[Paper Review] Closed String TCFT for Hermitian Calabi-Yau Elliptic Spaces

Kevin J. Costello, Thomas Tradler+1 more|ArXiv.org|Jul 18, 2008

Advanced Algebra and Geometry22 references3 citations

TL;DR

This paper constructs an explicit, geometric action of the chains on the moduli space of Riemann surfaces on the Hochschild complex of a Hermitian Calabi-Yau elliptic space, providing a chain-level refinement of string topology and B-model operations at all genera. The key contribution is a lax algebra structure over a prop of meta-graphs quasi-isomorphic to Segal's prop, yielding homological field theory operations on Hochschild homology that extend known string topology operations and match the higher-genus B-model partition function via a geometric, metric-dependent construction that is homotopy-invariant under metric choice.

ABSTRACT

We describe an explicit action of the prop of the chains on the moduli space of Riemann surfaces on the Hochschild complex of a Calabi-Yau elliptic space. One example of such an elliptic space extends the known string topology operations, for all compact simply-connected manifolds, to a collection indexed by the de Rham currents on the moduli space. Another example pertains to the B-model at all genera.

Motivation & Objective

• To extend the open TCFT construction for Calabi-Yau elliptic spaces to a closed TCFT in a geometric, non-algebraic way, avoiding homological algebra replacements of differential forms with cohomology.
• To provide an explicit, geometric action of the chains on the moduli space of Riemann surfaces on the Hochschild complex of a Calabi-Yau elliptic space, using a Hermitian metric as a geometric input.
• To show that this construction yields operations on Hochschild homology that are equivalent to known string topology and higher-genus B-model operations, but at the chain level.
• To establish a direct link between the geometric construction and the algebraic constructions of Kontsevich-Soibelman and Costello, proving homotopy equivalence.

Proposed method

• Introduce the prop of meta-graphs $MG$, which is weakly equivalent to Segal’s prop $\mathcal{S}$, and define its action on a model of the Hochschild chain complex.
• Construct a lax algebra over $C_*(MG)$, with structure maps $C_*(MG(n,m)) \otimes L_*(\mathcal{A})[n] \to L_*(\mathcal{A})[m]$ compatible with differentials, composition, and tensor products.
• Use the heat kernel and differential forms on metrized ribbon graphs as a piecewise linear model for the moduli space of Riemann surfaces.
• Apply the homological perturbation lemma to convert the Calabi-Yau elliptic space into a finite-dimensional cyclic $A_\infty$ algebra, enabling the TCFT construction.
• Establish a zig-zag of quasi-isomorphisms between the geometric construction and the algebraic one via the Poincaré model and heat kernel at time zero.
• Use spectral sequence arguments to show that the geometric and algebraic constructions are quasi-isomorphic, proving equivalence at the homology level.

Experimental results

Research questions

• RQ1 Can a closed TCFT be constructed for a Calabi-Yau elliptic space using a geometric, differential-geometric input (e.g., a Hermitian metric) rather than purely algebraic homological algebra?
• RQ2 Does the proposed construction yield operations on Hochschild homology that extend the classical string topology operations to the entire moduli space of Riemann surfaces?
• RQ3 Is the geometric action of the moduli space chains on the Hochschild complex equivalent to the algebraic constructions of Kontsevich-Soibelman and Costello, despite being more geometrically explicit?
• RQ4 Can this construction be applied to the higher-genus B-model, and does it recover the partition function as an integral over the fundamental chain of moduli spaces?
• RQ5 Does the construction remain independent of the choice of Hermitian metric up to homotopy, ensuring geometric invariance?

Key findings

• The paper constructs a lax algebra over the chain complex of the meta-graph prop $MG$, which is weakly equivalent to Segal’s prop $\mathcal{S}$, acting on a model of the Hochschild complex of a Calabi-Yau elliptic space.
• The resulting structure induces a homological field theory on Hochschild homology, with a map $H_*(\mathcal{S}(n,m)) \to \text{Hom}(HH_*(\mathcal{A},\mathcal{A})^{\otimes n}, HH_*(\mathcal{A},\mathcal{A})^{\otimes m})$.
• For the de Rham complex of a simply-connected, closed, oriented manifold $M$, the construction recovers string topology operations indexed by the homology of the full moduli space of Riemann surfaces.
• The construction is independent of the choice of Hermitian metric up to homotopy, ensuring geometric invariance.
• The operations commute with the string topology operations from the collapsed ribbon graphs $\tilde{\gamma}$, as shown by a commuting diagram involving $H_*(L_*(\Omega^*M)[n])$ and $H^*(\mathcal{L}M)^{\otimes n}$.
• For the B-model on a Calabi-Yau manifold $X$, the construction yields an explicit action of moduli space chains on the Hodge cohomology $H^{-\ast}(X, \Omega^\ast(X))$, which is dual to the space of extended deformations $\oplus H^i(X, \wedge^j TX)$, and provides a geometric realization of the partition function as an integral over the fundamental chain of moduli spaces.