Dual infrared limits of 6d $\cal N$=(2,0) theory

Compactifying type $A_{N-1}$ 6d ${\cal N}{=}(2,0)$ supersymmetric CFT on a product manifold $M^4\times\Sigma^2=M^3\times\tilde{S}^1\times S^1\times{\cal I}$ either over $S^1$ or over $\tilde{S}^1$ leads to maximally supersymmetric 5d gauge theories on $M^4\times{\cal I}$ or on $M^3\times\Sigma^2$, respectively. Choosing the radii of $S^1$ and $\tilde{S}^1$ inversely proportional to each other, these 5d gauge theories are dual to one another since their coupling constants $e^2$ and $\tilde{e}^2$ are proportional to those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU($N$) Yang-Mills theories on $M^4\times{\cal I}$ and on $M^3\times\Sigma^2$, where $M^4\supset M^3=\mathbb R_t\times T_p^2$ with time $t$ and a punctured 2-torus, and ${\cal I}\subset\Sigma^2$ is an interval. In the first case, shrinking ${\cal I}$ to a point reduces to Yang-Mills theory or to the Skyrme model on $M^4$, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on $M^3$ and employing the adiabatic method, we derive in the infrared limit a non-linear SU($N$) sigma model with a baby-Skyrme-type term on $\Sigma^2$, which can be reduced further to $A_{N-1}$ Toda theory.


Introduction and summary
The famous Alday-Gaiotto-Tachikawa (AGT) 2d-4d correspondence [1] relates Liouville field theory on a punctured Riemann surface Σ 2 and SU (2) super-Yang-Mills (SYM) theory on a fourdimensional manifold M 4 (see e.g. [2] for a nice review and references). This correspondence was quickly extended to 2d A N −1 Toda field theory and 4d SU(N ) SYM [3]. Since then various AGTlike correspondences between theories on n-and (6−n)-dimensional manifolds were investigated (see e.g. [4,5]) for reviews and references). One way to interpret these correspondences is to start from 6d N =(2,0) supersymmetric conformal field theory (CFT) on M n ×M 6−n . The theory on M n would appear as a low-energy limit when M 6−n shrinks to a point, while the theory on M 6−n would emerge when M n is scaled down to a point [2,4,5]. To be more precise, the AGT correspondence for n=4 relates the partition function of 6d CFT of type A N −1 on M 4 × Σ 2 with partition functions of theories on M 4 and Σ 2 , which are all equal due to the conformal invariance of the 6d theory. The standard way of establishing such correspondences consists of calculating partition/correlation functions on M n and M 6−n and to compare them. However, we are not aware of a geometric derivation of the AGT correspondence.
Recently, a derivation of the AGT correspondence via reduction of 6d CFT to M 4 and Σ 2 was discussed in [6,7,8,9]. A relation between A N −1 6d CFT compactified on a circle and 5d SU(N ) SYM was employed as well as further reduction to complex Chern-Simons theory on Σ 2 × I, where I ⊂ M 4 is an interval. Then a generalized version of the correspondence between 3d Chern-Simons theory and 2d CFT was used, with the Nahm-pole boundary conditions translating to the constraints reducing WZW models to Toda theories [7,8,9].
In this paper we consider SU(N ) Yang-Mills theories on manifolds M 4 × I and M 3 × Σ 2 , where M 4 = M 3 ×S 1 is Lorentzian and Σ 2 ∼ = S 1 ×I is a two-sphere with two punctures. The two manifolds agree in M 3 × I but differ in the additional circleS 1 versus S 1 . The supersymmetric extensions of Abelian gauge theories on these two 5d manifolds originate from type A N −1 6d N = (2, 0) supersymmetric CFT on M 4 × Σ 2 , compactified on S 1 orS 1 with radii R 1 = εR 0 orR 1 =εR 0 , respectively [10,11,12]. Choosingε = ε −1 the two 5d theories become dual to each other, and their gauge coupling constants e 2 orẽ 2 (in units of the reference scale R 0 ) will be inversely proportional to one another, which is consistent with the arguments from [13]. In order to have nontrivial vacuum solutions we take M 3 = R t × T 2 p with a temporal direction R t and a one-punctured 2-torus T 2 p . For further low-energy limits we introduce two scales,R and R, which determine the sizes of the two factors in the product manifolds M 4 R × I R and M 3 R × Σ 2 R , respectively. In the first case, via (R, R) = (R 0 , εR 0 ) and ε ≪ 1 we shrink I R to a point. Depending on the choice of reductiontranslational invariance along I R or adiabatic approach [14,15,16,17,18,4] -one obtains (N ≥ 2 extended) Yang-Mills theory or the Skyrme model on the manifold M 4 R 0 . In the second case, taking (R, R) = (εR 0 , R 0 ) andε ≪ 1 we scale down the metric on R t and T 2 p . The adiabatic method [15]- [22] then produces a non-linear sigma model with a baby-Skyrme-type term on Σ 2 R 0 . Finally, we briefly discuss a reduction of this 2d SU(N ) sigma model to Toda field theory on Σ 2 R 0 . To summarize, we propose a geometric background for establishing 4d-2d AGT correspondences between field theories on M 4 and Σ 2 . We argue that these correspondences depend not only on the topology of M 4 and Σ 2 but also on the method employed for deriving the low-energy effective field theories in the infrared (using symmetries, adiabatic limit, constraints etc.).
2 Two dual ways from 6d to 5d Yang-Mills theory 6d manifold. We consider the purported 6d Abelian N = (2, 0) CFT on the manifold is a temporal cylinder over a two-torus with sizeR and a puncture p. The radii of the circlesS 1 and S 1 and the length of the interval I are indicated. The metric on this space is taken as where a, b = 0, 1, 2, and all spatial coordinates in the second line range over [−π, π], with the Greek ones being periodic. The coordinate relations are Reduction to M 4 × I. Let us equateR 1 =R and scale down S 1 by choosing relative to some fixed radius R 0 . This reduction gives rise to a 5d super-Maxwell theory [13] on It has a non-Abelian SU(N ) extension consistent with gauge invariance and supersymmetry, whose pure-gauge sector we shall consider further. It is conjectured that this 5d SYM is the dimensional reduction of a non-Abelian 6d CFT. 1 Gauge fields on M 4 × I. The gauge potential A (connection) and the gauge field F (curvature) both take values in the Lie algebra su(N ). On M 4 R × I R we have For the generators I i in the fundamental N × N representation of SU(N ) we use the normalization tr( For the metric tensor (2.6) we have gâb R = ηâb and g zz R = R −2 . It follows that The standard Yang-Mills action functional with this metric takes the form where e is the gauge coupling constant. It is known [13] that e 2 is proportional to the S 1 compactification radius R 1 =εR 0 . A full supersymmetric extension of (2.9) can be found e.g. in [4].
Reduction to M 3 × Σ 2 . Alternatively, let us put R 1 = R and shrinkS 1 by taking This reduction produces the 5d super-Maxwell theory on with a metric Gauge fields on M 3 × Σ 2 . As before, we extend the gauge group to SU(N ) and restrict ourselves to the pure Yang-Mills subsector since supersymmetry plays no role in our discussion. Both gauge potential A and gauge field F take values in su(N ), and on M 3 Abbreviating the rescaled M 3 R coordinates as (t, α, β) = (y µ ) with µ = 0, 1, 2, for the metric tensor (2.12) we haveg µν R =R −2 η µν andgāb R = δāb. Hence, for the upper components of F we obtain (2.14) Then the standard Yang-Mills action functional on M 3 R × Σ 2 R will have the form whereẽ is the dual gauge coupling, whose square is proportional to theS 1 compactification radius R 1 =εR 0 . The reduction to M 3 R × Σ 2 R is dual to the one to M 4 R × I R in the sense that we may takeε = ε −1 so that ε ≪ 1 is related toε ≫ 1 and vice versa. Hence the Yang-Mills theories (2.9) and (2.15) are dual in agreement with the discussion in [13]. R × I R , discussed in the previous section as the first reduction from 6d, there exist two types of further reduction to four dimensions, both of which appear in the literature as low-energy limits: reduction with respect to the translation ∂ z [13] and reduction via the adiabatic approach (see e.g. [14,4,21,22]). Both reductions shrink I R to a point, via by saying that for R ≪R the momenta along I R are much larger than along M 4 . Then one can discard all higher modes of A a and Φ as well as of scalar and spinor fields of maximally supersymmetric Yang-Mills theory on M 4 × I R .
After substituting (3.2), the Yang-Mills action (2.9) reduces to where D a = ∂ a + [A a , · ]. Likewise, the full SYM theory on M 4 × I R passes to N = 2 or N = 4 super-Yang-Mills theory on M 4 , depending on twisting along Σ 2 and other assumptions (see e.g. [2,13]). The discussion simplifies because in the 6d to 5d reduction the two-punctured two-sphere Σ 2 ∼ = S 1 R 1 × I R gets deformed into a thin cylinder as in [4,14].
Skyrme model. A different infrared limit of SYM theory on M 4 × I R , discussed e.g. in [4,14,21], introduces where P denotes path ordering. The group element g is the holonomy of A along I. In the lowenergy limit, when the length of I becomes small, the 5d YM theory (2.9) reduces to the Skyrme model on M 4 , where L a = g −1 ∂ a g for g from (3.4), ς is the dimensionless Skyrme parameter, and f π is interpreted as the pion decay constant. Their relation to the dimensionful 5d gauge coupling e 2 and the infrared scale R is f 2 π 4 = π 4 e 2 R and 1 32 ς 2 = πR 120 e 2 . (3.6) Recall that e 2 is proportional to R [13] and R = εR 0 becomes small, so that f 2 π ∼ ε −2 and ς −2 ∼ ε 0 , and higher-order (in R) terms are suppressed.
The derivation of (3.5) does not impose (3.2). It is based on the adiabatic approach [15]- [18] which is equivalent to the moduli-space approximation with moduli given by g = g(x) from (3.4). The two-derivative term in (3.5) is the standard 4d non-linear sigma model, and its supersymmetric version was derived from 5d in [4]. The four-derivative term in (3.5) stabilizes solitons against scaling. This term was deduced from 5d SYM on M 4 × I R in [21], where its possible supersymmetrization, yet unknown, was briefly discussed.
Thus, in the infrared of 5d SYM theory on M 4 × I R one can find two different models on M 4 : N ≥ 2 SYM theory and the Skyrme model. The latter describes low-energy QCD by interpreting mesons as fundamental and baryons as solitons. Away from the infrared limit the Skyrme model gets extended by an infinite tower of heavy mesons beyond the leading Skyrme term displayed in (3.5) (cf. [25,26]).

Moduli space of YM vacua on M 3
For the remainder of the paper we focus on the second kind of 6d to 5d reduction presented in Section 2, namely the dual action (2.15), and its own infrared simplification from M 3 R × Σ 2 R to Σ 2 . In a manner dual to (3.1), we put which scales down the metric (2.12) in the M 3 R direction. We drop the reference to the (now fixed) scale of Σ 2 R . Hence, forR ≪ R the 5d YM theory reduces to an effective 2d field theory on Σ 2 which we now describe.
Flat connections on M 3 . Our derivation of the low-energy limit employs the adiabatic method [15]- [19] (for brief reviews and more references see e.g. [20,27]). In this approach one firstly restricts to M 3 R by putting and classifies the classical solutions A m (y n ) on M 3 R , independent of the Σ 2 coordinates xā. Secondly, one declares that the moduli X α , which parametrize such solutions, become functions of xā ∈ Σ 2 . Thirdly, one introduces small fluctuations δA m and Aā depending on y n ∈ M 3 R and on the moduli functions X α (xā), substitutes them into (2.15) and obtains an effective field theory for X α on Σ 2 . Since in the following we want to study small fluctuations around the vacuum manifold, we first take a look at the vacuum configurations.
The vacua on M 3 R are given by F µν = 0 , Flat connection on T 2 p . It is well known that gauge bundles over smooth tori T 2 (compact, without punctures) admit only reducible flat connections [28,18]. However, this theorem is not valid on Riemann surfaces with punctures or fixed points (see e.g. [29,30,31]). In particular, the moduli space M T 2 p of flat connections on a K-bundle over T 2 p is the gauge group K, where the group of gauge transformations G T 2 p = C ∞ (T 2 p , K) forms the fibres over the points in M T 2 p for the bundle π : We specialize to K = SU(N ).
. We have The variation of A m along T M T 2 p is then given by where ǫ α is a suitable gauge parameter which brings ∂ α A m back to π * T A T 2 p M T 2 p . This makes sure that the variation δ α A m obeys a linearization of the flatness condition (4.3), where {X α } is a set of local coordinates of a point X ∈ G/H = SU(N ) and ∂ α = ∂ ∂X α will denote derivatives with respect to them.
5 Toda theory on Σ 2 from YM on M 3 × Σ 2 Moduli space approximation. In the adiabatic approach, applicable forε ≪ 1, the moduli approximation assumes that the moduli X α vary with the coordinates xā ∈ Σ 2 [15]- [19]. In this way, the moduli of flat connections on T 2 p define a map so that X α = X α (xā) may be considered as dynamical fields. Admitting xā dependence only via X α ∈ SU(N ), our fields take the form A m = A m y n , X α (xb) and Aā = Aā y n , X α (xb) , where we stay with the gauge choice A 0 = 0.