Hep2001 Is Derived C M for the Exactly C M , Can Be Constructed for C M 6= 0, Where These Ideas Can Be B † =0And B Arsen@Pas.Rochester.Edu , and A

Home , Ashok Das

hep2001 is derived c M for the exactly c M , can be constructed for c M 6= 0, where these ideas can be B † =0and B [email protected] , and A. Melikyan Workshop on Integrable Field Theories, Solitons and Duality. ‡ In this talk, we show how the monodromy matrix, , J. Maharana ∗† [email protected] [email protected] PROCEEDINGS the two dimensional tree levelusing string effective its action. factorizability The property.non-trivially pole structure under It of the is non-compact showninvariant T-duality and that this group, can the which be monodromy leavesbackgrounds used matrix the to starting construct effective transforms from the action monodromy simpler matrix ones. for more complicated We construct, explicitly, Abstract: solvable Nappi-Witten model, both when directly checked. Speaker. ∗ University of Rochester, Rochester, NYE-mail: 14627-0171, USA. E-mail: Bhubaneswar -751005, India Institute of Physics Department of Physics and Astronomy † 1. Introduction The field theories in twothe space-time past dimensions few have decades. attractedtheories, considerable capture They attention several possess over salient a characteristicssuch variety of two four of dimensional dimensional interesting modelsturbative theories features. are properties and, used of therefore, Some as field of theories theoreticalThere these are laboratories. are much field simpler classes Moreover, to ofstructure: the study two in nonper- integrable two dimensional dimensional models theories models. which [1] are and endowed conformal with field a theories rich [2, symmetry 3] belong to this special ‡ Ashok Das Monodromy, Duality and Integrability ofDimensional Two String Effective Action hep2001 . ) ,R (1) (2 U SL + 16). For Ashok Das d, d ( O dimensions. The reduced d − –2– = 6, namely, in the case of reduction to four space-time dimensions, the field d Chandrasekhar and Xanthopoulos [22], in their seminal work, have shown that a vi- The string theories are abundantly rich in their symmetry content. The tree level -dimensional torus, from 10-dimensional space-time to 10 d Thus, the four dimensional theory possessessymmetries. the T-duality When as the well string asencounter the effective an S-duality enhancement action group in of is symmetry reducedThe as has string to been effective two studied space-time action byof dimensions describes several authors such we supergravity [7, theories theories 8, and 9, havemention 10]. the been that integrability investigated higher properties in dimensionaldimensional the theories, Einstein has recent theory, been past dimensionally studiedthe in [12, reduced the monodromy 13]. to past matrix [14]. effective Ittheories. which One two is of encodes An the worth effective approaches some while is twoaspects of to dimensional to of derive the action black naturally essential holeical appears features physics, models. when of colliding one integrable plane considershas Recently, waves field some connection we as with have well integrable as shownconstruct systems the special that from monodromy types the a matrix of for newwas two such cosmolog- perspective shown, dimensional theories while in with string investigating the wellodromy the effective defined sense collision matrix prescriptions that action of [15]. can one plane be It can [16]. fronted constructed stringy Subsequently, we explicitly were waves, for able that to aunder the give general given the mon- settings. procedures set for of It deriving isblack the background monodromy worth hole configurations matrix while physics to can mention bethere that described some is by of an an the effective interestingdimensional intimate two aspects space-time dimensional of relation with theory between two [18]. commuting colliding Moreover, Killing plane vectors. wavesolently and time the dependent descriptionof space-time of plane with four colliding a gravitationaldemonstrated pair waves. that of the Furthermore, Killing colliding Ferrari,ric plane vectors Ibanez wave to provides metric and the a can Bruni interiorconstructed be description [23] identified, of the have locally, a transformation to Schwarzschild which becolliding metric. provides isomet- waves and a In one connectionfrom corresponding another between the to a important study the metric step, of Schwarzschild describing collidingof black Yurtsever strings, plane hole. [24] there gravitational As is waves a is [25]ture curvature well corresponding singularity, singularity has known, to found in an the massless alternative future. states descriptionscenario in Recently, [26, the the context 27]. appearance of of Pre-Big Theuua the Bang incoming (PBB) fu- of plane the waves Universe, aresolutions to which correspond be collide to identified and Kasner as lead the type to initial metric the stringy and creation vac- the of exponents the fulfill Universe. the requirements Indeed, of the the case strength of the twoFurthermore, form the anti-symmetric tensor dilaton can and be axion traded can for be the pseudoscalar combined axion. to parameterize the coset theory is known to be invariant under the non-compact T-duality group, Workshop on Integrable Field Theories, Solitons and Duality. category among others. Underdimensional special theory circumstances, may in also the be presence described of by isometries, anstring a effective effective two four action, dimensional dimensionally theory. larged reduced symmetries to [4, lower 5, dimensions, 6].a is Let known us to consider, possess toroidal en- compactification of heterotic string on hep2001 2) , (2 O already c Ashok Das M , namely, the structure follow from the definition IV ). Therefore, it opens up c M , we recapitulate the form of d, d ( II O -dimensional string effective action . It was shown that the monodromy d D , for the string theoretic two dimen- T c M –3– , under a T-duality transformation, are derived once the -dimensional torus, d c M . V transforms under the group, whereas, if the underlying symmetry cor- for simple background configurations which still preserve some of the c c M M In this lecture, we review the works reported in [15, 16, 17]. We provide prescriptions When we focus our attention to address these problems in the frame work of string to the construction of the monodromy matrix for the problem under consideration. captures the stringy symmetryplicit in forms an of elegant manner. In this section,of we the also Nappi-Witten present model ex- inconclusion the in present Section context is analyzed in detail. We present a brief general features. We present an illustrative example in Section matrix transforms non-trivially under the duality group transformation from the initial backgrounds [28].for We the can construct new the set monodromy of matrix odromy backgrounds matrix using can our be prescriptions. constructedby On directly us. the by Indeed, utilizing other the we hand,two transformation explicitly, the demonstrated rules different new discovered that mon- routes the are monodromysymmetries matrices identical. of obtained the via It effectiveodromy the action is matrix play obvious and an from its importantTherefore, transformation the role properties it preceding in is under natural discussions the those toof symmetry construction that expect the transformations. of the intimate two the connections dimensional mon- between theory the and integrability the properties full stringyfor symmetry groups the of construction T and of S the dualities. monodromy matrix, and construction of thiscan matrix. derive how For example, if the action respects T-duality, thenthe one two dimensional action obtainedThen, by we present toroidal the compactification equationsodromy from of matrix, higher motion. is dimensions. the AIII coset key ingredient, space in reformulation the of derivationThe the of transformation reduced the rules action. mon- for Wematrix devote Section is constructed. An interesting observation is that the expression for sional effective action.general arguments. We The duality present transformation the properties of pole structure of the monodromyresponds to matrix S-duality from an appropriatebe transformation derived. rule for Our monodromy paper matrix is can organized also as follows: In Section the possibility of studyingtheories. integrable systems As which an might example,to appear an we in exact considered the conformal the context fieldodromy Nappi-Witten of theory matrix described [19] string for by model, the a which case WZWwell is model. when known that a the We an 2-form solution first anti-symmetric anti-symmetric obtained tensor the tensor background mon- is can set be to generated zero. through an It is also Workshop on Integrable Field Theories, Solitons and Duality. PBB conditions. Adimensional very effective action; important however, fact, thedimensional physical theory. in process this is effectively context, described is by a that two onetheory, it starts is from essential to a keepintegral in four part mind of the the special stringy symmetries,odromy symmetries. such matrix as We under have dualities, T-duality investigated which transformation [15] arewhen of the an the the behavior backgrounds two of under the dimensional a mon- general action setting is derived from a through compactification on a hep2001 ˆ µ ˆ ρ ˆ B (2.3) (2.2) (2.1) (2.4) (2.5) ˆ ν ∂ +), and Abelian -field in + , ˆ )andthe ν ˆ d B ˆ µ Ashok Das ˆ 2, then, the ··· B d, d , ˆ ρ ( − ∂ + ,whichcanal- O , + D ˆ ρ αβ − ) transformations . symmetric matrix ˆ ν ˆ ν = B , ˆ ˆ µ B d d ˆ d, d M ˆ 2 µ B ( ˆ ρ α ∂ is the two-dimensional ˆ ν × ∂ O ˆ µ 1 = d . Later, we will comment ˆ − ˆ 1 H αβ ρ ˆ ˆ ν ρ g x Abelian gauge fields coming ˆ ˆ ν µ M ,where ˆ µ d α ˆ d is a 2 H ˆ ∂ T H and ! 1 M 0 12 B x Tr 1 ¯ ij φ − 8 1 B − 2 G 1 -dimensions consisting of the graviton, → + − ˆ φ D BG 2 ¯ φ G ˆ ∂ Ω (2.6) ¯ − − φ , ∂ M log det 2). Finally, G + T 2 1 αβ ˆ 1 − G –4– is the dilaton and + Ω g 1 − − R ˆ − φ , and another set of D R ˆ is the corresponding two dimensional scalar curva- ν -dimensional torus, → → φ G ˆ µ = d BG ˆ φ ¯ ˆ , as a result of dimensional reduction [20] . Further- R φ = G − d ˆ αβ ν M

− . ˆ ¯ g µ φ ˆ ˆ Ge 1( = ge B αβ g − − − M √ q 1 x D dx =det d 0 , ..., D g 3 , Z dx = Z corresponds to a symmetric representation of the group =2 ˆ -dimensional metric in the string frame with signature ( S is the scalar curvature, = . Since we are in two space-time dimensions, we have dropped the gauge D M ˆ G i, j 1 are the two-dimensional space-time indices, S ˆ , αβ , R i B . x ˆ ν is the metric in the internal space, corresponding to the toroidally compactified =0 is the ˆ µ represents the moduli coming from the dimensional reduction of the ˆ ˆ ν G ij ˆ µ B G ˆ G α, β If we compactify this action on a The matrix =det -dimensions. In general, there will be additional terms in (2.2) associated with ˆ where ways be removed, if it depends only on the coordinates Here Here, space-time metric with dimensionally reduced action in (2.2) is invariant under the global on the gauge fields, whichin assume the a original significant string role, effective when action Abelian (2.1). gauge fields are present resulting dimensionally reduced actionaction, would which describe has the the two form dimensional [20, string 21], effective D where gauge fields arising from the metric, from the anti-symmetric tensor, more, there would also have beentensor terms field, involving the field strength of the two dimensional ture, while the shifted dilaton is defined as field terms and, in the same spirit, have not kept the field strength of coordinates G the dilaton and the anti-symmetric tensor field, is the field strength for the second-rank anti-symmetric tensor field, In this section, wedimensions, which will will form briefly the basis recapitulate forsimplicity, all the the our tree form subsequent discussions. level of Let string us the effective consider, string action for in effective action in two Workshop on Integrable Field Theories, Solitons and Duality. 2. Two dimensional effective action of the form hep2001 . η is a (2.7) ,the (2.13) M M Ω= ) metric η T Ashok Das d, d ( O =0=0 (2.8) (2.9) M M β β ) (2.12) ∂ ∂ 1 = 0 (2.14) = 0 (2.15) 1 1 =0 = 0 (2.16) =0 x ¯ − − φ − − M M e M 0 M M M needs some care since α α − + x ∂ ∂ − − + ∂ ( = 0 (2.11) ∂ ∂ ∂ ∂ 1 2 + M 1 1 1 1 − ∂ √ − − − Tr Tr M M αβ 8 1 β M M M = ¯ η φ αβ ∂ ) ! + − + g − + 1 − − . A simple method, for example, would ∂ ∂ ∂ d e = 0 (2.10) 8 1 0 − ¯ η φ ,x 1 ¯ φ β + + dimensions. Namely, Ω satisfies Ω d M − − x 0 = Tr Tr Tr ( D ¯ e 1 ∂ φ d α αβ F α β 8 8 8 1 1 1 –5– ,x e

+ ) D D gg ¯ 1 φ∂ + + + α = = − + x α ¯ ¯ φ φ , leads respectively to D F ∂ η MηM M √ − + + αβ − ¯ αβ − φ αβ g αβ 0 ∂ g ∂ − R g x F∂ F∂ 1 + e ( ∂ − + − − 2 ∂ ∂ 1 α ¯ − φ M √ ∂ ¯ β − − φ ¯ φ − ¯ ¯ = D φ φ − e α ∂ 2 2 + − + ∂ ∂ + D x + ∂ ∂ αβ g ) matrix satisfying , and the metric, ) is the global transformation matrix, which preserves the +2 ¯ φ R d, d d, d ( ( O O ∈ representing the identity matrix in d has the form of a conservation law 1 The equations of motion for the different fields follow from the dimensionally reduced We can further simplify these equations by working in the light-cone coordinates M In this case, eqs. (2.10) and (2.11) take respectively the forms effective action (2.2).shifted For dilaton, example, varying the effective action (2.2) with respect to the It follows from these that and choosing the conformal gauge for the two dimensional metric, namely, while eqs. (2.8) and (2.9) can be written explicitly as The variation of the effective action with respect to symmetric with Workshop on Integrable Field Theories, Solitons and Duality. where Ω involve adding the constraint tocase, the since effective action there through isEuler-Lagrange a no equation Lagrange potential multiplier. of term motion Into following in any from the the effective variation action of involving the the action with matrix respect hep2001 . ) d ij ( G O (2.20) (2.21) (2.17) = × ij ) ) d Ashok Das ( matrix (see E T O ) (2.23) x M E so that we can ( h α α ) transformations. Q ∂ d ) ( − x O ( 1 = ) transformations. Con- × − d ) T α h ( , which belongs to the Lie d ( O ,Q V O ) (2.22) )+ α × α β x T T P ) ∂ ( ) ) P d 1 ) rotation. h ( ) transformation α = V V − α d transforms like a gauge field, while P ), α α O ( Ω (2.19) V T d, d x ∂ ∂ Q α α α ( ( ) 1 1 O ! P M ) (2.18) Q x − − Q O ( T T x × 4Tr ( 0 Vh ( 1 V V + ) Ω E ( − − d α that 1 h ( Vh +( − P 1 = − −→ ) T O − −→ d V V = ( ) Ω V E α α −→ T . Correspondingly, it is convenient to introduce O –6– BE ), M ∂ ∂ ) V d,d × α d 1 1 ( d β ) α (

) ( d ∂ − − −→ O VV ∂ ( O 1 1 O d,d O V V = × − V ( − = ) ) rotation, the currents in (2.21) are invariant, while × d O V ( V 2 2 1 1 M )aswellasthelocal ) = 0), the action in (2.2) simply corresponds to a flat ,oftheform ) transformation, O M ,Q α d ) ) d d, d ( ¯ ∂ d )andalocal = = φ ( ( ( x d, d ) is the vielbein in the internal space so that ( , ( O ( O O O α α h d,d × d, d αβ O E ∈ P ( Q Tr ) α × ( η d ) O P ) ( O ) x ) transformation, d O ( = ( x d , we can construct the current ( h ( ∈ O 1 V O αβ − . Here, V g h T × ) )and ) and can be decomposed as d matrix is sensitive only to a global ( VV −→ O d, d α d, d = ( M ( P O O belongs to the Lie algebra of the maximally compact subgroup M ∈ belongs to the complement. Furthermore, it follows from the symmetric space α Q α P It is now straightforward to check that It is well known [20] that the moduli appearing in the definition of the From the matrix This brings out, naturally, the connection between the system under study and two transforms as belonging to the adjoint representation. It is clear, therefore, that (2.22) α Under a combined global Here, identify Namely, under a local algebra of Furthermore, under a global under a local P such that Namely, the where Ω a triangular matrix and automorphism property of the coset eq. (2.4)) parameterize the coset is invariant under the global Workshop on Integrable Field Theories, Solitons and Duality. dimensional integrable systems. First,dilaton let (namely, us if note that, in the absence of gravity and the sequently, the action in (2.2) is also invariant under the local hep2001 .In (2.27) (2.25) (2.26) (2.24) (2.28) αβ η Abelian ¯ φ , coupled e n ) Ashok Das d ( ) = O d,d × ( ) αβ d g O ( O Iαj . ˆ λ F J ˆ ρ ˆ I F , associated with the second αj ˆ ν F I ˆ µ H perms ˆ I i F . I a +2 i I IJ α (3) δ αβ ∂ ˆ λ (1) α (3) α I i I j F ˆ Iαβ ν ˆ I µ I i a a +cyc A A ˆ i g ˆ j a F A β β ˆ ρ − ˆ ν , which can be analyzed through a zero ˆ s ∂ ∂ µ βγ (1) α (1) I − ) αβ αβ ∂ I j ˆ g d F A − − F a F ( ) I j − i rs α I O (2) ˆ φ αβi a + η ∂ ˆ I ν d,d ¯ × φ − F ( I i ) r α (1) β (3) β ˆ I I i A − − d a O ( ˆ A A a A µ ˆ + –7– Ge I α (3) I i I α αβ O ge α α α ∂ 2 1 j ˆ ˆ ˆ 2 1 − A G A ∂ ∂ F ∂ − = − (1) αβ + q √ ======ˆ ν 1 F x I ˆ µ i βγ ij I I I I i I ˆ ij , needs to be redefined for gauge invariance as D αi I F a αβ dx B B d C (1) F 0 B (1) α (3) α (3) αβ α α F αβ ∂ − ∂ F A A F Z dx 4 1 = = = Z − 4 1 αij αβi αβγ = − H H ˆ H A and = ˆ S A S , ..., n 2 , -dimensional action in (2.1) involving the graviton, the dilaton and the anti- =1 D I,J So far, we have only discussed the two dimensional string effective action starting this case, the action will describe a sigma model, defined over the coset This action can alsoeffective action be takes dimensionally the reduced form [20] to two dimensions and the resulting to gravity. As wea will zero show curvature in condition the muchspectral next like parameter, section, the in this flat this system space case, can case, to also although be consistency be space-time requires analyzedfrom dependent. the through the symmetric tensor field. However, the action in (2.1) can be generalized by adding gauge fields, with the additionalstring action theory) of the form (such terms naturally arise in heterotic where we have defined where curvature condition with athe constant spectral next section). parameter (todilaton In be from the discussed the presence in action of more (2.2) gravity detail by as in choosing well the as particular the conformal dilaton, gauge we can eliminate the rank anti-symmetric tensor field, In the presence of the Abelian gauge fields, the field strength, Workshop on Integrable Field Theories, Solitons and Duality. space sigma model defined over the coset hep2001 ). ), × × V ) x n α ) αβγ ( d ∂ d ( H + ( 1 (2.35) (2.29) (2.30) (2.32) (2.33) − O Vh O V d, d → ( Ashok Das O V ,where ) (2.36) is not sensitive η x ( ) transformation, h M α n Ω= ∂ ) + η )andalocal x T n (    1 + d, d T − ( a T h O a T + 2) belonging to d, d 1 I ( a − T ) transformation, )+ 1 − (3) α O a x n − 1 ( A aG D G    − h + I i α a − d T G = ) satisfying Ω ( α 2 1 a and Q T 1+ n d ) O ) (2.34) Q    r α T + x x + Ω (2.31) 00 01 ( n × ( A j + E 0 (2) αi 1 1 CC ) ). However, the matrix β M a α − 1 d A n ∂ T (1) α d d, d Vh T ( h P − 1 ij β T ) and which can be decomposed as 00 ( + C 1 A 1 T + ∂ Ω − − n O T 1 B G O − = 1 ij d d Ω → C − r β T E 0 00 ( − + − 1 –8– 1 )matrix( + B ∈ T α G V A C − O aE n I j −→ E α α    C (2) βi + − a −→ − + ∂ ∂ × d, d + I i − A aG 1 ) ( M = αi a ). As in the earlier case, it is more convenient to α V d G − of the form = d ˆ 2 1 ( O n    η B ∂ ( 1 ) V 1 ,Q O r n × αβ − + = ) − 1 = = = ) + x n F d − ∈ d ( V ( ij + aG ) CG G (2) αi h ), O (2) d, d C αβ i ) transformations. We can now define the current x I α − ( d,d − A × ( ( F n ) P h O d (3) α ) O    ( + x O ( ,A = d belongs to the Lie algebra of the maximal compact subgroup 1 ( ∈ − α (2) αi . As before, under a combined global )and O h M Q V T n belongs to the complement. Under a global ) transformation ,A × i n + α ) is a symmetric −→ P VV + d (1) α ( α d, d A = M ( O P d, d are invariant, while under a local O ( =( M α O ∈ r α ) transformation Q ), while n A n + Once again, it is easy to see that, in two space-time dimensions, the field strength + and d d ( ( α which belongs to the Lie algebra of In the present case, and all the discussion for the earlier case can again be carried through. In this case, represents the metric for introduce a matrix such that where the parameter of transformation Ω where Ω O P can be set tostring zero. effective action Furthermore, can keeping be all shown to other have terms, the the same form complete as two in dimensional (2.2) with Workshop on Integrable Field Theories, Solitons and Duality. where Under a to the local O hep2001 2. = − (3.7) (3.1) .The ,H ) D =0)= n = + G/H d Ashok Das x, t ( is a constant d, d ˆ V ( t as O where G = ) d G ( ) )] = 0 (3.8) )] = 0 (3.2) O )where d,d ˆ × V V ( ) β β β d O x, t ( ∂ ∂ )and ( P 1 1 O d ˆ V − − ( =0 (3.3) αβ ˆ O V V ]=0 α ( ( β 2 × , , P t ) ) ) ] (3.4) t β ,P d belongs to the complement. The β ˆ 2 − V V ( α D α α α 1 α P O ,P ∂ ∂ − P Q α 1 1 + = ) = 0 (3.5) β − − Q + α ]+[ P ˆ V V M =0 (3.6) β α α P β +[ ,H P . Then, as we have noted in the last section, β 2 2 ) D ∂ β t t ,Q T P 1 . In this section, we will further analyze the , while = α P ) α − )+[( )+[( − d, d α n –9– Q H ( ) ˆ D V V V ∂ 1 1+ VV + M n d α belonging to the Lie algebra of α α ( O ( + αβ +[ ∂ = ∂ ∂ = α + O η 1 1 1 V = ∂ α β α d,d × − − − α ( ) M P Q αβ Q d G ∂ ˆ O V V V ( β α 1 η ( ( ∂ O = − β β D and ∂ ∂ V − ˆ V β α − − ∂ ) ) Q 1 ˆ α Abelian gauge fields are present in the starting string action, V V G/H − ∂ β β ˆ n V ∈ ∂ ∂ 1 1 − − V ˆ V V ( ( α α ∂ ∂ -dimensions, has a natural description of a sigma model defined on a coset, ). Let D n + belongs to the Lie algebra of d ( α O Q × Let us next introduce a one parameter family of matrices Let us consider a general sigma model in flat space-time, defined on the coset )and ) x d ( ( Then, it is straightforward to check that the integrability condition where the coset can be identified with On the other hand, if leads naturally to eqs.equation (3.3),(3.6). of Namely, motion the forcondition integrability the condition associated flat (3.3) with space as a sigma well potential model as which the depends are on obtained a from constant the spectral zero parameter. curvature Explicitly, this equation gives where we have defined The equations of motion for the flat space sigma model (see eq. (2.11)) integrability condition, following from this, corresponds to the zero curvature condition can be rewritten in the form V parameter (and not time), also known as the spectral parameter, such that we can decompose the current two cases of interest for us are when integrability properties of suchwith a the system system. and construct the monodromy matrix associated In the last section,reduced we from saw that thecoupled two dimensional to string gravity. effectivestarting action, string In dimensionally action, the the sigma case model when is defined there on are the coset no Abelian gauge fields present in the Workshop on Integrable Field Theories, Solitons and Duality. 3. Monodromy Matrix O hep2001 (3.13) (3.14) (3.11) (3.10) ) uniquely, Ashok Das x, t ( ˆ V ) (3.15) T x, t ( ˆ ) V t 1 α ∂ ) = 0 (3.9) )=0 ) x, α ( − − ) 1 M ) (3.12) ρ ρ t P 1 x, t β − ¯ − ) (3.16) ( φ ∂ 1 − − t ˆ x 1 − V + 1 ( − e t ω ω − ( − ( ˆ x, V α ¯ ρ ( φ √ √ ¯ φ M − T − ¯ D φ − − + e ˆ e V )) = − = )+ = 0 (3.17) ) e + + t 1 + ( ρ ρ T )= β x α β )= ∂ x, x, t M ( ∂ + + –10– ( ( ) P x α + ¯ ( t αβ φ 1 ˆ ˆ ∂ ω ω V V ρ αβ ρ ( − η √ √ 2 1 e η x, = ( ( − , eq. (2.11) takes the form )= α ) is a constant matrix, yield the same one parameter ˆ V x ω M D )= ( α αβ = ( )) = ρ x η ∂ t S ( ) αβ ¯ φ α t t e 1 x, t ∂ ( = x, ,η ˆ V ( ( 1 or αβ − ∞ g ,where η ˆ V ˆ V ) ω ( S and ˆ is the constant of integration, which can be thought of as a global spectral param- V ω There are several things to note from our discussion so far. First of all, the one pa- In the presence of gravity, however, the equation for the sigma model modifies [12, 13]. eter. It is clear that the solutions inrameter eq. family of (3.13) connections are (currents) double does valued not in determine nature. the potential where It can be shown, following from this, that family of connections.space Second, automorphism in can the be presence generalized of as the spectral parameter, the symmetric we note that the solutionas following from the equation for the shifted dilaton can be written namely, Given these, let us define As before, we canparameter introduce and a with one a parametercheck decomposition family that of of the the potentials zero form depending curvaturewell (3.7). condition on as in the However, a (3.8) in integrability spectral leads condition thisand provided to case, satisfies the the it spectral correct is parameter dynamical easy is equation to space-time as dependent With this, the solution to eq. (3.10) can be written as In the conformal gauge,simple equation. as Therefore, we defining have seen earlier in (2.14), the shifted dilaton satisfies a It follows now, from eq. (3.15), that In the conformal gauge Workshop on Integrable Field Theories, Solitons and Duality. hep2001 ! (3.21) (3.18) (3.24) (3.22) (3.19) (3.23) E α =0,this in (3.20) ∂ 0 1 t Ashok Das ± − P E )isknownas − ω P E ( α t t =0.Inthiscase, ∂ M 1 − 0 B − 1 1+ ) is diagonal, it follows ! E ! . Clearly, for x = ρ i ) − 0 ( E ) ω d ). Let us assume that ln ,d

1 d ψ x, t =0) =0) − ± 0 ( + B B = = 0 (3.20) V 1 ∂ ··· ( ( λ E are diagonal, as is relevant in , ( , − i i ˆ V 2 1 V ) t t 2

T T G , − ··· ) ) d ,e ± ∓ ∂ , = ψ =0) 1 2 1 1 + B =1 =0) =0) − ( ··· V λ ) B B , , =0) = ( ( V ) 1 =0)= ,e 2 i B ( V V t =0) ψ V )0 α α B ± + ··· ( ∂ ∂ x, t λ , ,i ( 1 1 ( ˆ 2 1 x, t V 2 1 − − ( ψ ( − i ) ) + =0) 0( ,e E ,V λ ) and, therefore, B –11– =0) . Furthermore, noting the form of i 1 ( =0) =0) ,e B ! ρ, ∂ i ψ + ˆ ( B B E V 1 t , d ( ( + , as adopted in [27]. =0) 0 t ψ G ln V λ + ( − V V B + G ( P 1 ± 2 1 − λ t

(( ∂ − t t e e t 0 V t t G +(( − i − . . i )= + ± ∓ t 1+ 1

d t log det 1 1 =0) =0) x, t = 1 d = B B ( ( ( = = i = V V t =diag =diag =0) =0) =0) α α V ± λ B B B ∂ ∂ ∂ ( G E ( ( 1 1 ˆ ˆ ) and is independent of space-time coordinates. − − V V M ) ) ω + . The other case can be studied in a completely analogous manner. To ( ) ∂ d 1 =0) =0) ( ) a diagonal matrix of the form ( ) M − B B O ) ( ( d,d × are diagonal matrices and ( x, t ) = V V ( d ( ( O =0sothat ± =0) ( i B P O ( =0) M is the spectral parameter corresponding to the constant ψ 2 2 1 1 ˆ B i V i ( ( t = = P V Since In this case, it follows that (see (2.21)) Let us next describe how the monodromy matrix is constructed for such systems. For α α P Q and recalling that the spectral parameters satisfy leads to the diagonal elements of Let us further assume that the matrix that we can write with so that we have with where start with, let us set the anti-symmetric tensor ﬁeld to zero, namely, we can write Workshop on Integrable Field Theories, Solitons and Duality. Namely, the monodromy matrix for thesuch system as under the study conserved and quantities encodes associated properties with of the integrability simplicity, system. we will considerthe the coset action in (2.2), which describes a sigma model deﬁned on the study of colliding plane waves. Namely, let us parameterize hep2001 . i E (3.29) (3.31) (3.25) (3.26) (3.27) (3.28) (3.30) (3.32) Ashok Das 1 − ± i ! -duality rotation. P E ) t t T ω ln ( 1 ± ∓ ± − 1 1 ∂ t t i M = i + i t ± ∓ )0 d i + t ! 1 1 =0. ω d ω − i ( 0 ω − E i t + B = − t d t ± M − t ∂ -duality transformations. In par- i ω 0 − 1 6=0througha i ω

i + − T t 1 t − d i + t 1 t − t B d E t i − )= ω ω i + 2 − t 1 E - duality, should be encoded in the mon- t )= d r − + − i t ± t 1 i i T x, − ∂ E = ( ω ω 1 − ln t 0 x, T 1 t − = ( ± ) = 0, it is possible, in some models (such as the i − i i ∂ E i i )= t V + i E + –12– B t − =0) ) t d d i ω t t t B ± ( i +

i t d ˆ 1 x, t − V t t = t ( i ,inwhichcase,wehave − M )( i i simple poles - one pair for every diagonal element ∓ ± ∓ − t i V + ω 1 t d 1 1 d ω x, t − i − have opposite signatures following from the double valued i ω ( i − = i + E )= t = d − + t ω ω =0) d i ln ( ω t B i =0) + ( ± )has2 = d B ∂ ˆ ( i V ω M ˆ and x, t V V = = ( i ± i t ∂ V =0) 1 =0) ± B − B ( ∂ ( 1 ˆ V i − c M =0) V B ) is diagonal with ( ω ˆ V ( M Thus, we see that, in the present case, Let us note next that we are dealing with a sigma model, obtained through dimen- Furthermore, it is simple to check from eq. ( 3.13) that the spectral parameters satisfy so that we can determine the monodromy matrix to be of the form We note that the doubleeters valued allows relation us between to the choose global and the local spectral param- provided we identify ( In that case, we can write and the matrix where ticular, starting from a backgroundNappi-Witten where model), to generate backgrounds with odromy matrix as well.(of For example, the it string is theory) known starting that from one given can ones generate through new backgrounds This determines the monodromy matrix for the casesional when reduction of asymmetries higher present in dimensional the tree string theory, level such string as eﬀective action. Therefore, the nature of the solutions in (3.13)) Workshop on Integrable Field Theories, Solitons and Duality. it is easy to verify that hep2001 ) ) ). x and ( d, d d, d h ( ( (3.36) (3.35) (3.34) ) Z O O − x, t ( = Ashok Das ˆ V T under an in- → = 0), it follows )Ω (3.33) c ) M Y,Z ω ( − x, t x, t ( c M ( ) is only sensitive to = ˆ V T ˆ ω V T ( T T Z Z Y X c )= M ) rotation. Since the one 22 X 12 12 x ( 22 c c M )Ω = Ω c d, d M M V t ( 1 c M + + − O x, − X ( 12 22 T 22 ! 21 c c ! ˆ M M V c c is only sensitive to the global ) ) M M W Z Z 12 22 much like T ) transformation, X + − − x, t x, t c c d M M ) ( ( 1+ ( − d 21 11 ( 12 ˆ ˆ ) V V VV O 11 21 O 12 T T c XY c c M d,d M M × = × c c ( Ω Ω ) Z M M c T T d M X –13– O ) ( d 1+

X X Y O M ( − matrix. Inﬁnitesimally, we can write [20] −→ −→

+ = O + − ∈ 11 d ) ) ) Y Y X t 1 c c × M M 11 21 11 Ω= x, t d ) transforms non-trivially under a combined global x, ( Y x, t ( ( x c c c ˆ M M − M V ( T ˆ V ˆ V V -duality transformation, within the context of string theory = = = = ) T ) rotation 12 21 22 11 x, t ) transformation. In a similar manner, ( c c c c d, d M M M M d ˆ ( V ( δ δ δ δ O O )= × ) transformation. Let us denote ) rotation. We will check this explicitly in the case of the Nappi-Witten -duality properties, which can be used as a powerful tool in determining ω ) ( d T ( d, d d, d c ( M ( O O O . Under such an infinitesimal transformation, it follows from (3.35) that T X − For completeness, we record here the transformation properties of Letusnotethatthe = W Let us also recall that, under a local where the infinitesimal parameters of the transformation satisfy where each element represents a transformations even though and a local the global solutions. model in the nexting section. connection between For the theaction integrability moment, properties and let of its us the note two dimensional that string this effective brings out an interest- finitesimal that under a global 4. Application The ideas presentedFor in example, the if earlier we are section considering can collision be of applied plane-fronted waves, to which various correspond physical to systems. Workshop on Integrable Field Theories, Solitons and Duality. It is natural, therefore,transformations, for, to then, examine we how canbackgrounds starting determine the from the monodromy simpler monodromy matrix ones. matrix for transforms more under complicated such (without Abelian gauge fields),parameter corresponds family to of matrices a global Therefore, the only local transformationsodromy which matrix will are preserve the the onesWe global that have nature do of already not the seen mon- depend that on the the local matrix spectral parameter explicitly. hep2001 (4.4) (4.3) (4.5) (4.1) (4.2) ) and, ) 2 y,z Ashok Das xdz 2) symmetry. 2 , x (2 sin 2 O 1 τ 2 tan 2 ! )= 0 2 +4sin 0 ξ ψ 2 xdz 1 2 0 ! − ξ , the string tension, there are 0 0 xdy α

λ =0) 2 ! 0 B = +tan ( 2 ξ 2 cos G ! 2 1 ξ ξ 0 τ ﬁeld. In this case, therefore, we have 2 1 dy .) = exp( 2 (0) − τ 1+ τ B ψ 2 ) . 2 − ) 2 0 = 1 ξ x 0 1 =0) 0 (0) ξ ξ 0 (4 cos 2 λ tan x B ) ( e x ) 1+ x ,ξ –14– + sin 2 G x ( τ 2 (0)

τ 2 ψ cos 2

1 dx + cos 2 ) = 0 τ cos 2 x = τ tan 1 ) (0) + − λ B 2 ( e τ =0) cos 2 G log(sin 2 )= cos 2 dτ cos 2

B 0 ( − τ − − ψ − = 1 M (cos 2 = = + (1 + =0 cos 2 0 2 φ B 2 λ = − ds G b dx = + 2 log(1 21 = exp( 2) rotation. In the language of our earlier discussion, we note that we can Wess-Zumino-Witten model and describes a closed expanding universe in 1 2 B dτ 1 , ξ field, of the form − − − (2 (2) (1) B O = = = U SU 2 φ 12 × ds B ) 1) , Note that the backgrounds do not depend on two of the coordinates, namely, ( ,R (1 (2 SO SL where the superscript “0” denotes the vanishing with On the other hand, it is easy to check that we can write several solutions to thebackgrounds. string One of equations these solutions, following studied by from Nappi and (2.1) Witten, corresponds that to a constitute gauged exact CFT Here, we have set anto arbitrary zero constant for parameter appearing simplicity. in the Nappi-Witten solution consequently, following from our earlier discussions, the system has an It is known that these backgroundsa can vanishing be obtained from a much simpler background, with through an write 3+1 dimensions. The backgrounds consist oftensor the metric, fields the of dilaton and the the forms anti-symmetric (Here, we identify Workshop on Integrable Field Theories, Solitons and Duality. massless states ofdescribed closed effectively strings, by because acarried of two over the [16]. dimensional isometries theorywhere In in our and this results the all can section, problem, be our we explicitly this earlier will verified and can discussions4.1 discuss prove be can The one quite useful. Nappi-Witten be other Model class ofThe physical Nappi-Witten phenomena modelthe [19] string is theory. an Let example us of note that, a to cosmological leading solution order following in from hep2001       ) 2 ξ 2 (4.13) 0 ξ ± ∂ ( ) 1 (4.8) (4.6) (4.7) ξ 1 (4.14) (4.12) (4.11) (4.10) 00 ξ ± 2anti- ∂ ( × ) Ashok Das 2 ξ 2 ξ ± 000 ∂ ( ), have to satisfy − is the 2 ) 1 x, t ξ 1 ( ξ ± 000 00 0 ∂ ( =0) ! =0) B − ( B B ( ± ˆ V       1 P − ! ) = t t B ) 1 B ± ∓ ! − ( =0) ) 1 1 0 ) G B ! b ( ( =0) B Ω (4.9) 2 ( B E B ξ = ( 0 )= G ! 1 =0) ( √ − E ! − B 2 2 ) − 1 ± ( x, t ) ξ ξ I ( ξ I B ∂ 1 1 0 2 identity matrix while ( 1 ξ ξ 10 M 0 √ 0 =0)

01 − G T × =0) − ) B − 2

( B 1 1 ( 1+ 1 1

–15– − √ E = =0) ˆ ) − =Ω V ( − ) ) 2) rotation. = B ) ) ± ( , B

( ∂ B B = =0) ( E ( ) (2 Ω= G B = ( ( G O B M ( x, t E B ( =0) =0) 1

B B − ( ( ) = E V represents the 2 ) ± =0) I B ∂ B ( 1 ( 0( − ˆ M V ) ( =0) 2 anti-symmetric matrix B ( × 2). Here, E , ( =0) (2 ﬁeld through a global − O

=0) B B is the 2 ( ± is the space-time dependent spectral parameter. = =0 t Q It follows from eq. (4.3) that we can write =0) =0) B B ( ± ( ± P Q where (since It is now a simple matter to check that and it follows that (see eq. (3.20)) so that we have where [28] where so that belongs to In this case, therefore, the one parameter family of potentials, Workshop on Integrable Field Theories, Solitons and Duality. symmetric matrix deﬁned inanti-symmetric (4.7). tensor ﬁeld Thisvanishing shows can that be the generated backgrounds from with a a nontrivial much simpler background with a hep2001 (4.20) (4.15) (4.16) (4.19) (4.18) (4.22) (4.17) ) 4 2 t t Ashok Das − − t t 2 4 ! t t 1 ! −      − 1 ) q 4 2 , ) − ω ω 2 3 1 ) 1 t t B 0 ) ξ − − ( ) − − ω ω B (2) and is of the form t t B ) ( E ( = 1 3 O t t B 3 1 )( E ( E 2 ± ω ω × 00 )( − B ± − P − − ) ∂ ± ω ω t t B q 1 , ∂ (2) ± 2 4 0 ( − 2 4 =0) ± ∓ t t ∂ ! ) 1 ω ω O − − B ( ) 1 1 000 ) (4.21) t t   − ( 2 − − 1 B ) b x bI ∈ 4 2 ω ω ! 2 ( ) − t t E ( + ξ ) ) ) 2 B 1 h E − ) ) − 1 3 ( ξ ξ x B ) ( ω ω 2 ( 0 B ( B 1+ 1 0 =( q x 000 00 0 ( ± ( E , − − h 1+ ( ( E √ 1 3 √ 2 4 ω ω      Q E t t 1 2 t t ( 1 − − q ω ω t t 1 1 2 =0) − − b      bI + − 3 1 ) − 2 − 0 B t t )= 2 2 ξ ) 1 1 ( ) 1 − = 1 ) . ω ω − − − B ξ ξ V 2 ( ) ) 1+ B 2 1 0 x, t ( ) ) q T ! ω ω ω ) ( E 1+ ( √ √ B B ) –16– ) , ( + − E B 00 . − ( ( ( 1 1 ( 1 B ω q

1 B ( ω ω ( ξ E E E = 1 ˆ 2 V )( )( )=Ω   ± 4 1 ω ω − = ∂ ± x √ B B − + 000 ( 1 2 ∂ = = 2 2 ,ω ) M ± ± ) 0 − 1 ω ω − ) ) )=diag B ∂ ∂ ) ) )0 ( ω ) ( ( 4 )= B B ω ω ( ( 1 1 ω x, t B x V − V ( ( ( 0 − + − − ( =0) 000 00 0 E V , 1 1 1 ) ) h B E = ) ) ω ω − ( 1 M ( ) 6= 0 and the one parameter family of potentials has to satisfy B B 3 ··· ) ( ( E − ,     

ω ± B 1 E E ( ( ( Q = = =( ˆ V 1 1 2 2 V ( ( 1 3 . 2) matrix deﬁned in (4.10) and t t

, =0) − B = = (2 ( ) in (4.15) does indeed satisfy (4.14). The monodromy matrix, in this case, ) O ± c B M ( 6= 0, we can similarly obtain ± Q x, t )=diag ( P B x, t =0) ( B ( =0) ˆ V When Following our earlier construction (see (3.28)), we can determine B ( ˆ V It is straightforward to check that where It can be checked explicitly,that using the equation satisfied by the spectral parameter, (3.24), In this case, therefore, It follows now that where Ω is the where we have identified follows to be Workshop on Integrable Field Theories, Solitons and Duality. hep2001 Ω =0) - (4.25) B σ ( c M (4.24) T Ashok Das )Ω = Ω t 1 ) in (4.17). This      ω 4 4 x, -model arises from ( ( σ V V T M ) α α 0 0 − + =0)      1 1 B 1 1 ( V V − 2 − 2 ˆ V β β ) (4.23) )( 3 M 3 x x 0 0 ( = 0 through a combined global V ( + −M V h α ) is not in general and that both T ) 1 1 α B h + ) x, t + M M x 2 x, t ( ( 2 ( ) 1 V h B 1 V − 1 ) ( β − 1 =0) β ˆ V − B M ( x, t M 3 ( ˆ 3 V + V + T 2 V β =0) 2 –17– β B ( + 00 00 M + ˆ 2 V −M 2 )=Ω T V 1 1 V are the two diagonal elements of α 1 1 − − , satisfies the defining relation in (4.20). We note here x, t α b ω ω ( − ) M M − + ) is triangular, 4 4 )=Ω 2 2 B 00 00 ( x ω ω t + + 1 1+ V V ( ˆ (2) transformation. Furthermore, as was pointed out earlier, ) V 2 2 β β √ = x, B O ( ( 0 0 + − 2 M M = T V × 1 1 ) 1 1 ) β M V V 2 2 − − B ( α α (2) M ˆ V O and and      0 0 )( + −M 2) rotation), the corresponding monodromy matrices are also related in 1 2 b ω ω , 1 1 are related to their counterparts with − + − (2 x, t 1 1 = ) ( 6= 0, while ω ω M M 1 ) O B ( B √      ( B = ˆ V ˆ 1 2 1 V = α M = = and ) 2) and a local One of our principal objective was to take into account the symmetries associated with ) , B ( B ( (2 c where where V shows explicitly that, for backgroundsexample, related an by a duality transformation (in the present a simple manner, as was pointed out in the last section. that,when O the local transformationquite does crucial, not for depend it immediately explicitly leads on to theM spectral parameter. This is In this talk, wematrix have for described two the dimensionalfollowed prescriptions string in effective for the action. the construction construction We of of adopted the the the monodromy procedure monodromy matrix commonly for a class of two dimensional 5. Summary and Discussion Workshop on Integrable Field Theories, Solitons and Duality. It can also be checked that models in curved space. Asthe mentioned dimensional earlier, reduction in of most higher ofspace-time dimensional the Einstein-Hilbert due cases, action the to to twoapproach the dimensional was presence adopted of in isometries.with dimensionally the reduced In past models the to in context construct gravity. of the string monodromythe string theory, matrix effective a action as and similar was constructabout the these the monodromy symmetries. case matrix We which have containsof succeeded information in the introducing a monodromy procedure matrix forfactorizability the and under construction presence general of grounds isolatedmonodromy poles. with matrix transforms Furthermore, some non-trivially we under mild have thethe demonstrated requirements non-compact that two T-duality such the group dimensional as when string effective action respects that symmetry. We feel, this is an hep2001 Ashok Das 6= 0. Then, as a B as an application of our meth- (1994) 77, hep-th/9401139. following our rules of the trans- IV =0 C244 B c M from B –18– c . This is a good testing ground for duality trans- M B (1994) 374, hep-th/9402016. B428 One of us (AD) would like to thank the organizers of the workshop under duality. Indeed, it is found that the monodromy matrix computed c M = 0. Subsequently, we have also constructed it for the case B Models in (1 + 1)-dimensional Quantum Field Theory, Les Houches Lectures,Field 1982. Theory, World Scientific, Singapore(1991). Polchinski, String Theory, Cambridge University Press. It is worth while to mention that all our results are derived for the case of classical We have discussed an illustrative example in Section [1] A. Das, Integrable Models, World Scientific, Singapore (1989); L. D.[2] Faddeev, Integrable E. Abdalla and M. C. B. Abdalla, Nonperturbative[3] Methods in M. Two-dimensional B. Quantum Green, J. H. Schwarz and E. Witten,[4] String A. theory, Giveon, Cambridge M. University Press; Porrati and J. [5] E. A. Rabinovici, Sen, Phys. Rep. Developments in Superstring[6] Theory, hep-th/9810044. J. Maharana, Recent Developments in[7] String Theory, I. hep-th/9911200. Bakas, Nucl. Phys. on integrable models forGrant the No. kind hospitality. DE-FG 02-91ER40685. This work is supported in part by US DOE formations of using the two different ways,the mentioned above, black coincide. hole Although solutions weNamely, have in the not heterotic monodromy discussed, matrix stringtheory for theory the can can ‘seed’ be also Schwarzschild constructed be blackthis, and hole described since that in in for the heterotic charged this T-duality stringalready black transformations way known. hole [17]. which solutions generate can charged be black obtained hole from solutionstwo are dimensional effective theoryrived as from is higher the dimensionalsystematically case Einstein-Hilbert the for action. construction effective It oftheory. two might the We dimensional be hope monodromy theories the interesting matrix work de- one to and presented encounters explore its here effective will properties two find dimensional in applications models quantum in in diverse the directions context where ofAcknowledgment: string theory. consistency check, we have derived the References formation properties of thecase monodromy matrix. We have constructed this matrix for the Workshop on Integrable Field Theories, Solitons and Duality. interesting and important result.monodromy The matrix, procedure, once adopted a by set us,if of allows we string know us background monodromy configurations to matrix are construct for known.directly the a As obtain given a set the result, of corresponding simplebackgrounds, string monodromy vacuum if matrix backgrounds, we for can the anotherbackgrounds. latter set of can more be complicated derived by dualityods. transformations We from considered the theand Nappi-Witten simpler non-vanishing model two form which potential is exactly solvable for both vanishing hep2001 (1996), Ashok Das (1995) 427, A11 (2000) 3599, B454 17 (1985) 223. (1992), hep-th/911105. (1993) 51, hep-th/9209052. D45 A398 (1988) 1706. B296 (2001) 306, hep-th/0107229. (2002) 126001. D38 (1987) 215; F. J. Ernst, A. Garcia and (1978) 985; H. Nicolai, Schladming (1987) 1053. 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