Exact Solutions of 2d Supersymmetric gauge theories

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1 Exact Solutions of 2d Supersymmetric gauge theories Abhijit Gadde, IAS w. Sergei Gukov and Pavel Putrov

2 UV to IR Physics at long distances can be strikingly different from the physics at short distances Even the notion of fundamental particles may be different QED and cooper pairs Kinks in the Ising model in strong magnetic fields Yang Mills theory and mass gap Massless QCD and pions Given a microscopic theory, finding its manifestation at long distances is of great practical importance.

3 Low energy theory At very low energy scales (i.e. when all the mass scales are taken to infinity), the spectrum could be of one out of two types: Gapped Degenerate vacuum Gapless Nontrivial theory of gapless modes; No scale Nontrivial CFT Tool: Anomaly matching

4 Seiberg duality A success of anomaly matching Supersymmetric QCD SU(N) gauge theory with Nf quarks dual SU(Nf-N) gauge theory with Nf magnetic quarks and Mesons Non-trivial superconformal field theory Only supersymmetric checks

5 (0,2) Supersymmetric QCD Generically, have global symmetries with non-vanishing anomalies CFT As we will see, they exhibit Seiberg type duality (actually a triality). Power of infinite dimensional conformal invariance, anomaly matching and modular invariance Solution of the theory Hence proving triality.

6 Motivation from 4-manifolds Compactification of 6d (2,0) theory on d- manifold 6-d dim SCFT For d=2, complex structure of the Riemann surface becomes the coupling constant 4d N=2 field theory Partition functions can be computed from 2d For d=4 4-manifolds Vafa Witten partition function 2d (0,2) theories Elliptic genus

7 (0,2) Multiplets Chiral multiplet: D + =0 = + p i + + Complex scalar Complex right-moving fermion Fermi multiplet: D + =0 = p 2 + G i + + Complex left-moving fermion Vector multiplet: Gauge invariant d.o.f.: Fermi multiplet

8 (0,2) SQCD Similar to 4d N=1 SQCD, but 2 types of matter U(Nc) gauge theory with Nb Chiral and Nf Fermi + Gauge anomaly cancellation + Normalizable vacuum g YM h 2N c -N b +N f Z Z QCD P h d P +t d N b N ª c N f 2 0

9 Anomalies J µ J µ 2N c -N b +N f N b N c P N b N ª c N f N c N b + N f 2 N c

10 Central charge c R =3TrR 2 2N c -N b +N f N f N c + N f N c P N b + N f N c + N f N b N ª c N f 0 N b N c N c + N f 2 0 c R =3 N c(n b N c )(N c N b + N f ) N c + N f

11 Dual frames 2N c N b + N f N b N c 2N c N b + N f N b N c N b N c N c N b +N f N b N f N b N f N c N b + N f N c N c N b + N f N c

12 (0,2) SQCD N 1 N 2 + N 3 2 Symmetric labeling n 2 N 1 + N 2 N 3 2 N 1 + N 2 N 3 2 n 3 N 1 + N 2 + N 3 2 n 1 Triality is invariance of the fixed point under permutations of N s In addition, there are 3 abelian symmetries

13 Low energy physics Poincare symmetry Virasoro (0,2) N=2 superconformal Affine H 3Y i=1 SU(N i ) ni U(1) NNi The central charges can be determined from c-extremization and gravitational anomaly c R = 3 4 c R = 3Tr 3 RR, c R c L =Tr 3 ( N 1 + N 2 + N 3 )(N 1 N 2 + N 3 )(N 1 + N 2 N 3 ) N 1 + N 2 + N 3 c L = c R 1 4 (N N N 2 3 2N 1 N 2 2N 2 N 3 2N 3 N 1 )+2

14 Low energy solution H = M H L H R Integrable modules Modules of N=2 Modules of H Sugawara central charge = c L Immense simplification: rational CFT Modular invariance of the partition function helps fix H R

15 NS-NS partition function Z(, ) :=Tr H e 2 i( L 0 L 0 ) Affine character Invariant under S and T 2 Z(, ) = X ( )K ( ) N=2 character

16 Partition function (contd) Use S invariance: Z(, ) =Z( 1, 1 )! S µ µ S S = I ) K! S µ K µ K is NOT the anti-holomorphic affine character of H Note that characters of level-rank dual transform with S H t H t = Y SU(n i ) Ni U(1) Nni

17 To summarize K is an N=2 character with central charge cr It transforms as a character of holomorphic under modular S-transformation H t Singlet under all affine symmetries K is a character of the Kazama- Suzuki coset [G]/[H t ] ( For appropriate G)

18 Intermission: SUSY WZW [g] k is SUSY extension of WZW model g at level k It is obtained by adding free adjoint fermions to J a = Jbos a i k f bc a a k = k bos + h _ c g c [g] = c g dim g Matching the right-moving central charge with that of the coset: c [G] = N 2

19 Combined with the condition G H t [G] =[U(N)] N =[U(1)] N 2 [SU(N)] N Bosonic level 0 Only bosonic part U(1) N 2 Coset character C is a branching function U(1) N 2 module C, t SO(dim G/H t ) module H t module

20 Solution We pick modular invariant combinations ( 0, 0) Then K = X t L, t C 0, 0 t has all the desired properties! H = M, t L, t H L H t R Module of H Module of [G]/[H t ] Matches with the UV computation of the index

21 Example H = 3 SU(2) 1 U(1) [G]/[H t ]=[U(3)] 6 3 /[U(1) 3 ] 3 c=1 minimal model

22 Z T222 = N (0,0) =2 (, ) 0,0,0 ( )+ 1,1,1 ( )+ 1, 1, 1 ( ) + N =2 ( 1 6, 1 )(, ) 1,0, 1 ( )+ 1,1,0 ( )+ 0, 1,1 ( ) 3 + N =2 ( 1 6, 1 )(, ) 1,0,1 ( )+ 1, 1,0 ( )+ 0,1, 1 ( ) 3 where a,b,c (, 1, 2, 3 ):= a (, 1 ) b (, 2 ) c (, 3 ) 1 (, ) := SU(2) 1 U(1) 6 (, 1) (, )+ SU(2) 1 U(1) 6 (,2) (, ) 0 (, ) := SU(2) 1 U(1) 6 (,0) (, )+ SU(2) 1 U(1) 6 (,3) (, ) 1 (, ) := SU(2) 1 U(1) 6 (,1) (, )+ SU(2) 1 U(1) 6 (, 2) (, ). Remarkably Z T222 = N (0,0) =2 (, ) (E 6) 1 (, i )+ N =2 ( 1 6, 1 )(, ) (E 6) 1 (, i )+ N =2 ( 1 3 6, 1 )(, ) (E 6) 1 3 (, i ),

23 Triality and enhancement

24 Solution to a general quiver N 1 N 3 N 2 N 4 N 5 N 7 N 6 N 8 N 9 N 10

25 Thank you!