Dual Infrared Limits of 6D N=(2,0) Theory

Physics Letters B 793 (2019) 297–302

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Physics Letters B

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Dual infrared limits of 6d N = (2, 0) theory ∗ Olaf Lechtenfeld a,b, , Alexander D. Popov a a Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany b Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany a r t i c l e i n f o a b s t r a c t

4 2 3 ˜ 1 Article history: Compactifying type AN−1 6d N =(2, 0) supersymmetric CFT on a product manifold M × = M × S × Received 19 November 2018 S1 × I either over S1 or over S˜ 1 leads to maximally supersymmetric 5d gauge theories on M4 × I or Accepted 25 February 2019 on M3 × 2, respectively. Choosing the radii of S1 and S˜ 1 inversely proportional to each other, these Available online 3 May 2019 5d gauge theories are dual to one another since their coupling constants e2 and e˜2 are proportional to Editor: M. Cveticˇ those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU(N) 4 × I 3 × 2 4 ⊃ 3 = × 2 Yang–Mills theories on M and on M , where M M Rt T p with time t and a punctured 2-torus, and I ⊂ 2 is an interval. In the first case, shrinking I to a point reduces to Yang–Mills theory or to the Skyrme model on M4, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on M3 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 2, which can be reduced further to AN−1 Toda theory. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction and summary them. However, we are not aware of a geometric derivation of the AGT correspondence. The famous Alday-Gaiotto-Tachikawa (AGT) 2d-4d correspon- Recently, a derivation of the AGT correspondence via reduction dence [1] relates Liouville field theory on a punctured Rie- of 6d CFT to M4 and 2 was discussed in [6–9]. A relation between 2 mann surface and SU(2) super-Yang–Mills (SYM) theory on AN−1 6d CFT compactified on a circle and 5d SU(N) SYM was a four-dimensional manifold M4 (see e.g. [2]for a nice review employed as well as further reduction to complex Chern-Simons and references). This correspondence was quickly extended to theory on 2 × I, where I ⊂ M4 is an interval. Then a generalized 2d AN−1 Toda field theory and 4d SU(N) SYM [3]. Since then version of the correspondence between 3d Chern-Simons theory various AGT-like correspondences between theories on n-and and 2d CFT was used, with the Nahm-pole boundary conditions (6 − n)-dimensional manifolds were investigated (see e.g. [4,5]) translating to the constraints reducing WZW models to Toda theo- for reviews and references). One way to interpret these correspon- ries [7–9]. dences is to start from 6d N = (2, 0) supersymmetric conformal In this paper we consider SU(N) Yang–Mills theories on man- n 6−n n field theory (CFT) on M × M . The theory on M would appear ifolds M4 × I and M3 × 2, where M4 = M3 × S˜ 1 is Lorentzian 6−n ∼ as a low-energy limit when M shrinks to a point, while the the- and 2 = S1 × I is a two-sphere with two punctures. The two 6−n n ory on M would emerge when M is scaled down to a point [2, manifolds agree in M3 × I but differ in the additional circle S˜ 1 = 4,5]. To be more precise, the AGT correspondence for n 4 relates versus S1. The supersymmetric extensions of Abelian gauge the- the partition function of 6d CFT of type A − on M4 × 2 with N 1 ories on these two 5d manifolds originate from type AN−1 6d 4 2 partition functions of theories on M and , which are all equal N = (2, 0) supersymmetric CFT on M4 ×2, compactified on S1 or due to the conformal invariance of the 6d theory. The standard ˜ 1 ˜ S with radii R1 = εR0 or R1 = ε˜ R0, respectively [10–12]. Choos- − way of establishing such correspondences consists of calculating ing ε˜ = ε 1 the two 5d theories become dual to each other, and n 6−n partition/correlation functions on M and M and to compare their gauge coupling constants e2 or e˜2 (in units of the reference scale R0) will be inversely proportional to one another, which is consistent with the arguments from [13]. In order to have non- Corresponding author. * trivial vacuum solutions we take M3 = R × T 2 with a temporal E-mail addresses: [email protected] (O. Lechtenfeld), t p 2 [email protected] (A.D. Popov). direction Rt and a one-punctured 2-torus T p . https://doi.org/10.1016/j.physletb.2019.02.051 0370-2693/© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 298 O. Lechtenfeld, A.D. Popov / Physics Letters B 793 (2019) 297–302

For further low-energy limits we introduce two scales, R˜ and R, Gauge fields on M4 × I. The gauge potential A (connection) and which determine the sizes of the two factors in the product man- the gauge field F (curvature) both take values in the Lie algebra 4 3 2 4 ifolds M × IR and M × , respectively. In the first case, via × I R˜ R˜ R su(N). On M ˜ R we have ˜ R (R, R) = (R0, εR0) and ε  1we shrink IR to a point. Depend- A = A aˆ + A ing on the choice of reduction – translational invariance along IR aˆ dx z dz and (2.7) or adiabatic approach [14–18,4]– one obtains (N ≥ 2 extended) 1 aˆ bˆ aˆ F = Fˆ ˆ dx ∧ dx + Fˆ dx ∧ dz . Yang–Mills theory or the Skyrme model on the manifold M4 . In 2 ab az R0 ˜ For the generators I in the fundamental N × N representation the second case, taking (R, R) = (ε˜ R0, R0) and ε˜  1we scale i 1 2 of SU(N) we use the normalization tr(Ii I j) =− δij for i, j = down the metric on Rt and T p . The adiabatic method [15–22]then 2 2 i i produces a non-linear sigma model with a baby-Skyrme-type term 1, ..., N − 1 and write A = A Ii and F = F Ii . ˆ ˆ ˆ ˆ − on 2 . Finally, we briefly discuss a reduction of this 2d SU(N) For the metric tensor (2.6)we have gab = ηab and gzz = R 2. It R0 R R sigma model to Toda field theory on 2 . follows that R0 To summarize, we propose a geometric background for estab- ˆ ˆ ˆ ˆ ˆ − ˆ 4 Fab = Fab Faz = 2Faz lishing 4d-2d AGT correspondences between field theories on M R and R R . (2.8) and 2. We argue that these correspondences depend not only on The standard Yang–Mills action functional with this metric takes the topology of M4 and 2 but also on the method employed for the form deriving the low-energy effective field theories in the infrared (us-  R   ing symmetries, adiabatic limit, constraints etc.). =− 4 F Faˆbˆ + 2 F Fazˆ S5 d x dz tr aˆbˆ 2 azˆ , (2.9) 8e2 R 2. Two dual ways from 6d to 5d Yang–Mills theory M4×I where e is the gauge coupling constant. It is known [13] that e2 6d manifold. We consider the purported 6d Abelian N = (2, 0) 1 is proportional to the S compactification radius R1=εR0. A full CFT on the manifold supersymmetric extension of (2.9)can be found e.g. in [4]. 4 × 2 = 3 × ˜ 1 × 1 × I 3 = × 2 M M ˜ S ˜ S R R , where M ˜ Rt T p 3 2 R R1 1 R Reduction to M × . Alternatively, let us put R1 = R and shrink (2.1) S˜ 1 by taking is a temporal cylinder over a two-torus with size R˜ and a punc- ˜ R1 = ε˜ R0  R0 with ε˜  1 . (2.10) ture p. The radii of the circles S˜ 1 and S1 and the length of the interval I are indicated. The metric on this space is taken as This reduction produces the 5d super-Maxwell theory on

2 a b 3 2 4 2 5 2 3 2 2 1 ds = η dx dx + (dx ) + (dx ) + (dx ) M ˜ × = Rt × T × S × IR (2.11) 6 ab  R R p R ˜ 2 2 2 2 = R −(dt) + (dα) + (dβ) (2.2) with a metric ˜ 2 2 2 2 2 2   ¯ + R (dϕ) + R (dθ) + R (dz) , 2 ˜ 2 2 2 2 a¯ b 1 1 ds˜ = R −(dt) + (dα) + (dβ) + δ¯ ¯ dx dx R˜ ab (2.12) = ¯ where a, b 0, 1, 2, and all spatial coordinates in the second line with a¯, b ∈{4, 5} . range over [−π, π], with the Greek ones being periodic. The coor- dinate relations are Gauge fields on M3 × 2. As before, we extend the gauge group x0 = Rt˜ , x1 = R˜ α , x2 = R˜ β, to SU(N) and restrict ourselves to the pure Yang–Mills subsector (2.3) since supersymmetry plays no role in our discussion. Both gauge x3 = R˜ ϕ , x4 = R θ, x5 = Rz. 1 1 potential A and gauge field F take values in su(N), and on M3 × R˜ 2 one has 4 ˜ ˜ 1 R Reduction to M × I. Let us equate R1 = R and scale down S by choosing a a¯ A = Aa dx + Aa¯ dx and ¯ ¯ ¯ (2.13) F = 1 F dxa ∧ dxb + F ¯ dxa ∧ dxb + 1 F ¯ dxa ∧ dxb R1 = εR0  R0 with ε  1 (2.4) 2 ab ab 2 a¯b . Abbreviating the rescaled M3 coordinates as (t, α, β) = (yμ) with relative to some fixed radius R0. This reduction gives rise to a 5d R˜ μν − super-Maxwell theory [13]on μ = 0, 1, 2, for the metric tensor (2.12)we have g˜ = R˜ 2ημν R˜ ¯ ¯ ¯ ¯ 4 2 ˜ 1 and g˜ ab = δab. Hence, for the upper components of F we obtain M × I = R × T × S × I (2.5) R˜ R˜ R t p R˜ R ¯ ¯ ¯ ¯ with a metric μν ˜ −4 μν μb ˜ −2 μb a¯b a¯b F = R F , F = R F and F ˜ = F . R˜ R˜ R 2 = aˆ bˆ + 2 2 ˆ ˆ ∈{ } dsR ηaˆbˆ dx dx R (dz) with a, b 0, 1, 2, 3 . (2.6) (2.14) It has a non-Abelian SU(N) extension consistent with gauge in- Then the standard Yang–Mills action functional on M3 × 2 will R˜ R variance and supersymmetry, whose pure-gauge sector we shall have the form consider further. It is conjectured that this 5d SYM is the dimen-  1 ˜ 3  sional reduction of a non-Abelian 6d CFT. ˜ R 3 2 1 μν S5 =− d y d x tr ˜ FμνF 8e˜2 R4 M3×2 (2.15) 1 There are various mathematical approaches to such 6d non-Abelian CFTs, see ¯ ¯  2 μb a¯b e.g. [23,24]and references therein. + F ¯ F + F¯ ¯ F , R˜ 2 μb ab O. Lechtenfeld, A.D. Popov / Physics Letters B 793 (2019) 297–302 299

˜ f 2 π 1 π R where e is the dual gauge coupling, whose square is proportional π = and = . (3.6) ˜ 1 ˜ 3 2 2 2 to the S compactification radius R1 = ε˜ R0. The reduction to M × 4 4 e R 32 ς 120 e R˜ 2 4 is dual to the one to M × IR in the sense that we may take 2 R R˜ Recall that e is proportional to R [13]and R = εR0 becomes ˜ = −1  ˜ 2 ∼ −2 −2 ∼ 0 ε ε so that ε 1is related to ε 1 and vice versa. Hence the small, so that fπ ε and ς ε , and higher-order (in R) Yang–Mills theories (2.9) and (2.15)are dual in agreement with the terms are suppressed. discussion in [13]. 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- 3. Extended SYM and Skyrme model on M4 from SYM on M4 × I 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 4 4 SYM on M . In the SYM model on M × IR , discussed in the pre- model, and its supersymmetric version was derived from 5d in [4]. R˜ vious section as the first reduction from 6d, there exist two types The four-derivative term in (3.5) stabilizes solitons against scaling. 4 ×I of further reduction to four dimensions, both of which appear in This term was deduced from 5d SYM on M R in [21], where its the literature as low-energy limits: reduction with respect to the possible supersymmetrization, yet unknown, was briefly discussed. 4 Thus, in the infrared of 5d SYM theory on M × IR one can translation ∂z [13] and reduction via the adiabatic approach (see find two different models on M4: N ≥ 2SYM theory and the e.g. [14,4,21,22]). Both reductions shrink IR to a point, via Skyrme model. The latter describes low-energy QCD by interpret- ˜ ing mesons as fundamental and baryons as solitons. Away from R = εR0  R0 and R = R0 for ε  1 , (3.1) the infrared limit the Skyrme model gets extended by an infinite but the final result is different. We drop the subscript on M4 and tower of heavy mesons beyond the leading Skyrme term displayed R˜ treat both cases in turn. in (3.5)(cf.[25,26]). Let us impose translation invariance along IR on all fields, 4. Moduli space of YM vacua on M3 A = = := A = 1 A ∂z a 0 ∂z for 5 R z , (3.2) For the remainder of the paper we focus on the second kind ˜ by saying that for R  R the momenta along IR are much larger of 6d to 5d reduction presented in Section 2, namely the dual 4 3 2 than along M . Then one can discard all higher modes of Aa and action (2.15), and its own infrared simplification from M × R˜ R as well as of scalar and spinor fields of maximally supersym- 2 4 to . In a manner dual to (3.1), we put metric Yang–Mills theory on M × IR . After substituting (3.2), the Yang–Mills action (2.9)reduces to R˜ = ε˜ R  R and R = R for ε˜  1 , (4.1)  0 0 0   3 π R 4 ab a which scales down the metric (2.12)in the M ˜ direction. We drop S4 =− d x tr FabF + 2Da D , (3.3) R 4e2 2 ˜  the reference to the (now fixed) scale of R . Hence, for R R the M4 5d YM theory reduces to an effective 2d field theory on 2 which 4 where Da = ∂a +[Aa, · ]. Likewise, the full SYM theory on M ×IR we now describe. passes to N = 2or N = 4 super-Yang–Mills theory on M4, de- pending on twisting along 2 and other assumptions (see e.g. [2, Flat connections on M3. Our derivation of the low-energy limit 13]). The discussion simplifies because in the 6d to 5d reduction employs the adiabatic method [15–19](for brief reviews and more ∼ the two-punctured two-sphere 2 = S1 × I gets deformed into R1 R references see e.g. [20,27]). In this approach one firstly restricts to a thin cylinder as in [4,14]. M3 by putting R˜

4 A = ¯ ∈{ } Skyrme model. A different infrared limit of SYM theory on M × a¯ 0fora 4, 5 (4.2) IR , discussed e.g. in [4,14,21], introduces n 3 and classifies the classical solutions Am(y ) on M , independent R˜ 2 a¯ z  of the coordinates x . Secondly, one declares that the mod- α = P A = = ∈ uli X , which parametrize such solutions, become functions of h(z) exp ydy and g h(z π) SU(N), a¯ 2 x ∈ . Thirdly, one introduces small fluctuations δAm and Aa¯ ¯ −π depending on yn ∈ M3 and on the moduli functions Xα(xa), sub- R˜ (3.4) stitutes them into (2.15) and obtains an effective field theory for Xα on 2. Since in the following we want to study small fluc- where P denotes path ordering. The group element g is the holon- tuations around the vacuum manifold, we first take a look at the omy of A along I. In the low-energy limit, when the length of vacuum configurations. I becomes small, the 5d YM theory (2.9)reduces to the Skyrme 3 The vacua on M ˜ are given by model on M4, R  F = 0 , (4.3) f 2 μν S =− d4x π ηab tr(L L ) eff a b meaning flat connections on M3 . The consideration of flat connec- 4 R˜ 4 M (3.5) tions (4.3) guarantees that the action (2.15)will be nonsingular in   the limit ˜ → 0(R˜ → 0). One can always choose the gauge A = 0, 1 ac bd ε 0 + η η tr [La, Lb][Lc, Ld] , 32 ς 2 which simplifies (4.3)to

−1 where L = g ∂ g for g from (3.4), ς is the dimensionless Fmn = 0 ⇔ F 2 = 0form,n ∈{1, 2} , (4.4) a a T p Skyrme parameter, and fπ is interpreted as the pion decay con- 2 2 describing flat connections A 2 on the punctured torus T = stant. Their relation to the dimensionful 5d gauge coupling e and T p p the infrared scale R is T 2\{p}. 300 O. Lechtenfeld, A.D. Popov / Physics Letters B 793 (2019) 297–302

2 Flat connection on T p . It is well known that gauge bundles over There is the natural projection smooth tori T 2 (compact, without punctures) admit only reducible : −→H flat connections [28,18]. However, this theorem is not valid on Rie- q G G/H , (4.14) mann surfaces with punctures or fixed points (see e.g. [29–31]). and on the principal H-bundle (4.14)over G/H = SU(N) there ex- In particular, the moduli space M 2 of flat connections on a T p ists a one-parameter family of connections 2 K -bundle over T p is the gauge group K ,  ıˆ ıˆ α α A =  e Iˆ =  e Iˆ dX =: A dX for  ∈ R {flat connections} SU(N) ı α ı α M 2 = N 2 /G 2 = { } = K , (4.5) T p T p T p gauge transformations (4.15) ∞ 2 where the group of gauge transformations G 2 = C (T , K ) forms T p p with curvature the fibres over the points in M 2 for the bundle T p ˆ  1 ıˆ jˆ k 1 α β F = ( − 1) f ˆ e ∧ e = Fαβ dX ∧ dX . (4.16) SU(N) 2 jˆk 2 G 2 T p π : N 2 −→ M 2 = K . (4.6) i ıˆ ı¯ T p T p The one-forms e = (e , e ) on G/H are pulled back from G. We choose We specialize to K = SU(N).

m = m+ ≡ (m, 0) ⊂ su+(N) ⊕ su−(N), (4.17) A A = A m Flat variation of 2 . Any solution 2 mdy of (4.4)is T p T p α M = so that the Maurer-Cartan equations have the form parametrized by the coordinates X on 2 SU(N), i.e. T p ¯ ¯ ˆ ¯ ¯ ¯ ¯ deı =−f ı ej ∧ ek − 1 f ı ej ∧ ek and n α ˆk¯ 2 ¯k¯ Am = Am(y , X ). (4.7) j j ˆ (4.18) ıˆ 1 ıˆ jˆ k A N de =− f ˆ e ∧ e . In general, flat connections 2 belong to the space 2 fibred 2 jˆk T p T p M over 2 , as we have the bundle (4.6). Hence, flat variations live T p The second relation in (4.18)was used to derive (4.16). For more N in the tangent bundle T 2 , defined as the fibration details and references see [32]. T p : N −→ M 2 3 2 π∗ T 2 T 2 (4.8) 5. Toda theory on  from YM on M ×  T p T p ∼ with fibres T G 2 = Lie G at any point A ∈ N . We have A 2 T p T 2 T 2 T 2 Moduli space approximation. In the adiabatic approach, applicable T p p p p for ε˜  1, the moduli approximation assumes that the moduli Xα N = ∗ M ⊕ G =∼ ⊕ G a¯ ∈ 2 TA 2 π TA T 2 TA 2 m+ Lie 2 . (4.9) vary with the coordinates x [15–19]. In this way, the moduli T 2 T p 2 p T 2 T p T p p T p p 2 of flat connections on T p define a map A M The variation of m along T 2 is then given by T p X : 2 → SU(N), (5.1) δαAm = ∂αAm − Dmα , (4.10) ¯ so that Xα = Xα(xa) may be considered as dynamical fields. Ad- a¯ α where α is a suitable gauge parameter which brings ∂αAm back mitting x dependence only via X ∈ SU(N), our fields take the ∗ M A to π TA T 2 . This makes sure that the variation δα m obeys a form 2 p T p  ¯   ¯  linearization of the flatness condition (4.3), n α b n α b Am = Am y , X (x ) and Aa¯ = Aa¯ y , X (x ) ,

DmδαAn − DnδαAm = 0 . (4.11) (5.2) where we stay with the gauge choice A = 0. SU(N) as coset space. We would like to realize SU(N) as a coset 0 We are interested in the low-energy effective action for Xα G/H. The simplest way is 2 on , generated by small fluctuations (δAm, Aa¯ ) around the vac- 2 G = SU+(N) × SU−(N) and uum (4.2) and (4.3) which respect the flatness on T p , i.e. obey (4.12) (4.11). This condition keeps the infrared-singular first term in (2.15) H = diag (SU+(N) × SU−(N)) . removed. We need to compute the field strength components Fam¯ = ⊕ = F Accordingly, g h m, where h Lie H. At any point in G/H the and a¯b¯ of the vacuum deviations (5.2). The mixed ones yield tangent space is isomorphic to m. Let {Ii } be a basis of the Lie   F = A − A = α A − A algebra g realized as 2N × 2N block-diagonal matrices with nor- am¯ ∂a¯ m Dm a¯ (∂a¯ X ) ∂α m Dm α 1 2   malization tr(Ii I j) =− δij for i, j = 1, ..., 2(N − 1). We choose α 2 = (∂a¯ X ) δαAm + Dm(α − Aα) (5.3) bases {Iıˆ} for h and {Iı¯} for m so that {Ii } ={Iıˆ, Iı¯}, and both ıˆ and 2 α ı¯ range from 1 to N − 1. Note that in general G/H is not symmet- = (∂a¯ X )δαAm , ric, and we consider this case of group manifolds with torsion. = A F ¯ where (4.10)was employed, and we put α α to keep am¯ The coset G/H = SU(N) supports an orthonormal frame {eı } of tangent to the vacuum moduli space M 2 . The moduli-space con- T p one-forms locally providing the G-invariant metric A α = A nection αdX SU(N) has been computed in (4.15). The fact ¯ 2 = ı¯ j¯ = ı¯ j α β =: α β that we choose  dXα equal to A which is constant on T 2, dsSU(N) δı¯j¯ e e δı¯j¯ eαeβ dX dX gαβ dX dX α SU(N) p (4.13) i.e. it does not depend on xm, does not matter because in the in- for α,β = 1,...,N2 − 1 , frared limit R˜  1only the zero mode in a Fourier expansion of { α} ∈ = m 2 where X is a set of local coordinates of a point X G/H the x dependence on T p will survive. As a result, we arrive at the = ∂ SU(N) and ∂α ∂ Xα will denote derivatives with respect to them. infrared approximation O. Lechtenfeld, A.D. Popov / Physics Letters B 793 (2019) 297–302 301

A = α A = α ıˆ = −1 = a¯ (∂a¯ X ) α (∂a¯ X ) eα Iıˆ  g ∂a¯ g  La¯ , models. However, if we take N=2 and restrict g ∈ SU(2) to the ˆ 1 = ⊂ α β ıˆ jˆ k coset C P SU(2)/U(1) SU(2), then exactly this term appears F¯ ¯ = ( − 1)(∂a¯ X )(∂¯ X ) f ˆe e Iˆ = ( − 1)[La¯ , L ¯ ] , ab b jˆk α β ı b in the baby Skyrme model in two dimensions [33]. (5.4) Toda model. By choosing  = 1in (5.8)one obtains the standard ∈ 2 ∼ 1 ˜ where g SU(N) and we abbreviated principal chiral model on = S × I, i.e. Seff = Skin. In [7]6d N =(2, 0) CFT was reduced to Chern-Simons theory on 2 × I and = α −1 = −1 La¯ (∂a¯ X ) g ∂α g g ∂a¯ g . (5.5) then to a sigma model with torsion (WZW model) on 2. Finally, by imposing current constraints, one can reduce the WZW model F = α A Non-linear sigma model. Substituting am¯ (∂a¯ X ) δα m into the to Toda field theory [34,7–9]. Relations between 4d SYM theories action (2.15), the second term becomes and 2d WZW theories were considered earlier in [35,36]. T   The standard principal chiral model (5.6)can also be reduced to ˜ 2 R 2 2 am¯ Toda field theory on . The 2d sigma model equations are equiva- Skin =− dt d y d x tr(Fam¯ F ) 4 e˜2 lent to the two-dimensional self-dual Chern-Simons (CS) equations 0 2 2 T p which by imposing some constraints can be reduced to the affine  Toda equations [37,38]. Thus, we have the correspondence ˜ ¯ RT 2 a¯b α β = d x δ ∂¯ X ∂¯ X g (5.6) ˜2 a b αβ 2 e 2d sigma model ⇔ 2d self-dual CS −→ 2d Toda theory . 2  ˜ (5.11) RT 2 a¯b¯ =− d x δ tr(La¯ L ¯ ), 4 e˜2 b This correspondence allows one, in particular, to construct solu- 2 tions of the Toda field equations from uniton solutions of the where we have compactified the time direction with a finite SU(N) sigma model as was discussed in [38]. ˜ length T R = ε˜ TR0 and  Acknowledgements 1 2 mn ı¯ j¯ g =− d y δ tr(δ A δ A ) = δ¯ ¯ e e αβ 2 α m β n ıj α β This work was partially supported by the Deutsche Forschungs- T 2 (5.7) gemeinschaft grant LE 838/13. It is based upon work from COST p  − − Action MP1405 QSPACE, supported by COST (European Cooperation =−1 tr (g 1∂ g)(g 1∂ g) 2 α β in Science and Technology). is the moduli-space metric (4.13)on the group SU(N). Thus, this part of the action (2.15)reduces to the standard non-linear sigma References model on 2 with SU(N) as target space. [1] L.F. Alday, D. Gaiotto, Y. Tachikawa, Liouville correlation functions from four- dimensional gauge theories, Lett. Math. Phys. 91 (2010) 167, arXiv:0906 .3219 Baby-Skyrme-type term. It remains to evaluate the last term in [hep -th]. the action (2.15). Substituting (5.4)we obtain [2] Y. Tachikawa, A brief review of the 2d/4d correspondences, J. Phys. A 50 (2017) 443012, arXiv:1608 .02964 [hep -th]. T   [3] N. Wyllard, A − conformal Toda field theory correlation functions from con- ˜ 3   N 1 R ¯ ¯ formal N = 2SU(N) quiver gauge theories, J. 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