[Paper Review] The Geometry of Calorons

Thomas M.W. Nye|arXiv (Cornell University)|Nov 22, 2003

Algebraic and Geometric Analysis33 references19 citations

TL;DR

This paper generalizes the Nahm transform—a duality between anti-self-dual (ASD) connections and solutions to Nahm's equations—from the 4-torus to the caloron setting, i.e., periodic instantons on $S^1 \times \mathbb{R}^3$. By analyzing families of Dirac operators with boundary conditions on a 3-ball, the author establishes a one-to-one correspondence between calorons and Nahm data on $S^1$, proving the transform is invertible and constructing calorons from Nahm data while computing their instanton charge.

ABSTRACT

Calorons (periodic instantons) are anti-self-dual (ASD) connections on S^1 imes R^3 and form an intermediate case between instantons and monopoles. The ADHM and Nahm constructions of instantons and monopoles can be regarded as generalizations of a correspondence between ASD connections on the 4-torus, often referred to as the Nahm transform. This thesis describes how the Nahm transform can be extended to the case of calorons. It is shown how calorons can be constructed from Nahm data similar to that for monopoles, but defined over the circle. The inverse transformation, from the caloron to the Nahm data, is also described.

Motivation & Objective

• To extend the Nahm transform, originally defined for ASD connections on the 4-torus, to the case of calorons—periodic instantons on $S^1 \times \mathbb{R}^3$.
• To provide a geometric and analytical construction of calorons from Nahm data, analogous to the monopole construction via Nahm's equations.
• To define and rigorously impose boundary conditions on $S^1 \times \overline{B}^3$ that ensure the Dirac operators in the transform are Fredholm.
• To compute the instanton charge of a caloron via topological obstructions arising from the boundary gauge structure.
• To prove the Nahm transform is invertible, establishing a duality between calorons and solutions to Nahm’s equations on $S^1$.

Proposed method

• Regard $\mathbb{R}^3$ as the interior of the closed 3-ball $\overline{B}^3$, and impose fixed gauge behavior on the boundary $S^1 \times S^2$.
• Define caloron boundary conditions by requiring the connection to resemble a pull-back of a $U(n)$ monopole in a fixed boundary gauge $f$.
• Use a family of Dirac operators $\Delta(x)$ parameterized by $x \in S^1 \times \mathbb{R}^3$ to define the Nahm transform.
• Deform $\Delta(x)$ to a model operator $\tilde{\Delta}(x)$ for which boundary behavior and instanton charge can be computed, showing the deformation preserves boundary data.
• Apply an index theorem for Dirac operators on $S^1 \times \mathbb{R}^3$ to compute the rank of the Nahm data.
• Use Nakajima’s method to analyze singularities of Nahm data at isolated points on $S^1$, showing prescribed singular behavior.

Experimental results

Research questions

• RQ1 How can the Nahm transform be generalized from the 4-torus to the caloron setting on $S^1 \times \mathbb{R}^3$?
• RQ2 What boundary conditions on $S^1 \times \overline{B}^3$ ensure the Dirac operators in the Nahm transform are Fredholm?
• RQ3 How is the instanton charge of a caloron determined from its boundary gauge structure?
• RQ4 Can the Nahm data constructed from a caloron be shown to have the correct singularities and rank at isolated points on $S^1$?
• RQ5 Is the Nahm transform for calorons invertible, and does it yield a complete construction of calorons from Nahm data?

Key findings

• The Nahm transform for calorons is defined via a family of Dirac operators on $S^1 \times \mathbb{R}^3$, with boundary conditions ensuring Fredholmness.
• The boundary conditions require the caloron to locally resemble a monopole in a fixed gauge on $S^1 \times S^2$, introducing a topological obstruction known as the instanton charge.
• The construction of a caloron from Nahm data is achieved by deforming the Dirac operator family to a model for which boundary behavior and instanton charge are computable, and showing this deformation preserves the boundary data.
• The rank of the Nahm data is computed using an index theorem for Dirac operators on $S^1 \times \mathbb{R}^3$, which accounts for the dimension of the kernel of $\Delta(x)$.
• The Nahm data exhibit prescribed singularities at isolated points on $S^1$, consistent with expectations from monopole theory, as analyzed via Nakajima’s method.
• The transform is invertible: every set of Nahm data on $S^1$ gives rise to a unique caloron, and every caloron determines a unique set of Nahm data, establishing a duality.