Seiberg duality as momentum conjugation

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Four-dimensional N = 2 gauge theories

This section is about four-dimensional gauge theories with N = 2 super-symmetries, that is, theories invariant two sets of 4 supercharges (see the reviews [Tac13; Tes14]). The aim is to prepare for the AGT correspondence (in Section 1.4), thus many properties of N = 2 theories are skipped, most importantly Seiberg–Witten curves.
The field content of an N = 2 gauge theory decomposes into vector multiplets (gauge multiplets) and hypermultiplets (matter multiplets). Both types of N = 2 supermultiplets can be split into supermultiplets of an N = 1 subalgebra. A hypermultiplet is composed of a pair of chiral multiplets, thus of complex scalars and their spinor superpartners, all in the same representation of a gauge group. An N = 2 vector multiplet is composed of an N = 1 vector multiplet and a chiral multiplet, in other words a gauge boson and its superpartners in the adjoint representation of the gauge group. Lagrangian couplings of these supermultiplets which preserve N = 2 supersymmetry are more restricted than in N = 1 theories, and boil down to the so-called holomorphic prepotential for the vector multiplet.
The prime example of an N = 2 gauge theory is SU(N) SQCD (super quantum chromodynamics) with Nf flavours, which consists of an N = 2 vector multiplet with gauge group SU(N) coupled to Nf hypermultiplets in the fundamental (dimension N) representation of SU(N). The one-loop beta function of the gauge coupling constant is proportional to 2N − Nf , and non-renormalization theorems imply that the exact beta function also is. The theory is thus asymptotically free for Nf < 2N, exactly conformal for Nf = 2N (in the absence of mass terms), and it is not UV complete for Nf > 2N.
For N > 2, the flavour symmetry of N = 2 SU(N) SQCD with Nf flavours is U(Nf ). For N = 2, each hypermultiplet splits into two half-hypermultiplets which are both representations of the N = 2 superalgebra, and the flavour symmetry enhances to SO(2Nf ) ⊃ U(Nf ). Technically, such a splitting and symmetry enhancement occurs whenever a hypermultiplet transforms in a pseudo-real representation of the gauge group.
Seiberg and Witten [SW94a; SW94b] worked out in 1994 the quantum vacua of N = 2 SU(2) SQCD with 0 ≤ Nf ≤ 4 fundamental flavors. These authors determined the exact prepotential of N = 2 SU(2) SQCD, from which one can extract for instance masses of W-bosons and dyons. They found that the Nf = 4 SQCD Lagrangian exhibits S-duality: the theory can be described by Lagrangians written in terms of different sets of fundamental degrees of freedom.
In terms of the (complexified) gauge coupling τ = 8πi/g2 + ϑ/π, S-duality states that the Nf = 4 Lagrangians with a given coupling τ and with the dual coupling τD = −1/τ describe the same physics. In this way, S-duality provides a weakly coupled description (τD → ∞, g D → 0) of a region of the parameter space where the initial theory is strongly coupled (τ → 0, g → ∞). The duality generalizes to give τD = aτcτ++db for any matrix ac db in SL(2, Z).
The S-duality group SL(2, Z) also acts by automorphisms of the flavour symmetry group SO(2Nf ) = SO(8). This action is conveniently described by splitting the hypermultiplets into two pairs, each with an SO(4) ˜ SU(2)2 flavour symmetry. In every S-dual description of the theory, the manifest SO(4) ×SO(4) symmetry results from pairing the factors of SU(2)4 ⊂ SO(8) in one of three possible ways. Concretely, one can include masses mA,B,C,D for the four SU(2) factors. In some Lagrangian description the Nf = 4 hypermultiplets have masses |mA ± mB| and |mC ± mD|. After S-duality, masses are |mA ± mC | and |mB ± mD|, or |mA ± mD| and |mB ± mC |.
Gaiotto [Gai09a] generalized S-duality to a wide class of four-dimensional N = 2 theories Tg,n, now called class S theories.
Flavour symmetry groups of an N = 2 theory can be gauged by a vector multiplet in the same way as global symmetries of non-supersymmetric theories are gauged by a gauge boson. Thus, any of the four SU(2) flavour symmetry groups of SU(2) SQCD with Nf = 4 can be promoted to a gauge group with an SU(2) vector multiplet. The additional vector multiplet can be coupled to a pair of hypermultiplets to keep the theory conformal, and these come with SU(2) × SU(2) flavour symmetry. Repeating the procedure with the new SU(2) symmetries generates a large number of N = 2 superconformal Lagrangians: their mass deformations describe SU(2) class S theories.
Both pairs of hypermultiplets in Nf = 4 SU(2) SQCD transform in the trifundamental representation of SU(2)3 flavour and gauge groups. The Nf = 4 theory can thus be regarded as trifundamental hypermultiplets of SU(2)A × SU(2)B × SU(2)G and SU(2)G × SU(2)C × SU(2)D whose common SU(2)G symmetry is gauged by a vector multiplet. As described above, S-dual Lagrangians group the SU(2)A,B,C,D in three possible pairings, N = 2∗ if the hypermultiplet is massive, and it has N = 4 supersymmetry in the massless case. The graphs on the right represent two vector multiplets coupled to two hypermultiplets in different ways. In fact, the two Lagrangians turn out to be S-dual, and describe the same theory.
depicted as graphs in Figure 1.1. Each trifundamental hypermultiplet is represented as a vertex connected to three line representing SU(2) symmetry groups. External lines are flavour symmetries, while internal lines are gauge groups. Using this dictionary, any trivalent graph (three edges per vertex) corresponds to a superconformal Lagrangian composed of trifundamental hypermultiplets and vector multiplets (see Figure 1.2 for examples).
Consider a trivalent graph representing a Lagrangian. In the limit where all SU(2) gauge couplings except one (corresponding to an internal edge) are vanishingly small, the remaining SU(2) gauge theory is simply Nf = 4 SQCD and it obeys S-duality. The duality reconnects in any pairing the four edges touching the chosen internal edge. This is expected to hold even when other gauge couplings are non-vanishing. Reconnecting edges through S-dualities leads from a trivalent graph to any other with the same numbers of internal and external lines. In other words, all graphs with g loops and n external lines correspond to Lagrangians which describe the same theory Tg,n in terms of different degrees of freedom.
Properly keeping track of how S-duality acts on the gauge coupling constants (τ 7→ −1/τ for SQCD) requires more structure than the graphs. The correct data to describe couplings of Tg,n is in fact a Riemann surface Cg,n with genus g and n punctures. The surface is obtained from any graph with g loops and n external edges by “fattening” the graph, in other words by replacing each edge by a tube and each trivalent vertex by a smooth trinion (three-punctured sphere) joining the three cylinders. The length and twisting angle of each tube encode the coupling constant for the gauge group attached to this edge of the graph, so that a long cylinder corresponds to a weakly coupled vector multiplet. S-duality is then retrieved by noting that Cg,n can be cut into tubes and trinions in many ways, labelled by the various trivalent graphs (see Figure 1.3). Each decomposition of Cg,n corresponds to a Lagrangian description of Tg,n.
Figure 1.3: Two decompositions of the Riemann surface C0,5 into tubes (drawn as ellipses) and trinions (three-punctured spheres between the tubes and external punctures), and their trivalent graphs (in dotted lines). The two corresponding Lagrangians describe the same theory T0,5.
Several theories in this class are worth noting. T0,3 is the theory of 4 free hypermultiplets, with no vector multiplet since its graph has no internal edge. T0,4 is SU(2) SQCD with Nf = 4, and depends on a single complexified coupling. The four-punctured sphere C0,4 is a tube joining two trinions. Its complex structure only depends on the cross-ratio q of the four punctures, and changing trinion decomposition maps q 7→1−q or 1/q. As a last example, T1,1 describes an SU(2) vector multiplet gauging two SU(2) flavour symmetries of a single trifundamental hypermultiplet: this results in a hypermultiplet in the adjoint representation of the gauge group and in the fundamental representation of the last SU(2). The theory, called N = 2∗ SYM (super Yang–Mills) when the hypermultiplet is massive, has an enhanced N = 4 supersymmetry when the hypermultiplet is massless.
As argued in [Gai09a] using Seiberg–Witten curves, the theory Tg,n is the four-dimensional reduction on Cg,n of the mysterious A1 (2, 0) six-dimensional superconformal field theory with some boundary conditions at the punctures. This theory is not known directly, but its reductions to various lower dimensions are known. For instance, its reduction on a circle is the maximally supersymmetric SU(2) Yang–Mills theory in five dimensions, which reduces further to the four-dimensional N = 2 vector multiplet associated to each cylinder of Cg,n in the description above.
In M-theory, the A1 (2, 0) theory is the world-volume theory of two coincident M5-branes. These branes are then wrapped around the Riemann surface Cg,n, whose punctures are realized by transverse M5-branes. Brane constructions give useful intuitions in Section 1.4 on extended operators of N = 2 theories.
A close cousin of A1 (2, 0) is the six-dimensional AN−1 (2, 0) supercon-formal field theory, the world-volume theory of N coincident M5-branes. Compactifying it on a Riemann surface Cg,n with some boundary conditions at punctures yields a four-dimensional N = 2 gauge theory with SU(N) gauge groups. The set of all such N = 2 theories is dubbed class S. In the absence of mass terms these theories are superconformal.

The AN−1

The standard example of a class S theory is N = 2 SU(N) SQCD with 2N fundamental hypermultiplets, obtained when Cg,n = C0,4 is the four-punctured sphere. For N > 2 the flavour symmetry of SQCD is U(2N), with an SU(N) × U(1) × U(1) × SU(N) subgroup made manifest by the six-dimensional construction. An important difference with the N = 2 case is that the four factors are not identical: correspondingly the punctures on C0,4 come with different boundary conditions. Two punctures carry an SU(N) flavour symmetry and are called full, and the other two carry a U(1) flavour symmetry and are called simple. Many other types of punctures exist.
As before, S-dual descriptions are labelled by trinion decompositions of Cg,n. A trinion with one simple and two full punctures is associated to N2 hypermultiplets and makes a U(1) × SU(N) × SU(N) flavour symmetry explicit. Joining full punctures of two trinions corresponds to gauging the two SU(N) flavour symmetries diagonally. In the case of SQCD, the two decompositions of C0,4 where both trinions have a simple puncture correspond to descriptions as an SU(N) vector multiplet coupled to two sets of N fundamental hypermultiplets. When the two full punctures belong to the same trinion, there is no Lagrangian description: one is still coupling two theories by gauging a common flavour symmetry, but the building block corresponding to one of the trinions is non-Lagrangian. More generally, while all SU(2) class S theories are described by Lagrangians, class S theories for N > 2 only have a Lagrangian description in duality frames where every trinion involves a simple puncture.
Since a class S theory only depends on the complex structure of Cg,n and on data at each puncture, any observable of the four-dimensional the-ory can in principle be derived from a computation on Cg,n. In practice, the identification is typically worked out by computing four-dimensional observables and finding a matching two-dimensional calculation. The AGT correspondence [AGT09] (Section 1.4) consists in a concrete dictionary be-tween several observables obtained through supersymmetric localization on spheres (Section 1.3) and correlators in the Toda CFT (Section 1.2) on Cg,n.

Toda conformal field theory

Toda theory is a two-dimensional CFT whose symmetry algebra WN is an extension of the Virasoro algebra by higher spin currents. The A1 Toda theory (N = 2) is the well-known Liouville CFT, and W2 is the Virasoro algebra. This section recalls basic notions of two-dimensional CFT (reviewed in [Rib14]) up to the braiding kernel of primary operators. It then describes effects of the WN symmetry, and the explicit proposal (5.3.27) for the braiding kernel of some WN primary operators. This introduction to the Toda CFT is enough to read the thesis, which concludes with a detailed study of the theory in Chapter 5.
Two-dimensional conformal symmetry implies an action of (two copies of) the Virasoro algebra on states of the theory. The two copies are due to holomorphic and antiholomorphic conformal transformations, and can be treated independently. Conformal symmetry also implies that the state-operator correspondence is a bijection between operators φ and states |φi obtained by acting on the vacuum.
The Virasoro algebra has generators Ln for n ∈ Z subject to L†n = L−n and the commutation relations [Lm, Ln] = (m − n)Lm+n + 12c (m3 − m)δm+n. A highest-weight state is |hi such that Ln |hi = 0 for n > 0 and L0 |hi = h |hi, and the corresponding operator is called a primary operator of dimension h. Acting with L−n for n > 0 yields a Verma module: a representation of the Virasoro algebra whose states are linear combinations of L−n1 • • • L−np |hi for nj > 0. Such a state is called a descendant of |hi at level Pj nj. A primary operator and its descendants form a conformal family. For convenience, the central charge is parametrized as c = 1 + 6q2 with q = b + 1/b, and the dimension h = α(q − α) of a primary operator Vα is expressed in terms of a momentum α ∈ C. Conformal symmetry expresses correlators of descendant operators in terms of correlators of primary operators. It forces sphere two-point functions of primary operators to vanish unless the two operators have the same dimension. It also fixes the coordinate dependence of sphere three-point functions hVαVβVγi, but not an overall factor C(α, β, γ). All n-point functions of primary operators on the sphere are then fixed as follows in terms of the three-point functions C(α, β, γ), also called structure constants.
Any pair of primary operators can be replaced by their OPE (operator product expansion), a linear combination of primary operators and descen-dants whose coefficients are fixed in terms of the structure constants by conformal symmetry. The n-point function gets recast as an integral (or sum) over conformal families of a structure constant multiplied by an (n − 1)-point function and by a factor keeping track of descendant contributions. Repeating the procedure expresses any n-point function as an integral of products of n − 2 three-point functions multiplied by a factor that is fixed by conformal symmetry. This conformal factor factorizes as a conformal block F holomorphic in the positions of operators times an antiholomorphic conformal block. Glossing over details such as inverse two-point functions, hVα1 • • • Vαn i = Z dβ3 • • • dβn−1 C(α1, α2, β3) • • • C(βn−1, αn−1, αn) F » α 2 α 3 αn 2 αn 1 # 2 (1.2.1) • α 1 β3 −βn − 1 − αn .
The trivalent graph describes which OPEs were performed, and keeps track of the resulting momenta. There is one structure constant for each vertex of this trivalent graph. The momenta αi are called external momenta, while βi are internal momenta and are integrated over. In a different channel, that is, a choice of which operators to pair into OPEs represented by another trivalent graph, the expression involves completely different structure constants and conformal blocks F. Crossing symmetry states that the expressions must be equal, as they both compute the same n-point function.
In fact, crossing symmetry is implied by its simplest case, four-point functions. A global conformal transformation places the operators at 0, x, 1, and ∞. Taking the OPE of the operator at x with that at 0, 1, or ∞ yields expressions in terms of s-, t-, and u-channel conformal blocks, respectively:
It turns out that crossing symmetry and the holomorphic/antiholomorphic factorization imply that holomorphic conformal blocks in one channel are linear combinations of holomorphic conformal blocks in another channel, after analytic continuation in x. The linear combinations are expressed as a fusion kernel Fαα0 and a braiding kernel Bαα0: Fα(s) = dα0 Fαα0 Fα(t)0 = dα0 Bαα0 Fα(u)0 . (1.2.3)
These kernels are related by a permutation of the αi, and were determined in [PT99] as an integral of ratios of Barnes double sine functions.
One last word on theories with Virasoro symmetry. A Verma module whose momentum is one of αr,s = (1 − r)b/2 + (1 − s)/(2b) or q − αr,s for integers r, s ≥ 1 contains a null-vector at level rs, namely a zero-norm descendant state that is orthogonal to the whole representation, hence the module is reducible. These momenta are called degenerate. Correlation functions which include degenerate primary operators simplify because of null-vectors. Using the level rs null-vector of Vαr,s , the three-point functions hVαr,s VβVγi are found to vanish unless γ (or q − γ) is one of the rs values β + jb + k/b with j = 1−2r , . . . , r−21 and k = 1−2s , . . . , s−21 . These non-zero three-point functions constrain what conformal families can appear in the* OPE of Vαr,s with Vβ: the fusion rule is (r−1)/2 (s−1)/2 X− X− (1.2.4) Vαr,s × Vβ = [Vβ+jb+k/b] j=(1 r)/2 k=(1 s)/2 where brackets denote contributions from descendants, structure constants are omitted, and sums run in steps of 1. Describing the Toda CFT requires some Lie algebra notations. The Cartan subalgebra h of AN−1 = su(N) is identified to h∗ using its Killing form. The weights hs (1 ≤ s ≤ N) of the fundamental representation of AN−1 sum to zero and form an overcomplete basis of h. Simple roots are ek = hk − hk+1.
The AN−1 Toda Lagrangian describes a scalar field ϕ ∈ h with a back-ground charge and an exponential potential term. More precisely, the potential term is PN−1 ebhek,ϕi in terms of a parameter b, and the background k=1 charge Q is a fixed element of h multiplied by q = b + 1/b. Much more important than the Toda Lagrangian is its invariance under (two copies of) the WN algebra, a higher-spin generalization of the Virasoro algebra. This algebra has N − 1 sets of generators Wn(p) for 2 ≤ p ≤ N, with Wn(2) = Ln. Primary operators Vα of the WN algebra are labelled by the eigenvalues of all W0(p) expressed in terms of a momentum α ∈ h. Permuting the components hα − Q, hsi of the momentum does not change the eigenvalues of W0(p): this Weyl symmetry generalizes the α 7→q − α invariance of Virasoro primary operators. In the Toda CFT, an appropriate normalization Vbα (5.4.3) of Vα is invariant under Weyl symmetries.
Three types of momenta play a role in the present work. Generic mo-menta α are such that the Verma module constructed by acting with W−(pn), n > 0, on |αi = Vα |vacuumi has no null-vector. Semi-degenerate momenta take the form {h1 (up to Weyl symmetries), and their Verma modules have some null-vectors. Degenerate momenta −bω − ω0/b are characterized by two dominant weights ω, ω0 of AN−1, and Verma modules have a maxi-mal number of null-vectors. For N = 2, there is no distinction between generic and semi-degenerate momenta, and degenerate momenta reproduce the degenerate momenta −r−21 b − s−21 b−1 of Virasoro.
Two-point functions hVαVβi of primaries vanish unless the two operators have equal eigenvalues of all W0(p) up to a sign (−1)p. In terms of momenta, β = 2Q − α or a Weyl permutation thereof. Three-point functions with one degenerate primary operator vanish in most cases: accordingly the OPE of a degenerate and a generic primaries is V−bω−ω0/b × Vα = [Vα−bh−h0/b] , (1.2.5) h∈R(ω) h0∈R(ω0) the natural extension of the OPE (1.2.4) of Virasoro primaries. Here, sums run over weights h of the representation R(ω) with highest weight ω, and similarly for h0. Another useful fusion rule is V−bh1 × V{h1 = [V({−b)h1 ] + [V{h1−bh2 ] (1.2.6) and its generalization (5.5.24) to the fusion of a semi-degenerate operator with any degenerate V−bω. All of these fusion rules are confirmed in the Toda CFT through the Coulomb gas formalism, but the author does not know of a proof using only WN symmetry.
A major difference between the Virasoro algebra and WN for N ≥ 3 is that correlators of WN descendants are not fixed in terms of correlators of their WN primary operators. Sphere n-point functions of primary operators can still be decomposed in terms of three-point functions of primary and descendant operators, but do not decompose further to three-point functions of primaries multiplied by factorized conformal blocks. To solve a WN – invariant theory, it is thus not enough to find all three-point functions of WN primaries. Of course, knowing the three-point functions of all Virasoro primaries suffices, but these are much more numerous.
Despite this difficulty, conformal blocks exist if enough primary operators are semi-degenerate (or degenerate). The three-point function of a semi-degenerate and two generic operators fixes all three-point functions of their descendants, hence conformal blocks exist whenever each vertex of the trivalent graph defining the channel has a semi-degenerate momentum. For instance, the n-point function (1.2.1) of Virasoro primaries keeps essentially the same form for WN primaries (replacing momenta by vectors) if all α2, . . . , αn−1 are taken to be semi-degenerate and α1, αn and the βi to be generic.
Consider the four-point function hVα∞(∞)Vλh1 (1)V−bh1 (x, x¯)Vα0 (0)i with two generic momenta α0 and α∞, a semi-degenerate λh1, and a degenerate −bh1 labelled by the fundamental representation R(h1) of AN−1. Operators are placed at 0, x, 1 and ∞ through a global conformal transformation. This four-point function was originally determined in [FL07] by working out using null-vectors of W3 that conformal blocks obey a hypergeometric differential equation (up to some factors), then writing the correct generalization for all N. Section 5.2.1 directly attacks the general N case through a bootstrap approach since null-vectors are not known explicitly for WN .

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Table of contents :

1F Présentation des travaux (French summary) 
1F.1 Théories de jauge N = 2 à quatre dimensions
1F.2 Théorie conforme des champs de Toda
1F.3 Localisation supersymétrique sur S2
1F.4 Correspondance AGT et opérateurs étendus
1F.5 Dualités N = (2, 2) à deux dimensions
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
Bibliography

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