Derivative corrections to Dirac-Born-Infeld and Chern-Simon actions from Non-commutativity

a r X i v :h e p -t h /0108104v 4 6 S e p 2001IP/BBSR/2001-21hep-th/0108104Derivative corrections to Dirac-Born-Infeld and Chern-Simon actions from Non-commutativityShesansu Sekhar Pal e-mail:shesansu@iopb.res.inInstitute of Physics Bhubaneswar -751005,IndiaAbstractWe show that the higher order derivative α′corrections to the DBI and Chern-Simon action is derived from non-commutativity in the Seiberg-Witten limit,and is shown to agree with Wyllard’s (hep-th/0008125)result,as conjec-tured by Das et al.,(hep-th/0106024).In calculating the corrections,we haveexpressed ˆFin terms F,ˆA in terms of A up to order O (A 3),and made use of it.1IntroductionThere are several interesting developments has taken place in recent years,one of them is non-commutativity in position coordinates.In string theory,if we take a Dp brane in a flat background metric g ij and suspend it in a constant second rank an-tisymmetric tensor B ij field background,then one realizes a non-commutative string theory[1,2],i.e.the ends of the open string that ends on Dp brane satisfy the follow-ing non-commutative algebra,[X i ,X j ]=iθij .Where X i ’s are the coordinate of the open string,and θij is a function of the g ij ,B ij [2],the background fields.In string theory with the above mentioned background,we know that there exists two different kind of descriptions,namely,commutative and non-commutative theory.These different kind of theories arises depending on the kind of regularisation schemes that we adopt.This can be seen as:the interaction of the gauge field with the string world sheet is gauge invariant,under the gauge transformation δA i =∂i λ,at the tree as well as the loop level in the Pauli-Villar regularisation scheme.If we shall adopt the Point-Splitting regularisation scheme instead of the Pauli-Villar regularisation scheme then the above mentioned interaction is gauge invariant only if the form ofthe gauge transformation is ˆδˆλˆA i =∂i ˆλ+i [ˆλ,ˆA i ]∗[2].Moreover,it is well-knownthat,in quantum field theory,different regularisation schemes do not yield different S-matrix elements,also,the S-matrix element is unchanged under field redefinition in the effective action.Although,we are dealing with two seemingly different kind of descriptions but actually they are equivalent as discussed in [2],which can be realized by the well-known Seiberg-Witten map,and it implies that the actions described by the above two descriptions are related up to total derivative modulo field redefinition,i.e.ˆS−S =Total derivative +O (∂F ),in the DBI approximation.So,these two ways of describing the same theory can be written in a more general way[2,3],in which the parameters of the open string,g,B,g s ,and closed strings,G,θ,G s ,are related as,1G +2πα′Φ+θdet (G +2πα′Φ)B,the value of G s and G can be determined from the above equation.To find the gauge invariant coupling of the bulk modes with the gauge fields one1needs to introduce Wilson lines[6].The Wilson line is defined as:W(x,C)=P∗e i 10dσ∂σξi(σ)ˆA i(x+ξ(σ))(1.2) Whereξi(σ)=θij k jσ,i.e.a straight Wilson line C,and P∗is the path ordering with respect to∗product,andˆA i is the gauge potential in non-commutative space. In passing,we should mention that comparison between the R-R couplings in differ-ent descriptions yields the Seiberg-Witten map and the other topological identities[8].We know that the low energy limits of the string theory on the brane is described by an effective theory in theα′→0limit of the string theory and the effective action has two parts,DBI and Chern-Simon actions.Whenα′=0then one expects to have α′corrections to the action,and these corrections might be useful in the study of dualities.In this paper,we shall verify the conjecture made by Das et al.,that the deriva-tive corrections to the Chern-Simon and the DBI action can be derived from non-commutativity.Recently,it has been conjectured[10]that derivative corrections to DBI and Chern-Simon action can be found from non-commutativity,and the calculation has already been done to check it,to some order in F.In this report,we have not only extended this calculation but have presented the form of the Seiberg-Witten map to O(A3), namely,we have expressedˆF in terms of F andˆA in terms of A to order A3.We should mention en passant that it’s important to know the Seiberg-Witten map,be-cause the non-commutative action is written in non-commutative variables,ˆF andˆA, but to make a comparison with[9],we have to express all the terms in commutative variables i.e.in terms of F and A.As we shall try to check this conjecture by calculating the4-derivative correc-tions to the F3term for DBI action and4-form4-derivative corrections at F4,6-form 4-derivative corrections at F4,8-form8-derivative corrections at F4to the Chern-Simon action.Since Das et al.,has already derived the4-derivative corrections at F2to DBI action and4-form4-derivative corrections at F3,6-form6-derivative cor-rections at F3,and8-form8-derivative corrections at F4to the Chern-Simon action. Evaluating the derivative corrections to Chern-Simon and DBI action,calculated by Wyllard[9],in the Seiberg-Witten limit shows an agreement of result coming from non-commutativity.The plan of the paper is as follows:In section2,we shall calculate the4-derivative corrections at F3along with the4-derivative corrections at F2to DBI action,and in section3,we shall calculate the derivative corrections to Chern-Simon action,and2conclude in section4.We shall derive the Seiberg-Witten map and the kernels of the ∗n product of n functions in position space in appendix A and B respectively.2Corrections to the DBI actionThe correction to the DBI action,has been calculated[9]using boundary state tech-nique,and it is:S DBI=1det(g+2πα′(B+F))[1+(2πα′)42h ij h kl S jk S li)](2.3)Where S npjk=∂n∂p F jk+2.2πα′h rs∂n F jr∂p F ks and S jk=h mn S jkmn,and h ij= (1det(g+2πα′(B+F))= det(1+2πα′NF)(2.4) Where N is defined in eq.(2.10).Let’s evaluate the above action in the Seiberg-Witten limit,where the Seiberg-Witten limit is defined in eq.(2.8).Using2πα′h|SW= (1+θF)−1θ,and keeping terms to order A3i,we get:S DBI|SW=−1det(g+2πα′B)[θijθklθmnθpq{∂n∂p F jk∂q∂m F li−12F rs∂n∂p F jk∂q∂m F li−1can also be derived from non-commutativity,as conjectured by Das et.al.The DBI action in the commutative and non-commutative theory are[7]:S DBI=T9P fθdetθ.Here,we are dealing with a space-filling brane,to avoid the appearance of scalars through pull-back.The Seiberg-Witten limit is defined as,α′∼√g s .Hence Eq.(2.6)can bewritten as1det(g+2πα′B)g+2πα′B )ij=θijG+2πα′Φ)ij(2.10)In the Seiberg-Witten limit,N ij|SW→θijg s det(1+θF)(2.11)Where as the corresponding non-commutative DBI action eq.(2.7)can be rewritten as:1det(1−θˆF) det(1+2πα′NˆF1g s L∗[Therefore,in order tofind the derivative corrections we have tofind the difference between eq.(2.13)and eq.(2.11).But its not easy tofind the difference,since one of the equation is written using non-commutative variables where as the other one is in commutative variables.In order tofind the difference we shall use the Seiberg-Witten map,to convert the non-commutative variables into its commutative form.Eq.(2.11)can be written up to O(A3),asS DBI|SW=1det(g+2πα′B)[1+14tr(θF)2+16tr(θF)3−148(tr(θF))3](2.14) Rearranging the terms,we get:S DBI|SW=1det(g+2πα′B)[1−θij∂i A j−θijθkl6(2∂j A k∂l A m∂n A i−2∂j A k∂l A m∂i A n−2∂k A j∂l A m∂n A i +6g s 2θijθkl∂j∂l ˆA i,ˆA k+1The non-commutative,and commutative DBI action to the order we are working in is:ˆSDBI|SW=1det(g+2πα′B)[1+θij∂j A i−θijθklg s2(∂j A k∂l A i−∂j A k∂i A l−∂i A j∂k A l)](2.17)and the difference between them is :ˆSDBI |SW −S DBI |SW =1det (g +2πα′B )[−θij θkl2F jk ,F li −14F ij ,F kl +1g s2{θmn θpq2∂m ∂p F ij ∂n ∂q F kl )+θmn θpq θrs θuv2∂m ∂p ∂r ∂u F ij ∂n ∂q ∂s ∂v F kl )}](2.19)The difference between these two actions at the next order,at A 3i is:ˆS DBI |SW −S DBI |SW =1det (g +2πα′B )[θij θkl θmn θpq θrs 2∂p ∂r F jk ∂q ∂s F lm F ni +18F ij ∂p ∂r F lm ∂q ∂s F nk −14F pm ∂r ∂n F lk ∂q ∂s F ji +12F pk ∂q ∂s F jm ∂l ∂r F ni −∂p F kj ∂r F ml ∂q ∂s F in }](2.20)We can see that the equation (2.5)is same as the corrections that we found from non-commutativity,namely,the sum of equation (2.19)and equation (2.20),up to 4-derivative in F 3.3Corrections to the Chern-Simon actionThe Chern-Simon action in two different descriptions,namely in the θ=0,Φ=B and Φ=−B,θ=1g snC(n )∧e 2πα′(B +F )(3.21)ˆSCS =1P fθnC(n )∧e2πα′Q −1W (x,C )](3.22)The Non-commutative Chern-Simon action is written in momentum space,butwhile calculating the difference between the above two action,we shall do so in6position space.To make comparison with the results of[9],we shall parametrise the Chern-Simon action with correction as follows:1S CS=2W4∧W4of ours.For completeness,we shall mention the W’s that we shall use to calculate the corrections,and they are:W4≡W48π2h ij h kl S jk∧S liW6≡W624π3h ij h kl h mn S lm∧S jk∧S niW8≡W864π4h ij h kl h mn h pq S np∧S lm∧S jk∧S qi(3.24) Where the S ij is a two form and defined as1S ij=The difference between the two actions(3.21)and(3.22)for this case gives rise to a topological identity,which is[8]:δ(k)= dxL∗[Onfinding the difference here,between(3.21)and(3.22)gives us the Seiberg-Witten map[4,8].F(k)= dxL∗[Interaction with C (6):2g s L ∗[P fQ2g sC (6)∧2πα′(B +F )∧2πα′(B +F )(3.28)The difference between these two actions,in position space,to order O (A 4)is:∆S CS ≡∆S CS8g sC (6)[ ˆFab ,ˆF cd +2θij ˆF ab ,ˆF ci ,ˆF jd +18ˆFij ,ˆF kl ,ˆF ab ,ˆF cd −12∂j ∂l ˆFab ,ˆF cd ,ˆA i ,ˆA k +2∂j ˆF ab ,ˆF ck ,ˆF ld ,ˆA i +18g sC (6)∧[ F ab ,F cd −F ab F cd−2θmn { F ab , A m ,∂n F cd + F ab , F cm ,F nd − F ab ,F cm ,F nd − F ab ,∂n F cd ,A m }]dx a ∧dx b ∧dx c ∧dx d (3.30)Where (in general)the expression f, g,h ∗2 ∗2is written as f, g,h and f,g,h ∗3as f,g,h to avoidclumsiness 3.On substituting the expression for ∗2,∗2within ∗2and for ∗3into the above equation,we get:∆S CS =124∂i ∂k F ab ∂j ∂l F cd −θij θkl θmn2F im ∂k ∂n F cd ∂j ∂l F ab +∂i F cm ∂k F nd ∂j ∂l F ab }]dx a ∧dx b ∧dx c ∧dx d (3.31)Let’s compare this with the result that will come from W 4in the Seiberg-Witten limit,i.e.the 4-form 4-derivative correction to the Chern-Simon action in the Seiberg-Witten limit is:W 4|SW =12The Seiberg-Witten map is derived in Appendix A.3The exact form of ∗2,∗3,and ∗2within ∗2,is derived in Appendix B.8On inclusion of tension as well as the integration,we can very easily see that both the above equations are same.Interaction with C (4):6g s L ∗[P fQ6g sC (4)∧2πα′(B +F )∧2πα′(B +F )∧2πα′(B +F )(3.33)In position space,the difference between the above two actions,to order O (A 4),is :∆S CS ≡∆S CS48g sC (4)∧{ ˆFab ,ˆF cd ,ˆF ef −F ab F cd F ef −148g sC (4)∧{ F ab ,F cd ,F ef −F ab F cd F ef −3θgh A g ,∂h F ab , F cd ,F ef+3θgh F ag ,F bh ,F cd ,F ef +148{ F ab ,F cd ,F ef −F ab F cd F ef −3θgh A g ,∂h F ab , F cd ,F ef+3θgh F ag ,F bh ,F cd ,F ef +14These expressions are written in Appendix B.9Interaction with C(2):4!g s L∗[P fQ4!g s C(2)∧2πα′(B+F)∧2πα′(B+F)∧2πα′(B+F)∧2πα′(B+F)(3.38) Using the Seiberg-Witten map,the difference between them,to order F4,becomes:∆S CS≡∆S CS4!gs C(2)∧{ F∧F∧F∧F −F∧F∧F∧F}(3.39) Using eq.(3.23)the corrections to the Chern-Simon action is parametrized as:112W4∧W4+14! F∧F∧F∧F −F∧F∧W4|SW−12∂i∂k F ab∂m∂p F cd∂j∂n F ef∂l∂q F gh dx a∧...∧dx h(3.42)5760It is easy to check that the W8of eq.(3.24)in the Seiberg-Witten limit reproducesthe above result with the same coefficient.4ConclusionWe have demonstrated the conjecture that the derivative corrections to the com-mutative theory can be found from non-commutativity in the Seiberg-Witten limit.It’s important to take this limit because in this limit all the corrections to the non-commutative action vanishes and left with the derivative corrections to the commu-tative theory.Also,to do the calculation at higher order infield strength we needto know the Seiberg-Witten map,i.e.the expression ofˆF ij in terms of F ij andˆA i in10terms of A i .Moreover,we shall explain the 4-form 4-derivative corrections to the Chern-Simon action at F 4.Let’s explain it in detail.The corrections to the Chern-Simon action at this order,fromeq.(3.24)is :θij θkl θmn θpq8g sC (6)∧θef θgh [− ∂g ∂e F cd ,A h ,A f ,F ab−2 ∂g F cd ,∂e A h .A f ,F ab − F cd ,∂e A h ,p g A f ,F ab −4 F cg ,∂e F dh ,A f ,F ab −12F ge ,F hf ,F ab ,F cd −2 A g ,∂h A e ,∂f F ab ,F cd − ∂f A g ,∂h A e ,F ab ,F cd + A g ,∂e A h ,∂f F ab ,F cd +14F fg ,F he ,F ab ,F cd + ∂f ∂h F ab ,F cd ,A g ,A e + ∂f F ab ,∂h F cd ,A g ,A e +2 F ab ,∂f F cd ,A g ,∂h A e +1192[4∂i F am ∂k F bn ∂j F cp ∂l F dq −4F pm ∂i F cn ∂k F dq ∂j ∂l F ab+8∂l F dq ∂j F cp F mi ∂k ∂n F ab −F qi F kn ∂j ∂l F ab ∂m ∂p F cd +2∂i ∂p F cd F mq F kn ∂j ∂l F ab −2∂i ∂k F ab A m ∂j ∂p A n ∂l ∂q F cd ]dx a ∧...∧dx d (4.45)11We can see that to order θ4,the result of the calculation from non-commutativity matches with that of eq.(4.43),i.e.we reproduced all the terms that appear in eq.(4.43),but along with these terms,we have an extra term in eq.(4.45),from non-commutativity,and this extra term vanishes due to symmetry arguments.AcknowledgementsWe would like to thank S.R.Das,J.Maharana,S.Mukhi,S.Mukherji,N.Surya-narayana and N.Wyllard for useful discussions and correspondence,and to Niclas Wyllard for pointing out an error in eq.(4.43).5Appendix AIn this section we shall derive the Seiberg-Witten map,namely,expressing ˆFij in terms of F ij and ˆAi in terms of A i ,by solving the equation(3.27)along with the expression of A i in terms of ˆAi [5].Moreover,it’s easy to check that the expression of ˆFij in terms of F ij is consistent with the form of ˆA i in terms of A i .Let’s expand the eq.(3.27)to order A 3,and writing in position space,we get the field strength as:F ab =ˆFab +θij {∂j ˆA i ,ˆF ab +12θij θkl {∂i ∂k ˆFab ,ˆA l ,ˆA j −∂k ˆF ij ,ˆF ab ,ˆA l +2∂k ˆF ai ,ˆF bj ,ˆA l }−θij θkl {18ˆFab ,ˆF kl ,ˆF ij −12θij ˆAi ,(∂j ˆA b +ˆF jb ) +12θcd θef [∂c ∂e F ab ,A d ,A f −∂e F cd ,F ab ,A f +2∂e F ac ,F bd ,A f ]+θcd θef [18F ab ,F cd ,F ef −12∂d A e ,∂c A f ,F ab +1andˆA b =A b −θij A i ,∂j A b +12θij θkl [−2 A i ,∂k A b ,∂j A l + A i ,∂k A b ,∂l A j + ∂k ∂i A b ,A j ,A l +2 ∂k A i ,∂b A j ,A l ]+θij θkl [ A i ,∂j A k ,∂l A b + A k ,∂l A i ,∂j A b −12A k ,∂i A l ,∂j A b−14A i ,∂b A k ,∂j A l −14A k ,∂i A l ,∂b A j −12)2f 1g 2|1=2(6.50)Where ∂1∧∂2=∂1i θij ∂2j ,and f 1=f (x 1).It’s infinitesimal form,up to 8-derivative is:f,g =fg −θij θkl1920∂i ∂k ∂m ∂p f∂j ∂l ∂n ∂q g −...(6.51)The form of ∗3is:f,g,h ={sin(∂2∧∂32)2∂1∧(∂2+∂3)2)sin(∂2∧(∂1+∂3)(∂1+∂2)∧∂32}f 1g 2h 3(6.52)The infinitesimal form of this,up to 8-derivative is:fgh −θij θkl1920(∂i ∂k ∂m ∂p f∂j ∂l ∂n ∂q gh +∂i ∂k ∂m ∂p fg∂j ∂l ∂n ∂q h+f∂i ∂k ∂m ∂p g∂j ∂l ∂n ∂q h )+1720(∂i ∂k ∂p f∂j ∂m ∂q g∂l ∂n h −∂i ∂k ∂p f∂j ∂m g∂n ∂l ∂q h+∂i ∂k f∂j ∂m ∂p g∂l ∂n ∂q h )}...(6.53)13The expression of the ∗4is:f,g,h,p =sin(∂1∧∂4(∂1+∂2+∂3)∧∂42)sin((∂1+∂3+∂4)∧∂2(∂1+∂2+∂4)∧∂32+sin(∂2∧∂32)2(∂1+∂4)∧(∂2+∂3)2)2(sin(∂1∧∂32)2((∂1+∂3)∧(∂2+∂4)2)sin(∂1∧(∂2+∂3+∂4)(∂1+∂2+∂4)∧∂32)+sin(∂3∧∂4(∂1+∂2+∂3)∧∂42)sin((∂1+∂3+∂4)∧∂2(∂1+∂2)∧(∂3+∂4)2+sin(∂2∧(∂3+∂4)2)2∂1∧(∂2+∂3+∂4)24{∂i ∂k f∂j ∂l ghp +∂i ∂k fg∂j ∂l hp +∂i ∂k fgh∂j ∂l p +f∂i ∂k g∂j ∂l hp+f∂i ∂k gh∂j ∂l p +fg∂i ∂k h∂j ∂l p }+θij θkl θmn θrs576{∂i ∂k ∂m ∂r f∂j ∂l g∂n ∂s p +∂i ∂k ∂m ∂r f∂j ∂l gh∂n ∂s p +∂i ∂k f∂j ∂l ∂n ∂s g∂m ∂r hp +∂i ∂k f∂j ∂l ∂n ∂s gh∂m ∂r p +∂i ∂k f∂j ∂l g∂m ∂r h∂n ∂s p +∂i ∂k ∂m ∂r fg∂j ∂l h∂n ∂s p +∂i ∂k f∂m ∂r g∂j ∂l ∂n ∂s hp +∂i ∂k f∂m ∂r g∂j ∂l h∂n ∂s p +∂i ∂k fg∂j ∂l ∂n ∂s h∂m ∂r p +∂i ∂k f∂m ∂r g∂n ∂s h∂j ∂l p +∂i ∂k f∂m ∂r gh∂j ∂l ∂n ∂s p +∂i ∂k fg∂m ∂r h∂j ∂l ∂n ∂s p +f∂i ∂k ∂m ∂r g∂j ∂l h∂n ∂s p +f∂i ∂k g∂j ∂l ∂n ∂s h∂m ∂r p +f∂i ∂k g∂m ∂r h∂j ∂l ∂n ∂s p }+θij θkl θmn θrs 720{∂i ∂k f∂m ∂r g∂j ∂n h∂l ∂s p−∂i ∂k f∂j ∂m g∂n ∂r h∂l ∂s p +∂i ∂k f∂j ∂m g∂l ∂r h∂n ∂s p }(6.55)From now onwards,we shall write down only the infinitesimal form of ∗n within ∗m for n ≤m .The infinitesimal form of f,g ,h ,up to 4-derivative is: f,g ,h =fgh −1+2(∂i f)(∂k g)(∂j∂l h)]+...(6.56) The infinitesimal form of f,g,h ,Q ,up to4-derivative is:θijθklfghQ−24{∂i∂k f∂j∂l ghQ+∂i∂k fg∂j∂l hQ+∂i∂k fgh∂j∂l Q+f∂i∂k g∂j∂l hQ+f∂i∂k gh∂j∂l Q+fg∂i∂k h∂j∂l Q+2∂i∂k f∂j g∂l hQ+2∂i∂k f∂j gh∂l Q+2∂i∂k fg∂j h∂l Q+2f∂i g∂k h∂j∂l Q}(6.58) The infinitesimal form of f,g , 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