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Tomohisa Takimi ( 基研 )

A non-perturbative study of Supersymmetric Lattice Gauge Theories. Tomohisa Takimi ( 基研 ). Contents. 1. Introduction (our purposal) 2. Our proposal for non-perturbative study 3. Topological property in the continuum theory

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Tomohisa Takimi ( 基研 )

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  1. A non-perturbative study ofSupersymmetric Lattice Gauge Theories Tomohisa Takimi (基研)

  2. Contents 1. Introduction (our purposal) 2. Our proposal for non-perturbative study 3. Topological property in the continuum theory 3.1 BRST exact form of the model 3.2 Partition function (Witten index) 3.3 BRST cohomology (BPS state) 4. Topological property on the lattice 4.1 BRST exact form of the model 4.2 Partition function (Witten index) 4.3 BRST cohomology (BPS state) 5. Construction of new class of model 6. Summary K.Ohta, T.T (To appear in Prog.Theor.Phys Vol.117, No2) (K.Ohta, T.T(2007))

  3. 1. Introduction Supersymmetric gauge theory • One solution of hierarchy problem • Dynamical SUSY breaking Lattice study may help to get deeper understanding but lattice construction of SUSY field theory is difficult. SUSY algebra includes infinitesimal translation which is broken on the lattice.

  4. Fine-tuning problem in present approach • Standard action • Plaquette gauge action + Wilson or Overlap fermion action • Violation of SUSY for finite lattice spacing. • Many SUSY breaking terms appear; • Fine-tuning is required to recover SUSY in continuum. • Time for computation becomes huge. Difficult to perform numerical analysis ex. N=1 SUSY with matter fields gaugino mass, scalar mass fermion mass scalar quartic coupling

  5. Exact supercharge on the lattice for a nilpotent (BRST-like) supercharge in Extended SUSY Lattice formulations free from fine-tuning

  6. Twist in Extended SUSY (E.Witten, Commun. Math. Phys. 117 (1988) 353, N.Marcus, Nucl. Phys. B431 (1994) 3-77 • Redefine the Lorentz algebra by a diagonal subgroup of the Lorentz and the R-symmetry • in the extended SUSY • ex. d=2, N=2 • d=4, N=4 There are some scalar supercharges under this diagonal subgroup. If we pick up the charges, they become nilpotent supersymmetry generator which does not include infinitesimal translation in their algebra.

  7. After the twist, we can reinterpret the extended supersymmetric gauge theory action as an equivalent topological field theory action Extended Supersymmetric gauge theory action Nilpotent scalar supercharge is extracted from spinor supercharges Twisting Topological Field Theory action Supersymmetric Lattice Gauge Theory action lattice regularization is preserved

  8. Models utilizing nilpotent SUSY from Twisting • CKKU models (Cohen-Kaplan-Katz-Unsal) • 2-dN=(4,4),N=(2,2),N=(8,8),3-d N=4,N=8, 4-d N=4 • super Yang-Mills theories • ( JHEP 08 (2003) 024, JHEP 12 (2003) 031, JHEP 09 (2005) 042) • Catterall models (Catterall) • 2 -d N=(2,2),4-d N=4 super Yang-Mills • (JHEP 11 (2004) 006, JHEP 06 (2005) 031) • Sugino models • 2 -d N=(2,2),N=(4,4),N=(8,8),3-dN=4,N=8, 4-d N=4 • super Yang-Mills (JHEP 01 (2004) 015, JHEP 03 (2004) 067, JHEP 01 (2005) 016 Phys.Lett. B635 (2006) 218-224) We will treat 2-d N=(4,4) CKKU’s model and 2-d N=(2,2) Catterall’s model among these.

  9. Do they really have the desired continuum theory with full supersymmetry ? • Perturbative investigation They have the desired continuum limit CKKU JHEP 08 (2003) 024, JHEP 12 (2003) 031, Onogi, T.T Phys.Rev. D72 (2005) 074504 • Non-perturbative investigation Sufficient investigation has not been done ! Our main purpose

  10. Our proposal for the • non-perturbative study - ( Topological Study ) -

  11. We look at that the lattice model actions are lattice regularization of topological field theory action equivalent to the target continuum action Extended Supersymmetric gauge theory action Topological Field Theory action Supersymmetric Lattice Gauge Theory action continuum limit a 0 lattice regularization

  12. And the target continuum theory includes a topological field theory as a subsector. So if the theories recover the desired target theory,even including quantum effect,topological field theory and its property must be recovered Extended Supersymmetric gauge theory Supersymmetric lattice gauge theory continuum limit a 0 Witten index BPS states Topological field theory Must be realized in a 0

  13. Partition function( Witten index) • BRST cohomology(BPS state) Topological property (action ) these are independent of gauge coupling Because • We can obtain these valuenon-perturbatively • in the semi-classical limit.

  14. The aim of this thesis A non-perturbative study whether the lattice theories have the desired continuum limit or not through the study of topological property on the lattice

  15. 3. Topological property in the continuum theories - (Review) 3.1 BRST exact action 3.2 Partition function 3.3 BRST cohomology In the 2 dimesional N = (4,4) super Yang-Mills theory

  16. 3.1 BRST exact form of the action (Dijkgraaf and Moore, Commun. Math. Phys. 185 (1997) 411) Equivalent topological field theory action : covariant derivative (adjoint representation) : gauge field

  17. BRST transformation BRST transformation change the gauge transformation law BRST BRST partner sets BRST transformation is not homogeneous : linear function of : not linear function of

  18. 3.2 Partition function (Witten index) • || :Partition function of continuum theory explicit form (Gerasimov and Shatasvli hep-lat/0609024) It should be checked whether the partition function of lattice theory realizes this in the continuum limit

  19. 3.3BRST cohomology in the continuum theory (E.Witten, Commun. Math. Phys. 117 (1988) 353) The following set of k –form operators, (k=0,1,2) satisfies so-calleddescent relation Integration of over k-homology cycle ( on torus) homology 1-cycle becomes BRST-closed

  20. They are not BRST exact Although they are BRST transformation of something ,and are not gauge invariant due to the inhomogeneous term in gauge transformation They are BRST cohomology composed by

  21. 4.Topological property on the lattice （K.Ohta，T.T (2007)) 4.1 BRST exact action 4.2 Partition function 4.3 BRST cohomology We investigatein the 2 dimensionalN = (4,4) CKKU supersymmetric lattice gauge theory

  22. - 4.0 The model 2 dimensional N=(4,4) CKKU model - (Cohen-Kaplan-Katz-Unsal JHEP 12 (2003) 031) • Dimensional reduction of 6 dimensional super Yang-Mills theory • Orbifolding by in global symmetry 2-dimensional lattice structure in the field degrees of freedom • Deconstruction kinetic term in 2-dim

  23. 4.1 BRST exact form of the lattice action （K.Ohta，T.T (2007)) To investigate the topological properties we rewrite the N=(4,4) CKKU action as BRST exact form .

  24. BRST transformation on the lattice They are Linear functions of partner In continuum theory, it is not homogeneous transformation of BRST partner sets If we split the field content as Bosonic field Fermionic field is not included in Homogeneous transformation of So the transformation can be written as tangent vector

  25. Location on the Lattice *BRST partners sit on same links or sites *Gauge transformation law does not change under BRST

  26. 4.2 Partition function (Witten index) (K.Ohta, T.T (2007)) (1) We will compare this with that of the target continuum theory :Partition function of continuum theory Problem: How do we carry out the path integral (1) ?

  27. Exact integration by Nicolai Map • (1) is exactly obtained by the semi-classical limit. action can be simplified in semi-classical limit as • By the change of variables ( ) ( ) Integration over becomes Gaussian integration over (Nicolai Map) this is first time to discover the Nicolai map in supersymmetric lattice gauge theory

  28. Then we can simplify the (1) as We only have to perform the last integral and compare with continuum results

  29. 4.3 BRST cohomology on the lattice theory (K.Ohta, T.T (2007)) No-go theorem The BRST closed operators on the N=(4,4) CKKU lattice model must be the BRST exact except for the polynomial of

  30. proof 【1】the BRST transformation : and following fermionic operator Compose the number operator as counting the number of fields within 【2】 commute with the number operator since is homogeneous about 【3】Any field function can be written as

  31. 【4】 From in 【2】, 【5】 , from 【1】, for 【6】 transformation commute with gauge transformation : gauge invariant : gauge invariant 【7】 From 【5】【6】 , BRST closed eigenfunction : must be BRST exact . ) BRST closed function including the field in Must be BRST exact

  32. From 【7】, BRST cohomology in BRST closed function in 【4】 must come from zero eigenstates which does not contain any field in namely a term composed only of can be BRST cohomology(End of proof)

  33. Essence of the No-go theorem • Lattice BRST transformation is homogeneous about We can define the number operator of by using another fermionic transformation • Lattice BRST transformation does not change the representation under the gauge transformation We cannot construct the gauge invariant BRST cohomology by the BRST transformation of gauge variant value

  34. on the lattice BRST cohomology must be composed only by disagree with each other BRST cohomology are composed by in the continuum theory * BRST cohomology on the lattice Not realized in continuum limit ! * BRST cohomology in the continuum theory

  35. Result of topological study on the lattice We have found a problem in the 2 dimensional N=(4,4) CKKU model Extended Supersymmetric gauge theory action Supersymmetric lattice gauge theory Really ? continuum limit a 0 Topological field theory on the lattice Topological field theory

  36. (K.Ohta, T.T (2007)) 5. Construction of new class of model In the continuum theory we can obtain N=(2,2) from N =(4,4) From N = (4,4) Truncating and Half degree of freedom of N = (2,2) And their BRST partner Is the N=(2,2) supersymmetric lattice model obtained from N=(4,4) lattice model by using analogous method ? Non-trivial

  37. Since we find the BRST exact form of the N=(4,4) CKKU action, we can utilize analogous method in the lattice theory. * N= (2,2) lattice model can be obtained by the suitable truncation of fields in N=(4,4) CKKU lattice model. The N=(2,2) model can preserve same BRST charge.

  38. * We find that the N= (2,2) lattice model is equivalent to N=(2,2) lattice model proposed by Catterall.(JHEP 11 (2004) 006) It is not expected since Catterall model does not originally use the matrix model construction Since Catterall model is obtained from N=(4,4) CKKU model, Topological analysis on N=(4,4) would be utilized in N=(2,2) Catterall model to judge whether the Catterall model work well. (future work)

  39. 6. Summary • We have proposed that • the topological property • (like as partition function, • BRST cohomology) • should be used as • a non-perturbative criteria to judge whether supersymmetic lattice theories • which preserve BRST charge on it • have the desired continuum limit or not.

  40. We apply the criteria to N= (4,4) CKKU model • *The model can be written as BRST exact form. • *BRST transformation becomes homogeneous transformation on the lattice. • *We discover Nicolai Map and calculate the partition function to compare with the continuum result. • *The No-go theorem in the BRST cohomology on the lattice. It becomes clear that there is possibility that N=(4,4) CKKU model does not work well ! This becomes clear by using this criteria. (We do not know this in perturbative level.) It is shown that the criteria is powerful tool.

  41. h: number of genus Parameter of regularization h-independent constant depend on Parameter which decide the additional BRST exact term Weyl group

  42. Prospects • Applying the criteria to other models • (for example Sugino models ) • to judge whether they work as supersymmetric lattice theories or not. • Clarifying the origin of impossibility to define the BRST cohomology on N=(4,4) CKKU model to construct the model which have desired continuum limit. • (Idea: to study the deconstrution)

  43. Since N=(2,2) Catterall model can be obtained from N=(4,4) CKKU model, it would be judged by utilizing the topological analysis in N=(4,4) CKKU model

  44. Hilbert space Hilbert space of extended super Yang-Mills: Hilbert space of topological field theory: Topological field theory is obtained from extended super Yang-Mills as a subsector Hilbert space of topological field theory Hilbert space of extended super Yang-Mills

  45. Possible virtue of this construction We might be able to analyze the topological property of N=(2,2) Catterall model by utilizing that of topological property on N=(4,4) CKKU

  46. If the theory lead to desired continuum limit, continuum limit must permit the realization of topological field theory • There we pick up the topological property on the lattice which enable us non-perturbative investigation.

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