Relations among Supersymmetric Lattice Gauge Theories

1. Relations among Supersymmetric Lattice Gauge Theories So Matsuura @ Niels Bohr Institute based on the works arXiv:0704.2696 arXiv:0706.3007 arXiv:0708.4129 arXiv:0709.4193 with P.H.Damgaard Isaac Newton Institute

2. Introduction Lattice Gauge Theory • Constructive definition of a gauge theory • Non-perturbative analysis by numerical simulations If supersymmetric gauge theories are constructed on a lattice, • It gives a “definition” of the theory. • We can compute any physical observable even if it is not restricted by the SUSY algebra. • We can compare results in strong coupling region directly with, say, the AdS/CFT correspondence. Isaac Newton Institute

3. Difficulty It seems impossible to construct a SUSY invariant theory on a lattice. SUSY invariant action in continuum space-time Suppose an action is written as ; superfield Essentially, a SUSY generator can be represented as Variation of the action Leibniz rule Isaac Newton Institute

4. continuum theory lattice theory difference operator differential operator deformed Leibniz rule Leibniz rule It seems impossible to keep all SUSY on a lattice. QUESTION Can we keep a part of SUSY on a lattice? Yes! Isaac Newton Institute

5. Map of lattice theories with SUSY on a lattice A.Cohen, E.Katz, D.Kaplan, M.Unsal, S. Catterall Orbifold lattice theories Catterall's lattice theories an extension P.H.Damgaard, S.M. equivalent some reduction P.H.Damgaard, S.M. T.Takimi Lattice theories by the “link approach” Sugino’s lattice theories F.Sugino A. D'Adda, I. Kanamori, N. Kawamoto, K. Nagata Isaac Newton Institute

6. Contents • Introduction • Classification of Orbifold Lattice Gauge Theories • Exact Vacuum Energy of Orbifold Theory • Relation with Catterall’s Supersymmetric Lattice Gauge Theory • Equivalence between the Orbifolding and the Link Approach • Conclusion Isaac Newton Institute

7. Classification of Orbifold Lattice Gauge Theories OUT LINE Mother Theory A supersymmetric Yang-Mills matrix theory STEP 1 orbifold projection Orbifolded Matrix Theory A matrix theory with “scalar supercharges” (a lattice formulation without kinetic terms) STEP 2 deconstruction Orbifold Lattice Theory A lattice theory with scalar supercharges Isaac Newton Institute

8. Yang-Mills matrix with 4 SUSY Yang-Mills matrix with 8 SUSY mother theory 1 mother theory 2 ・・・・・ ・・・・・ lattice ∞? lattice 8 lattice 1 lattice 1 lattice 1 lattice 2 lattice 2 Yang-Mills matrix with 16 SUSY Unknown mother theory 4 mother theory 3 ・・・・・ ・・・・・ lattice 1024 lattice 1 lattice 2 lattice ?? lattice 1 lattice 2 Isaac Newton Institute

9. [1] construction of Q=4 orbifold lattice theory STEP 0 Mother theory with 4 SUSY A matrix theory that is obtained by dimensional reduction of Euclidean 4D N=1 SYM theory with a gauge group . : four hermitian matrices : a Majorana spinor Symmetries maximal U(1) subgroup 1) global symmetry 2) gauge symmetry Isaac Newton Institute

10. Equivalent expression in which the U(1) symmetries are manifest: U(1) charges We can take any linear combination. where Isaac Newton Institute

11. supersymmetry with The variation of the action is zero if and only if the SUSY parameters are trivial; Isaac Newton Institute

12. Orbifold projection STEP1 We consider a transformation generated by where and : clock matrix We keep only components that are invariant under this transformation. simple example z(1) projection of a matrix with U(1) charge 1 z(2) projection by z(3) z(4) each block is an MxM matrix Isaac Newton Institute

13. 1 1 Orbifolded action Substituting the projected field, we obtain Projection of the supersymmetry The supersymmetry parameters have definite U(1) charges: They become non-trivial after orbifolding. The only preserved supersymmetry is the one corresponding to k. Isaac Newton Institute

14. Deconstruction STEP2 We introduce kinetic terms and a lattice spacing by Finally, we get an action: where Isaac Newton Institute

15. Classification of the theories The lattice action depends on • two vectors and . • two real numbers and . How the theory depends on them? a physical interpretation a space-time lattice an abstract lattice : linear mapping Impose ・・・☆ The continuum theory should be Lorentz invariant Isaac Newton Institute

16. The kinetic terms in the continuum limit The condition ☆ determines the linear mapping f as The lattice theory is unique and on a square lattice. the continuum theory 2D N=(2,2) SYM theory with the gauge group U(M) Isaac Newton Institute

17. [2] construction of Q=8 orbifold lattice theory Mother theory with 8 SUSY A matrix theory that is obtained by dimensional reduction of Euclidean 6D N=1 SYM theory. : six hermitian matrices : independent four-component spinors Symmetries maximal U(1) subgroup 1) global symmetry 2) gauge symmetry Isaac Newton Institute

18. Equivalent expression in which the U(1) symmetries are manifest: U(1) charges linear combinations of Isaac Newton Institute

19. 1) Orbifold projection 2) Deconstruction the lattice action: Isaac Newton Institute

20. Classification of the theories generates the lattice Dimensionality of the lattice the dimensionality of the lattice the number of linearly independent vectors in Preserved supersymmetry on the lattice The supercharges corresponding to scalar fermions are preserved. • At least one SUSY corresponding to his preserved. • SUSY enhances if another U(1) charges of a fermion becomes zero. Isaac Newton Institute

21. (1) 3D lattice with 1 SUSY (three linearly independent ) continuum limit 3D SUSY Yang-Mills theory with 8 SUSY Isaac Newton Institute

22. (2-1) 2D lattice with 2 SUSY (2-2) 2D lattice with 2 SUSY (2-3) 2D lattice with 2 SUSY (3) 2D lattice with 1 SUSY (an example) The common continuum limit is 2D N=(4,4) SUSY Yang-Mills theory. Isaac Newton Institute

23. N.B. • There are additional three kinds of 2D lattice theories obtained by shifting only two bosons, say, and as The continuum theory is the same. • We can construct 4D, 3D and 2D lattice theories from the mother theory with sixteen supercharges (IKKT matrix theory). In particular, the 4D theory is a lattice formulation of the 4D N=4 SYM theory. Isaac Newton Institute

24. Exact Vacuum Energy of the Orbifold Theories The classical moduli space of the orbifold lattice theories are parametrized by the vacuum expectation values of the potential terms up to gauge transformations. QUESTION Can we estimate quantum corrections to the vacuum energy? It seems non-trivial since the supersymmetry is almost broken. • contributions from higher-loops • non-perturbative contributions We can estimate the exact vacuum energy in this case! Isaac Newton Institute

25. key points • The orbifold theories have a BRST symmetry on a lattice. • The actions can be written in Q-exact forms. The partition function does not depend on the coupling constant; The vacuum energy estimated in the 1-loop level is exact. Isaac Newton Institute

26. The second order actions For both the case it is easy to show that the 1-loop contribution to the partition function is equal to 1. The vacuum energy of the orbifold lattice theories constructed from the mother theories with 4 and 8 SUSY never receive quantum corrections. Isaac Newton Institute

27. Relation with Catterall’s Supersymmetric Lattice Gauge Theory Brief review of Catterall’s formulation Starting with the topologically twisted 2D N=(2,2) SYM, where Q is a BRST charge acting the fields as nilpotent up to a gauge transformation The lattice theory is obtained by the following three steps. Isaac Newton Institute

28. STEP 1 The theory is latticised by putting fields on a lattice corresponding to the tensor structures; tensors vectors scalars STEP 2 Complexified fields are introduced to make the action real; tensors vectors STEP 3 The field strength and the covariant derivatives are replaced by Isaac Newton Institute

29. Then we obtain a lattice action, • A BRST symmetry is preserved on a lattice. • The path-integral is carried out along the real line, • The other supersymmetries are shown to be restored in the continuum limit by numerical simulations. • By restricting the complexified fields in a different way, we obtain Sugino’s lattice formulation. S.Catterall 2006 F.Sugino 2004 T.Takimi 2007 Isaac Newton Institute

30. Claim This prescription is automatically reproduced by orbifolding. The only assumption is a complexification. Let us consider the mother theory with 4 SUSY in a Q-exact form; where Q acts on the fields as We complexified the matrices and extend the action as where d is a auxiliary field and we have also doubled h as . Isaac Newton Institute

31. By this extension, the theory acquires extra U(1) symmetries, and the charge assignment for the fields is Then we can define the corresponding orbifold projection. The orbifold projected action is obtained by substituting into the action. Furthermore, instead of shifting and , we replace them as Isaac Newton Institute

32. Then we obtain a lattice action, This is equivalent with Catterall’s formulation. Isaac Newton Institute

33. N.B. • The BRST symmetry is enhanced by the complexification; with which satisfies . • This method can be applied to other SUSY gauge theories. (ex) 4D N=2 SYM theory, etc... Isaac Newton Institute

34. Equivalence between the Orbifolding and the Link Approach Recall the U(1) charges of the fields in the mother theory with 4 SUSY new U(1) charges three-component vectors obtained from We can carry out the orbifold projection using these U(1) charges. Isaac Newton Institute

35. The result is A typical example This action is completely the same with the one obtained by the link approach. A. D'Adda, I. Kanamori, N. Kawamoto, K. Nagata (2005) The lattice action given by the link approach is obtained by orbifolding procedure. Isaac Newton Institute

36. Supersymmetry of this theory The SUSY is completely broken because of the discussion given above. but They claim that all the supercharges are preserved on a lattice in adeformed sense. deformed SUSY in the mother theory Consider the “supersymmetry transformation” in the mother theory with non-trivial Instead of the usual Leibniz rule, a matrix made from the shift matrix. let us impose a modified Leibniz rule by hand, for each of and F.Bruckmann, S.Catterall, M.de Kok (2006) They satisfy although there is some discussion in whether this is consistent or not…. Isaac Newton Institute

37. Conclusion • We classified the lattice theories constructed from the mother theories with 4 and 8 supercharges. • We showed that the vacuum energy of the orbifold lattice theories does not receive any quantum correction. • We showed that the formulation given by Catterall can be understood in terms of the orbifolding procedure. • We showed that the SUSY lattice theories obtained by the link approach are equivalent to the orbifold lattice theories. Isaac Newton Institute

38. Future Problems • Lattice theories constructed from the mother theory with 16 supercharges (IKKT matrix theory) • Numerical simulations • Connection to the superstring theory • classification of the theories • structure of the quantum vacuum • including 4D N=4 SYM theory • AdS/CFT correspondence in terms of lattice theories? • recovering of the supersymmetries in the continuum limit • comparison with exact results • non-BPS operators orbifold D-instantons Isaac Newton Institute