Yangians In Deformed Super Yang-Mills Theories

Transcription

1 Yangians In Deformed Super Yang-Mills Theories arxiv: [hep-th] JHEP 04:051, 2008 Jay N. Ihry UNC-CH April 17, 2008 Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 1 / 2704:0

2 Outline 1 Background 2 Algebra 3 Hamiltonian 4 Deformed Hamiltonian 5 Twisted Coproducts and Broken Symmetry Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 2 / 2704:0

3 Symmetry Algebra N = 4, D = 4 Super Yang-Mills has PSU2,2 4 superalgebra. A Yangian extension exists in the planar limit of the SUN gauge group. Marginal deformations have been used to deform to N = 1, D = 4 SYM theories, SU2, 2 1 U1 U1. Deformations beta and twists have been shown to maintain integrability. Twisted theories have a non-standard coproduct. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 3 / 2704:0

4 Marginal Deformation We break the N = 4 to a N = 1 superconformal theory by the addition of the marginal deformation with the superpotential W = ihtr e iπβ Φ 1 Φ 2 Φ 3 e iπβ Φ 1 Φ 3 Φ 2 + ih 3 Tr Φ Φ3 2 + Φ3 3. For an exact marginal deformation, h e iπβ e iπβ 2 + h 2 N2 4 N 2N 2 = g 2. We left h = 0 and β real in the large N limit. Then h = g. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 4 / 2704:0

5 The Deformed Lagrangian The Lagrangian of a deformed N = 4, D = 4 SYM: L = 1 g 2 Tr 1 4 F µν F µν + D µ Φ i D µ Φ i 1 2 [Φ i, Φ j ] Cij [ Φ i, Φ j ] Cij [Φ i, Φ i ][Φ j, Φ j ] + λ A σ µ D µ λ A i[λ 4, λ i ] B4i Φ i +i[ λ 4, λ i ] B4i Φ i + i 2 ɛijk [λ i, λ j ] Bij Φ k + i 2 ɛ ijk[ λ i, λ j ] Bij Φ k where [Φ i, Φ j ] Cij = e ic ij Φ i Φ j e ic ij Φ j Φ i and [λ A, λ B ] BAB = e ib ABλ A λ B e ib ABλ B λ A. Here 1 i, j 3 and 1 A, B 4. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 5 / 2704:0

6 SU2 3 SU2 3 sector, a subset of states of the PSU2, 2 4 theory. The field content is: Φ J = {φ 1, φ 2, φ 3 ; ψ 1, ψ 2 }. φa = c ac 4 0 ψα = a αc 4 0 The generators of the SU2 3 superalgebra are R a b = c b ca 1 3 δa b c cc c, L α β = a β aα 1 2 δα β a γa γ, D = c cc c a γa γ, S γ c = c ca γ, Q c γ = a γc c. Maximal subalgebra Large enough for interesting structural features to arise. Higher order length fluctuations. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 6 / 2704:0

7 Yangian Algebra YSU2 3: J A, Q A,... Defining relations [ J A, J B} = f AB CJ C, [ J A, Q B} = f AB CQ C, [ Q [A, [ Q B, J C}}} = αf AG Df BH Ef CK F f GHK J {D J E J F } The J A are SU2 3 generators. The tree-level, first nonlocal Yangian generator is Q0 A = f A CB J0 B ijc 0 j. i<j Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 7 / 2704:0

8 Standard Coproducts A coproduct is a holomorphic map : A A. Introduces the idea of single site, double site, etc... representations for the algebra A. The coproduct for the ordinary SU2 3 generators J A = J A J A. and the coproduct to create the two-site Yangian generators Q A = Q A Q A f A CBJ B J C will be used to construct tree level representations. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 8 / 2704:0

9 Hamiltonian The two-site Hamiltonian in terms of oscillators is H1, 2 = c a1c b 2 c b 1c a2 c b 2c a 1 + c a1a α2 + a α1c a2 a α 2c a 1 + a α1c a2 + c a1a α2 c a 2a α 1 + a α1a β 2 + a β 1a α2 a β 2a α 1. The commutation relations of the oscillators are {c a i, c b j} = δa b δ ij and [a α i, a β j] = δα β δ ij. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String [hep-th][-10pt] Seminar JHEP 9 / 2704:0

10 Quadractic Casimir The quadratic Casimir of the subalgebra SU2 3 is g AB J A J B = 1 3 D Lα βl β α 1 2 Ra br b a 1 2 [Qc γ, S γ c]. A Casimir of any algebra has the property [g AB J A J B, J C ] = 0 When acting on any two-particle state Φ I Φ J the quadratic Casimir and the two-site Hamiltonian are equivalent, H1, 2 Φ I Φ J = g AB J A J B Φ I Φ J. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 10 / 2704:0

11 Eigenstates The two-particle eigenstates of the Hamiltonian form two towers, 13 symmetric H 12 Φ 1 Φ 2 + = 0 Φ 1 Φ 2 + and 12 antisymmetric H 12 Φ 1 Φ 2 = 2 Φ 1 Φ 2 eigenstates. ab ± = c a1c b 2 ± c b 1c a2 c 4 1c 4 2 0, aβ ± = c a1a β 2 a β 1c a2 c 4 1c 4 2 0, αβ ± = a α1a β 2 a β 1a α2 c 4 1c Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 11 / 2704:0

12 Analysis of Yangian Symmetry A property of the dilatation generator is [D, J A ] = dimj A J A. Also, [D, Q A ] = dimj A Q A. Expanding both the dilatation generator and the Yangian, [D, Q A ] = [D 0 + g 2 YM D 2 +, Q A 0 + g YMQ A 1 + g2 YM QA 2 + ]. Group in powers of the Yang-Mills coupling to Og 2 dimj A Q A 0 + g YMQ A 1 + g2 YM QA 2 + g2 YM [D 2, Q A 0 ] dimja Q A. That means [D 2, Q0 A ] must give zero or approximately. In the large N planar limit D 2 is our spin chain Hamiltonian, H. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 12 / 2704:0

13 Edge Effects In PSU2, 2 4, an explicit check of the commutator gives the lattice derivative or edge effects of the system, [D 2, Q A 0 ] = qa 0, q A 1L = JA 1 J A L. Introduce the identity, Q A 12 = 1 4 [ ] g BC J1 B J2 C, q12 A, Recall, the quadratic Casimir is equivalent to the Hamiltonian when acting on states. For the two-site case, [ ] H1, 2, Q A 12 ΦI Φ J = 1 [ 4 H1, 2 2 q12 A + qa 12H1, 22 2H1, 2q A 12 H1, 2] Φ I Φ J. = q A 12 Φ IΦ J Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 13 / 2704:0

14 Yangian on Two-Particle States H1,2 =0 ab H1,2=2 ab u1 su2 su3 supercharges Key J A Q A 0 aβ aβ 0 αβ αβ Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 14 / 2704:0

15 Deformed Hamiltonian The deformed R matrix, a solution to the Yang-Baxter equation is R IJ KL u = 1 ue ib IJ IIJ KL u + i + ipkl IJ. The identity and projection operators are IIJ KL = δk I δj L and PKL IJ = δi LδK J. The deformed monodromy matrix is T J;β 1...β L I;α 1...α L = R b L 1β L L 2 β L 1 Iα Rb L b L 1 α L 1 R L i 1 b 1β 2 b 2 α RJβ 1 2 b 1 α 1 exp iπ [α i ] + [β i ][α j ] i=1 j=1 The deformed transfer matrix is T u = [J] T J J u. The deformed Hamiltonian is derived as H = i T u 1 d du T u u=0 Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 15 / 2704:0

16 Deformed Two-Site Hamiltonian The two-site transfer matrix is T u = R b 1β 2 aα 2 Raβ 1 b 1 α 1 exp [iπ[α 2 ] + [β 2 ][α 1 ]]. Using the prescribed method, we derive the deformed Hamiltonian to be H = δ β 1 α 1 δ β 2 α 2 δ β 2 α 1 δ β 1 α 2 e ibα 1 α 2 + δ β 1 α 1 δ β 2 α 2 δ β 2 α 1 δ β 1 α 2 e ibα 2 α 1 = H β 1β 2 α 1 α 2 + H β 2β 1 α 2 α 1. Define the deformed two-site Hamiltonians, H1, 2 H β 1β 2 α 1 α 2 and H2, 1 H β 2β 1 α 2 α 1. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 16 / 2704:0

17 Deformed Hamiltonian The oscillator representation of the deformed Hamiltonian H 12 = c a1c b 2 e ib abc b 1c a2 c b 2c a 1 + c a1a α2 + e ibaα a α1c a2 a α 2c a 1 + a α1c a2 + e ibαa c a1a α2 c a 2a α 1 + a α1a β 2 + e ib αβa β 1a α2 a β 2a α 1. B IJ is a real, antisymmetric matrix. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 17 / 2704:0

18 Problems Due to twisting, the Hamiltonian is no longer a Casimir of the SU2 3 superalgebra [ ] H1, 2, J A 12 0, nor do we generally have the edge effects [ H1, 2, Q A 12 ] q A 12, when using the previous coproduct construction for J A 12 and QA 12. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 18 / 2704:0

19 Reshitikhin Twist The Reshitikhin twist is generated by a deforming function F. The R-matrix deforms as Ru = FRuF 1. The coproduct receives a deformation as well, F = F F 1. The deforming function has the definition [ i F = exp B IJ E II E JJ E JJ E II] 2 I<J Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 19 / 2704:0

20 Twisted Coproducts Twisted coproducts J A B = K AB J A B + J A B K BA. R a b = K ab R a b + R a b K ba, L α β = K αβ L α β + L α β K βα, Q c γ = K cγ Q c γ + Q c γ K γc, S γ c = K γc S γ c + S γ c K cγ, D = 1 D + D 1. The twisting is brought about by [ i K IJ = exp 2 ] 5 B IK B JK E KK. K =1 We define matrices E IJ KL = δ IK δ JL which satisfy [E IJ, E KL ] = δ KJ E IL δ IL E AJ. The E IJ are the generators of U2 3. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 20 / 2704:0

21 Yangians Via Twisted Coproducts The first Yangian construction via coproducts is Q I J K IJ Q I J + Q I J K JI K =1 J I K K KJ K KI J K J K IK J K J J I K K JK. An example two-site Yangian is Q R a b a b = K ab Q R a b a b + Q R a b a b K ba Ra ck cb K ca R c b K ac R c b R a ck bc Q a γ K γb K γa S γ b + K aγ S γ b Q a γk bγ δa b Qc γk γc K γc S γ c + K cγ S γ c Q c γk cγ Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 21 / 2704:0

22 It Works Acting on two particle states, the deformed Hamiltonian is equivalent to the deformed Casimir J A B J B A Φ I Φ J = H1, 2 Φ I Φ J. We check the one-loop calculation of the dilatation generator again, [ H1, 2, Q A 12B ] = q A 12B, where the edge effect term, q A 12B, has a deformation dependence q A 12B = J A B K AB K BA J A B. We find for an infinite length spin chain when JB A1, JA B L 0, then q A B 0. Recall, this is what we want! Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 22 / 2704:0

23 Deformations Maintaining N = 1 SCFT: Case 1 Residual SU2 U1 3 symmetry. This is the beta deformation of Lunin-Maldacena. B 13 = B 21 = B 32 = γ. Residual Symmetry Remaining Symmetry L α β = 1 L α β + L α β 1, R a b = K ab R a b + R a b K ba, D = 1 D + D 1, Q c γ = K cγ Q c γ + Q c γ K γc, R c c = 1 R c c + R c c 1, S γ c = K γc S γ c + S γ c K cγ. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 23 / 2704:0

24 Deformations Maintaining N = 1 SCFT: Case 2 Residual SU2 1 U1 2 symmetry. B 12 = B 13 = B 23 = B 1α = B 2α = γ. Residual Symmetry Remaining Symmetry L α β = 1 L α β + L α β 1, R a b = K ab R a b + R a b K ba, Q 3 γ = 1 Q 3 γ + Q 3 γ 1, Q c γ = K cγ Q c γ + Q c γ K γc, S γ 3 = 1 S γ 3 + S γ 3 1, S γ c = K γc S γ c + S γ c K cγ. D = 1 D + D 1, R c c = 1 R c c + R c c 1, Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 24 / 2704:0

25 Conclusion Constructed a Yangian algebra for SU2 3 The deformed theory corresponds to a deformed R-matrix Introduced twisted coproducts corresponding to the twisted R-matrix Showed that twisted coproducts could be used to describe broken symmetry in L The deformed Hamiltonian one-loop dilatation generator still has SU2 3 Yangian symmetry but with twisted coproduct Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 25 / 2704:0

26 Yangians Via Twisted Coproducts Q R a b a b = K ab Q R a b a b + Q R a b a b K ba h Ra ck cb K ca R c b K ac R c b R a ck bc h Q a γk γb K γa S γ b + K aγ S γ b Q a γk bγ 1 6 hδa b Qc γk γc K γc S γ c + K cγ S γ c Q c γk cγ, Q L α β α β = K αβ Q L α β α β + Q L α β α β K βα h Lα γk γβ K γα L γ β K αγ L γ β L α γk βγ h Sα K cβ K cα Q c β + K αc Q c β S α ck βc 1 4 hδα β Sγ ck cγ K cγ Q c γ + K γc Q c γ S γ ck γc, Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 26 / 2704:0

27 Yangians Via Twisted Coproducts Q Q a α a α = K aα Q Q a α a α + Q Q a α a α K αa h Qa γk γα K γa L γ α K aγ L γ α Q a γk αγ h Ra ck cα K ca Q c α K ac Q c α R a ck αc, Q S α a α a = K αa Q S α a α a + Q S α a α a K aα h Sα ck ca K ca R c a K αc R c a S α ck ac h Lα γk γa K γα S γ a K αγ S γ a L α γk aγ, Q D = 1 Q D + Q D h Sγ ck cγ K cγ Q c γ + K γc Q c γ S γ ck γc. Jay N. Ihry UNC-CH Yangians In Deformed Super Yang-Mills Theories arxiv: UNC String[hep-th][-10pt] Seminar JHEP 27 / 2704:0