sigma.tex

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\title{$\sigma$-models in the BV formalism}
\maketitle

\section{The Riemannian $\sigma$-model} 

Suppose that $(M,h)$ is a Riemannian manifold. 
The way a physicist writes the action functional for the $\sigma$-model of maps $\phi \colon \Sigma \to M$ is
\[
\int_{\Sigma}  \phi \triangle \phi .
\]
Let us rewrite this in the first-order formalism. 
Introduce a field $A \in \Omega^1(\Sigma, \phi^* \T_M^*)$ and consider the action
\[
\int_{\Sigma} A \d \phi - \frac12 \int_\Sigma h^{-1} (A, \star A).
\]
Here, $\star$ is the Hodge star operator on $\Sigma$ and $h^{-1}$ is dual to the metric $h$.

\subsection{$L_\infty$ space}

Let $(M, \fg_M)$ be the $L_\infty$ space with $\clie^\bu(\fg_M) \cong \Omega^\bu(M, J \ul{\CC})$. 
There is a quasi-isomorphism
\[
j_\infty \colon C^\infty_M \to \clie^\bu(\fg_M) 
\]
induced by taking the $\infty$-jet of a smooth function. 

Given any vector bundle $E \to M$ we obtain an $L_\infty$ $\fg_M$-module that we denote by the same symbol $E$. 
There is an identification of $\Omega^\bu_M$-modules 
\[
\Omega^\bu(M , J E) = \clie^\bu(\fg_M ; E) 
\]
and hence a quasi-isomorphism 
\[
j_\infty \colon \cE \xto{\simeq} \clie^\bu(\fg_M ; E) .
\]

As an example, consider the bundle $E = {\rm Sym}^2 (\T^*_M)$. 
At the level of $L_\infty$ spaces this is the $\fg_M$-module $\Sym^2(\fg^*_M[-1])$. 
The metric $h$ defines a section $h^{-1}$ of ${\rm Sym}^2 (\T^*_M)$. 
Thus, we can view 
\[
j_\infty h^{-1} \in \clie^\bu\left(\fg_M ; \Sym^2(\fg^*_M[-1]) \right) .
\]

\subsection{The BV model} 

Given any local $L_\infty$ algebra $\cG$ together with a local cocycle 
\[
\alpha \in \cloc^\bu(\cG) 
\]
of cohomological degree zero we obtain a BV theory which we denote by
\[
\T^*_\alpha [-1] (\cG[1]) .
\]
The fields are $\cG[1] \oplus \cG^*[-2]$ and the action is 
\[
S_0 + \alpha
\]
where $S_0$ is the standard BF theory action. 

Consider the following local $L_\infty$ algebra on $\Sigma$:
\[
\cG = C^\infty_\Sigma \otimes \fg_M \ltimes \Omega^1_\Sigma \otimes \fg_M^* [-2] ,
\]
We denote the elements by $(\phi, A)$. 
Notice that the local $L_\infty$ structure is completely independent of the metric.
We define a local cocycle $\alpha$ on $\cG$ by the formula
\[
\alpha (\phi, A) = \int_\Sigma A \d \phi + ?? + \int_\Sigma j_\infty h^{-1} (A \star A ; \phi) .
\]

\begin{dfn}
The perturbative Riemannian $\sigma$-model of maps $\Sigma \to M$ (near the constant map) is the BV theory $\T^*_\alpha [-1](\cG)$ with $\cG$ and $\alpha$ as above. 
\end{dfn}

Since $\alpha$ depends on the inverse of the Riemannian metric, we have the following heuristic 
\[
\lim_{vol(M) \to \infty} \T^*_\alpha [-1](\cG) = \T^* [-1](\cG') .
\]
In other words, the large volume limit of the Riemannian $\sigma$-model is BF theory for (a different) local $L_\infty$ algebra $\cG'$. 

\begin{eg}
Let's consider a (non flat) example.
We take $M$ to be affine space $\RR^2$ equipped with the metric 
\[
h_\lambda = \lambda e^{-y^2} (\d x^2 + \d y^2) 
\]
where $\lambda > 0$ is a parameter.

In this case the $L_\infty$ space is simply $(\RR^2, \fg_{\RR^2} = \RR^2 [-1])$, where $\RR^2 [-1]$ is equipped with the trivial $L_\infty$ structure.
Therefore, we can take as the fields
\begin{align*}
\phi_1, \phi_2 & \in C^\infty_\Sigma \\
A_1 , A_2 & \in \Omega^1_\Sigma .
\end{align*}
The anti-fields are
\begin{align*}
\phi_1^+, \phi_2^+ & \in \Omega^2_\Sigma [-1] \\
A_1^+ , A_2^+ & \in \Omega^1_\Sigma [-1] ,
\end{align*}
though, they will not appear in the action. 

We read off the Taylor components of the inverse metric $h^{-1}$ as
\[
j_\infty h_\lambda^{-1} (A \star A ; \phi) = (A_i \star A_i) \sum_{n \geq 0} \frac{1}{n!} \phi^{2n} .
\]

Thus, the full BV action is 
\[
\int_\Sigma A_i \d \phi_i + \lambda^{-1} \sum_{n \geq 0} \frac{1}{2n!} \int_{\Sigma} (A_i \star A_i)  \phi^{2n} .
\]
The large volume limit corresponds to taking $\lambda \to 0$, in which case it is clear that we recover the standard BF type action.

In this example, the local $L_\infty$ algebra $\cG'$ is simply the cochain complex 
\[
(C^\infty_\Sigma)^{\oplus 2} [-1]_{\phi_i} \xto{\d} (\Omega^1_\Sigma)^{\oplus 2} [-2]_{A_i^+} .
\]
\end{eg}
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