Abstract
Most of the non-Abelian string-vortices studied so far are characterized by two-dimensional \cpn models with various degrees of supersymmetry on their world sheet. We generalize this construction to "composite" non-Abelian strings supporting the Grassmann $\mathcal{G}(L,M)$ models (here $L+M=N$). The generalization is straightforward and provides, among other results, a simple and transparent way for counting the number of vacua in ${\mathcal N}=(2,2)$ Grassmannian model.
Highlights
The 2D CP(N − 1) nonlinear sigma model has recently undergone much analysis, in particular appearing as worldsheet theories on the simplest non-Abelian string vortices [1,2,3,4] including its heterotic versions [9]
We have introduced extra degrees of freedom on the string: these will turn out to live inside a Grassmannian sigma model once worldsheet fluctuations are considered for all moduli of the string
D at the same time, this produces dt dz |(1i j ∂ ̃ −i(A)i j )XA j |2 +Di j. This verifies our previous assertion: despite the fact that X exists in a linear representation of SU(N)× U(L), U(L) is not a global invariance of the ansatz we presented above but a local one
Summary
The 2D CP(N − 1) nonlinear sigma model has recently undergone much analysis, in particular appearing as worldsheet theories on the simplest non-Abelian string vortices [1,2,3,4] (see Refs. [5,6,7,8] for reviews) including its heterotic versions [9]. Where the first term in the final decomposition is the overall number of the orientational moduli, while 2L2 describes 2L relative positions and orientations of L components (i.e., the elementary strings with the unit flux) Attempts to reproduce these results in field theory do not lead to a transparent description of the worldsheet theory, see Ref. [see Eq (2)] coincides with the dimension of the Grassmannian In this setup, we construct explicit multi-string solution and derive U(L) gauge linear sigma model on the string worldsheet. Needless to say that in this construction the symmetry of N elements of the U(N ) Cartan subalgebra under permutations is broken This will lead us to a non-Abelian string with the G(L, M ) Grassmann model on the worldsheet.
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