Topological vertex/anti-vertex and supergroup gauge theory

Abstract

We propose a new vertex formalism, called anti-refined topological vertex (anti-vertex for short), to compute the generalized topological string amplitude, which gives rise to the supergroup gauge theory partition function. We show the one-to-many correspondence between the gauge theory and the Calabi-Yau geometry, which is peculiar to the supergroup theory, and the relation between the ordinary vertex formalism and the vertex/anti-vertex formalism through the analytic continuation.

Quantum integrable systems from supergroup gauge theories

“…On the other hand, the five-brane web formalism verifies that the instanton partition function itself does not depend on such an ordering [38] (which can be T-dual to D3-branes), so that the spectral curve, to be identified with the SW curve, and the classical Hamiltonians are consequently independent of the brane ordering. This implies that the apparently different classical Hamiltonians are isomorphic to each other under a suitable linear transformation, whereas the quantum Hamiltonians may depend on the brane ordering due to the surface defect insertion.…”

“…Let us consider a set of three positive and three negative branes for definiteness, their different arrangements correspond to the different Dynkin diagrams with the odd roots labeling representation under sl(3|3) superalgebra. See also [38] for a related argument in the topological string setup. For instance, consider the Dynkin diagram with a single odd root at the 3rd/middle vertex [2,11,12]:…”

More on topological vertex formalism for 5-brane webs with O5-plane

“…Furthermore, web diagrams each of which are made by gluing three or four (dual) toric diagrams have been constructed in [19] and the method developed there computes the partition functions of SO(2N ) gauge theories and also the pure E 6 , E 7 , E 8 gauge theories. Recently the topolgical vertex formalims has been also extended in other directions [20][21][22][23][24][25][26][27][28][29].…”

Defects at the Intersection: the Supergroup Side

“…Furthermore, we considered yet another Higgsing of the 5d U(1|1) theory, and the partition function of the resulting single component defect theory resembles that of refined Chern-Simons on L(2, 1). It is well-known that the large rank expansion of a matrix model and its supergroup version are equivalent up to non-perturbative effects [11] (in the unrefined case, this fits with the generic Higgsing being sensitive only to r − c): it would be interesting to retrace and adapt the existing analysis around supergroups, large rank dualities and non-perturbative effects in our refined setup, also in view of the vertex/anti-vertex formalism [40]. The relations between supergroup-like and ordinary theories was instrumental for understanding non-perturbative effects in topological strings [10], most notably on the local P 1 × P 1 geometry, the closed dual to Chern-Simons on L(2, 1): a deeper understanding of the subject reviewed in this note may help in shedding more light on seemingly different proposals [41].…”