[Paper Review] String Topology: Background and Present State

Dennis Sullivan|ArXiv.org|Oct 22, 2007

Homotopy and Cohomology in Algebraic Topology50 references17 citations

TL;DR

This paper establishes a 2D field theory for closed strings on the free loop space of an oriented manifold, using equivariant chains and a master equation $dX + X * X = 0$ to define a quantum Lie bialgebra structure. The construction arises from compactified moduli spaces of Riemann surfaces and provides a chain-level generalization of string topology operations, resolving the involutive identity up to homotopy.

ABSTRACT

The data of a "2D field theory with a closed string compactification" is an equivariant chain level action of a cell decomposition of the union of all moduli spaces of punctured Riemann surfaces with each component compactified as a pseudomanifold with boundary. The axioms on the data are contained in the following assumptions. It is assumed the punctures are labeled and divided into nonempty sets of inputs and outputs. The inputs are marked by a tangent direction and the outputs are weighted by nonnegative real numbers adding to unity. It is assumed the gluing of inputs to outputs lands on the pseudomanifold boundary of the cell decomposition and the entire pseudomanifold boundary is decomposed into pieces by all such factorings. It is further assumed that the action is equivariant with respect to the toroidal action of rotating the markings. A main result of compactified string topology is the Theorem (closed strings): Each oriented smooth manifold has a 2D field theory with a closed string compactification on the equivariant chains of its free loop space mod constant loops. The sum over all surface types of the top pseudomanifold chain yields a chain X satisfying the master equation dX + X*X = 0 where * is the sum over all gluings. This structure is well defined up to homotopy. The genus zero parts yields an infinity Lie bialgebra on the equivariant chains of the free loop space mod constant loops. The higher genus terms provide further elements of algebraic structure called a "quantum Lie bialgebra" partially resolving the involutive identity. There is also a compactified discussion and a Theorem 2 for open strings as the first step to a more complete theory. We note a second step for knots.

Motivation & Objective

• To formalize a 2D field theory with closed string compactification using equivariant chains on the free loop space modulo constant loops.
• To extend classical string topology operations (loop product, bracket, cobracket) to the chain level with homotopy coherence.
• To resolve the involutive identity in the Lie bialgebra structure up to first homotopy, introducing the concept of a 'quantum Lie bialgebra'.
• To unify geometric, algebraic, and homotopical perspectives in string topology via moduli space compactification and gluing.
• To provide a foundation for open string topology and applications to symplectic topology using resolution techniques in homotopy theory.

Proposed method

• Construct a cell decomposition of the moduli space of punctured Riemann surfaces, compactified as a pseudomanifold with boundary.
• Define an equivariant chain-level action using labeled inputs (with tangent directions) and weighted outputs (summing to unity).
• Implement gluing operations along the pseudomanifold boundary, ensuring all codimension-one strata are captured via factorization.
• Use the Poincaré dual cocycle to define a diffuse intersection product at the chain level, generalizing the classical intersection product.
• Formulate the master equation $dX + X * X = 0$, where $X$ is the sum of top-dimensional chains over all surface types and $*$ denotes gluing.
• Apply homotopy theory of $E_inity$ and $L_inity$ structures to resolve algebraic identities (e.g., Jacobi, co-Jacobi, involutive) up to homotopy.

Experimental results

Research questions

• RQ1 How can the classical loop product and bracket in homology be lifted to a chain-level structure with full homotopy coherence?
• RQ2 What algebraic structure emerges from the sum of top-dimensional chains over all compactified moduli spaces of Riemann surfaces?
• RQ3 How is the involutive identity in the Lie bialgebra structure resolved up to homotopy in the chain-level construction?
• RQ4 In what way does the master equation $dX + X * X = 0$ encode the full 2D field theory for closed strings?
• RQ5 How can the framework be extended to open strings and symplectic topology via resolutions of dgOAs and Kuranishi-style models?

Key findings

• The genus-zero part of the master equation yields an $L_inity$ bialgebra structure on the equivariant chains of the free loop space modulo constant loops.
• The full structure, including higher genus contributions, defines a 'quantum Lie bialgebra' that resolves the involutive identity up to first homotopy.
• The master equation $dX + X * X = 0$ is well-defined up to homotopy and governs the entire 2D field theory on the loop space.
• The chain-level loop product is infinitely chain homotopy graded commutative and associative, realizing an $E_inity$ structure.
• The cobracket and co-Jacobi relations are realized at the chain level with explicit chain homotopies, ensuring homotopy coherence.
• The framework provides a resolution of the algebraic structure via free triangular dgOAs, with homotopy equivalence preserving the master equation package.