[Paper Review] Topological Field Theory on a Lattice, Discrete Theta-Angles and Confinement

Anton Kapustin, Ryan Thorngren|arXiv (Cornell University)|Aug 13, 2013

Black Holes and Theoretical Physics9 references4 citations

TL;DR

This paper formulates a lattice topological quantum field theory (TQFT) for confining phases of 4D gauge theories using a discrete 2-form gauge field valued in a finite abelian magnetic gauge group Π₂. It shows that theta-angles in such theories are quantized and classified by quadratic functions on Π₂, generalizing discrete theta-angles found in continuum Yang-Mills theory. The key result is that the TQFT is not dual to an ordinary gauge theory when theta-angles are non-zero, as its partition function depends on the 4-manifold's signature, distinguishing it from standard topological gauge theories.

ABSTRACT

We study a topological field theory describing confining phases of gauge theories in four dimensions. It can be formulated on a lattice using a discrete 2-form field talking values in a finite abelian group (the magnetic gauge group). We show that possible theta-angles in such a theory are quantized and labeled by quadratic functions on the magnetic gauge group. When the theta-angles vanish, the theory is dual to an ordinary topological gauge theory, but in general it is not isomorphic to it. We also explain how to couple a lattice Yang-Mills theory to a TQFT of this kind so that the 't Hooft flux is well-defined, and quantized values of the theta-angles are allowed. The quantized theta-angles include the discrete theta-angles recently identified by Aharony, Seiberg and Tachikawa.

Motivation & Objective

• To formulate a topological quantum field theory (TQFT) that describes confining phases of 4D gauge theories on a lattice.
• To identify and classify the allowed theta-angles in such TQFTs, showing they are quantized and labeled by quadratic functions on the magnetic gauge group Π₂.
• To establish a lattice formulation of Yang-Mills theory with quantized theta-angles that match the low-energy TQFT, resolving the absence of a lattice counterpart for continuous theta-angles.
• To clarify the microscopic origin of discrete theta-angles and their role in the spectrum of line operators, as identified by Aharony, Seiberg, and Tachikawa.

Proposed method

• Formulate a lattice TQFT using a discrete 2-form field h with values in a finite abelian group Π₂, the magnetic gauge group.
• Define the topological action via the Pontryagin square of h, which maps cohomology classes in H²(K, Π₂) to H⁴(K, Γ(Π₂)) using a quadratic refinement of the cup product.
• Show that the partition function depends on the 4-manifold's signature when theta-angles are non-zero, distinguishing the theory from standard topological gauge theories.
• Establish duality between the B-field and a gauge field in the zero-theta case via the Pontryagin dual group Π₂*, showing electric-magnetic duality holds only when theta-angles vanish.
• Construct a lattice Yang-Mills theory coupled to the TQFT such that the 't Hooft flux is well-defined and quantized, with theta-angles matching those in the low-energy TQFT.
• Use the ∪¹ cup product operation in simplicial cochains to define the Pontryagin square in general, ensuring consistency under gauge transformations and cohomology class dependence.

Experimental results

Research questions

• RQ1 How can a confining phase of a 4D gauge theory be consistently formulated on a lattice using only a discrete 2-form field?
• RQ2 What is the mathematical classification of allowed theta-angles in such a lattice TQFT, and how do they relate to the magnetic gauge group Π₂?
• RQ3 Why is the TQFT with non-zero theta-angles not equivalent to an ordinary topological gauge theory, and how does its partition function depend on the 4-manifold's topology?
• RQ4 Can discrete theta-angles identified in continuum Yang-Mills theory by Aharony, Seiberg, and Tachikawa be realized in a lattice formulation?
• RQ5 Why is there no satisfactory lattice counterpart for the continuous theta-angle, and what role does the instanton number play in this obstruction?

Key findings

• The allowed theta-angles in the lattice TQFT are classified by quadratic functions on the magnetic gauge group Π₂, with the topological action given by the Pontryagin square of the discrete 2-form field h.
• For Π₂ = Z_r with r even, the discrete theta-angle q is an integer modulo 2r, and the action is Stop(h) = 2πiq/r (˜h ∪ ˜h)[T⁴] modulo 2r, with only q mod r relevant due to the even intersection form.
• For Π₂ = Z_r with r odd, the discrete theta-angle q is an integer modulo r, and the action is Stop(h) = 2πiq/r (˜h ∪ ˜h)[T⁴], with the Pontryagin square defined via f ∪ f.
• When theta-angles vanish, the TQFT is dual to a topological gauge theory with gauge group Π₂*, but for non-zero theta-angles, the theory is not isomorphic to any ordinary gauge theory due to its dependence on the 4-manifold's signature.
• The lattice formulation allows all discrete theta-angles identified by Aharony, Seiberg, and Tachikawa to be realized, but no lattice counterpart exists for the continuous theta-angle due to the absence of a well-defined instanton number on the lattice.
• The partition function weight is not positive-definite when theta-angles are non-zero, but it can be made positive by fixing the cohomology class [h], allowing Monte Carlo simulations via summing over cohomology classes.