin{defn}
\label{OCDGCAlgebras}\hypertarget{OCDGCAlgebras}{}
\textbf{([[dgc-algebras]] over $\mathcal{O}(C)$)}

Write

\begin{displaymath}
\begin{aligned}
    cdgAlg_{\mathcal{O}(C)} 
      & \coloneqq 
    CMon(\mathcal{O}(C) Mod)
    \\
      & \simeq
    \mathcal{O}(C)/_{cdgAlg_k}
  \end{aligned}
\end{displaymath}
for the [[category of monoids|category of commutative monoids]] in $\mathcal{O}(C)$-modules, equivalently the [[coslice category]] of $cdgAlg_k$ under $\mathcal{O}(C)$.

\end{defn}
\begin{prop}
\label{SymOCAdjunction}\hypertarget{SymOCAdjunction}{}
There is a [[model category]] structure on $cdgAlg_{\mathcal{O}(C)}$ (def. \ref{OCDGCAlgebras}) whose fibrations and weak equivalences are those of the underlying $\mathcal{O}(C)$-modules such that the [[free-forgetful adjunction]]

\begin{displaymath}
cdgAlg_{\mathcal{O}(C)}
   \underoverset
     {\underset{U}{\longrightarrow}}
     {\overset{Sym_{\mathcal{O}(C)}}{\longleftarrow}}
     {\bot}
  \mathcal{O}(C) Mod
\end{displaymath}
is a [[Quillen adjunction]].

This is

\begin{enumerate}%
\item [[combinatorial model category|combinatorial]];


\item [[proper model category|proper]].



\end{enumerate}
\end{prop}
\begin{proof}
This follows with the general discussion at \emph{[[dg-geometry]]}. We indicate how to see it directly.

We observe that the adjunction exhibits the [[transferred model structure]] on the left. By the statement discussed there, it is sufficient to check that

\begin{enumerate}%
\item $\mathcal{O}(C) Mod$ is a [[cofibrantly generated model category]].

This follows because the [[nLab:model structure on dg-modules]] (as discussed there) is itself transferred along

\begin{displaymath}
U' \colon \mathcal{O}(C) Mod \to Ch^\bullet(k)
\end{displaymath}
from the cofibrantly generated [[model structure on chain complexes|model structure on cochain complexes]].


\item $U$ preserves [[filtered colimits]].

This follows from the general fact $U : CMon(\mathcal{C}) \to \mathcal{C}$ creates filtered colimits for $\mathcal{C}$ closed symmetric monoidal (see \href{category+of+monoids#FilteredColimits}{there}) and that $A Mod$ is [[closed monoidal category|closed]] [[symmetric monoidal category|symmetric monoidal]] (see \href{model+structure+on+dg-modules#Properties}{there}).

To check this explicitly:

Let $A_\bullet \colon D \to cdgAlg_k$ be a [[filtered category|filtered diagram]]. We claim that there is a unique way to lift the underlying colimit $\lim_\to U A_\bullet$ to a dg-algebra cocone: for $a \in A_i \to \lim_\to U A_\bullet$ and $b \in A_j \to \lim_\to U A_\bullet$ there is by the assumption that $D$ is filtered a $A_i \to A_l \leftarrow A_j$. Therefore in order for the cocone component $U A_l \to \lim_{\to} U A_\bullet$ to be an algebra homomorphism the product of $a$ with $b$ in $\lim_\to U A_\bullet$ has to be the image of this product in $A_l$. This defines the colimiting cocone $A_l \to \lim_\to A_\bullet$.


\item The left hand has functorial fibrant replacement (this is trivial, since every object is fibrant) and functorial [[path objects]].

This follows by the same argument as for the path object in $cdgAlg_k$ (\href{model+structure+on+dg-algebras#PathObjectsForUnboundedCommutative}{here}) this can be taken to be $(-)\otimes_k \Omega^\bullet_{poly}(\Delta[1])$.



\end{enumerate}
\end{proof}
\hypertarget{formal_cotangent_bundle}{}\subsubsection*{{Formal cotangent bundle}}\label{formal_cotangent_bundle}

Given $C \in dgcAlg_k^{op}$ as \hyperlink{BasicDgGeometry}{above} we want to consider its \emph{formal [[cotangent bundle]]} $T^\ast_f c$, i.e. the [[infinitesimal neighbourhood]] around the [[zero section]] of the would-be actual [[cotangent bundle]]

\begin{defn}
\label{}\hypertarget{}{}
Write

\begin{displaymath}
Der(\mathcal{O}(C)) \in \mathcal{O}(C) Mod
\end{displaymath}
for the [[automorphism ∞-Lie algebra]] of $A$ whose underlying cochain complex is

\begin{displaymath}
\itexarray{
    \cdots
      \to
    Der(\mathcal{O}(C))_k
      \overset{[d_{\mathcal{O}(C)},-]}{\longrightarrow}
    Der(\mathcal{O}(C))_{k+1}
      \longrightarrow
    \cdots
  }
  \,.
\end{displaymath}
where $Der(\mathcal{O}(C))_k$ is the module of [[derivations]]

\begin{displaymath}
v \colon \mathcal{O}(C)^\bullet \to \mathcal{O}(C)^{\bullet + k}
\end{displaymath}
of degree $k$ and $[d_{\mathcal{O}(C)}, -]$ is the graded commutator of derivations with the [[differential]] of $\mathcal{O}(C)$ regarded as a degree 1 derivation $d_{\mathcal{O}(C)} \colon \mathcal{O}(C) \to \mathcal{O}(C)$.

We say that $\mathcal{O}(C)$ is \emph{smooth} if $Der(\mathcal{O}(C))$ is cofibrant as an object on $\mathcal{O}(C) Mod$.

Write

\begin{displaymath}
\mathcal{O}(T^*_f C)
    \coloneqq
  Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C))
   \in 
  cdgAlg_{\mathcal{O}(C)}
\end{displaymath}
for the free $\mathcal{O}(C)$-algebra over $Der(\mathcal{O}(C))$.

We write

\begin{displaymath}
T^*_f C \in cdgAlg^{op}_k/_C
\end{displaymath}
for its formal dual.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Every $S \in \mathcal{O}(C)$ defines a morphism

\begin{displaymath}
d S \colon C \to T^*_f C
\end{displaymath}
in $dgcAlg_k^{op}$ which is dually given by

\begin{displaymath}
\mathcal{O}(C) 
    \leftarrow 
   Sym_{\mathcal{O}(C)} Der(\mathcal{O}(C))
    \;\colon\;
   Sym_{\mathcal{O}(C)} [\hat S , -]
  \,,
\end{displaymath}
where $\hat S : \mathcal{O}(C) \to \mathcal{O}(C)$ is the $k$-linear multiplication operator defined by $S$ and where for $v \in Der(\mathcal{O}(C))$ we set

\begin{displaymath}
[\hat S, v] = v(S)
  \,,
\end{displaymath}
which may be regarded as the multiplication operator given by the commutator of $k$-linear endomorphisms of $\mathcal{O}(C)$ as indicated.

\end{defn}
\hypertarget{derived_critical_locus}{}\subsubsection*{{Derived critical locus}}\label{derived_critical_locus}

\begin{defn}
\label{DerivedCriticalLocusWithFormalCotangentBundle}\hypertarget{DerivedCriticalLocusWithFormalCotangentBundle}{}
\textbf{(derived critical locus)}

The \textbf{derived critical locus} of a morphism $S \colon C \to \mathbb{A}^1$ in dgcAlg\_k is the [[homotopy pullback]] $C_{\{d S = 0\}}$ in $cdgAlg^{op}/_{C}$

\begin{displaymath}
\itexarray{
    C_{\{d S = 0\}} &\to& C
    \\
    \downarrow &\swArrow& \downarrow^{\mathrlap{0}}
    \\
    C &\stackrel{d S}{\to}& T^*_f C
  }
  \,.
\end{displaymath}
\end{defn}
\begin{prop}
\label{PresentationByFreeDGCAlegbraOnMappingCone}\hypertarget{PresentationByFreeDGCAlegbraOnMappingCone}{}
\textbf{(presentation by free dgc-algebra on mapping cone)}

If $C$ is smooth in the sense that $Der(\mathcal{O}(C)) \in \mathcal{O}(C) Mod$ is cofibrant, then the derived critical locus (def. \ref{DerivedCriticalLocusWithFormalCotangentBundle}) is presented by

\begin{displaymath}
\mathcal{O}(C_{\{d S = 0\}})
   \simeq
  Sym_{\mathcal{O}(C)}
  \left(
     Cone\left( 
       Der(\mathcal{O}(C)) \stackrel{[\hat S , -]}{\to} \mathcal{O}(C)
    \right)
  \right)
    \underset{Sym_{\mathcal{O}(C)}(\mathcal{O}(C))  }{\otimes}
  \mathcal{O}(C)
  \,,
\end{displaymath}
where on the right we have the free $\mathcal{O}(C)$-algebra over the [[mapping cone]] of $[\hat S, -]$ with [[extension of scalars]] along $\mathcal{O}( C \overset{(id,0)}{\to} C \times \mathbb{A}^1 )$.

\end{prop}
\begin{proof}
Using the [[pasting law]] we may decompose the homotopy pullback into a [[pasting]] of two homotopy pullback squares as follows:

\begin{displaymath}
\itexarray{
    C_{\{d S = 0\}} &\longrightarrow& &\longrightarrow& C
    \\
    \downarrow &\swArrow& \downarrow &\swArrow& \downarrow^{\mathrlap{0}}
    \\
    C 
      &\underset{(id,0)}{\longrightarrow}& 
    C \times \mathbb{A}^1 &\underset{}{\longrightarrow}& T^*_f C
  }
  \,.
\end{displaymath}
First consider the square on the right:

By prop \ref{SymOCAdjunction} the functor $Sym_{\mathcal{O}(C)}$ is left Quillen. Hence if $Der(\mathcal{O}(C))$ is cofibrant in $\mathcal{O}(C) Mod$ then the homotopy pushout corresponding to the square on the right may be computed as the image under $Sym_{\mathcal{O}(C)}$ of the homotopy pushout in $\mathcal{O}(C) Mod$.

By the disucssion at [[model structure on dg-modules]], for these the homotopy cofibers are given by the ordinary [[mapping cone]] construction for chain complexes.

\begin{displaymath}
\itexarray{
    Cone\left(
      Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C)
    \right)
    &\leftarrow&
    Cone\left(
      Der(\mathcal{O}(C)) \stackrel{Id}{\to} Der(\mathcal{O}(C))
    \right)
    \\
    \uparrow && \uparrow
    \\
    \mathcal{O}(C)
    &\stackrel{[\hat S, -]}{\leftarrow}&
    Der(\mathcal{O}(C))
  }
  \,.
\end{displaymath}
More in detail, write

\begin{displaymath}
Cone\left(
    Der(\mathcal{O}(C)) \stackrel{Id}{\to} Der(\mathcal{O}(C))
  \right)
   \in 
  \mathcal{O}(C) Mod
\end{displaymath}
for the [[mapping cone]] on the identity:

\begin{displaymath}
\itexarray{
     \cdots
     & 
     Der(\mathcal{O}(C))_k &\stackrel{-[d_{\mathcal{O}(C)}, -]}{\to}&
     Der(\mathcal{O}(C))_{k+1}
     \\
     & \oplus &\searrow^{\pm \mathrlap{Id}}& \oplus & \cdots
     \\
     \cdots
     & Der(\mathcal{O}(C))_{k-1} 
        &\stackrel{[d_{\mathcal{O}(C)}, -]}{\to}&
     Der(\mathcal{O}(C))_k    
     &
     \cdots
  }
  \,.
\end{displaymath}
Then the [[mapping cone]] $Cone\left(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to}  \mathcal{O}(C) \right)$ is

\begin{equation}
\itexarray{
     \cdots
     & 
     Der(\mathcal{O}(C))_k &\stackrel{-[d_{\mathcal{O}(C)}, -]}{\to}&
     Der(\mathcal{O}(C))_{k+1}
     \\
     & \oplus &\searrow^{\pm [\hat S, -]}& \oplus & \cdots
     \\
     \cdots
     & \mathcal{O}(C)_{k-1} 
        &\stackrel{[d_{\mathcal{O}(C)}, -]}{\to}&
     \mathcal{O}(C)_k    
     &
     \cdots
  }
  \,.
\label{TheBVComplex}\end{equation}
If we extend the graded commutators in the evident way we may write the [[nLab:differential]] in $Cone(Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C))$ as

\begin{displaymath}
d  
   = 
  \left[
    \hat S 
      + 
    d_{\mathcal{O}(C)}
    \;,\; 
    -
  \right]
  \,.
\end{displaymath}
Here the second term will be the differential of the [[nLab:BRST-complex]] of $\mathfrak{c}$, whereas the sum is of the type of a differential in a [[nLab:BRST-BV complex]].

For that to happen, however the two copies of $\mathcal{O}(C)$ in $Sym_{\mathcal{O}(C)}(\mathcal{O}(C))$ need to be identified, this is achived by the remaining homotopy pushout corresponding to the square on the left

\begin{displaymath}
\itexarray{
    \mathcal{O}(C_{\{d S = 0\}})
    &\longleftarrow&
    Sym_{\mathcal{O}(C)}
    \left(
      Cone\left(
        Der(\mathcal{O}(C)) \stackrel{[\hat S, -]}{\to} \mathcal{O}(C)
      \right)
    \right)
    \\
    \uparrow && \uparrow
    \\
    \mathcal{O}(C)
       &\underset{}{\longleftarrow}& 
    Sym_{\mathcal{O}(C)}(\mathcal{O}(C))
  }
  \,.
\end{displaymath}
Since here the morphism on the right is the pushout of a cofibration, it is itself still a cofibration, and by assumption $Sym_{\mathcal{O}(C)}(\mathcal{O}(C))$ is cofibrant. Therefore this homotopy pushout is given by the ordinary pushout, and that yields the [[tensor product]] as in the claim.

\end{proof}
\hypertarget{ComparisonToBVBRSTExample}{}\subsubsection*{{BV-BRST complex}}\label{ComparisonToBVBRSTExample}

Traditionally the [[BV-BRST complex]] of a [[Lagrangian field theory]] is obtained by

\begin{enumerate}%
\item choosing a [[Koszul-Tate complex]] $s_{KT}$ resolving the [[shell]];


\item choosing a [[BRST complex]] $s_{BRST}$ exhibiting the [[gauge invariance]]


\item appealing to [[homological perturbation theory]], for extending the sum of the two differentials to a unified [[BV-BRST differential]]

\begin{displaymath}
s_{BV} = s_{KT} + s_{BRST} + more
\end{displaymath}


\end{enumerate}
(e.g. \href{BV-BRST+formalism#Henneaux90}{Henneaux 90. around (50)})

Vie the concept of the derived critical locus this process is systematized: Given just $s_{BRST}$ and the Lagrangian, both $s_{KT}$ and ``more'' follows (if $s_{BRST}$ does capture all the relevant gauge symmetries, that is) and the appearance of the [[antibracket]] finds its conceptual explanation.

Let $\mathfrak{a}$ be a [[Lie algebroid]] over a space $X$, with [[Chevalley-Eilenberg algebra]]

\begin{displaymath}
\mathcal{O}(\mathfrak{a})
  \;=\;
  \left(
    Sym_{\mathcal{O}(X)}(\langle c^a\rangle)
    \;,\;
    d_{\mathfrak{a}}
  \right)
\end{displaymath}
with differential given by

\begin{displaymath}
d_{\mathfrak{a}} 
    \;:\; 
  f \mapsto c^a R^i_a \frac{\partial}{\partial x^i} f
\end{displaymath}
\begin{displaymath}
d_{\mathfrak{a}} \;:\; 
   c^a 
    \mapsto 
   \frac{1}{2} C^{a}{}_{b c} c^b \wedge c^c 
  \,.
\end{displaymath}
for functions $f \in \mathcal{O}(X)$, [[infinitesimal gauge symmetries]] $R^i_a \frac{\partial}{\partial x^i}$, gauge symmetry structure functions $C^{a}{}_{b c}$ and [[ghost]] generators $c^a$.

The ``algebra of vector fields/[[derivations]]'' $Der(\mathcal{O}(\mathfrak{a}))$ on $\mathfrak{a}$ is the [[automorphism ∞-Lie algebra]] whose underlying cochain complex is

\begin{displaymath}
\itexarray{
    \left\langle 
      \frac{\partial}{\partial c^a} 
    \right\rangle
    &
    \overset{[d_{\mathfrak{a}}, -]}{\to}
    &
    \left\langle 
      \frac{\partial}{\partial x^i} 
    \right\rangle    
    \oplus
     \left\langle 
       c^a \frac{\partial}{\partial c^b} 
     \right\rangle
    \\
    -1 && 0
  }
  \,.
\end{displaymath}
We check on generators that

\begin{displaymath}
\begin{aligned}
    \left[d_{\mathfrak{a}}, \frac{\partial}{\partial c^a} \right]  
     = 
     R_a^i \frac{\partial}{\partial x^i} 
      + 
     C^b{}_{a c} c^c \frac{\partial}{\partial c^b}
  \end{aligned}
\end{displaymath}
and

\begin{displaymath}
\begin{aligned}
    \left[
      d_{\mathfrak{a}}, \frac{\partial}{\partial x^i}
    \right]
     =
    c^a \frac{\partial R_a^j}{\partial x^i} \frac{\partial}{\partial x^j}
  \end{aligned}
  \,.
\end{displaymath}
Now let

\begin{displaymath}
S \;\colon\; \mathfrak{a} \longrightarrow \mathbb{R}
\end{displaymath}
be a morphism, dually a dgc-algebra homomorphism of the form

\begin{displaymath}
\mathcal{O}(\mathfrak{a}) 
    \longleftarrow  
   \mathcal{O}(\mathbb{R}) 
     \;\colon\;  
   S^*
  \,.
\end{displaymath}
This is equivalently a function

\begin{displaymath}
S \;\colon\; X \longrightarrow \mathbb{R}
\end{displaymath}
which is [[gauge invariance|gauge invariant]]

\begin{displaymath}
\begin{aligned}
    d_{\mathfrak{a}} S 
     & = 
    c^a R_a^i \frac{\partial}{\partial x^i} S 
    \\
     & = 0
   \end{aligned}
  \,.
\end{displaymath}
We have a contraction homomorphism of $\mathcal{O}(\mathfrak{a})$-[[modules]]

\begin{displaymath}
\iota_{d S} 
  \;\colon\; 
  Der(\mathcal{O}(\mathfrak{a})) 
    \longrightarrow 
  \mathcal{O}(\mathfrak{a})
  \,.
\end{displaymath}
and may form its [[mapping cone]] \eqref{TheBVComplex}

\begin{displaymath}
\itexarray{
    \left\langle 
      \frac{\partial}{\partial c^a}
    \right\rangle
       &\stackrel{[d_{\mathfrak{a}}, -]}{\longrightarrow}&
    \left\langle 
       c^a \frac{\partial}{\partial c^b}
    \right\rangle
      \oplus
    \left\langle 
       \frac{\partial}{\partial x^i} 
    \right\rangle 
    \\
      &&
      &\searrow^{\mathrlap{\iota_{d S}}}&
    \\
      && 
      &&
    \mathcal{O}(X)
      &\stackrel{d_{\mathfrak{a}}}{\to}&
    \left\langle 
      c^a 
    \right\rangle
   \\
   -2 && -1 && 0 && 1
  }
  \,.
\end{displaymath}
On the free algebra of this

\begin{displaymath}
Sym_{\mathcal{O}(X)}
  \left(
    Der(\mathcal{O}(\mathfrak{a}))[-1] \stackrel{\iota_{d S}}{\to} \mathcal{O}(X)\oplus \langle c^a\rangle)
  \right)
\end{displaymath}
we have the differential given on generators by

\begin{displaymath}
\frac{\partial}{\partial c^a} 
   \mapsto 
  R_a^i \frac{\partial}{\partial x^i} + C^b{}_{a c} c^c 
  \frac{\partial}{\partial c^b}
\end{displaymath}
\begin{displaymath}
\frac{\partial}{\partial x^i} \mapsto \frac{\partial S}{\partial x^i}
    + 
    c^a \frac{\partial R_a^j}{\partial x^i} \frac{\partial}{\partial x^j}
\end{displaymath}
\begin{displaymath}
x^i \mapsto c^a R_a^i
\end{displaymath}
\begin{displaymath}
c^a \mapsto \frac{1}{2}C^a{}_{b c} c^b \wedge c^c
\end{displaymath}
and similarly after tensoring in order to identify the extra copy of $\mathcal{O}(X)$ with the base $\mathcal{O}(X)$.

If $\langle R_a\rangle$ is the full kernel of $\iota_{d S} : Der(C^\infty(X)) \to C^\infty(X)$ and there are no further relations, then this is the full [[BRST-BV complex]] of $S$.

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

The above material is adapted from

\begin{itemize}%
\item [[Urs Schreiber]]: \emph{[[schreiber:derived critical locus]]} Seminar notes, March 2011

\end{itemize}
(taking into account a correction provided by [[Vincent Schlegel]])

aimed at providing proof for the claim in

\begin{itemize}%
\item [[Kevin Costello]], [[Owen Gwilliam]], section 4.8.1 of \emph{[[Factorization algebras in perturbative quantum field theory]]} (\href{https://pdfs.semanticscholar.org/0eb5/652340bb414869764cf5b30feed90597558d.pdf}{pdf})

\end{itemize}
See also

\begin{itemize}%
\item [[Gabriele Vezzosi]], \emph{Derived critical loci I - Basics}, \href{http://arxiv.org/abs/1109.5213}{arxiv/1109.5213}


\item [[Tony Pantev]], [[Bertrand Toen]], M. Vaquie, G. Vezzosi, \emph{Quantization and derived moduli spaces I: shifted symplectic structures}, \href{http://arxiv.org/abs/1111.3209}{arxiv/1111.3209}


\item [[Kevin Costello]], \emph{Notes on supersymmetric and holomorphic field theories in dimension 2 and 4} (\href{http://www.math.northwestern.edu/~costello/sullivan.pdf}{pdf})



\end{itemize}
[[!redirects derived critical loci]]



\end{document}