Abstract
Using an explicit path integral approach we derive non-abelian bosonization and duality of 3D systems in the large N limit. We first consider a fermionic U(N) vector model coupled to level k Chern-Simons theory, following standard techniques we gauge the original global symmetry and impose the corresponding field strength F μν to vanish introducing a Lagrange multiplier Λ. Exchanging the order of integrations we obtain the bosonized theory with Λ as the propagating field using the large N rather than the previously used large mass limit. Next we follow the same procedure to dualize the scalar U (N) vector model coupled to Chern-Simons and find its corresponding dual theory. Finally, we compare the partition functions of the two resulting theories and find that they agree in the large N limit including a level/rank duality. This provides a constructive evidence for previous proposals on level/rank duality of 3D vector models in the large N limit. We also present a partial analysis at subleading order in large N and find that the duality does not generically hold at this level.
Highlights
One of the most remarkable concepts in theoretical physics is duality – the fact that the same physical theory admits more than one description in terms of different degrees of freedom
We have considered integration in various limits but most prominently the large N limit employing some of the standard techniques
U(N) vector models we were able to identify the two dual theories and show that they agree in the large N limit exhibiting explicitly a level/rank duality for these systems
Summary
One of the most remarkable concepts in theoretical physics is duality – the fact that the same physical theory admits more than one description in terms of different degrees of freedom. As a path integral equivalence, two-dimensional bosonization was completely formulated in [4, 5] wherein through a set of transformations the fermionic partition function was mapped into the bosonic partition function These serve as one of the very few examples where a duality between two theories can be proved as an equivalence of partition functions using a series of transformations in the path integral rather that making a conjecture and checking some of its implications. For many physically important consequences the conjecture and check approach is completely satisfactory as the AdS/CFT correspondence abundantly shows. Conceptually this approach lacks some clarity and finality. Equivalence, in a certain limit, between very massive fermions and a Chern-Simons theory was established [6, 7]
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