Triality of Two-dimensional (0,2) Theories

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1 J.Guo,B.Jia,E.Sharpe,arXiv: April 11, 2015

2 Outline 1 2d N = (0,2) Gauge Theories 2 Triality Proposal Checks Low Energy Description 3 Chiral Rings Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem 4 Examples

3 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ +, θ +

4 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ +, θ + Chiral Multiplet Φ: D + Φ = 0 Φ = φ + 2θ + ψ + iθ + θ + + φ

5 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ +, θ + Chiral Multiplet Φ: D + Φ = 0 Φ = φ + 2θ + ψ + iθ + θ + + φ Fermi Multiplet Ψ: D + Ψ = 2E(Φ) Ψ = ψ 2θ + G iθ + θ + + ψ 2θ + E

6 2d N = (0,2) Gauge Theories Two fermionic coordinates: θ +, θ + Chiral Multiplet Φ: D + Φ = 0 Φ = φ + 2θ + ψ + iθ + θ + + φ Fermi Multiplet Ψ: D + Ψ = 2E(Φ) Ψ = ψ 2θ + G iθ + θ + + ψ 2θ + E Vector Multiplet: V = v 2iθ + λ 2iθ + λ + 2θ + θ + D

7 2d N = (0,2) Gauge Theories J-term: dθ + Ψ i J i (Φ) θ + =0

8 2d N = (0,2) Gauge Theories J-term: dθ + Ψ i J i (Φ) + θ =0 Fayet-Illiopoulos term: t 4 dθ + Λ + θ =0,where t = ir + θ 2π

9 2d N = (0,2) Gauge Theories J-term: dθ + Ψ i J i (Φ) + θ =0 Fayet-Illiopoulos term: t 4 dθ + Λ + θ =0,where t = ir + θ 2π Potential for the scalars: V = e2 2 Da D a + a E i (φ) 2 + a J i (φ) 2

10 Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov ]

11 Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov ] Matter Content Φ Ψ P Γ Ω U(N c ) 1 det SU(N 1 ) SU(N 2 ) SU(N 3 ) SU(2) N c = (N 1 + N 2 N 3 )/2

12 Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov ] Matter Content Φ Ψ P Γ Ω U(N c ) 1 det SU(N 1 ) SU(N 2 ) SU(N 3 ) SU(2) N c = (N 1 + N 2 N 3 )/2 J-term: L J = dθ + Tr(ΓΦP) θ + =0

13 Proposal Checks Low Energy Description Triality Triality: IR equivalence of three 2d (0,2) gauge theories [Gadde,Gukov,Putrov ] Matter Content Φ Ψ P Γ Ω U(N c ) 1 det SU(N 1 ) SU(N 2 ) SU(N 3 ) SU(2) N c = (N 1 + N 2 N 3 )/2 J-term: L J = dθ + Tr(ΓΦP) θ + =0 Triality under permutation of N 1, N 2, N 3

14 Proposal Checks Low Energy Description Quiver Diagram

15 Proposal Checks Low Energy Description Checks Non-abelian Flavor Anomalies: k SU(N1 ) = 1 4 ( N 1 + N 2 + N 3 ) k SU(N2 ) = 1 4 (+N 1 N 2 + N 3 ) k SU(N3 ) = 1 4 (+N 1 + N 2 N 3 ) Invariant under cyclic permutations of N 1, N 2, N 3

16 Proposal Checks Low Energy Description Checks Central Charge: c R = 3 ( N 1 + N 2 + N 3 )(N 1 N 2 + N 3 )(N 1 + N 2 N 3 ) 4 N 1 + N 2 + N 3 c L = c R 1 4 (N2 1 + N N 2 3 2N 1 N 2 2N 2 N 3 2N 3 N 1 ) + 2 Both are invariant under the permutations of N 1, N 2, N 3

17 Proposal Checks Low Energy Description Checks Central Charge: c R = 3 ( N 1 + N 2 + N 3 )(N 1 N 2 + N 3 )(N 1 + N 2 N 3 ) 4 N 1 + N 2 + N 3 c L = c R 1 4 (N2 1 + N N 2 3 2N 1 N 2 2N 2 N 3 2N 3 N 1 ) + 2 Both are invariant under the permutations of N 1, N 2, N 3 Elliptic Genus is also invariant under the permutations.

18 Proposal Checks Low Energy Description General Quiver Gauge Theories The triality transformation can be applied to any gauge node in a general quiver gauge theory [Gadde,Gukov,Putrov ]

19 Proposal Checks Low Energy Description Geometry of the Triality S N 3 Q N 1 G(n 3, N 2 ) = S N 1 Q N 3 G(n 1, N 2 ) + - T N1,N 2,N 3 T N3,N 1,N 2 S N 3 Q N 2 G(n 3, N 1 ) - T N2,N 3,N S N 1 Q N 2 G(n 1, N 3 ) S N 2 Q N 3 G(n 2, N 1 ) S N 2 Q N 1 G(n 2, N 3 ) N = N 1+N 2 +N 3 2, n i = N N i

20 Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 kg

21 Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 kg Chirality of the affine algebra is determined by the sign of k G

22 Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 kg Chirality of the affine algebra is determined by the sign of k G Left-moving affine algebra is: R = 3 i=1 SU(N i) ni U(1) NNi

23 Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 kg Chirality of the affine algebra is determined by the sign of k G Left-moving affine algebra is: R = 3 Sugawara central charge for R is c R = 3 i=1 i=1 SU(N i) ni U(1) NNi ( ) ni (Ni 2 1) n i +N i + 1

24 Proposal Checks Low Energy Description IR Fixed Point At the fixed point, the simple global symmetry G is enhanced to the affine symmetry G 2 kg Chirality of the affine algebra is determined by the sign of k G Left-moving affine algebra is: R = 3 Sugawara central charge for R is c R = 3 i=1 i=1 SU(N i) ni U(1) NNi ( ) ni (Ni 2 1) n i +N i + 1 The full spectrum H = λ Hλ L H λ R, λ runs over all integrable representations of R

25 Proposal Checks Low Energy Description The partition function Z(τ, ξ i ; τ, η) = λ χ λ(τ, ξ i )K λ (τ, η)

26 Proposal Checks Low Energy Description The partition function Z(τ, ξ i ; τ, η) = λ χ λ(τ, ξ i )K λ (τ, η) K λ transforms as a character of holomorphic R t symmetry under modular transformation, where R t = 3 i=1 SU(n i) Ni U(1) Nni

27 Proposal Checks Low Energy Description The partition function Z(τ, ξ i ; τ, η) = λ χ λ(τ, ξ i )K λ (τ, η) K λ transforms as a character of holomorphic R t symmetry under modular transformation, where R t = 3 i=1 SU(n i) Ni U(1) Nni Right-moving sector = Supersymmetric Kazama-Suzuki coset model G/R t

28 Proposal Checks Low Energy Description The partition function Z(τ, ξ i ; τ, η) = λ χ λ(τ, ξ i )K λ (τ, η) K λ transforms as a character of holomorphic R t symmetry under modular transformation, where R t = 3 i=1 SU(n i) Ni U(1) Nni Right-moving sector = Supersymmetric Kazama-Suzuki coset model G/R t By comparing the right-moving central charge, G = U(N) N

29 Proposal Checks Low Energy Description The partition function Z(τ, ξ i ; τ, η) = λ χ λ(τ, ξ i )K λ (τ, η) K λ transforms as a character of holomorphic R t symmetry under modular transformation, where R t = 3 i=1 SU(n i) Ni U(1) Nni Right-moving sector = Supersymmetric Kazama-Suzuki coset model G/R t By comparing the right-moving central charge, G = U(N) N H = λ,λ L λ,λ H λ L,WZW H λ R,KS [Gadde,Gukov,Putrov ]

30 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Chiral States in (0,2) Theories The set of all states annihilated by a right-moving supercharge forms a ring

31 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Chiral States in (0,2) Theories The set of all states annihilated by a right-moving supercharge forms a ring The massless states in the (R,R) sector have the form bī1,,ī n,a 1,,a m λ a 1 λam ψī1 + ψīn + 0

32 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Chiral States in (0,2) Theories The set of all states annihilated by a right-moving supercharge forms a ring The massless states in the (R,R) sector have the form bī1,,ī n,a 1,,a m λ a 1 λam ψī1 + ψīn + 0 The Fock vacuum is defined by ψ i + 0 = λā 0 = 0, it transforms as a section of (det E) 1/2 K +1/2 X

33 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Chiral States in (0,2) Theories The set of all states annihilated by a right-moving supercharge forms a ring The massless states in the (R,R) sector have the form bī1,,ī n,a 1,,a m λ a 1 λam ψī1 + ψīn + 0 The Fock vacuum is defined by ψ i + 0 = λā 0 = 0, it transforms as a section of (det E) 1/2 K +1/2 X The massless chiral states realize the sheaf cohomology groups ( ) H n X, ( m E) (det E) 1/2 K +1/2 X

34 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Bott-Borel-Weil Theorem Let β and γ be some dominant weights of U(k) and U(n k) respectively, if there is a way to transfer α = (β, γ) into a dominant weight α of U(n), then the cohomology H (G(k, n), K β S K γ Q ) = K α V δ,l(α), where V is the fundamental representation of U(n), and l(α) is the number of elementary transformations performed. Otherwise, H (G(k, n), K β S K γ Q ) = 0.

35 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Elementary transformation: (, a, b, ) (, b 1, a + 1, )

36 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Elementary transformation: (, a, b, ) (, b 1, a + 1, ) Example 1: Sym 2 S Sym 2 Q over G(2, 4) (2, 0, 2, 0) (2, 1, 1, 0) H m (G(2, 4), Sym 2 S Sym 2 Q ) = 0 for m 1 H 1 (G(2, 4), Sym 2 S Sym 2 Q ) = K (2,1,1,0) C 4

37 Chiral Rings in (0,2) Theories Bott-Borel-Weil Theorem Elementary transformation: (, a, b, ) (, b 1, a + 1, ) Example 1: Sym 2 S Sym 2 Q over G(2, 4) (2, 0, 2, 0) (2, 1, 1, 0) H m (G(2, 4), Sym 2 S Sym 2 Q ) = 0 for m 1 H 1 (G(2, 4), Sym 2 S Sym 2 Q ) = K (2,1,1,0) C 4 Example 2: H (G(2, 4), 2 S Sym 2 Q ) = 0

38 First Example N 1 = 2, N 2 = 3, N 3 = 3, N c = 1

39 First Example N 1 = 2, N 2 = 3, N 3 = 3, N c = 1 Positive Phase: S 3 Q 2 S 2 P 2 Negative Phase: S 3 Q 3 S 2 P 1

40 First Example N 1 = 2, N 2 = 3, N 3 = 3, N c = 1 Positive Phase: S 3 Q 2 S 2 P 2 Negative Phase: S 3 Q 3 S 2 P 1 Global Symmetry: SU(2) SU(3) SU(3) SU(2) U(1) 3

41 States shared between the two phases: States as representations of r >> 0 r << 0 SU(2) SU(3) SU(3) SU(2) E H (P 2 ) E H (P 1 ) U(1) 3 charges (1,1,1,1) (+3, 0, -3) (1,3,1,2) (+2, -1/2, -3/2) (1,3,1,1) (+1, +2, -3) (2,1,3,1) (+2,0,-2). (2,1,3,1) (-2,0,+2) (1,3,1,1) (-1,-2,+3) (1,3,1,2) (-2,+1/2,+3/2) (1,1,1,1) (-3,0,+3)

42 States not shared between the two phases: States as representations of r >> 0 r << 0 SU(2) SU(3) SU(3) SU(2) E H (P 2 ) E H (P 1 ) U(1) 3 charges (1,6,1,1) (+1, -1, 0) (2,8,1,1) (0, 0, 0) (1,6,1,1) (-1, +1, 0) (1,6,1,1) (+1, -1,0) (2,8,1,1) (0,0,0) (1,6,1,1) (-1, +1, 0) (4,1,1,1) (0,0,0) (4,1,1,1) (0,0,0)

43 The massless spectrum is invariant under Serre duality

44 The massless spectrum is invariant under Serre duality The discrepancy between the two phases may due to nonperturbative corrections [McOrist,Melnikov ]

45 The massless spectrum is invariant under Serre duality The discrepancy between the two phases may due to nonperturbative corrections [McOrist,Melnikov ] The unshared states do not contribute to elliptic genera and none of them is integrable [Guo,Jia,Sharpe ]

46 The massless spectrum is invariant under Serre duality The discrepancy between the two phases may due to nonperturbative corrections [McOrist,Melnikov ] The unshared states do not contribute to elliptic genera and none of them is integrable [Guo,Jia,Sharpe ] The third geometry is obtained by switching N 2 and N 3 same as the negative phase in this example

47 Second Example N 1 = 4, N 2 = 2, N 3 = 4, N c = 1

48 Second Example N 1 = 4, N 2 = 2, N 3 = 4, N c = 1 Positive Phase: S 4 Q 4 S 2 P 1 Negative Phase: S 4 Q 2 S 2 P 3

49 Second Example N 1 = 4, N 2 = 2, N 3 = 4, N c = 1 Positive Phase: S 4 Q 4 S 2 P 1 Negative Phase: S 4 Q 2 S 2 P 3 Global Symmetry: SU(4) SU(2) SU(4) SU(2) U(1) 3

50 States shared between the two phases: States as representations of r >> 0 r << 0 SU(4) SU(2) SU(4) SU(2) E H (P 1 ) E H (P 3 ) U(1) 3 charges (1,3,1,1) (+4, 0, -4) (1,2,4,1) (+4, -1, -3) (4,2,1,1) (+3, +1, -4) (1,4,1,2) (+2,0,-2). (1,4,1,2) (-2,0,+2) (4,2,1,1) (-3,-1,+4) (1,2,4,1) (-4,+1,+3) (1,3,1,1) (-4,0,+4)

51 States not shared between the two phases: States as representations of r >> 0 r << 0 SU(4) SU(2) SU(4) SU(2) E H (P 1 ) E H (P 3 ) U(1) 3 charges (1,5,1,1) (0, 0, 0) (1,5,1,1) (0, 0, 0) (10,1,1,1) (+2, -2, 0) (10,1,1,1) (-2, +2, 0). (20,1,1,1) (0,0,0) (20,1,1,1) (0,0,0) (1,20,1,2) (+1,-1,0) (1,20,1,2) (-1,+1,0)

52 Third Example N 1 = 2, N 2 = 2, N 3 = 2, N c = 1, global symmetry SU(2) SU(2) SU(2) SU(2)

53 Third Example N 1 = 2, N 2 = 2, N 3 = 2, N c = 1, global symmetry SU(2) SU(2) SU(2) SU(2) Both Phases: S 2 Q 2 S 2 P 1

54 Third Example N 1 = 2, N 2 = 2, N 3 = 2, N c = 1, global symmetry SU(2) SU(2) SU(2) SU(2) Both Phases: S 2 Q 2 S 2 P 1 Symmetry Enhancement in the Infrared: For N c = 1, Ψ transforms in the same way as Ω, SU(N 3 ) and SU(2) combine to form SU(N 3 + 2)

55 Third Example N 1 = 2, N 2 = 2, N 3 = 2, N c = 1, global symmetry SU(2) SU(2) SU(2) SU(2) Both Phases: S 2 Q 2 S 2 P 1 Symmetry Enhancement in the Infrared: For N c = 1, Ψ transforms in the same way as Ω, SU(N 3 ) and SU(2) combine to form SU(N 3 + 2) For N 1 = 2, N 2 = 2, N 3 = 2, N c = 1, the flavor symmetry is enhanced to E 6 at level 1 in the infrared

56 Under the (SU(2) 4 )/Z 2 subgroup of E 6, 27 = (2, 2, 1, 1) + (2, 1, 2, 1) + (2, 1, 1, 2) +(1, 2, 2, 1) + (1, 2, 1, 2) + (1, 1, 2, 2) + 3(1, 1, 1, 1)

57 Under the (SU(2) 4 )/Z 2 subgroup of E 6, 27 = (2, 2, 1, 1) + (2, 1, 2, 1) + (2, 1, 1, 2) +(1, 2, 2, 1) + (1, 2, 1, 2) + (1, 1, 2, 2) + 3(1, 1, 1, 1) There are 54 states shared between the two phases and they form two copies of the decomposition above [Guo,Jia,Sharpe ]

58 Summary Triality: IR equivalence of three 2d (0,2) gauge theories

59 Summary Triality: IR equivalence of three 2d (0,2) gauge theories It can be described at the level of GLSM and NLSM

60 Summary Triality: IR equivalence of three 2d (0,2) gauge theories It can be described at the level of GLSM and NLSM The IR fixed point can be deduced by symmetry and properties of the elliptic genera

61 Summary Triality: IR equivalence of three 2d (0,2) gauge theories It can be described at the level of GLSM and NLSM The IR fixed point can be deduced by symmetry and properties of the elliptic genera Massless chiral states realize the sheaf cohomology groups of vector bundles over the target space

62 Summary Triality: IR equivalence of three 2d (0,2) gauge theories It can be described at the level of GLSM and NLSM The IR fixed point can be deduced by symmetry and properties of the elliptic genera Massless chiral states realize the sheaf cohomology groups of vector bundles over the target space Bott-Borel-Weil theorem is an effective tool for computing sheaf cohomology groups

63 Summary Triality: IR equivalence of three 2d (0,2) gauge theories It can be described at the level of GLSM and NLSM The IR fixed point can be deduced by symmetry and properties of the elliptic genera Massless chiral states realize the sheaf cohomology groups of vector bundles over the target space Bott-Borel-Weil theorem is an effective tool for computing sheaf cohomology groups Examples agree with the triality proposal

64 Summary Triality: IR equivalence of three 2d (0,2) gauge theories It can be described at the level of GLSM and NLSM The IR fixed point can be deduced by symmetry and properties of the elliptic genera Massless chiral states realize the sheaf cohomology groups of vector bundles over the target space Bott-Borel-Weil theorem is an effective tool for computing sheaf cohomology groups Examples agree with the triality proposal The mismatched states become massive along RG flow

65 Summary Triality: IR equivalence of three 2d (0,2) gauge theories It can be described at the level of GLSM and NLSM The IR fixed point can be deduced by symmetry and properties of the elliptic genera Massless chiral states realize the sheaf cohomology groups of vector bundles over the target space Bott-Borel-Weil theorem is an effective tool for computing sheaf cohomology groups Examples agree with the triality proposal The mismatched states become massive along RG flow Enhanced IR global symmetry

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