[Paper Review] Chern-Simons-Matter Theory and Mirror Symmetry

Daniel L. Jafferis, Xi Yin|ArXiv.org|Oct 7, 2008

Black Holes and Theoretical PhysicsPhysics and Astronomy23 references19 citations

TL;DR

This paper proposes that certain abelian N=4 Chern-Simons-matter (CSM) theories with U(1)×U(1) gauge group at level (k,−k) describe the same superconformal field theory (SCFT) as the infrared fixed point of N=4 QED with Nf flavors, due to quantum-corrected hypermultiplet moduli spaces matching those of the dual QED. Mirror symmetry in the CSM theory manifests as exchange of the two Higgs branches, and quantum corrections restore moduli spaces lifted classically in N=2 CSM theories, yielding symplectic quotients of the form C^{M+1}//U(1).

ABSTRACT

In this paper we study supersymmetric Chern-Simons-matter (CSM) theories with several Higgs branches. Two such theories at small Chern-Simons level are conjectured to describe the superconformal field theory at the infrared fixed point of N = 4 QED with N_f = 2, 3. In particular, the mirror symmetry which exchanges the Coulomb and Higgs branches of N = 4 QED with N_f = 2 is manifest in the Chern-Simons-matter description. We also study the quantum corrections to the moduli space of a class of N = 2 CSM theories.

Motivation & Objective

• To identify abelian N=4 Chern-Simons-matter theories that flow to the same SCFT as N=4 QED with Nf=2,3 flavors.
• To understand how quantum corrections to hypermultiplet moduli spaces in CSM theories differ from those in N=4 Yang-Mills theories.
• To establish a correspondence between the moduli spaces of CSM theories and those of N=4 QED via brane constructions and quantum corrections.
• To explore how quantum effects restore moduli spaces in N=2 CSM theories that are classically lifted by D-terms and Chern-Simons levels.
• To generalize the duality to nonabelian N=4 SQCD and investigate the role of 't Hooft operators in enhancing global symmetries at small k.

Proposed method

• Construct N=4 CSM theories with U(1)×U(1) gauge group at level (k,−k), coupled to hypermultiplets with specific U(1) charges.
• Compute the D-term and F-term scalar potentials after integrating out auxiliary fields, leading to a potential that depends on field norms.
• Use one-loop effective action to compute quantum corrections to the hypermultiplet moduli space, showing it becomes a symplectic quotient C^{M+1}//U(1).
• Apply brane constructions to realize the CSM theories as fractional M2-brane systems in N=4 orbifolds, providing evidence for duality with N=4 QED.
• Analyze the role of 't Hooft operators in enhancing global symmetries at small Chern-Simons levels, particularly for k=1.
• Use the supersymmetric completion of the effective Lagrangian to show that moduli spaces are lifted unless bare CS levels are tuned to −M1/2 and −M2/2.

Experimental results

Research questions

• RQ1 Do N=4 CSM theories with U(1)×U(1) gauge group at level (k,−k) flow to the same SCFT as N=4 QED with Nf=2 and Nf=3 flavors?
• RQ2 How do quantum corrections modify the hypermultiplet moduli space in CSM theories, and why do they differ from N=4 Yang-Mills theories?
• RQ3 Can mirror symmetry in the CSM theory be understood as the exchange of two Higgs branches, and how is it related to the mirror symmetry of N=4 QED?
• RQ4 Why are moduli spaces restored in N=2 CSM theories despite being classically lifted by D-terms and Chern-Simons levels?
• RQ5 What is the role of 't Hooft operators in enhancing global symmetries at small Chern-Simons levels in CSM theories?

Key findings

• The quantum-corrected Higgs branch of the N=4 CSM theory with U(1)×U(1) gauge group at level (k,−k) is isomorphic to the Coulomb and Higgs branches of the IR fixed point of N=4 QED with Nf flavors.
• For Nf=2 and Nf=3, brane constructions confirm that the CSM theory describes the same SCFT as the IR limit of N=4 QED, with the moduli space becoming a symplectic quotient C^{M+1}//U(1).
• The moduli space of the N=2 CSM theory is restored quantum mechanically, with one-loop corrections yielding a Kähler metric isomorphic to C^{M+1}//U(1), provided the bare Chern-Simons levels are tuned to −M1/2 and −M2/2.
• A two-loop contribution to the effective Lagrangian lifts the moduli space if the bare CS levels are not tuned, showing that quantum corrections are essential for moduli space restoration.
• Enhanced global symmetries at small k arise via 't Hooft operators, which allow the construction of new conserved currents, and are crucial for the duality at k=1.
• The duality extends to N=4 U(N) SQCD for Nf=1,2,3, suggesting that CSM theories may describe the IR SCFTs of nonabelian N=4 SQCD, though explicit constructions remain open for SU(N) gauge groups.