1. 9/10/2007Isaac Newton Institute1 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

2. 9/10/2007Isaac Newton Institute2 Introduction Lattice Gauge Theory Constructive definition of a gauge theory If supersymmetric gauge theories are constructed on a lattice, We can compute any physical observable even if it is not restricted by the SUSY algebra. It gives a “definition” of the theory. We can compare results in strong coupling region directly with, say, the AdS/CFT correspondence. Non-perturbative analysis by numerical simulations

3. 9/10/2007Isaac Newton Institute3 Difficulty It seems impossible to construct a SUSY invariant theory on a lattice. Suppose an action is written as Essentially, a SUSY generator can be represented as ; superfield SUSY invariant action in continuum space-time Variation of the action Leibniz rule

4. 9/10/2007Isaac Newton Institute4 differential operator difference operator Leibniz rule deformed Leibniz rule continuum theorylattice theory It seems impossible to keep all SUSY on a lattice. Can we keep a part of SUSY on a lattice? Yes! QUESTION

5. 9/10/2007Isaac Newton Institute5 Map of lattice theories with SUSY on a lattice Orbifold lattice theories Catterall's lattice theories Sugino’s lattice theories Lattice theories by the “link approach” A.Cohen, E.Katz, D.Kaplan, M.Unsal,S. Catterall F.Sugino A. D'Adda, I. Kanamori, N. Kawamoto, K. Nagata some reduction T.Takimi equivalent P.H.Damgaard, S.M. an extension P.H.Damgaard, S.M.

6. 9/10/2007Isaac Newton Institute6 Contents 1.Introduction 2.Classification of Orbifold Lattice Gauge Theories 3.Exact Vacuum Energy of Orbifold Theory 4.Relation with Catterall’s Supersymmetric Lattice Gauge Theory 5.Equivalence between the Orbifolding and the Link Approach 6.Conclusion

7. 9/10/2007Isaac Newton Institute7 Classification of Orbifold Lattice Gauge Theories Mother Theory OUT LINE A supersymmetric Yang-Mills matrix theory Orbifolded Matrix Theory STEP 1 orbifold projection Orbifold Lattice Theory STEP 2 deconstruction A matrix theory with “scalar supercharges” A lattice theory with scalar supercharges (a lattice formulation without kinetic terms)

8. 9/10/2007Isaac Newton Institute8 mother theory 1 lattice 1 lattice 2 lattice ∞? ・・・・・ mother theory 2 lattice 1 lattice 2 lattice 8 ・・・・・ mother theory 3 lattice 1 lattice 2 lattice ?? ・・・・・ mother theory 4 lattice 1 lattice 2 lattice 1024 ・・・・・ Yang-Mills matrix with 4 SUSYYang-Mills matrix with 8 SUSY Yang-Mills matrix with 16 SUSYUnknown lattice 1

9. 9/10/2007Isaac Newton Institute9 A matrix theory that is obtained by dimensional reduction of Euclidean 4D N=1 SYM theory with a gauge group. STEP 0 : four hermitian matrices : a Majorana spinor maximal U(1) subgroup Mother theory with 4 SUSY 1) global symmetry 2) gauge symmetry [1] construction of Q=4 orbifold lattice theory Symmetries

10. 9/10/2007Isaac Newton Institute10 Equivalent expression in which the U(1) symmetries are manifest: U(1) charges where We can take any linear combination.

11. 9/10/2007Isaac Newton Institute11 supersymmetry with The variation of the action is zero if and only if the SUSY parameters are trivial;

12. 9/10/2007Isaac Newton Institute12 STEP1 We consider a transformation generated by where and : clock matrix projection of a matrix with U(1) charge 1 z(1) z(2) z(3) z(4) We keep only components that are invariant under this transformation. Orbifold projection projection by simple example each block is an MxM matrix

13. 9/10/2007Isaac Newton Institute13 Projection of the supersymmetry The supersymmetry parameters have definite U(1) charges: The only preserved supersymmetry is the one corresponding to  Orbifolded action Substituting the projected field, we obtain 1 1 They become non-trivial after orbifolding.

14. 9/10/2007Isaac Newton Institute14 STEP2 Deconstruction We introduce kinetic terms and a lattice spacing by Finally, we get an action: where

15. 9/10/2007Isaac Newton Institute15 Classification of the theories The lattice action depends on two vectors and. two real numbers and. How the theory depends on them? an abstract lattice a physical interpretation a space-time lattice : linear mapping The continuum theory should be Lorentz invariant Impose ・・・☆

16. 9/10/2007Isaac Newton Institute16 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. 2D N=(2,2) SYM theory with the gauge group U(M) the continuum theory

17. 9/10/2007Isaac Newton Institute17 [2] construction of Q=8 orbifold lattice theory 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 Mother theory with 8 SUSY 1) global symmetry 2) gauge symmetry

18. 9/10/2007Isaac Newton Institute18 Equivalent expression in which the U(1) symmetries are manifest: U(1) charges linear combinations of

19. 9/10/2007Isaac Newton Institute19 1) Orbifold projection the lattice action: 2) Deconstruction

20. 9/10/2007Isaac Newton Institute20 generates 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  is preserved. SUSY enhances if another U(1) charges of a fermion becomes zero. Dimensionality of the lattice Classification of the theories

21. 9/10/2007Isaac Newton Institute21 (1) 3D lattice with 1 SUSY (three linearly independent ) 3D SUSY Yang-Mills theory with 8 SUSY continuum limit

22. 9/10/2007Isaac Newton Institute22 (2-3) 2D lattice with 2 SUSY (2-1) 2D lattice with 2 SUSY(2-2) 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.

23. 9/10/2007Isaac Newton Institute23 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.

24. 9/10/2007Isaac Newton Institute24 Exact Vacuum Energy of the Orbifold Theories The classical moduli space of the orbifold lattice theories are parametrized by the vacuum expectation values of up to gauge transformations. the potential terms 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!

25. 9/10/2007Isaac Newton Institute25  The orbifold theories have a BRST symmetry on a lattice.  The actions can be written in Q-exact forms. key points The partition function does not depend on the coupling constant; The vacuum energy estimated in the 1-loop level is exact.

26. 9/10/2007Isaac Newton Institute26 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.

27. 9/10/2007Isaac Newton Institute27 Relation with Catterall’s Supersymmetric Lattice Gauge Theory 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. Brief review of Catterall’s formulation

28. 9/10/2007Isaac Newton Institute28 The theory is latticised by putting fields on a lattice corresponding to the tensor structures; scalars vectors tensors STEP 1 STEP 2 Complexified fields are introduced to make the action real; vectors tensors STEP 3 The field strength and the covariant derivatives are replaced by

29. 9/10/2007Isaac Newton Institute29 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

30. 9/10/2007Isaac Newton Institute30 This prescription is automatically reproduced by orbifolding. The only assumption is a complexification. Claim 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  as.

31. 9/10/2007Isaac Newton Institute31 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

32. 9/10/2007Isaac Newton Institute32 Then we obtain a lattice action, This is equivalent with Catterall’s formulation.

33. 9/10/2007Isaac Newton Institute33 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...

34. 9/10/2007Isaac Newton Institute34 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.

35. 9/10/2007Isaac Newton Institute35 The result is 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. A typical example

36. 9/10/2007Isaac Newton Institute36 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 a deformed sense. Consider the “supersymmetry transformation” in the mother theory with non-trivial Instead of the usual Leibniz rule, let us impose a modified Leibniz rule by hand, for each of and a matrix made from the shift matrix. They satisfy although there is some discussion in whether this is consistent or not…. F.Bruckmann, S.Catterall, M.de Kok (2006) deformed SUSY in the mother theory

37. 9/10/2007Isaac Newton Institute37 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.

38. 9/10/2007Isaac Newton Institute38 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? orbifold D-instantons recovering of the supersymmetries in the continuum limit comparison with exact results non-BPS operators