The ABCD of topological recursion

Abstract

Kontsevich and Soibelman reformulated and slightly generalised the topological recursion of math-ph/0702045, seeing it as a quantization of certain quadratic Lagrangians in $T^*V$ for some vector space $V$. KS topological recursion is a procedure which takes as initial data a quantum Airy structure -- a family of at most quadratic differential operators on $V$ satisfying some axioms -- and gives as outcome a formal series of functions in $V$ (the partition function) simultaneously annihilated by these operators. Finding and classifying quantum Airy structures modulo gauge group action, is by itself an interesting problem which we study here. We provide some elementary, Lie-algebraic tools to address this problem, and give some elements of classification for ${\rm dim}\,V = 2$. We also describe four more interesting classes of quantum Airy structures, coming from respectively Frobenius algebras (here we retrieve the 2d TQFT partition function as a special case), non-commutative Frobenius algebras, loop spaces of Frobenius algebras and a $\mathbb{Z}_{2}$-invariant version of the latter. This $\mathbb{Z}_{2}$-invariant version in the case of a semi-simple Frobenius algebra corresponds to the topological recursion of math-ph/0702045.

Figures (6)

• Figure 1: $F_{g,n}(i_1,\ldots,i_n)$ is represented as a surface of genus $g$ with $n$ boundaries, carrying the labels $i_1,\ldots,i_n$. In this graphical language, the terms appearing in the recursion \ref{['TRForm']} are all the topologies resulting from the removal of a pair of pants $P$ bounding the first boundary. The weight of $P$ is a $B$ or a $C$ depending on whether it has one or two external boundary components.
• Figure 2: Unfolding \ref{['TRForm']} gives EORev a set $\mathfrak{G}_{g,n}(1)$ of pairs $(G,T)$ where $G$ is a trivalent graph with first Betti number $g$ and $n$ leaves, with cyclic order at each vertex, and $T$ is a spanning tree rooted at the first leaf, having the extra property that edges which are not in $T$ must connect vertices $v$ and $v'$ which are parent. This means that the geodesic in $T$ from the root to $v$ contains (or is contained) in the geodesic from the root to $v'$. Vertices incident to a loop are assigned a $D$, vertices incident to one external leg are $B$s, vertices incident to two external legs are $A$s. Internal vertices can be $A,B,C$ as prescribed by the recursive construction of the graph --- which is remembered by the spanning tree rooted at the first leg. We have listed these graphs for low values of $(g,n)$. The $\tfrac{1}{2^p}$ is the symmetry factor which arises from the repetition of factor of $\tfrac{1}{2}$ in the $C$-term of \ref{['TRForm']}. A $(k)$ indicates that there are $k$ such graphs, which differ by the labeling $2,\ldots,n$ of the legs. When two such graphs are related by an exchange of the two legs outgoing from a $C$, the two graphs give the same contribution to $F_{g,n}$, and we listed it as a single graph with a factor of $\tfrac{1}{2}$ less.
• Figure 3: A $\Box$ indicates a $B^{i_1}_{i_2,i_3}$ with incident edges labeling as indicated by the numbers. A $\boldsymbol{\times}$ indicates an $A^{*}_{**}$. A $\bullet$ represents a $C^{i_1}_{**}$, with first incident edge carrying the upper index. The three relations are that these combinations are symmetric with respect to permutation of the two left legs.
• Figure 4: Each edge carries an index. Indices carried by dashed edge are summed over. $\bullet$ means symmetrisation of the indices of the edge incident at that vertex, while $\circ$ means antisymmetrisation. The arrow indicates which index is placed first, and thus defines the order of composition of the two operators.
• Figure 5: Genus $0$ graphs without inner trivalent vertices.

Theorems & Definitions (73)

• Definition 2.1
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Remark 2.2
• Definition 3.1
• Definition 3.2
• Lemma 3.1
• Lemma 3.2