[Paper Review] On fermionic representation of the framed topological vertex

Fusheng Deng, Jian Zhou|arXiv (Cornell University)|Nov 2, 2011

Black Holes and Theoretical Physics3 references3 citations

TL;DR

This paper proposes and proves the Framed ADKMV Conjecture, a fermionic representation of the framed topological vertex analogous to the ADKMV Conjecture for the standard topological vertex. It establishes a simple fermionic formula via Bogoliubov transforms of the vacuum state, proving the one- and two-legged cases and deriving a determinantal formula for the three-legged case under the conjecture, extending fermionic methods to framed Calabi-Yau geometries.

ABSTRACT

The Gromov-Witten invariants of \mathbb{C}^3 with branes is encoded in the topological vertex which has a very complicated combinatorial expression. A simple formula for the topological vertex was proposed by Aganagic et al in the fermionic picture. We will propose a similar formula for the framed topological vertex and prove it in the case when there are one or two branes.

Motivation & Objective

• To generalize the ADKMV Conjecture—originally for the standard topological vertex in the fermionic Fock space—to the framed topological vertex in toric Calabi-Yau threefolds with branes.
• To provide a simple fermionic expression for the framed topological vertex via Bogoliubov transforms of the vacuum, analogous to the original ADKMV Conjecture.
• To prove the one-legged and two-legged cases of the Framed ADKMV Conjecture rigorously using fermionic techniques.
• To derive a determinantal formula for the three-legged framed topological vertex based on the conjecture, extending known results to the framed case.

Proposed method

• Propose the Framed ADKMV Conjecture as a fermionic representation of the framed topological vertex, expressed as a vacuum expectation value of a Bogoliubov-transformed fermionic operator.
• Use the boson-fermion correspondence to map symmetric functions and Schur functions in the bosonic picture to fermionic Fock space states.
• Apply the Koszul sign convention and signed permutations to track signs in fermionic operator products, particularly in the expansion of the vertex operator.
• Derive a determinantal formula for the three-legged framed topological vertex by decomposing the fermionic vacuum expectation value into block matrices involving Schur, complete homogeneous, and elementary symmetric functions.
• Utilize Frobenius notation for partitions and the identity $ \kappa_{\mu} = \sum_{i=1}^k m_i(m_i+1) - n_i(n_i+1) $ to express fermionic matrix elements.
• Verify the conjecture in the one- and two-legged cases by direct computation using fermionic creation/annihilation operators and vacuum expectation values.

Experimental results

Research questions

• RQ1 Can the ADKMV Conjecture’s fermionic simplicity be extended to the framed topological vertex in toric Calabi-Yau threefolds with branes?
• RQ2 What is the explicit fermionic representation of the framed topological vertex in terms of Bogoliubov transforms of the vacuum state?
• RQ3 How do the one- and two-legged cases of the framed topological vertex behave under the fermionic formalism, and can they be rigorously proven?
• RQ4 What is the determinantal structure of the three-legged framed topological vertex under the Framed ADKMV Conjecture?
• RQ5 How do the signs and combinatorics of fermionic matrix elements arise in the general case, and how are they tracked via Koszul conventions?

Key findings

• The one-legged and two-legged cases of the Framed ADKMV Conjecture are rigorously proven using fermionic Fock space techniques and vacuum expectation values.
• The three-legged framed topological vertex admits a determinantal formula derived from the conjecture, expressed as a sum over partitions with signed products of Schur, complete homogeneous, and elementary symmetric function matrices.
• The sign factors in the determinantal formula are fully tracked using the Koszul sign convention and permutation signs of ordered sets of integers.
• The formula involves block matrices with zero blocks and non-zero entries in Schur, $ \bar{h} $, and $ \bar{e} $ functions, reflecting the combinatorial structure of the vertex.
• The derivation confirms that the framed topological vertex can be expressed as a product of fermionic operators acting on the vacuum, generalizing the ADKMV framework.
• The result generalizes previous formulas (e.g., (37) and (69)) to the framed case, providing a systematic method for computing Gromov-Witten invariants in the presence of branes.