Two-dimensional gauge theory dualities

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Introduction

Crustal‐scale balanced cross‐section restorations synthesize our knowledge about the structural‐kinematic history of a mountain belt. Although the method had been used previously, the basic principles of creating a balanced cross section were thoroughly discussed for the first time by Dahlstrom (1969). Crustal‐scale restorations require understanding not just the present‐day geology and the history of the area in question, but also the processes acting on the deep structures as well as those playing a role in the formation of topography. The synthesis of all available geological and geophysical data is essential to allow maximum constraint on the restoration. Section restorations based on deep seismic profiles have been published for the Alps (e.g. Roure et al., 1996; Schmid et al., 1996), the Canadian Rocky Mountains (e.g. Cook et al., 1992) and the Pyrenees (e.g. Muñoz, 1992; Séguret & Daigniéres, 1986) while, more recently, restorations based mainly on geologic data have been compiled for the Andes (e.g. McQuarrie, 2002; McQuarrie et al., 2008) and the Himalaya (e.g. DeCelles et al., 2001; Long et al., 2011; Robinson et al., 2006), among others.
Additional and independent constraints on orogen kinematics are provided by thermochronology data, which record exhumation of rocks in response to tectonic thickening, rock uplift and erosion (e.g., Braun et al., 2006; Reiners and Brandon, 2006). Such data can therefore, in principle, be used to discriminate between competing kinematic models for orogens (e.g., Herman et al., 2010; Robert et al., 2011) or to provide additional temporal constraints on balanced section restorations (e.g., Long et al., 2012; McQuarrie et al., 2008). Conversely, the kinematics implied by such restorations can be used to predict thermal history and thermochronologic age patterns in, for instance, foreland fold‐and‐thrust belts ( Lock and Willett, 2008; ter Voorde et al., 2004).
However, these two approaches to decipher orogen kinematics generally remain detached from each other and no quantitative, internally consistent framework exists as yet to fully integrate the two. In this study, we propose such a methodology and quantitatively evaluate the consistency of a balanced cross‐section restoration of the central Pyrenees, based on Muñoz (1992) and Beaumont et al. (2000), with independent thermochronological data for a range of systems with different closure temperatures. The central Pyrenees constitute an ideal test case to validate our methodology, as the existing balanced cross‐section is well constrained by a wide range of geological and geophysical data but does not incorporate the more recently collected extensive thermochronological database (see details and references in the next section). The method can be applied to other orogenic belts where fewer constraints exist to evaluate the consistency of proposed cross‐section restorations with available thermochronology data or to provide additional temporal constraints on such restorations.

The Pyrenean orogen

The Pyrenees (Figure 1) are a collisional orogen formed by Late‐Cretaceous (90 Ma) to Early‐ Miocene (20 Ma) convergence between the Iberian and European plates (Roest and Srivastava, 1991; Rosenbaum et al., 2002; Vissers and Meijer, 2012). According to these plate‐kinematic analyses, the convergence rate reached its peak during the Eocene to Oligocene (50‐20 Ma). The range is dominated by inversion tectonics (Muñoz et al., 1986) owing to thrusting along pre‐existing extensional structures (Figure 2). These structures were originally formed during Triassic to Cretaceous rifting and transtension associated with anticlockwise rotation of the Iberian plate with respect to Europe and consequent opening of the Bay of Biscay (Roest and Srivastava, 1991; Rosenbaum et al., 2002).
The asymmetrical Pyrenean orogen resulted from northward underthrusting of the Iberian crust below the European crust, resulting in a wide southern pro‐wedge and a narrower northern retro‐wedge (Figure 2). In the centre of the belt, the Axial Zone comprises a south‐ vergent antiformal stack of upper‐crustal thrust sheets (Muñoz, 1992). The Axial Zone is flanked by the South Pyrenean Unit towards the south and the North Pyrenean Unit towards the north, respectively. The South Pyrenean Unit is a fold‐and‐thrust belt consisting of Mesozoic and Cenozoic sedimentary rocks, while in the North Pyrenean Unit, basement and cover rocks form north‐vergent thrust sheets and pop‐up structures (Muñoz, 1992; Capote et al., 2002). The Pyrenees are flanked by the Ebro and the Aquitaine foreland basins towards the south and north, respectively (Figures 1 and 2).
Our study‐area is located in the central Pyrenees, along the ECORS deep seismic profile (ECORS
Pyrenees Team, 1988) (Figure 1). Although we have used the entire section to constrain the velocity fields in the structural‐kinematic model, the thermo‐kinematic models cover only the Axial Zone and the North Pyrenean Unit, as the bulk of the published thermochronological data has been collected in these areas.
Figure 1. Geological map of the Pyrenees. The black line represents the ECORS deep seismic profile along which the balanced cross section of Figure 2 was constructed, and the box indicates the boundaries of the thermo‐kinematic model. AR: Arize block, MM: Marimaña massif, ML: Maladeta massif, NZ: Nogueres Zone, GT: Gavarnie‐thrust B: Bóixols, M: Montsec, SM: Sierras Marginales (redrawn after Fillon and van der Beek, 2012).

Structural evolution of the Pyrenees

The kinematics are well constrained in the South Pyrenean Unit as the tectonic evolution is exceptionally well recorded by syntectonic sediments (e.g. Muñoz, 1992; Puigdefabregas et al., 1986, 1992; Vergés and Muñoz, 1990). The evolution of the Axial Zone has been inferred from correlation of its thrust sheets with those in the South Pyrenean Unit, while that of the North Pyrenean Unit is less well constrained.
The South Pyrenean Unit is made up of three thin‐skinned thrust sheets (from south to north, the Sierras Marginales, Montsec and Bóixols; Figure 2), which have been emplaced on top of a detachment located in Upper Triassic evaporites in a southward‐propagating deformation sequence (Muñoz, 1992). The Bóixols thrust was activated during the Maastrichtian (from ~70‐ 65 Ma), as indicated by the overlying syn‐tectonic sequences. Deformation stepped onto the Montsec thrust during the Ypresian (~55 Ma). Finally, the Sierras Marginales unit was activated between the Early and Late Eocene (50‐40 Ma). The forward propagation of deformation in the South Pyrenean Unit was modified during the last (Late Eocene) stage of the thrust belt evolution by break‐back reactivation of the older thrusts and by the development of new, minor out‐of‐sequence thrusts, possibly in response to rapid accumulation of syntectonic sediments (Capote et al., 2002; Fillon et al., 2013). The northern fault contact of the Bóixols thrust sheet is the Morreres backthrust, which has been interpreted as a passive‐roof thrust (Muñoz, 1992).
The antiformal stack of the Axial Zone involves upper to middle crustal rocks and consists of three thrust sheets: Nogueres, Orri and Rialp (Figure 2). These units were initially juxtaposed and separated by extensional faults before the onset of thrusting (Muñoz, 1992). The highly eroded Nogueres thrust sheet is the uppermost of the three units. Its frontal tip has been preserved in the southern limb of the antiformal stack and is known as the Nogueres Zone, while its root‐zone crops out in the northern part of the Axial Zone (Muñoz, 1992).
Figure 2. Four time slices (between 50 Ma and Present) of the crustal‐scale area‐balanced cross‐section restoration of the Central Pyrenees along the ECORS deep‐seismic profile by Muñoz (1992) and Beaumont et al. (2000). NPF: North Pyrenean Fault; NPFT: North Pyrenean Frontal Thrust. Figure modified from Beaumont et al. (2000).
Thrusting of the Nogueres unit over the Orri unit took place between the Late Cretaceous and the Early Eocene (90‐50 Ma). Deformation subsequently stepped onto another Mesozoic extensional fault, causing the Orri unit to overthrust the Rialp unit during the Middle to Late Eocene (50‐36 Ma). Finally, deformation shifted again to create the Rialp thrust sheet between the Middle Eocene and the Late Oligocene (36‐20 Ma) (Capote et al., 2002; Metcalf et al., 2009).
North of the Axial Zone in the North Pyrenean Unit, very steep, north‐vergent thrust sheets and pop‐up structures involve basement and Mesozoic cover rocks in both their foot‐ and hanging‐ walls. Most of the orogenic displacement in the North Pyrenean Unit occurred along the North Pyrenean Frontal Thrust, while the steep faults observed in the basement were probably not strongly reactivated during the Pyrenean orogeny. They may represent either contractional Hercynian structures or younger, extensional pre‐Pyrenean fault zones (Capote et al., 2002). As the timing of the deformation along the sub‐vertical faults is very difficult to assess, the structural history of North Pyrenean Unit remains controversial (ECORS Pyrenees Team, 1988).

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Thermochronology data

Altogether, 74 published thermochronometric ages have been used from 48 different locations to test the predictions of the thermo‐kinematic models: 40 apatite fission‐track (AFT) ages, 27 apatite (U‐Th)/He (AHe) ages, 4 zircon fission‐track (ZFT) ages and 3 time‐temperature models extracted from K‐feldspar 40Ar/39Ar degassing experiments (Fitzgerald et al., 1999; Gibson et al., 2007; Metcalf et al., 2009; Sinclair et al., 2005). The samples were collected along two age‐ elevation profiles in the Maladeta massif, one profile in the Barruera block, one profile in the Marimaña massif, and one profile in the Lacourt massif (Figures 1 and 3). Individual samples were also collected from the Nogueres zone along the southernmost edge of the Axial Zone. The Maladeta and Barruera profiles are located in the Orri thrust sheet, whereas the Marimaña profile is located in the Nogueres thrust sheet, in the hanging‐wall of the Gavarnie thrust (Figure 1). The Nogueres samples come from the southern frontal zone of the Nogueres thrust sheet. The Lacourt profile is located within the southern part of the Arize block (Figure 3) and is the only profile located in the North Pyrenean Unit. The samples in the Maladeta, Barruera, Marimaña and Lacourt profiles were collected from Hercynian granitic plutons while the samples in the Nogueres Zone are from Cambrian to Triassic volcanic rocks.

Table of contents :

1 Introduction and summary 
1.1 Four-dimensional N = 2 gauge theories
1.2 Toda conformal field theory
1.3 Supersymmetric localization on S2
1.4 AGT correspondence and extended operators
1.5 Two-dimensional N = (2, 2) dualities
2 Two-dimensional N = (2, 2) gauge theories 
2.1 Introduction
2.2 N = (2, 2) gauge theories on S2
2.3 Localization of the path integral
2.4 Coulomb branch
2.5 Higgs branch representation
2.6 Gauge theory/Toda correspondence (omitted)
2.7 Seiberg duality (omitted)
2.8 Discussion
2.A Notations and conventions
2.B Supersymmetry transformations on S2
2.C Supersymmetric configurations
2.D One-loop determinants
2.E One-loop running of FI parameter
2.F Factorization for any N = (2, 2) gauge theory
2.G Vortex partition function
2.H SU(N) partition function in various limits (omitted)
3 AGT for surface operators 
3.1 Introduction and conclusions
3.2 Surface operators as Toda degenerates
3.3 SQED and Toda fundamental degenerate
3.4 SQCD and Toda antisymmetric degenerate
3.5 SQCDA and Toda symmetric degenerate
3.6 Quivers and multiple Toda degenerates
4 Two-dimensional gauge theory dualities
4.1 Introduction
4.2 Seiberg duality as momentum conjugation
4.3 SQCDA dualities: crossing and conjugation
4.4 Dualities for quivers
4.A SQCD vortex partition functions
4.B SQCDA vortex partition functions
5 Toda conformal field theory 
5.1 W algebra
5.2 Braiding matrices
5.3 Braiding kernel
5.4 Toda CFT correlators
5.5 Fusion rules
5.6 Irregular punctures

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