Abstract
In this paper we present various 4d mathcal{N} = 1 dualities involving theories obtained by gluing two E[USp(2N)] blocks via the gauging of a common USp(2N) symmetry with the addition of 2L fundamental matter chiral fields. For L = 0 in particular the theory has a quantum deformed moduli space with chiral symmetry breaking and its index takes the form of a delta-function. We interpret it as the Identity wall which identifies the two surviving USp(2N) of each E[USp(2N)] block. All the dualities are derived from iterative applications of the Intriligator-Pouliot duality. This plays for us the role of the fundamental duality, from which we derive all others. We then focus on the 3d version of our 4d dualities, which now involve the mathcal{N} = 4 T[SU(N)] quiver theory that is known to correspond to the 3d S-wall. We show how these 3d dualities correspond to the relations S2 = −1, S−1S = 1 and STS = T−1S−1T−1 for the S and T generators of SL(2, ℤ). These observations lead us to conjecture that E[USp(2N)] can also be interpreted as a 4d S-wall.
Highlights
In this paper we present various 4d N = 1 dualities involving theories obtained by gluing two E[USp(2N )] blocks via the gauging of a common USp(2N ) symmetry with the addition of 2L fundamental matter chiral fields
For L = 0 in particular the theory has a quantum deformed moduli space with chiral symmetry breaking and its index takes the form of a delta-function
We show how starting from the so-called braid duality [4], which involves two copies of the E[USp(2N )] theory glued with the insertion of two chirals, we can obtain in 3d a duality related to the ST S = T −1S−1T −1 property
Summary
We study the gluing of two E[USp(2N )] theories corresponding to commonly gauging a diagonal combination of one USp(2N ) symmetry. Where the indices L/R distinguish the field and operators of the left and right E[USp(2N )] blocks, for example HL and HR denote the H operators in the antisymmetric representation of their manifest USp(2N ) symmetries which we identify and call USp(2N )z, and Trz is taken over such gauged USp(2N )z. In the limit cd → 1 the pairs of poles coming from the second product collide to the points xi = yj±1 This implies that if we integrate the index of E[USp(2N )] over the variables xi against a test function f (x), the colliding poles pinch the integration contour and we should take the corresponding residues as discussed in [27]. We show that the residue at these poles gives the delta-functions
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