nLab open-closed homotopy algebra

Definition

Let $\mathcal{H}=\mathcal{H}_o\oplus\mathcal{H}_c$ be a $\mathbb{Z}$-graded vector space, and $\mathfrak{k}$ a collection of $n$-ary maps that make $(\mathcal{H}_c,\mathfrak{l})$ an $L_\infty$-algebra. If there is a collection of maps

$$\mathfrak{n} \;=\; \big\{ n_{k,l} \colon (\mathcal{H}_c ^{\otimes k})\otimes (\mathcal{H}_o )^{\otimes l} \to \mathcal{H}_o \big\}_{k,l\geq 0}$$

each graded-symmetric for $(\mathcal{H}_c ^{\otimes k})$ satisfying the identities

$$\begin{array}{l} \sum_{p+r=n} \sum_{\sigma\in S_n} \frac{(-1)^{\epsilon(\sigma)}}{p! r!} n_{1+r,m}(l_p (c_{\sigma (1)} , \cdots , c_{\sigma (p)}),c_{\sigma (p+1)}, \cdots, c_{\sigma (n)}; o_1, \cdots, o_m) \\ \;+\; \sum_{p+r=n} \sum_{i+s+j=m} \sum_{\sigma\in S_n} \frac{(-1)^{\mu_{p,i}(\sigma)}}{p! r!} n_{p,i+1+j} (c_{\sigma (1)}, \cdots , c_{\sigma (p); o_1, \cdots, o_i, n_{r,s}(c_{\sigma (p+1), \cdots, c_{\sigma(n)};o_{i+1}, \cdots, o_{i+s}}}), o_{i+s+1}, \cdots, o_m) \\ \;=\; 0 \,, \end{array}$$

where the sign exponent $\mu_{p,i}(\sigma)$ is defined as

$$\mu_{p,i}(\sigma) = \epsilon(\sigma)+ (c_{\sigma (1)}+ \cdots + c_{\sigma(p)}) + (o_1+ \cdots + o_i) + (o_1+ \cdots + o_i)(c_{\sigma (p+1}+ \cdots + c_{\sigma (n)})$$

then $(\mathcal{H},\mathfrak{l},\mathfrak{n})$ is called a weak open-closed homotopy algebra (weak OCHA).

If furthermore one has that $l_0=n_{0,0} =0$ then this is an open-closed homotopy algebra (OCHA).

Properties

By definition, $(\mathcal{H}_c,\mathfrak{l})$ is a Lie-infinity-algebra. Furthermore, given a (weak) OCHA one can show that $(\mathcal{H}_o,\{n_{0,k}\})$ is an (weak) A-infinity algebra. These are interpreted as the closed, and open sectors, respectively (see string field theory for more on this).

References

General:

• Hiroshige Kajiura, Jim Stasheff. Open-closed homotopy algebra in mathematical physics. J. Math. Phys. 47, 023506 (2006). (doi).