A Schematic Model of Gauged S-Duality Spontaneous Breakdown
aa r X i v : . [ h e p - t h ] M a r A SCHEMATIC MODEL OF GAUGED S-DUALITYSPONTANEOUS BREAKDOWN
EIJI KONISHI
Abstract.
We present a schematic model of the \ sl (2 , R ) Kugo-Ojima physi-cal state condition of gauged S-duality using modular forms of the Ramanujan∆ function type. The properties of the solutions with spontaneously brokengauged S-duality with their dimensionality of space-time and the gauge sym-metries of low energy theory are studied. Dynamical gauged S-duality, proposed by the author, has been used to describethe dynamics of D-brane fields.[1, 2, 3] The gauged S-duality is defined only byits algebraic structure and, in this formalism, we cannot always evaluate the path-integrals. Nevertheless, we can obtain some non-perturbative properties of thetheory by solving the \ sl (2 , R ) Kugo-Ojima physical state condition[4] of gaugedand affinized S-duality.[5] The search for the solutions of the Kugo-Ojima physicalstate condition brings us back to Ramanujan’s ∆ function.[6] This function is oneof the most celebrated modular forms and is defined by(1) ∆[ q ] = q ∞ Y ℓ =1 (cid:16) − q ℓ (cid:17) , q ∈ exp(2 πi h ) , for the Poincar´e upper half plane h .We stress that Ramanujan’s ∆ function is a robust prototype of the theory,which, as seen in the following, indicates the 24 dimensional spatial structure ofspace-time. This is a very important point of view, especially when we constructstring theory by using category theory,[5] since adopting the category itself in theformulation of the string theory specifies only the formalism, and in the derivedcategory theory there is no robust prototype with a physically nontrivial meaning.The situation of this ∆ function is same as that of the coordinate frame in Euclideanspace, which was the most clear expression of the space in classical Newtonianphysics and the starting point on its road to the theory of relativity.This ∆ function results from the discrete modular symmetry originating in theaffine Lie symmetry of the model and incorporates the quantization of strings, asseen from the early discoveries in the dual resonance model.[7] We construct thetheory based on this ∆ function by introducing the full \ sl (2 , R ) symmetries, whichmatch the degrees of freedom of the model.It is worth pointing out that the modularity of this ∆ function is related to themodulus of the world sheet as seen in, for example, the Gliozzi-Scherk-Olive projec-tion in superstring theory[8] and the Mandelstam-type open/closed string dualityin the partition function of bosonic string theory. The number 24 differs from thecritical dimension of bosonic string theory by the dimension of the tachyonic states. Date : January 5, 2019.
Since the Kugo-Ojima physical state condition on the physical states ψ for the \ sl (2 , R ) Becchi-Rouet-Stora-Tyutin (BRST) charge Q (2) Qψ = 0 , is \ sl (2 , R ) covariant, the space of solutions must have SL (2 , Z ) modular symmetry.The dynamics is reduced on the Riemann surface of the moduli space of affinizedpart of \ sl (2 , R ) with an infinite number of time variables due to an infinite numberof gauge symmetry generators of \ sl (2 , R ) and Eguchi-Kawai large N reduction.[5, 9]Kugo-Ojima physical state condition of the gauged S-duality is infinitesimal de-scription of the moduli space of vacua, so the subject of this note is focused onthe individual vacuum in this moduli space. Namely, we will not refer the non-perturbative transitions between different vacuum configurations.So, we schematically deduce the properties of solutions of Kugo-Ojima physicalstate condition in terms of modular forms.Our schematic model of internal gauge symmetries and space-time requires:(1) The solution in modular form such that the weight , sum of cusps and index of the congruence group are 2 n , 24 m and 24 n for integers n and m . Theinteger 24 comes from the critical dimension of bosonic string theory minustwo.(2) The cusps of the modular form correspond to the classical saddle-pointsolution with spontaneous S-duality breakdown. The cusp i ∞ correspondsto gravity and is always included. The number of gauge symmetries andtheir ranks determine the internal space according to the Kaluza-Kleinreduction. The standard model demands three kinds of gauge interactionsbesides gravity. These gauge interactions would be unified in the higherrank terms of the physical state ψ . In the modular form, gauge symmetriesare encoded as ideals (a notion defined in number theory) in an infiniteproduct expression. An ideal(3) I ℓ = ℓ Z , consists of the ℓ -th saddle point ( ℓ = 1 , , · · · , ∞ ) of a factor χ Nn/ Q ℓ (1 − N n χ ℓ ) N in the physical state (see Eq.(7)). We identify each prime idealwith an independent gauge symmetry. In the analogy with a scalar fieldwith mass µ , i.e., ( (cid:3) − µ ) φ = 0, the logarithm of space-time dimension atsuch saddle points, i.e., P ℓ ln(1 − χ ℓ ) = P ℓ ( − χ ℓ − χ ℓ / − · · · ), depends onthe vacuum energy ,[5] corresponding to the Casimir operator of \ sl (2 , R ).(3) The genus of the gauged S-duality moduli space is desired to be a low one.This requires the axion-dilaton degenerations of T -dual type IIB stringtheory from the original M-theory.The schematic form of the solution when the gauged S-duality is not broken is(4) ψ [ χ ] = χ ∞ Y ℓ =1 (cid:16) − χ ℓ (cid:17) . The variable of the wave function ψ is the physical state χ corresponding to theChan-Paton factor of one D-brane which satisfies(5) Q ℓ χ ℓ = 0 SCHEMATIC MODEL OF GAUGED S-DUALITY SPONTANEOUS BREAKDOWN 3 for the U ( ℓ ) BRST charge Q ℓ . The wave function χ ℓ , which needs to have no massdimension, can be expanded in terms of the number of ghosts(6) χ ℓ = X ~ℓ χ ~ℓ ℓ Y c , where the coefficients χ ~ℓ are independent of the Grassmann numbers c of the ghostsand the anti-ghosts.The degeneracy of vacua in ψ [ χ ], that leads to the flat ten dimensionality ofspace-time, breaks due to the existence of the vacuum energy in the BRST charge.[5]We call this gauged S-duality spontaneous breakdown (GSSB).The schematic form of the solution of GSSB that satisfies these requirements is ψ (0) [ χ ] = χ ∞ Y ℓ =1 Y n k ∈ c Γ − (cid:20) n O χ (cid:21) ℓ ! k . (7)The form of the solution ψ (0) has the symmetry group Γ of subgroup of SL (2 , Z ).The cusps of this group c Γ are denoted by n k with k orbits and width n .We make important remarks about two fundamental structures of these schematicsolutions.On the dimensionality of space-time.Eq.(7) has the following division of the spatial dimensionality:(8) 24 = X n k ∈ c Γ ( k · n ) . We remark that GSSB breaks the flat directions of the moduli space of vacua non-perturbatively since the flatness of each direction of space-time is perturbativelystable so that a ∆[ q ] type solution is possible.We also draw attention to the fact that our formulation is completely backgroundindependent, and even the dimensionality is not assumed.We note that the factor χ r in Eq. (7) arises from the consistency with mod-ularity. The product of Dedekind η functions Q ki =1 η ( r i χ ) is a modular form withweight k/ P i r i = 24.On the gauge symmetries.However tempting it might be to regard the width of a cusp as the rank of acorresponding gauge symmetry, this will not work. The rank of the gauge symmetrydoes not appear in each solution explicitly. The gauge symmetries are based onideals and if one ideal is included by two other ideals, i.e,(9) I ⊂ I ∩ I , the first interaction unifies the other two interactions.In this note, we can only conclude the number of gauge symmetries and theirrelations with each other since we may need to solve the Kugo-Ojima physical statecondition exactly including some self-consistency conditions to specify the concreteform of the variable χ . EIJI KONISHI
References [1] E. Konishi, Prog. Theor. Phys. , 1125 (2009), arXiv:0902.2565 [hep-th].[2] T. Yoneya, JHEP , 28 (2005), arXiv:hep-th/0510114.[3] T. Yoneya, Prog. Theor. Phys. , 135 (2007), arXiv:0705.1960 [hep-th].[4] T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. , 1 (1979).[5] E. Konishi, Int. J. Mod. Phys. A 26 , 4785 (2011), arXiv:1001.3382 [hep-th].[6] S. Ramanujan, Trans. Cambridge Philos. Soc. , 159 (1916).[7] J. H. Schwarz, Phys. Rep. C 8 , 269 (1973).[8] F. Gliozzi, J. Scherk and D. Olive, Nucl. Phys.
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