I've been reading advanced string theory text but I came across with a paper that I definitely don't understand at all, what it means?, this text is Higgs branches of 5d rank-zero theories from geometry (Andrés Collinucci, a Mario De Marco, b Andrea Sangiovannic, d and Roberto Valandroc)Higgs branches, what do you mean by higgs branch at 5d and columb branch?,I saw the video https://tube.switch.ch/switchcast/unibe.ch/events/325eaef5-8c03-44a0-866a-cee9b9c2d382, but it talks about supersymmetry in QCD, but I still don't understand what the main idea is about it.
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asked Dec 4, 2021 at 5:06
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3$\begingroup$ Do you understand Higgs and Coulomb branches in 4d? If not, arxiv.org/abs/1312.2684 might help. $\endgroup$– Connor BehanCommented Dec 4, 2021 at 11:14
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$\begingroup$ @ConnorBehan i can undestand the basics for superconformal field theories I think it is part of Seiberg-Witten theory but I do not understand why they are using an expected value in the void of a monduli space, nor that it is doing the mirror symmetry in the monduli space, nor that it is a Topological twist, I'm good in topology but I can't see the complete map of everything or the general idea. $\endgroup$– Juan Carlos Dominguez Solis.Commented Dec 4, 2021 at 22:30
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$\begingroup$ @JuanCarlosDominguezSolis. The definition of the Higgs Branch is the same for QCD in any dimension (the region of the moduli space of the theory where mesons and baryons adquire VEVs). What could be confusing about the 5d case is that most 5d SQCD theories are analized using $(p,q)-5$ webs or other constructions whose dynamics are restricted to the plane (or lines as in the Hannany-Witten construction in $3d$) where you require specific knowledge to understand the geometrization of the dynamics. $\endgroup$– Ramiro Hum-SahCommented Dec 5, 2021 at 16:37
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$\begingroup$ @JuanCarlosDominguezSolis. So, the recommendation is: Read Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams) where all the required definitions are discussed in depth. If you are comfortable with algebraic-geometry, you may find interesting: On Correspondences Between Toric Singularities and (p,q)-webs; the idea is that by identifying some web constructions with toric singularities you would be able to rigorously define the Higgs and Coulomb branches of a given $5d$ theory in terms of ... $\endgroup$– Ramiro Hum-SahCommented Dec 5, 2021 at 16:42
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$\begingroup$ @JuanCarlosDominguezSolis. ... honest moduli spaces of non-compact toric Calabi-Yau manifolds. $\endgroup$– Ramiro Hum-SahCommented Dec 5, 2021 at 16:43
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There is a nice and simple geometric way to understand the Higgs branch of a 5d SQCD.
Consider the case of a 5d theory taking place along four common spacelike directions (plus time) of a $D5$-brane and an $NS5$ brane. Now suppose you are interested in the geometry of the branes in the remaining direction (following Webs of (p,q) 5-branes, Five Dimensional Field Theories and Grid Diagrams).
The point is to ask about the meaning of the three 'unused' directions of the type IIB theory in the aforementioned situation. In SQCD theories, there are two general ways to move into the Higgs branch: giving a VEV to a meson, and giving a VEV to a baryon. The way this can be done in the context of ($p,q$)-5 webs is by separating the branes along the 'unused' directions (broking the $SO(3)$ symmetry group of the 'unused' directions).
Turning on VEVs for fields can be seen equivalently to produce local deformations of the ($p,q$)-5 webs (see page 29 in [1] to understand the geometry of the problem). In short: The Higgs branch of 5d SQCDs can be understood as the space of all possible local deformations of the web diagramas.
Observation: The above considerations are not unique to the five-dimensional case, they are similiar in other dimensions; what is unique of the 5d case is the particularly nice geometric visualization of the Higgs branch
answered Dec 4, 2021 at 22:40