The followings are from Mnev's paper about BV formalism.

Example 4.15(Definition of split supermanifold)Let $E \to M$ be a rank $m$ vector bundle over $n$-manifold $M$, then there exists a split $(n|m)$-supermanifold $\Pi E$ with body $M$ and structure sheaf $\mathcal O_{\Pi E} = \Gamma(M ,\bigwedge^{\bullet}E^*)$.

Section4.2.4 (Definition of Berezin line bundle of split supermanifold)Let $\mathcal M = \Pi E$ be the split $(n|m)$-supermanifold of vector bundle $E \to M$.

Berezin line bundle of $\mathcal M$ is $\text{Ber}(\mathcal M) := \bigwedge^{n} T^*M\otimes \bigwedge^m E$ over $M$.

For $T^*M \to M$, we have $\text{Ber}(\Pi T^*M) \cong (\bigwedge^{n} T^*M)^{\otimes 2}$. This is (161).

Then in (162), it writes

Similarly, for $\mathcal N$ a supermanifold, one has $\text{Ber}(\Pi T^*\mathcal N)|_{\mathcal N} \cong\text{Ber}(\mathcal N)^{\otimes 2}$.

Here we understand $\text{Ber}(\mathcal N)$ as a line bundle over $\mathcal N$ and the l.h.s. is a pullback of a line bundle over $\Pi T^* \mathcal N$ to $\mathcal N$.

I have difficulty in understanding the above sentence.

Suppose $\mathcal N$ is a $(k|n-k)$-supermanifold of body $N$, then $T^*\mathcal N$ has dimension $(2k|n-2k)$ and $\Pi T^*\mathcal N$ has dimension $(n|n)$.

**Question 1:**

What is the expression of $\text{Ber}(\Pi T^*\mathcal N)$ for supermanifold $\mathcal N$?

**Question 2:**

From Batchelor's theorem, we can assume $\mathcal N = \Pi{B}$ for some bundle $B \to N$.

By definition, $\text{Ber}(\mathcal N) = \bigwedge^{k}T^*N \otimes \bigwedge^{n-k} B$, so how can we show $$\text{Ber}(\Pi T^*\mathcal N)|_{\mathcal N} \cong \text{Ber}(\mathcal N)^{\otimes 2} ?$$

**Update:** My senior told me that this is a conclusion of adjunction formula for supermanifold, though I'm not very familiar with this and I can't understand the followings. Here's the proof:

$\text{Ber}$ can be defined on any locally free sheaf. Especially, when we say $\text{Ber}(X)$ we actually mean $\text{Ber}(T^*_X)$, here $T^*_X$ means cotangent sheaf on $X$.

$X \hookrightarrow Y$ is closed embedding of smooth supermanifold, ideal of $X$ is $\mathcal I$, then $$0 \to \mathcal I/\mathcal I^2 \to \Omega^{1}_{Y}|_{X} \to \Omega^{1}_{X} \to 0$$ is exact. This implies $\text{Ber}(Y)|_{X} \cong \text{Ber}(X) \otimes \text{Ber}(\mathcal I / \mathcal I^2)$.

Take $Y = V(\Pi T_{X})$ (here $V(E)$ means $E^*$ bundle and $T_X$ means tangent sheaf), then $\mathcal I/\mathcal I^2 = \Pi T_X$ and $$\text{Ber}(\Pi T_X) = (\text{Ber}(T_X))^{-1} = \text{Ber}(X)$$ here $(\text{Ber}(T_X))^{-1}$ means dual bundle of $\text{Ber}(T_X)$.

Or is there any reference for this?

I really can't find too much information of **Berezin bundle**.

Thanks for your time and effort.