[Paper Review] TQFT string operations in open-closed string topology

Hirotaka Tamanoi|ArXiv.org|Mar 7, 2008

Algorithms and Data Compression13 references3 citations

TL;DR

This paper establishes that most degree-zero string operations in open-closed string topology vanish, identifying a complete and finite list of open-closed cobordism surfaces—characterized by genus zero, zero or one window, and specific boundary string configurations—where nontrivial operations can occur. The key contribution is a classification of all possible non-vanishing string operations via topological invariants such as genus, number of windows, and incoming/outgoing string counts on boundary components.

ABSTRACT

To open-closed cobordism surfaces, open-closed string topology associates topological quantum field theory (TQFT) operations, namely string operations, which depend only on homeomorphism types of surfaces and which satisfy the sewing property. We show that most TQFT string operations vanish in open-closed string topology. We describe those open-closed cobordisms with vanishing string operations, and give a short list of open-closed cobordisms with possibly nontrivial string operations.

Motivation & Objective

• To determine which open-closed cobordism surfaces in string topology support nontrivial degree-zero string operations.
• To classify all such cobordisms where string operations may be non-vanishing, based on topological invariants like genus, number of windows, and boundary string configurations.
• To extend previous results on closed string topology vanishing to the open-closed setting, providing a complete characterization of nontrivial operations.

Proposed method

• Analyzes string operations as TQFT maps between homology groups of free loop spaces and path spaces of D-branes.
• Applies transfer maps and cohomological techniques to study degree-lowering operations in homology, particularly focusing on comodule structures.
• Uses vanishing results for five key operations: open window, genus one, handle attachment, double saddle, and double closed window operations.
• Employs degree arguments and module structure properties (e.g., φIJ(a·b) = φII(a)·b) to prove vanishing of operations under specific topological conditions.
• Applies the sewing property of TQFTs to decompose complex cobordisms into simpler components whose operations are known to vanish.
• Derives a complete list of exceptional cobordisms by eliminating all cases where any of the five vanishing conditions apply.

Experimental results

Research questions

• RQ1 Which open-closed cobordisms in string topology can support nontrivial degree-zero string operations?
• RQ2 Under what topological conditions do string operations vanish in the open-closed setting?
• RQ3 What is the complete and finite list of cobordism types where nontrivial string operations may occur?
• RQ4 How do genus, number of windows, and boundary string types (incoming/outgoing open/closed) affect the non-vanishing of string operations?
• RQ5 Can the structure of H∗(LM)-comodule maps on path spaces P_{IJ} be used to prove vanishing of certain string operations?

Key findings

• All string operations vanish if the cobordism has genus g(Σ) ≥ 1.
• All string operations vanish if the number of windows ω(Σ) ≥ 2.
• All string operations vanish if the number of boundary components with both incoming and outgoing open strings t(Σ) ≥ 2.
• All string operations vanish if there are q(Σ) ≥ 3 outgoing closed strings.
• All string operations vanish if s(Σ) ≥ 1 and s(Σ)+q(Σ) ≥ 2, i.e., if there is at least one boundary with only outgoing open strings and at least one outgoing string overall.
• The only possible nontrivial string operations occur for cobordisms with genus 0, at most one window, and specific boundary configurations: (1) one outgoing closed string with one window, (2) one incoming closed string and two outgoing closed strings with one window, (3) one boundary with both incoming and outgoing open strings and no outgoing closed or purely outgoing open strings, (4) one incoming closed string and one outgoing closed string with no open strings, and (5) one incoming closed string and one outgoing closed string with no open strings and no windows.