Relations between 6D N = (2, 0) super conformal field theory and 5D, 4D gauge theories

Abstract

The six-dimensional (6D), N = (2,0) super conformal field theory (SCFT), which contains a tensor multiplet, is considered to govern some of the lower dimensional supersymmetric gauge theories. After a general introduction to the 6D, N = (2, 0) theory with sixteen supercharges and supersymmetric Yang-Mills theory in 4D and 5D, there follows a description of the partition function approach for a single M5-brane of which the world volume theory is the abelian 6D, N = (2, 0) SCFT. We introduce the conjecture by Michael Douglas and Neil Lambert that the (2, 0) SCFT on S1 is equivalent to the 5D maximally supersymmetric Yang-Mills theory. S-duality is an important property first found in Maxwell theory and later generalized to different supersymmetric gauge theories, such as 4D, N = 4 super Yang-Mills and 4D supersymmetric QCD. We briefly discuss the origin of the S-duality of the 4D abelian gauge theory with an theta angle from the 6D tensor theory. By computing and comparing the explicit formulas for the partition functions, we will show that the 4D and 5D abelian gauge theories share fundamental properties with the 6D tensor theory. In Chapter 2, we give our preliminary test of the conjecture of Douglas and Lambert by using the partition functions computation. We give an explicit computation of the partition function of a five-dimensional abelian gauge theory on a five-torus with a general flat metric using the Dirac method of quantizing with constraints. We compare this with the partition function of a single fivebrane compactified on S1 times T5, which is obtained from the six-torus calculation of Dolan and Nappi. The radius R1 of the circle S1 is set to the dimensionful gauge coupling constant. We find the two partition functions are equal only in the limit where R1 is small relative to T5, a limit which removes the Kaluza-Klein modes from the 6D sum. This suggests the (2, 0) tensor theory on a circle is an ultraviolet completion of the 5D gauge theory, rather than an exact quantum equivalence. In Chapter 2, we compute the partition function of four-dimensional abelian gauge theory on a general four-torus T4 with flat metric using Dirac quantization. In addition to an SL(4;Z) symmetry, it possesses SL(2;Z) symmetry that is electromagnetic S-duality. We show explicitly how this SL(2;Z) S-duality of the 4D abelian gauge theory has its origin in symmetries of the 6D (2; 0) tensor theory, by computing the partition function of a single fivebrane compactified on T2 T4, which has SL(2;Z) SL(4;Z) symmetry. If we identify the couplings of the abelian gauge theory with the complex modulus of the T2 torus, , then in the small T2 limit, the partition function of the fivebrane tensor field can be factorized, and contains the partition function of the 4D gauge theory. In this way the SL(2;Z) symmetry of the 6D tensor partition function is identified with the S-duality symmetry of the 4D gauge partition function. Each partition function is the product of zero mode and oscillator contributions, where the SL(2;Z) acts suitably. For the 4D gauge theory, which has a Lagrangian, this product redistributes when using path integral quantization.