[Paper Review] Quantum Background Independence In String Theory

Edward Witten|ArXiv.org|Jun 23, 1993

Black Holes and Theoretical PhysicsPhysics and Astronomy18 references199 citations

TL;DR

This paper argues that the holomorphic anomaly in topological string theory—originally identified by Bershadsky, Cecotti, Ooguri, and Vafa—provides a fundamental explanation for the failure of background independence in string theory. It shows that while background independence fails order-by-order in perturbation theory, the exact partition function encodes a background-independent state in an auxiliary quantum Hilbert space, offering a novel form of 'quantum background independence'.

ABSTRACT

Not only in physical string theories, but also in some highly simplified situations, background independence has been difficult to understand. It is argued that the ``holomorphic anomaly'' of Bershadsky, Cecotti, Ooguri, and Vafa gives a fundamental explanation of some of the problems. Moreover, their anomaly equation can be interpreted in terms of a rather peculiar quantum version of background independence: in systems afflicted by the anomaly, background independence does not hold order by order in perturbation theory, but the exact partition function as a function of the coupling constants has a background independent interpretation as a state in an auxiliary quantum Hilbert space. The significance of this auxiliary space is otherwise unknown.

Motivation & Objective

• To resolve the persistent challenge of background independence in string theory, particularly in topological string models.
• To understand why the mirror map in Calabi-Yau compactifications appears to depend on a base-point choice in the B-model moduli space.
• To clarify the origin of background dependence in the space-time effective field theory of the closed B-model topological string.
• To investigate whether the genus-zero partition function vanishing at a base-point implies it vanishes identically, challenging background independence.
• To explore whether the holomorphic anomaly provides a mechanism for quantum background independence in string theory.

Proposed method

• Analyzes the structure of moduli spaces in the A and B models of topological string theory, particularly focusing on the affine structure of the A-model and the complex structure dependence of the B-model.
• Introduces special coordinates on the B-model moduli space via a base-point complex structure and holomorphic three-form, which induce a flat structure despite the non-flat geometry.
• Applies the holomorphic anomaly equation—derived from the cohomology of b0 and b̄0 operators—to describe the non-trivial t–t̄ dependence of the partition function.
• Identifies the partition function as a section of a prequantum line bundle, with its transformation properties under change of base-point revealing quantum background independence.
• Demonstrates that the partition function, though background-dependent in perturbation theory, defines a single state in an auxiliary Hilbert space, implying a deeper background-independent interpretation.
• Uses the genus-one term F₁ as a probe of the prequantum line bundle, highlighting its role in the quantum structure of the theory.

Experimental results

Research questions

• RQ1 Why does the flat structure of the A-model moduli space correspond to a base-point at infinity in the mirror B-model, and what is the analog of this in the A-model?
• RQ2 What causes the background dependence in the space-time effective field theory of the closed B-model topological string, and how can it be reconciled with background independence?
• RQ3 Does the vanishing of the genus-zero partition function at a base-point imply it vanishes identically, and if not, how is this compatible with background independence?
• RQ4 How does the holomorphic anomaly equation encode a form of quantum background independence despite perturbative background dependence?
• RQ5 Can the auxiliary Hilbert space constructed from the partition function be interpreted as a quantum phase space, and what would this imply for cosmological initial conditions?

Key findings

• The holomorphic anomaly equation provides a fundamental explanation for the failure of background independence in perturbative string theory, particularly in the B-model.
• Although the partition function is background-dependent order-by-order in perturbation theory, the exact partition function defines a single state in an auxiliary quantum Hilbert space, implying a hidden form of background independence.
• The genus-zero partition function vanishes at the base-point, but this does not imply it vanishes identically; the anomaly allows for non-trivial dynamics despite this behavior.
• The connection used in the quantization of the affine moduli space is projectively flat, leading to a phase ambiguity in the wave function that is resolved by interpreting the partition function as a section of a prequantum line bundle.
• The partition function's dependence on the coupling constants is not perturbatively background-independent, but its full quantum definition is, suggesting a non-perturbative realization of background independence.
• The structure of the holomorphic anomaly closely resembles the equation for quantum background independence, indicating a deep and non-coincidental connection between the two.