Realizing Arbitrary BPS Decay Orders Via Attractor Flows

Introduction

In the realm of string theory, the study of BPS (Bogomol'nyi-Prasad-Sommerfield) states and their decay orders has been a topic of significant interest. BPS states are stable solutions to the equations of motion that preserve some of the supersymmetry of the theory. The decay order of a BPS state refers to the rate at which it decays into other states. In this article, we will explore the concept of realizing arbitrary BPS decay orders via attractor flows in the context of type IIA string theory compactified on a compact Calabi-Yau 3-fold.

Background and Motivation

Type IIA String Theory and Calabi-Yau 3-Folds

Type IIA string theory is a ten-dimensional superstring theory that includes both bosonic and fermionic degrees of freedom. When compactified on a compact Calabi-Yau 3-fold, $X$, the theory reduces to a four-dimensional effective theory. The compactification manifold, $X$, is a complex manifold with a Kähler metric and a holomorphic volume form.

Charge Lattice and Symplectic Pairing

The charge lattice, $\Gamma$, is defined as the third cohomology group of $X$ with integer coefficients, $H^{3}(X,\mathbb Z)$. The symplectic pairing, $\langle\,,\rangle$, is a non-degenerate bilinear form on $\Gamma$ that satisfies certain properties. The symplectic pairing plays a crucial role in the study of BPS states and their decay orders.

Attractor Flows and BPS Decay Orders

Attractor flows are solutions to the attractor equations, which describe the behavior of BPS states in the presence of a non-trivial moduli space. The attractor equations are given by:

$$\frac{\partial}{\partial \phi} \left( e^{-K} \right) = 0$$

where $K$ is the Kähler potential and $\phi$ is a moduli field. The attractor equations describe the flow of BPS states in the moduli space, and the decay order of a BPS state is related to the attractor flow.

Realizing Arbitrary BPS Decay Orders

The Main Result

The main result of this article is that arbitrary BPS decay orders can be realized via attractor flows in type IIA string theory compactified on a compact Calabi-Yau 3-fold. This result is based on the following theorem:

Theorem 1: Given a compact Calabi-Yau 3-fold, $X$, and a charge lattice, $\Gamma$, with symplectic pairing, $\langle\,,\rangle$, there exists an attractor flow that realizes any desired BPS decay order.

Proof of Theorem 1

The proof of Theorem 1 involves several steps. First, we need to show that the attractor equations can be solved for any given moduli field, $\phi$. This involves using the symplectic pairing and the Kähler potential to rewrite the attractor equations in a more tractable form.

Next, we need to show that the attractor flow can be used to realize any desired BPS decay order. This involves using the attractor flow to compute the decay rate of a BPS state and showing that it can be made arbitrarily small.

Implications of Theorem 1

Theorem 1 has several implications for our understanding of BPS states and their decay orders in type IIA string theory compactified on a compact Calabi-Yau 3-fold. First, it shows that arbitrary BPS decay orders can be realized via attractor flows, which provides a new tool for studying BPS states.

Second, it shows that the decay order of a BPS state is not fixed by the compactification manifold, $X$, but rather by the attractor flow. This has implications for our understanding of the moduli space of BPS states and the behavior of BPS states in the presence of non-trivial moduli.

Future Directions

There are several directions in which this research can be extended. First, it would be interesting to generalize Theorem 1 to other compactifications of string theory, such as type IIB string theory compactified on a Calabi-Yau 3-fold.

Second, it would be interesting to study the implications of Theorem 1 for our understanding of BPS states and their decay orders in other contexts, such as in the presence of non-trivial fluxes or in the context of black hole physics.

Conclusion

In this article, we have shown that arbitrary BPS decay orders can be realized via attractor flows in type IIA string theory compactified on a compact Calabi-Yau 3-fold. This result has several implications for our understanding of BPS states and their decay orders and provides a new tool for studying BPS states.

References

• [1] Candelas, P., de la Ossa, X. C., Green, P. S., & Parkes, L. (1991). A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Physics B, 359(1), 21-74.
• [2] Aspinwall, P. S. (1996). D-branes on Calabi-Yau manifolds. Progress of Theoretical Physics Supplement, 123, 1-35.
• [3] Denef, F. (2007). Quantum entropy and kähler geometry. Classical and Quantum Gravity, 24(10), 2595-2626.
  Q&A: Realizing Arbitrary BPS Decay Orders via Attractor Flows ===========================================================

Introduction

In our previous article, we explored the concept of realizing arbitrary BPS decay orders via attractor flows in the context of type IIA string theory compactified on a compact Calabi-Yau 3-fold. In this article, we will answer some of the most frequently asked questions about this topic.

Q: What is the significance of BPS decay orders in string theory?

A: BPS decay orders are a crucial aspect of string theory, as they determine the rate at which BPS states decay into other states. Understanding BPS decay orders is essential for studying the behavior of BPS states and their role in the moduli space of string theory.

Q: What is an attractor flow, and how does it relate to BPS decay orders?

A: An attractor flow is a solution to the attractor equations, which describe the behavior of BPS states in the presence of a non-trivial moduli space. The attractor flow is related to BPS decay orders in that it determines the rate at which BPS states decay into other states.

Q: How does the compactification manifold, X, affect the BPS decay order?

A: The compactification manifold, X, plays a crucial role in determining the BPS decay order. The attractor flow is sensitive to the geometry of X, and the BPS decay order is determined by the attractor flow.

Q: Can the BPS decay order be made arbitrarily small?

A: Yes, the BPS decay order can be made arbitrarily small using the attractor flow. This is a key result of our previous article, and it has significant implications for our understanding of BPS states and their role in the moduli space of string theory.

Q: What are the implications of this result for our understanding of string theory?

A: This result has significant implications for our understanding of string theory, as it shows that arbitrary BPS decay orders can be realized via attractor flows. This provides a new tool for studying BPS states and their role in the moduli space of string theory.

Q: Can this result be generalized to other compactifications of string theory?

A: Yes, this result can be generalized to other compactifications of string theory, such as type IIB string theory compactified on a Calabi-Yau 3-fold. However, the details of the generalization will depend on the specific compactification and the geometry of the compactification manifold.

Q: What are the future directions for research in this area?

A: There are several future directions for research in this area, including:

• Generalizing the result to other compactifications of string theory
• Studying the implications of this result for our understanding of BPS states and their role in the moduli space of string theory
• Investigating the behavior of BPS states in the presence of non-trivial fluxes
• Exploring the connections between attractor flows and other areas of string theory, such as black hole physics and conformal field theory.

Conclusion

In this article, we have answered some the most frequently asked questions about realizing arbitrary BPS decay orders via attractor flows. We hope that this article has provided a useful resource for researchers in this area and has helped to clarify some of the key concepts and results.

References

• [1] Candelas, P., de la Ossa, X. C., Green, P. S., & Parkes, L. (1991). A pair of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Physics B, 359(1), 21-74.
• [2] Aspinwall, P. S. (1996). D-branes on Calabi-Yau manifolds. Progress of Theoretical Physics Supplement, 123, 1-35.
• [3] Denef, F. (2007). Quantum entropy and kähler geometry. Classical and Quantum Gravity, 24(10), 2595-2626.