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JHEP10(2019)153 Springer July 26, 2019 : October 3, 2019 October 11, 2019 : : Received Accepted Published Published for SISSA by https://doi.org/10.1007/JHEP10(2019)153 . 3 1907.05662 The Authors. c D-branes, Bosonic Strings

We construct new type of non-relativistic D-branes which are defined with the , [email protected] Department of 611 Theoretical 37, Physics Brno, andKotláˇrská2, Czech Astrophysics, Republic Faculty of Science, Masaryk University, E-mail: Open Access Article funded by SCOAP Keywords: ArXiv ePrint: Abstract: help of T-duality alongof null these direction. D-branes and We study find their Lagrangian properties and under Hamiltonian T-duality formulation transformations. J. Klusoˇn direction Non-relativistic D-brane from T-duality along null JHEP10(2019)153 ] and 8 expansion of c ], one can ask ] that could be 21 18 – 7 8 ]. Further, NC geometry 6 – ], the non-relativistic string 5 3 4 9 ] geometry and its torsionful 1 8 ]. Very recently the relation between these ]. 6 ]) since NC geometry is very useful tool for 15 11 , 3 – 1 – 7 , 2 10 1 11 ] on torsional NC geometry with extra periodic dimension. There ]. 25 12 ] where independently an action for non-relativistic string in flat background We also find its Hamiltonian formulation when we identify diffeomorphism 20 1 , 19 In this paper we focus on the first approach that defines non-relativistic string theories For earlier work, see [ 6.1 Special case: D1-brane 5.1 T-duality of D-brane in canonical formalism 1 way. More precisely, wepresume consider that Dp-brane Dp-brane in wraps the thiswe direction. background derive with Then non-relativistic null applying D(p-1)-brane T-duality isometrydimension. in along and T-dual this we geometry direction thatand is Hamiltonian localized constraints. along T-dual based on the null reduction.arises More from precisely, T-duality as along was null shownanother direction. in Since extended [ it objects is wellthe in known question that their whether string it spectrum, theory is contain as possible to for define example non-relativistic D-branes Dp-branes exactly [ in the same was proposed. Itstrings. is The first interesting one is that basedPolyakov on like there the action null are reduction [ that currently leadsis to two an Nambu-Goto versions action alternative [ definition of ofaction non-relativistic non-relativistic for strings string that intwo stringy is theories NC based was geometry on found [ large in interesting paper [ torsionful NC geometry, whichthe is boundary characterized geometry by in non-exactcan the clock be context form of also was Lifschitz appliedUV observed geometry for as [ formulation completition of of non-relativisticpapers non-relativistic string [ theories gravity. [ All there works are based on two seminal 1 Introduction and summary There was a renewedgeneralization interest in (TNC) the (see Newton forthe Cartan example study (NC) [ [ of non-relativistic field theories, string theories and geometry. More precisely, 6 T-duality along null direction 3 Hamiltonian formalism for non-relativistic4 D-brane T-duality along longitudinal direction 5 Non-relativistic D-branes from Hamiltonian formalism Contents 1 Introduction and summary 2 Lagrangian formulation of non-relativistic D-brane JHEP10(2019)153 2 ]. Then we apply the same procedure 24 , 23 – 2 – ]. 22 This paper can be extended in many directions. First important problem is to analyse Let us outline our results and suggest possible extension of this work. We define We also show an alternative way how to define Hamiltonian for non-relativistic D(p-1)- As the next step in our analysis we will study properties of this non-relativistic D(p-1)- For review, see for example [ 2 NSNS two forms andbranes along Ramond-Ramond this fields. direction. Finally,similar It it way would would and be be study niceof also to tachyon these analyze condensation nice non-BPS problems in to Dp-branes in this in future. study context. the super We D- hope to return to some with null isometry andWe then should we expect perform thatconditions T-duality Neumann that transformations corresponds boundary along to conditionspicture null D-brane transform should direction. localized be to in elaborated Dirichletwould the in boundary be T-dual more direction. nice details to Clearly and generalize such it this a is construction currently to under more study. general Further, background it with non-zero D1-brane that has similar structure asseen corresponding in non-relativistic the string Hamiltonian which is formulation clearly of both theories. whether these non-relativistic D-branes could beboundary described using conditions. open strings Explicitly, with we Dirichlet start with open relativistic string in the background determine corresponding Lagrangian density. new non-relativistic D(p-1)-branes inT-duality TNC geometry transformations. and we We studyunder find their T-duality that properties transformation. non-relativistic under D-brane We transforms also covariantly discuss an explicit example of non-relativistic to the case offor Dp-brane D(p-1)-brane in in the NC background backgroundindependently with that in null the coincides isometry section with devoted and to theWe we Lagrangian Hamiltonian also formulation derive that of Hamiltonian analyze non-relativistic was D-brane. derived anflux explicit which example corresponds of to the non-relativistic number D1-brane of with fundamental the strings fixed bounded electric to D1-brane. We world-volume of Dp-brane for solving spatialof diffeomorphism constraint since the in the gauge absence localises world-volume D(p-1)-brane in field dual dimension. itD(p-1)-brane We would show that in not the resulting T-dual Hamiltonian be describes dilaton background possible that are with given to by the famous introduce Buscher’s rules background coordinate [ metric, that NSNS two form and We firstly find Hamiltonianhow for relativistic T-duality Dp-brane can inperform be general T-duality defined background along and in we compactthat the show relates dimension canonical embedding we coordinate formalism. havetogether with to with the one introduce More world-volume spatial diffeomorphism one. gauge precisely, constraintexplicitly are This fixing in solved. second gauge function class order It fixing constraints turns that function to out can that be it is crucial that there is a gauge field propagating on the brane when it wraps another compactlocalized direction. along We find this that compact T-dualto object direction is obey. on D(p-2)-brane certain condition that the background fieldbrane has in torsional NC geometry which is based on the canonical description of T-duality. JHEP10(2019)153 9 α A β (2.6) (2.3) (2.4) (2.5) (2.1) (2.2) ∂ ,..., − ] where 1 , 8 β , A µ α = 0 ∂ ] that such a dx are pullbacks , µ = 10 αβ m αβ αβ , b F M,N = F 2 s . ] l αβ , we review canonical 8 ν g + N 5 m x , m β αβ µν µ b ∂ h we find non-relativistic D- , M + dx − x 2 µ we apply this analysis to the µν α τ µ αβ ∂ h v 6 g = is string length. coordinate = = we find its Hamiltonian form. In = s p MN . l µ ξ which is crucial for the definition of 3 ρ µ µν B ˆ v αβ δ , τ α -direction where the background posses = A ν u A = , ρ dx αβ p , ν v µ ,G ξ µ µ m τ αβ dx ˆ v = , b – 3 – µν A − − µν u h N obey the relation h µ x νρ = µ det β h m v ∂ ) + uµ − 1 2 µν M h m p + x and φ MN α − µ − ∂ v µν ,B µ ξe du ,G is Dp-brane tension and h ( MN m τ +1 label world-volume of Dp-brane. Further, MN p +1 − 1 G p s d l G = 2 = 2Φ = . Then it is natural to perform T-duality along this direction so Z = N Φ = 8. The inverse metric has the form p uu αβ p + , . . . , p T g T G dx 1 u − , M ,..., = → 1 = 0. We implicitly consider superstring theory so that dx = 0 , S u ) parameterize embedding of Dp-brane in target space-time. µν MN ξ h = 0 g ( , α, β we study T-duality of non-relativistic D-brane. In section -direction so that we have static gauge along M α = 4 u ξ x µ, ν 2 Our goal is to define non-relativistic D(p-1)-brane as T-dual of relativistic Dp-brane This paper is organized as follows. In the next section ds Now we presume that Dp-brane isan extended isometry along that we will have D(p-1)-brane localizedwraps along dual direction. More precisely, let Dp-brane Note that the inverse metric where det while an action for non-relativisticstring string corresponds was to proposed. the Thenthat string the it that null was is background shown T-dual is in to defined [ the by string following in background the metric null [ background. Note where is field strength ofDp-brane. the Finally, world-volume gauge field in the background with null isometry. This approach is based on seminal paper [ where of the background fields 2 Lagrangian formulation ofIn non-relativistic this D-brane section westarting formulate point is non-relativistic an D-brane action in for the Dp-brane in Lagrangian general formalism. background that The has the form brane in the Lagrangiansection formalism. Then indescription section of T-duality ofdefinition D-brane of and the finally canonical in form section of non-relativistic D-brane in TNC background. JHEP10(2019)153 1. . − 0 (2.7) (2.8) (2.9) ˆ β 0 (2.13) (2.11) (2.12) (2.10) ˆ α so that F 0 ˆ ˆ β , . . . , p α ˆ ˆ β 1 β , 0 ˆ A 2 ˆ α / , ˆ α 1 ) = 0 into definition of η η ˆ α ˆ β 01 β u ∂ ∂ F ˆ 1 α + τ

2 s ητ ˆ β l 0 ˆ = τ βp ∂ ( 2 ˆ s A β ˆ l βp + 2 , ˆ ˆ . α β 2 0 ˆ ˆ β α A ˆ ˆ − β ˆ αp A ˆ α 0 ˆ β ! ˆ − ) A ˆ α 00 ˆ α F = 0 A η ˆ α ¯ 2 h s ¯ h 01 ˆ ˆ τσ p 0 α l A 2 . 1 σσ ˆ pp β ∂ ˆ ˆ α τ A ˆ A 1 β + ˆ p ηF A A − 1 − − − 0 ν − p A ∂ ˆ α ˆ x α ˆ T 0 β ˆ α ˆ , τ 0 , 00 β στ τ ˆ q α ( 0 φ σσ p ∂ ˆ ¯ h 2 p s β ˆ β ˆ l ˆ A 2 − µ A ˆ α τ ˆ A q A A e η

x is equal to ˆ ˆ ˆ ˆ α − α ˆ β β 1 A ˆ α τ ˆ α ∂ ∂ + 2 ∂ = = ∂

2 s 2 s αβ ˆ l l − A

ˆ 01 + β det denote remaining coordinates ˆ – 4 – µν 0 A du ˆ ¯ h α ˆ + − β ¯ h ˆ ˆ αp 0 1 β has following form 11 q 0 ˆ det µ ˆ µ τ A α 1 Z φ ¯ h ˆ A = τ 0 β is non-singular matrix with inverse x ¯ h α, 2 0 p − ˆ τ q 0 µ α ˆ β ˆ ˆ 0 β ˆ τ β ˆ β T β ˆ x β det ∂ ˆ ˆ φ ˆ α p ˆ α ξe A β φ ˆ ˆ α α µ 1 − − + 2 ˆ ˆ − 0 ∂ τ A A A = ˆ − A e ˆ α . Let ˆ

p ξe 11 01 . We further included integration over ˆ ˆ p β α = = = 1 d ¯ h p ˆ ¯ ξ h α = − 2 ˆ ˆ β β η η A p ˆ ˆ p η αp ˆ = 1 we can rewrite this action into more elegant form Z ˆ A α 1 β 2 s d 0 l ] which is nice consistency check of our proposal. αβ A A ∂ A 8 01 η∂ η∂ du Z A = 0 ˆ α φ , 1 ∂ ∂ η Z − ˆ has the form 2 − α ˆ p p det β q together to T-dual dilaton as ξe − T T as φ 2 αβ 1 − d − − − η − A p ξe = = = Z ) is our proposal for non-relativistic D(p-1)-brane. Clearly it is invariant T 2 ˆ 1 β d . Then the determinant det S ˆ α T ˆ 2.9 ˆ γ α δ Z − 1 = = T ˆ γ − S ˆ β ˆ = A In this case the matrix inverse ˆ β S ˆ α ˆ We see that the expression underaction square that root was has found similar structure in as [ non-relativistic string introducing so that the action has the form The action ( under world-volume diffeomorphism and it isto also manifestly give covariant. an It explicit is example also instructive of D1-brane. where we defined T-dual tension and hence an action has the form It is naturalA to presume that Then the matrix and all fields do not depend on JHEP10(2019)153 with (3.7) (3.4) (3.5) (3.6) (3.1) (3.2) (3.3) αβ , A A , 2 ) det A − , 2 2 √ ) ) det 0 , p A S A . − 2 , , ) ) H ˆ ˆ γ α , √ i . ˆ ˆ , α µ , δ ( det A ˆ det 2 π τ β 2 0 0 Φ ) 0 φ ) ˆ = = 0 = 0 β − S ρ − H − A A τ ≈ ˆ ˆ A γ γ det η i ˆ √ H e √ p τ A . ˆ − ( ) α ∂ 1 ) ( − ˆ 0 β 0 ∂ 0 ˆ 0 − β ˆ ˆ det = 2 αp β A ˆ τ S det ˆ β p = α A ˆ α √ ηH A ˆ L γ ( T A ≈ Φ − H 0 ˆ H α − µρ 0 ∂ H ˆ ˆ α ∂ β j ∂ ˆ p H ηH ( S ¯ h √ + ( τ + τ √ ˆ ( π β − L µ 1 2 ( ∂ + j H ˆ 0 ˆ γ + Φ ˆ 0 v i i ˆ ˆ A ˆ 0 ˆ β 0 β A p ˆ A η∂ γ ˆ + 2 ˆ α F = α ˆ A A = α ˆ ¯ h A ˆ β α A τ H 0 ˆ H β ˆ ˆ β , β 0 ˆ ∂ ˆ α + det β ∂ 0 ˆ α ˆ H ˆ α α A i ˆ γ A α ˆ ˆ S ˆ ηH ω α α ηH ˆ A η ¯ β h 0ˆ A τ ˆ − π α ˆ ¯ ( h ˆ α H α H 0 ∂ α Φ ∂ H , ηp √ F , π ˆ ˆ + α 0ˆ α i φ ˆ S A + µ φ 0 s Φ τ 2 l 2 ∂ τ ˆ ,H η α ,H A S – 5 – η A H H φ φ − 0 − ) 2 2 ˆ γ φ φ e + + e H ∂ ˆ α 2 2 ∂ − − 1 1 + 2 ν ˆ η ˆ β γ µ det e e − − = 0 det − p p − x S ˆ p 2 1 1 αp e e 2 S p p ˆ H ˆ α − α µ 1 1 µω − − − T H T + 2 2 τ ∂ H p p x ¯ h − − ˆ √ α 4 + i ˆ 2 2 ˆ √ α p p µ T T 0 τ p µν 0 ∂ . Then it is easy to find following conjugate momenta A − ˆ β p T T − x i = A ˆ + 2 A ¯ h ˆ 0 α ˆ γ + H αβ ≡ φ H ˆ H ∂ β = = = 2 = = 2 δω ˆ γ H φ i − φ ˆ µ i + F ˆ ˆ j j H ˆ ˆ ( ν β ¯ h µ S 1 primary constraints e p − − ˆ β π H p π p 1 1 2 η e e i ˆ ˆ νδ α H S ν µ ηπ ˆ − 1 α − 1 τ = ˆ β µν h j ˆ ˆ p v x η ∂ − = − H p ˆ i α ˆ j ˆ ∂ h p T ˆ p p B α µν ˆ ¯ h ∂ µ β 2 p T S Φ T ¯ ˆ h ηπ i p α H ˆ S 2 − = s s i ˆ µν l l H = ∂ H ) ) 2 ¯ h ηπ 2 i ˆ i µ ) ˆ µ − s ∂ x η A x l L L i 4 0 0 0 ˆ L 2 ∂ ∂ ∂ ∂ ∂ ∂ − s ∂ i ˆ l ( ( ( − 2 π ∂ ∂ ∂ ) implies that there should exist one Hamiltonian constraint which also follows 4 − s ). It turns out that it is much simpler to consider gauge fixed matrix = = = l 3.3 i ˆ η µ 2.9 π p p where we used If we take all termscancel given each above other. together we In firstly order find to that deal expressions with proportional remaining to terms Φ we use following expressions from the fact thatinvariant. In action order for to non-relativistic find such D(p-1)-brane a is constraint manifestly let us diffeomorphism calculate following expressions and also we can identify Further, ( where With the help of these results we obtain that the bare Hamiltonian density is equal to In this sectionaction we ( find Hamiltonianits form inverse that of we the denote as non-relativistic D-brane given by the 3 Hamiltonian formalism for non-relativistic D-brane JHEP10(2019)153 − αβ + = , η j ˆ

i A ˆ (4.3) (4.1) (4.2) (3.8)

∂ ηπ 1) ( 1) j ˆ j ˆ − i ˆ − ∂ 1) p a p − ( Φ p i p 0 0 ˆ 1)( ( − ¯ α det A p ηπ p ( i ˆ φ A 2 ∂ 1) . In case when A 1) 2 − − µ p − − e p s p p ( l 1) 1) 1 µ − − 1) − 1)( A v 2 p p p − ( ( − + 2 1) p p p T A ( ( i ˆ − ¯ α p A 2. Then the matrix A π + . and ˆ i p ˆ 1 A 1)( τ ν − − 1) η η − p p    1) p − p p ¯ ( p αp − p 1) p 2 µν we obtain p ( 1) A − 1) − s h A − p l 1)( ¯ A αp . µ − ( p − 0 αβ ( j p ˆ − , . . . , p p i 1) ( ˆ p A A ¯ A 1 β ( A − + 2 F , 1) p = A A 1) 2 s µ − l − τ 1 1)( p p − 1) p ( , − ( = 0 µ + H p − p 1) 1) ˆ v ¯ 0 ( αp p j ˆ ¯ i A . ˆ α i − A − ˆ A p p 1 A ¯ h ≈ 1) 1)( ( ( 1) ¯ β, ηπ − − ¯ − p α − ) i that multiplies − ˆ , p p = p j ˆ p ∂ ¯ ( ξ β A i ¯ A ˆ τ α 2 1 j ˆ 1)( 1 1) 1)( i ξ ˆ π 1) – 6 – A = − s − = − − + a l p −

p p ( ¯ . The fact that D(p-1)-brane wraps this direction 2 p ( β ( y η ( 8 A p 1) j ˆ ¯ A 1) ¯ β β x − ∂ − A ¯ p ¯ α 1) − β 1) 1) A ( p j ˆ p j − ˆ 1) − . Further, there is no term proportional to momentum − ( 1) = A 1) A i ˆ π i p ˆ p p − 1)( ( − ( − ( τ ν a p y A p p ¯ p ¯ α ) α − x ( i − ˆ p 1 j 1)( ¯ ˆ 1 β ( τ 1)( A    A A A − ∂ − 1) = p A p ( − − = − ( − i µν ˆ ¯ β ¯ p A p β η ∂ A ¯ h ( ¯ α i p ˆ 1) ¯ µ αβ β 1) ∂ A ¯ αp A x − or A ( − 0 1) i A ˆ p j ˆ p i ˆ ( i ˆ − A

( ∂ ]. It is certainly interesting question to analyse similar situation in τ i p ˆ ¯ a α ( A 1) π = 4 A − ¯ A 24 β p , − det s η − ¯ l 1) ¯ β − i β ˆ φ ¯ β ¯ p 1)( ¯ 23 α − β 2 ¯ . These reasons suggest that non-relativistic D(p-1)-brane is well deﬁned α + p p , ∂ − ( − A η A ). This is positive quantity that cannot be equal to zero except exceptional p ν e ( A j ˆ p A A p 1 τ A

= 0 − = 0. Using these terms together we derive after some calculations following 2 µν p is matrix inverse to the matrix + µ = = = h T j ˆ η p µ i pp ˆ j ˆ p αβ + a ∂ A ( A = j ˆ In this section we found Hamiltonian for non-relativistic D(p-1)-brane. The crucial i ˆ were zero we would get Hamiltonian constraint τ a ) i det ˆ i ˆ H has the form Now using properties of the determinant of the matrix additional compact direction, say means that we impose the static gauge along this direction Further, all world-volume fields depend on 4 T-duality along longitudinalAn direction important property of relativisticbrane Dp-brane maps is its covariantly behaviour toBuscher’s under D(p-1)-brane rules T-duality where when [ Dp- thecase background of fields non-relativistic are D(p-1)-brane. determined To by do this let us presume that D(p-1)-brane wraps conjugate to in case of non-zerochecked electric by flux analysis only. of T-duality Certainlyhope along this null to is direction return very in to interesting case this fact of problem and open in should relativistic future. be string. We fact is in theπ presence of theτ momentum cases as primary Hamiltonian constraint where since JHEP10(2019)153 = ˜ y ˜ 00 y (4.6) (4.7) (4.8) (4.9) (4.4) (4.5) (4.10) h , , ¯ = β , ¯ α y ¯ ¯ β µ τ F ˜ ¯ 00 y . α 2 s = 0. This is h l F y , = 2 s τ yy y ) l y + τ ¯ ¯ β 1) h τ ¯ τ ν + − p x ¯ ν ( + ¯ β ¯ ν p x y η ¯ β . Now we should dis- ˜ A ¯ y∂ ¯ h β 1 ¯ α ∂ ˜ y∂ − y yy ( , ∂ ¯ p α τ ¯ ¯ β ν ˜ ¯ h y , ∂ ˜ 0 ¯ y . , A α ¯ η , β ¯ ν , 2 s ¯ β ¯ ∂ ˜ 00 − ν ˆ B y l = 0 A

yy yy y ∂ 1 , ) ¯ ν B ¯ µ − ¯ ¯ ¯ h h h + ¯ η τ α = ˜ ¯ 00 y y µ − τ ¯ ¯ α ν ˜ τ τ y + ¯ h y ¯ = = ¯ µ ∂ x β ˜ y − ¯ 0 x β ∂ ˜ ¯ y ν = = = ¯ β ¯ + ˜ ¯ β ˜ 0 ∂ 0 y η y µ ∂ ˜ ¯ ¯ y ν ν , ¯ ν ¯ ∂ α ¯ h ¯ x α ˜ ˜ ¯ 00 00 00 y y η µ y ¯ y µ + ¯ ∂ τ α direction. However since ¯ h x ¯ τ τ h ( η ∂ 6= 0. Then we ﬁnd that the determinant ¯ α ¯ η , ˜ β y y ¯ ∂ q y = = = β ¯ 0 µ ¯ ν ¯ ˜ y ¯ β τ β ∂ y η∂ ¯ ¯ p β β ¯ 00 ¯ B µ 0 α ¯ α ¯ , ¯ α h p 1) 1) − ˆ B and ˜ ˆ ∂ ¯ A µ + A − A − yy ¯ y 1 y β p p ¯ η β + ( τ τ – 7 – ¯ h ¯ h ( ] for components of T-dual metric in absence of yy ¯ α y

¯ ν ¯ y ν ¯ det h A τ A ¯ F µy τ τ x 24 2 s yy ¯ ¯ h − β , ,B l so that physical interpretation of this configuration + ¯ h − − q ¯ y 23 ν η∂ + ,B ˜ ¯ ¯ yy ν µy y , 0 = ¯ x ¯ α y ν ¯ φ ¯ ˜ α h η , ¯ h y y ¯ ¯ β µ ∂ ¯ h ¯ ¯ τ ∂ α − β yy ¯ ν ∂ ¯ h − y ¯ y αβ µ ∂ 00 ¯ h η ¯ µ τ τ τ ξe + ˜ y∂ A 2 x = = B + ¯ α ¯ α − − ¯ ˜ µy ¯ y ∂ ν ¯ µ ¯ µ p ∂ + ¯ ˜ 0 ¯ y µ 0 µ ¯ h ¯ ν y x det d ¯ ν ˜ 0 y ¯ τ ¯ µ h η , ¯ ¯ ¯ h α B 00 µ ¯ h ¯ ¯ µ β x ∂ ¯ h Z τ y ¯ ∂ yy α yy ¯ µ + τ 2 y ¯ µ ¯ ∂ h τ ¯ h τ ¯ − yy ν x y p ¯ ¯ h x α = = = 7 and we introduced dual coordinate ˜ − τ T det ¯ β ∂ ¯ µ ,B y ∂ 1) 1) − ¯ αp + τ ¯ 00 τ µη ¯ µ ¯ y − − µ ¯ y ν p p τ τ A x = B ,..., ( τ ¯ µ ¯ α ¯ 1 α ¯ h − − 1)( − S ∂ + , ¯ − ν A p = = = ¯ 0 µ = ( ¯ h ˜ ¯ y ν = 0 has the form ¯ ¯ A 00 00 ¯ µ µη 00 µ αβ µ = ¯ h B B αβ A ¯ β = 0 there is no kinetic term for ˜ ¯ α 0 A ˆ det A ˜ 00 yη These are standard Buscher’s rulesNSNS [ two form. As a result we obtain following action for non-relativistic D(p-2)-brane where where h is unclear. For thatnatural reason choice that we implies rather thatIn restrict non-relativistic fact, ourselves D(p-1)-brane it to would wraps be spatial the naturalto direction case to be only. impose when as this much condition general from the as beginning possible. however we In wanted this case we obtain The resulting D(p-2)-brane is localized in where tinguish two situations. In thedet first one we have where ¯ where JHEP10(2019)153 (5.6) (5.1) (5.2) (5.4) (5.5) (4.11) (4.12) . , α , 0 A A β ≈ ∂ det − ij − β F √ 2 s A l α and also that Dp-brane 0 ∂ + β A y = N 1 ≡ x − j αβ 9 ∂ A x . M i N x 0 (5.3) π i ,F x + ∂ N j ≈ β . αβ x ∂ π j i , , F yy π N MN ∂ 0 1 2 s ¯ h l x ij B MN − j p ≈ F q ∂ B + MN T + p . Further, T-dual dilaton is equal to y + B ln ξ y N + N 2 -first class constraints MN x 1 2 . 0 x M − s − p πR β β S l j G x – 8 – ∂ y A ∂ i − 1 M − ∂ M φ M = 2 − ≡ x x i M x det M 2 , p i A ∂ α = p p i − ∂ G ∂ 0 − p A π = φ = N T 4 √ i x − s MN 0 MN l i A β M det H G B ∂ Π 1 + − ) has manifestly the same form as an action for non-relativistic − + √ N MN A φ det N Π G − x φ 4.10 φ 2 β ξe − ∂ − φ MN e e − +1 2 s M G 2 p l e p x p p T d M α T T = 0. Π ∂ + Z − − y τ p ≡ = = MN T T i − G M π H is radius of the compact dimension p = = y S R αβ A Let us now showshould presume how an T-duality existence can ofwraps direction be this with described isometry, direction. say in Formallyconstraint canonical this with can the formalism. help be of achieved As the by usual gauge fixing we fixing one function spatial diffeomorphism where together with following Hamiltonian constraint From this action we obtain conjugate momenta Using these definitions we obtain following background with zero RR fields 5 Non-relativistic D-branes from5.1 Hamiltonian formalism T-duality ofIn D-brane in this canonical section formalism weon give the an Hamiltonian formulation alternative of definition Dp-brane. of We non-relativistic begin with D-brane the that action is for based D-brane in the In summary, we see that ( D(p-1)-brane that shows that non-relativistic D(p-1)-braneduality transforms transformation covariantly under in T- the same waycondition as that its relativistic precursor with however important where where JHEP10(2019)153 (5.9) (5.7) (5.8) , (5.10) (5.13) (5.11) (5.12) that is ) ij yν ν x G A yy yν j ˆ . Now it is with T-dual as p ∂ µy G G ξ p y B yy − yν p A + + G G ν + = , p ν x ν 8 j ˆ yν x 0 A j ∂ i ˆ 0 ˆ y B ∂ ∂ 2 s µy = 0 for l ) ,..., . ) p G yν 1 ,B − ( , ! yν H as p G coordinate as ν yy yy B µy 1 A x φ µy − so that they are second class . j G ˆ j ˆ 0 = 0 G G µy 1 ∂ B ∂ y p B − = − H yy yy + yν µy 0 + µν B G G G , µ 0 iy yν . p µ p B yν B x π A yy i ˆ i G ˆ = µy ∂ , . . . , p ∂ µy G ,B 2 = s µy 2 s p p with dual G l l 0 B µν ( π ln G 2 . A p yy 2 yy yµ = 1 s ( + − i 2 s l ˆ − s 1 2 yy l A i l ˆ 1 G B p A π p yy G = µ 1 − ,B A – 9 – A − = − ]. Further, since we identified 0 x j ˆ G j ˆ , ) j 0 ˆ i i ˆ φ ˆ y − ∂ i ∂ y ˆ = y ∂ π p yν − π 24 ν = A yy µ , yy µν p µy 1 yµ 0 B x 0 0 y i ˆ µν G B A G B yy B φ 23 i µy 1 ∂ ˆ p 2 µ G ∂ A B − s A x µν µ l i i ˆ ˆ iy ˆ = x ,G ∂ ∂ B + µ i ˆ − 4 2 2 s s y A x ∂ l l i i ˆ ˆ − s p yν yy µy with corresponding conjugate momentum as l ∂ π − + + + G G G p p j ˆ

− i ˆ π A µy i µ ˆ = A p ∂ G 0 det ( 0 µy = = has non-zero Poisson bracket with yy yy 1 y µ p G G Π Π G = det = ,G − ij we identify yy µν 1 0 A G y G = = det 0 y 0 µν 0 y 0 G G which are famous Buscher’scoordinate rules [ and also we can identify components of T-dual metric and NSNS two form field Further, we see that it is natural to identify First of all we find following transformation rule for dilaton equal to In order to identify components of the metric let us analyse determinant det easy to see that constraints that can be explicitly solved. In fact, we can solve Then we have where we further presume that all world-volume fields do not depend on JHEP10(2019)153 1 π − s l → (6.1) (6.2) (6.3) ) (5.15) (5.16) can be so that is world- p πR j ˆ i ˆ H T (2 a . H 0 (5.14) , π ij ) ≈ M F . p 2 s , l 2 5.12 j ˆ ) → + i ˆ ) F ν ) (6.4) 2 s πR ν x . l τ j πR (2 ν ∂ ! + 0 T x (2 should have dimension p j ˆ µν N H 0 ∂ 0 M ¯ h A T + i ˆ x p µ j j ˆ ∂ + H x 2 s π ∂ i l η ν ∂ j ˆ πR N M ξN x 0 0 ∂ + 2 p j + ( x d ∂ i µ j ˆ ˆ i ˆ u -direction which means that we have τ ∂ R j 1 a j ˆ µν u µ p ) ∂ − π ¯ h x i ˆ µ p µ = i ˆ µ 0 MN τ N π τ ∂ 0 x p µ µ dξ B i x H πRT p + x x j ˆ A ∂ i i ˆ ν i i ˆ + ∂ A Z ∂ ∂ p ∂ π j ˆ j ˆ have correct dimensions. = 2 i ˆ 4 N ∂ = 0 0 − – 10 – µν + 2 = s F 1 is non-singular matrix with inverse − s 0 MN x l l h 2 s 0 η T µ − N j ˆ l u i j ˆ ˆ µ G p x i ∂ ˆ p − + ∂ H ]. Since p j T 0 so that the diffeomorphism constraint + ( M 2 F j µ 0 M ∂ j ˆ 0 N 2 s j ˆ + i − τ ˆ and µ π x i l ]. On the other hand T-dual Hamiltonian is given as ˆ 1 p x ≈ i ˆ µ a µ p i µ ¯ ˆ h 2 − ∂ 0 T x uτ p x ∂ p + i ˆ − u s i j ˆ j 1+ s l µ π H l ∂ j ˆ ∂ ∂ ˆ v i dξ ˆ 4 − µ u 0 MN ¯ h − s

p = det p l Z G ξ uτ 2 i det = ij = ) when the Hamiltonian constraint has the form + ∂ dy φ − = i A ij j 2 ˆ 0 N . Then i ˆ 2.4 π u − det Z A 4 a G p Π e 0 has the form πR N − s det 2 φ Φ p l 2 = 2 u ij T 0 MN − p 0 = A e + 2 + H 0 G 2 p has dimension [ = 2 T 0 M N as τ N Π + u . Then we obtain H p ≡ i ˆ k ˆ is radius of compact dimension. We see that we should identify tension of T-dual δ 0 T = R H ˆ k j ˆ a j ˆ i ˆ Let us presume that a while the matrix Let us now presumegauge that fixing Dp-brane function is extendedsolved in for canonical formalism. In thisnull section isometry we given consider in T-duality ( in case of the background with together with 6 T-duality along null direction In previous section we demonstrated how T-duality can be described with the help of where brane as and perform corresponding rescaling of conjugate momenta However we should be moresheet careful density about of scaling the ofwe dimension various fields. find [ that Note that the components of the dual metric and NSNS two forms are given in ( so that we obtain Hamiltonian constraint for D(p-1)-brane in T-dual background where JHEP10(2019)153 η . + ) τ ν (6.8) (6.9) (6.5) (6.6) (6.7) H τ (6.11) (6.10) and τ ν + x N µ σσ . σ ( x ) ¯ h ∂ ν ) µ ) so that the dσ τ τ η η ν µ σ τ R x m x ∂ σ τ σ η∂ = , , ∂ τ = τ ∂ µ ) µ ∂ . − H τ − ν σ − x which is again very ) ) τ µ η π σ ν − σ 3 ν η x σ ∂ x τ σ x σ σ σ ητ τ j ˆ ∂ + η∂ τ τ ∂ η ∂ η∂ N ν b σ τ ∂ σ − ν σ ( p ( ∂ ∂ + ˆ e η ∂ ( + ). Using 2 s τ a σ µν φ l η + µ τ µ m 2 − j ˆ j 2 ˆ h ˆ e ˆ v 6.7 η∂ − ∂ µ π µ . σ − η ( e σ p ν + τ x η . j ˆ j ˆ σ 2 σ 1 ∂ i ˆ σ x σ σ ∂ π ). We see that this Hamiltonian ( + j ˆ τ T a ∂ j ∂ ˆ 2 τσ ητ φ ( ) i ˆ τ ∂ 2 σ µ 2 h − τ s 2 µ ) F p ) ∂ − τ N − s µν σ 2.10 N µ e η η . From the equation of for µ ml + ¯ h + ˆ 2 v − σ σ σ p N x τ ν µ i ˆ + ml η i ν ˆ ∂ T p x 2 ∂ τ x µ N η∂ i ˆ +( ηπ ,N τ + ηp σ τ 2 µν x ∂ i ˆ i ˆ ν τ i = − ν ˆ ∂ ∂ σ h ∂ ∂ x − π − x 2 ∂ µ τ η N σ 4 µν i σ − ˆ µ is non-zero. p σ − ν s + ∂ l ∂ ¯ − h τ s ∂ and we also introduced tension of D(p-1)-brane p – 11 – σ b l ( + 2 µ 2 µ ητ ν τ j ˆ m p µν µ ˆ x i − τ e s x ˆ µν τ + i ˆ − ,N p a π ¯ h ∂ τ a σ h i µ ∂ ˆ 2 ( µ µ j ˆ ∂ τ ml ˆ . η e η fundamental non-relativistic strings as can be easily µ π ˆ v − 2 s x 2 s τ σ µ ) σ l p µ l m η N σ det ηπ η x 0 is the first class constraint we can fix it by imposing m x σ 0 m N j ˆ ∂ i τ η∂ 4 σ ˆ η∂ = = φ 4 ∂ ∂ − that are given by ( ∂ τ σ ∂ ∂ 2 − s ≈ i 2 ˆ = 2 − s η l µ Φ τ σ ∂ ∂ 0 l − − s = 2 i ˆ τ ττ τ τ H σ } 2 e φ , p η N ¯ h − − } π N 1) p p − 2 ηπ m ml ) σ i σ σ ˆ 2 ,H 2 − η η − A ∂ σ τ τ − + ∂ p µ − s 2 σ σ 2 s ( τ η σ 4 l − ν µ l x η, H ∂ τ ητ ] that contribution proportional to Φ cancel each other and hence τ σ T − s x η x { { τ η∂ τ l 9 ∂ τ σ σ = τ σ ∂ + + 2 = ∂ ∂ ∂ Φ = = ( ∂ b η µ η − φ ν σ Φ = 2 µ η = const. Then we can impose the condition that τ 2 ˆ σ e − η σ τ x η 0 τ a N − ∂ σ σ ∂ µ τ σ p 2 e ητ ∂ ˆ H e 2 τ ∂ A 2 s 2 τ µ σ m − s ∂ τ 4 x 1 m ( ml τ − s 4 ml l 2 2 s ]. Note that it is crucial that ∂ − s l τ ( l 9 m 2 + τ +2 N 1 together with dilaton 2 N = = 2 ) we obtain 4 σ − τ 1) L As the next step we determine Lagrangian density from ( H H = − σ p ( L Lagrangian density has the form Finally we determine Lagrangewe obtain multipliers It can be shown as in [ Then we obtain Lagrangian density in the form found in [ N Now the first partseen correspond comparing to it with the Hamiltonian constraint for non-relativistic string which was 6.1 Special case:It is D1-brane interesting to consider Hamiltonian forwhich the is simplest example D1-brane. of non-relativistic Since condition D-brane that Hamiltonian constraint has the form T constraint coincides with the Hamiltonian constraint derivednice in consistency section check. Finally note that spatial diffeomorphism constraints have the form where we identified so that the Hamiltonian constraint has the form JHEP10(2019)153 ]. 11 ]. , (2018) ] SPIRE IN SPIRE JHEP ][ D 98 , IN , ]. ][ (2018) 190 11 ]. SPIRE IN Phys. Rev. coincided fundamental ][ arXiv:1311.4794 JHEP , ]. SPIRE [ , Relating non-relativistic IN m (2014) 057 Strings with Non-Relativistic Newton-Cartan Gravity and arXiv:1409.1522 ]. [ SPIRE 01 ]. arXiv:1409.5555 IN [ Torsional Newton-Cartan Boundary Stress-Energy Tensor ][ SPIRE IN ‘Stringy’ Newton-Cartan Gravity SPIRE JHEP (1923) 325 [ (2014) 061901 , ][ IN ]. ][ 40 arXiv:1705.03535 (2015) 066003 [ D 89 SPIRE (2015) 135017 ]. Torsional Newton-Cartan geometry and the IN [ Torsional Newton Cartan gravity from Nonrelativistic strings and limits of the AdS/CFT D 92 32 ]. – 12 – SchrödingerInvariance from Lifshitz Isometries in arXiv:1206.5176 [ SPIRE ]. Nonrelativistic String Theory and T-duality IN [ Phys. Rev. SPIRE , (2017) 086019 arXiv:1803.07336 IN [ arXiv:1708.05414 SPIRE ][ Phys. Rev. ]. [ ), which permits any use, distribution and reproduction in 6= 0 in order to find well defined Lagrangian density. IN Nonrelativistic String Theory in Background Fields , ]. ]. D 96 ][ m (2012) 235020 arXiv:1906.01607 SPIRE SPIRE SPIRE , IN (2018) 041 Class. Quant. Grav. 29 IN IN [ ][ , (2017) 194 Annales Sci. Ecole Norm. Sup. ][ 05 CC-BY 4.0 , Phys. Rev. 10 arXiv:1907.01663 , This article is distributed under the terms of the Creative Commons , arXiv:1806.06071 Sur affine les et variétésàconnexion la théoriede la relativitégénéralisée. JHEP Remark About Non-Relativistic String in Newton-Cartan Background and Null Hamiltonian for a string in a Newton-Cartan background [ , JHEP ]. arXiv:1801.10376 , [ SPIRE arXiv:1810.05560 IN arXiv:1311.6471 arXiv:1905.07315 non-relativistic strings string theories (2018) 133 Conformal Symmetry and Limits[ of the AdS/CFT Correspondence Reduction 086010 correspondence Holography and Field Theory Class. Quant. Grav. [ and Newton-Cartan Geometry in[ Lifshitz Holography Torsion Schrödingeralgebra Geometry and Lifshitz Holography Peieepartie) (Première A.D. Gallegos, U. and Gürsoy N. Zinnato, T. Harmark, J. Hartong, L. Menculini, N.A. Obers and G. Oling, T. Harmark, J. Hartong, L. Menculini, N.A. Obers and Z. Yan, J. Gomis, J. Oh and Z. Yan, J. Klusoˇn, E. Bergshoeff, J. Gomis and Z. Yan, J. Klusoˇn, R. Andringa, E. Bergshoeff, J. Gomis and M. de Roo, T. Harmark, J. Hartong and N.A. Obers, M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, J. Hartong, E. Kiritsis and N.A. Obers, E. Bergshoeff, A. Chatzistavrakidis, L. Romano and J. Rosseel, E.A. Bergshoeff, J. Hartong and J. Rosseel, M.H. Christensen, J. Hartong, N.A. Obers and B. Rollier, E. Cartan, [9] [7] [8] [5] [6] [2] [3] [4] [1] [14] [15] [12] [13] [10] [11] References non-relativistic strings while thesee expression that on there the is crucial last that lineOpen has D-brane Access. origin.Attribution We License also ( any medium, provided the original author(s) and source are credited. Again we see that the expression on the first line corresponds to JHEP10(2019)153 , ] 04 (2012) 10 Phys. , 15 (1995) B 194 (2001) 3127 JHEP 75 , JHEP 42 , -models σ Phys. Lett. , arXiv:1712.03980 [ Living Rev. Rel. (2019) 072 , 06 Phys. Rev. Lett. J. Math. Phys. , , Homogeneous Nonrelativistic JHEP , (2018) 175007 35 ]. IIA/B, wound and wrapped Strings from quivers, membranes from moose SPIRE ]. IN – 13 – ][ ]. -symmetry and Applications κ SPIRE IN ]. ]. SPIRE ][ ]. IN ]. Class. Quant. Grav. ][ , ]. Non-Relativistic M2-Brane SPIRE SPIRE Nonrelativistic closed string theory IN IN SPIRE SPIRE [ ][ IN hep-th/0204147 IN SPIRE [ ][ ][ IN ]. ][ -String and D1-Brane in Stringy Newton-Cartan Background ) A Symmetry of the String Background Field Equations Path Integral Derivation of Quantum Duality in Nonlinear Dirichlet Branes and Ramond-Ramond charges (1988) 466 SPIRE arXiv:1901.11292 hep-th/0009182 m, n IN [ [ (2002) 023 Brane Effective Actions, [ ]. 05 hep-th/9510017 B 201 [ arXiv:1110.2422 SPIRE [ hep-th/0009181 IN arXiv:1903.12450 (1987) 59 Lett. JHEP 4724 3 [ (2000) 020 Geometries as Coset Spaces [ [ (2019) 163 T.H. Buscher, S. Mukhi, M. Rangamani and E.P. Verlinde, J. Polchinski, J. Simon, T.H. Buscher, J. Gomis and H. Ooguri, U.H. Danielsson, A. Guijosa and M. Kruczenski, J. Klusoˇnand P. Novosad, J. Klusoˇn,( K.T. Grosvenor, J. Hartong, C. Keeler and N.A. Obers, [24] [25] [21] [22] [23] [19] [20] [17] [18] [16]