A note on the S-dual basis in the free fermion system

Abstract

Free fermion system is the simplest quantum field theory which has the symmetry of Ding-Iohara-Miki algebra (DIM). DIM has S-duality symmetry, known as Miki automorphism which defines the transformation of generators. In this note, we introduce the second set of the fermionic basis (S-dual basis) which implement the duality transformation. It may be interpreted as the Fourier dual of the standard basis, and the inner product between the standard and the S-dual ones is proportional to the Hopf link invariant. We also rewrite the general topological vertex in the form of Awata-Feigin-Shiraishi intertwiner and show that it becomes more symmetric for the duality transformation.

Argyres-Douglas theories, S-duality and AGT correspondence

“…That would shed some light on the relation between the 4d D 2n (SU(2)) theory and the 5d D 2n (SU(2)) theory [7,52]. More generally, it would be interesting to study how our results are phrased in terms of the W 1+∞ algebra and the DIM algebra along the lines of [53][54][55][56][57][58][59][60][61][62][63][64][65]. 26 The "magnetic charge" here is simply associated with the SU(2) vector multiplet without mixing with those arising from the (A1, D4) sectors.…”

Quantum $W_{1+\infty}$ subalgebras of BCD type and symmetric polynomials

“…(2) k = W 0,k is diagonal. Roughly speaking, Miki's automorphism sends W m,n to W −n,m , thereby exchanging the two subalgebras [17,37].…”

“…This algebraic framework led to enormous progress in the field, e.g. by extending the topological vertex technique to various theories [21][22][23][24][25][26][27][28] and observables [29][30][31], by deriving proofs for the q-deformed AGT correspondence [32][33][34], or by describing the action of fiber-base duality [34][35][36][37] to name only a few. It should be emphasized that the algebra W plays the same role as the quantum toroidal gl(1) algebra in the self-dual limit of topological strings [17], and that it can be used in the same way to build the original topological vertex [38], thus providing the quantum algebraic framework behind the melting crystal construction of Okounkov, Reshetikhin and Vafa [39].…”

Intertwining operator and integrable hierarchies from topological strings

“…This observation led to a number of important results in this field. For example, we can mention the extension of the topological vertex technique to various theories [17][18][19][20][21][22][23][24] and observables [25][26][27], the derivation of proofs for the q-deformed AGT correspondence [28][29][30], or the description of the fiber-base duality [30][31][32][33].…”

“…The AFS intertwiner has been introduced directly in the context of the refined topological vertex [11]. However, it is relatively easy to perform the self-dual limit t → q of this object and obtain the formulation relevant to the unrefined case [33]. Since the quantum toroidal gl(1) algebra reduces to quantum W 1+∞ in this limit, we obtain an intertwiner between modules of the latter.…”