Jhep02(2021)047

JHEP02(2021)047 ’s σ Springer February 5, 2021 December 3, 2020 : December 28, 2020 : : whose ﬁeld strength is of the GL(5) of the 5 ) Received Published P Accepted 10 σ ( MN X Published for SISSA by https://doi.org/10.1007/JHEP02(2021)047 b [email protected] , . and Warren Siegel 3 a 1502.00510 The Authors. c F-Theory, M-Theory, p-branes, String Duality

). The current algebra gives a rederivation of the F-bracket. The background- We describe the worldvolume for the bosonic sector of the lower-dimensional F- , τ [email protected] College Park, MD 20742-4111, U.S.A. C.N. Yang Institute forStony Theoretical Physics, Brook, State NY University 11794-3840, of U.S.A. New York, E-mail: Center for String andUniversity Particle of Theory, Maryland Department at of College Physics, Park, b a Open Access Article funded by SCOAP Keywords: ArXiv ePrint: independent subalgebra of thenew Virasoro type algebra of gives section condition the followssolving usual from just Gauß’s section law, condition, the tying while the oldmanifestly worldvolume a T-dual condition to version spacetime: yields of the M-theory, string,also while find and a solving the covariant combination form only of produces the the the condition usual that new string. dimensionally one reduces We the gives string coordinates. the Abstract: theory that embeds 4D, N=1(5-brane) M-theory theory and the is 3D thatselfdual. Type II of Thus superstring. unlike a string The singlein worldvolume theory, the the 6D Hamiltonian spacetime formalism, gauge indices the 2-form (neglecting are spacetime coordinates tied are to a the worldsheet ones: William D. Linch III F-theory from fundamental five-branes JHEP02(2021)047 ]. In this paper 7 ]. For manifestly – 4 ] as “evidence” for 5 8 strings [ 8 E signature. (The modern use of 2+2 ] has described only the massless sector. 19 – 9 ] and heterotic , 3 2 – 1 – , 1 ] shows. Indeed, even the string worldvolume was 4 8 ] of how string theory (S-theory) relates to its mani- ]. (Generalized Geometry is a further simplification of 2 , 7 1 – 2 5 7 M,T,S 6 1 → 7 So far all the work on this F-theory [ In its original form, similar coordinates were conjectured for F-theory, as a review of to derive its symmetries,5-brane and is how a they dynamical actsimilar one, on to meant a to the massless be string. background.in first-quantized addition (Details (and This to maybe of fundamental the second) the targetwill in dynamics space contain a will coordinates infinitely manner and be many their massive left exceptional modes. for extensions, the the future.) spectrum In particular, will continue to apply the“non-critical” term versions as of originally that intendedthe idea). (and U-duality extend In group it particular, is to realized the lower-dimensional on models the will algebra be ofHowever, of world-volume T-theory F-theory currents. was type originally if derivedwe approach from F-theory string from current its formulation algebra as [ a fundamental 5-brane, using current algebra section 3 of the foundationalconjectured reference to [ be extended tothe fundamental brane term with often refersthe to type existence IIB of backgrounds such that a are spacetime- referred and to worldvolume-extended in theory.) [ In this paper, we to additional target space coordinates.arising in For M-theory, the this strong-coupling theT-duality-invariant limit theories familiar of they eleventh are dimension IIA the [ coordinatesmodes of of double compactified field strings theorythis for [ in the which winding the winding coordinates are truncated.) 1 Introduction We continue our considerations [ festly T-dual formulation (T-theory),manifestly M-theory, and STU-dual the version interpretation of of all F-theory as these. the These extensions of string theory all give rise 6 Conclusions 3 Algebras and gauge symmetries 4 Backgrounds 5 Sections: F Contents 1 Introduction 2 Currents and constraints JHEP02(2021)047 ]. 27 (2.1) (2.2) (2.3) (2.4) (2.5) become ] in that 26 X – 20 ] 28 σ, 5 d . The momentum con- mn . , . QRS P pq pq F n ∂ τmn X X F r r ) τm QRS σ ∂ ∂ 6 m = X d + mn τ, σ MNP mnpqr mnpqr P H dτ 1 3! MNP → 1 2 1 2 mnp Z F ± F M + − + σ and mn mn mnp MNP ] – 2 – P P F MNP σdτ F 5 F 1 = = mn d 12 Z = X ) + mn p − 1 [ 12 (+) ( τmn τmn ∂ X ) describing spacetime coordinates are gauge fields. Also, F F 1 2 τ mn ± = ∂ ( ) MNP P = S := := P , unlike treatments based on consideration of Wess-Zumino F mn σ mn X ( P P mn mn 1 2 mnp . 1 4 and . . ˜ x F MN ] Z X − Z . In Hamiltonian form ( MN = = X mn P P [ S H ∂ is as well as 1 2 σ ]. For the case of F-theory embedding the 3D string, this gauge field is the = mn 16 X – 14 MNP F As usual (cf. electromagnetism) the time components of the gauge field The action is then, in Lagrangian form, Our treatment of fundamental branes also differs from previous versions [ (actually the 5D indices remaining in a temporal gauge) thus become identified with while the antiselfdual field strengths are the symmetry currents Lagrange multipliers. After using them to identify the constraint (Gauß’s law), we eliminate The selfdual field strengths are the currents for the covariant derivatives [ where the field strengths in Hamiltonian language are with jugate to multiplier for selfduality have beenformalism, fixed, which we using find the convenient action for only our to analysis. define the Hamiltonian 2 Currents and constraints Covariant selfdual 6D field theoryFor has simplicity, been we start described here previously in a in “conformal terms gauge” of where an both action the 6D [ metric and Lagrange spacetime indices. Analysisspacetime of gauge fields the of current theF-bracket algebra resulting massless of sector from of their this F-theory, algebra,the theory their the gauge gauge naturally F-section field transformations, leads condition, the tie etc.adds to the a the Just worldvolume as constraint to the tothat spacetime, indices the reduces of so generalized does Virasoro the algebra 2-form’s as Gauß well law, as which a new section condition the worldvolume fields the currents are all linearterms in [ 6D gauge 2-form with selfdualX field strength. The worldvolume indices on this gauge field JHEP02(2021)047 . P m (for ◦ U (2.6) (2.7) (2.8) ) (2.9c) (2.10) (2.9a) (2.9b) σ ]: ( . 29 X pq . or x mn mn mn P p . n n . ∂ ∂ nq ’s. η .) This is symmetric σ = are := is a gauge field, as a mp (+) NPQ ) m η . (This requires treating m F (+) 1 8 U ττ MNP X ◦ . (It simplified using U F mn PQ m = (+) η ◦ S .) This is the subset that’s T T (+) , M ) or functions of m ) = 0 F σ , m τm 1 8 ∂ qr Bg pq . (+) )( . = qr qr T P p np . , Af mn . np np (+) MN ]. T-theory with doubled selfdual scalars . p p mn mn T = ( P in that formalism. Dimensional reduction 30 n (+) n ∂ mnpqr T ( P mnpqr mnpqr rmnpq ⇒ – 3 – m ABf := 1 16 1 4 1 8 p 1 16 m (for hitting with = := := mn U ) = η ⇒ m σ m m r NPQ ( S ◦ S S ◦ S : the latter allows them to commute with Virasoro.) It F X . ˜ , which arises because the 6D , as opposite to those on PQ or = 0 m of coordinate transformations for the 5 nq or M U σ . mn ]. T-theory with both selfdual and anti-selfdual scalars was F m . 1 4 AB ].) S X 31 mp , both dimensional reduction conditions can be written with = . 33 used in the manifestly T-dual version of the string that has doubled , pq R 32 η = 0 MN P 1 2 T ] − n ). − L ∂ P T m mn [ p ∂ mn η = that are functions of section condition Virasoro dimensional reduction replaces coordinates. (This reducesfor to doubled coordinates was inventedwas in considered [ in [ attempted in [ dimensional reduction condition sincehave a it’s new linear covariant in dimensionalSince reduction the condition string variables.replaced with We either also with the generators We treat Gauß’s law We generalized (the background independent part of) the string Virasoro algebra mn We thus have 3 types of conditions: For purposes of analyzing kinematics, we need consider only those constraints that are f, g T 2. 1. hitting with where the section conditions are to be interpreted as being applied as both for by replacing some string coordinates in these constraints with their zero-modes [ GL(5) covariant, that is, needcontravariant not indices involve the on SO(3,2) metric background independent, since theelement to background break is GL(5) to introduced SO(3,2). as Various section-like a conditions GL(5)/SO(3,2)GL(1) can then be introduced The Gauß constraint is momentum tensor for the selfdual field strength (The unusual normalization is consequenceand of traceless. our Its definition Hamiltonian of components them by choosing a temporal gauge. The Virasoro algebra is then defined by the energy- JHEP02(2021)047 – − 5 1 σ (3.7) (3.3) (3.4) (3.5) (3.6) (3.1) (3.2) ( (using . (The ) (5) i σ δ r pq qr ]), but we . . ˜ ˜ U . 28 pq and np − . p x in T-theory [ pq . . n p (2)] = 0 )( mnpqr mnpqr m f (2))] p 4 1 mn − X . mn ] ( mn = ∂ , is shorthand for n η ( . S 1 2 , f qr 2) (1) 2) , m P 2) [ p 2) (1) − ) so that, up to second-class 2) − ∂ U np − mnpqr h . ˜ p − (1 − , the Poisson brackets with the mn (1 δ ) 1 8 (1 2) δ . ˜ 2 (1 r (1 δ − fδ δ . ∂ X r ] mnpqr ( q ) = m n ∂ (1 that mutually restricts δ O f 4 1 δ (2)] i∂ 2) = [ m ◦ i p [ 2 U O mnpqr − mn − iδ (1) and [ ∂ i mnpqr ] − 2 − replaced by )( (0) + (1 i mn − f nr (1) + . fδ . – 4 – (2)] = = 0 = O q X mn p [ [ . To see this, note that mn ∂ δ m , f 1 2 σ (2)] = ∂ i∂ 2) (2)] = 2 ( mn (2) (2) pq , originally found by closure of gauge transformations − ◦ S ((1) + (2)) )(1) pq − pq pq X − 1 2 . ,X 1 2 m . . ˜ ˜ ) ) (1 ˜ but with S , , , n pq δ ], and a new one (1) ], the new constraint is linear in the string coordinates. S q 9 S − . ˜ (1) (1) (1) ∂ 2) (2))] = 27 mn m i 2 (0) + − S mn P mn mn X − f [ ( . [( pq . . ˜ ((1) + (2)) := [ (1 1 2 . , f ) is thus a second-class constraint (as for string scalars [ δ ) = O )( m (2)] = ( (1) f X = 0 ( i∂ mn commutation relations we find f mn . ˜ mn . ∂ . , ( [ ] = 0 .. generates translations in U (1) , S (2)] = p n S U mnpqr [ S , formally the same as 1 8 ].) ] = [ ˜ S (1) (see below) in F-gravity [ former condition reduces to the7 original section condition = U The section conditions include On functions From the m , f .) Selfduality ( S 3. S r [ ) [ 2 ∂ (with constraints, The worldvolume version of this is given by where we are defining currents give spacetime derivatives Straighforward calculation gives (Here “1” and “2” refer toσ points on the worldvolume, and saw above the covariant divisiona into section first-class condition). as Unlike [ Using the Poisson bracket we find the algebra 3 Algebras and gauge symmetries JHEP02(2021)047 : (3.8) (3.9) 2 (3.15) (3.11) (3.12) (3.13) (3.14) (3.10) , 1 . = , pq 2] i p , ] λ n pq pq . for . ∂ p ∂ . ] mn mn [1 i m [ λ pq V 2] ξ 1 4 pq V . + ] − ] mn q pq 2] pq 2] ∂ ] mn ]) closes n 2] λ λ 9 . λ mn pq [ ((1) + (2)) p [1 mn . ∂ pq 1 2 ) ∂ V ∂ ∂ p 1 8 ].) pq ξ X mn m 2 [ [1 [ ( mn 6 [ [1 p [1 V λ − + . λ λ , g r p 1 ) ] 16 1 2 pq mn λ 1 2] 1 n 16 . X mn V V pq . − − ( was derived by dimensionally reducing . m ∂ − ) f 2) S mn m mn 2] mn 2] pq ∂ [ ∀ − mn i ∂f m ξ λ λ . ξ λ p mn mnpqr (1 pq pq [1 ∂ pq 2] δ Z Z ∂ ∂ V r λ – 5 – 1 2 i i 2 pq [1 pq [1 ∂ + ( ) = 0 λ λ = mn = 2) ) g 1 2 1 4 ∂ ξ := mn δ − i . mn mn = = mnpqr mn [1 2.9c Λ ) ∂ (1 λ i p δλ )( 1 2 ξ iδ mn 12 p f λ ∂ n − ] = 2 ∂ Z ( i 2 pq isn’t exactly a tensor, but its integral is invariant. . ) = ( ] . with pq ] = 2 mn 2 . Λ ,V ) reparameterizations are generated by 12 , p [ σ 1 ξ mn ( [Λ . p ] = Λ ∂ ), we can compute the commutator of two vector fields ) implies that the gauge parameter itself has the gauge ambiguity 2 1 2 mn 1 Λ V 3.7 , 3.7 [ = 1 , we find for their commutator ) implies 2 [Λ , mn 1 3.3 . = ξ Using ( Worldvolume ( δ i Note that ( This shows that the algebra of gauge transformations (cf. ref. [ for modulo the second-class constraints(spacetime) (and gauge sectioning). transformations: We use defining this to check the algebra of This demonstrates that corresponding to a densitymation term terms, (integrable a on coordinate scalars), transformation, two and contravariant two index terms transfor- for integrability on non-scalars. Then ( (In T-theory the analogue(solving of the this second-class constraint) formula and for then oxidizing [ modulo the new section condition ( JHEP02(2021)047 . i ) mn ef ] . . . cd (4.7) (4.4) (4.5) (4.6) (4.1) (4.2) (4.3) [ ef abcde ↔ pq ] ) is thus an E ab n m [ ] ( a b m a cd E E − e E ] ] m d ↔ , a | of replacing [ ) becomes pq q ] . λ a E ab n e S 4.4 ] = det( λ pq , . = − ∂ a are representations of ab pq ] [ . Since | qm m ∂ c pq mn [ mn λ [ E 2) m c [ mn cd mnpqr ab pq p ab + − ab E ∂ ] E r ab E p d e E | ef a 1 8 (1 ] E mn ∂ b E δ E ∂ q i . + + ⇒ m d e + and ef ∂ mn E a pq [ e [ p mn a | . λ pq c m c ab m r λ [ λ a cd ab e pq E E E E mn pq c c ∂ E n pq 1 8 ∂ ((1) + (2)) . b ∂ + . In terms of these, ( − 2 1 ] ] E e pq mn m − ] f d abcde a d mn m pq ef ab ab δ mnpqr , a pq e . E ab [ cd E E e 2) c E 1 4 cd [ E 1 1 2 2 c E ∼ [ ab cd – 6 – − E 1 2 c ab + ab − + h c gives pq mn (1 c = mn ] m 1 2 δ de ab a ∂ e + mn mn =: ab E ab E ∂ E ] ] cd . ab ab . mn f d ] = 2) . pq , mn , and we have chosen the proportionality factor to be δ E E ab ∂ e cd e a [ ] bc − abcde pq pq E m pq m , ∂ ∂ ∂ E ab ) [ . The anholonomy coefficients λ (1 = [Λ E ) again, one computes that e E ab pq pq δ m 1 2 ) c 2) ∂ r λ λ a mn h ab [ m abcde ∂ 1 2 1 2 ∂ E = − c . 3.11 [ 2) n λ 1 4 b representations, each of which has been expressed in terms of the E m = = (1 δ − a E δ det( ab i mnpqr i (1 ]. (There is also a nonderivative Sp(4) gauge transformation on the 8 and we have rewritten m ∂ 10 mn δ δE 9 a ⇒ i 2 i 2 ab m − +2 E = ( ∂ E a a and λ := ) in terms of the fundamental field, it is useful to flatten the indices on the ef .) δ m m 5 does not generate an independent symmetry. This is essentially the statement E E ab is the inverse of (2)] = (2)] = 4.4 S a, b ∂ m ab cd ) applied to cd cd S c := m in . . a ) in the final step for convenience. Thus a ) to find its gauge transformation: , ∼ , E ab 3.11 m ∂ is a tensor under transformation by the vielbein, and thus the vielbein is an element a a . 4.2 (1) S (1) Alternatively, using ( Factorization of the vielbein follows from requiring that the result E ab ab . . [ [ where To rewrite ( derivatives reduce to Then the first termunconstrained has matrix, to it’s be more proportionaluse convenient to ( to use as the fundamental field. We therefore where det( GL(5) in the other above. with that of GL(5): introducing linear dependence through a new vielbein in agreement with [ flat indices Background fields are introduced as usual through the covariant derivatives Then ( 4 Backgrounds JHEP02(2021)047 m S (5.2) (5.3) (5.1) 2) , 1 3) , , 0 2 ) , l , 1 1 1 − , − X ) = as before, but also the k k 3 = 0 1 i , ∂ ( i − m 2 ( )( , ◦ S j X l 1 1 3 1 , i ij i − 1 ∂ gives 0 − 1 p , − kl i − X ◦ X 1 1 U ijk i = p P 2) k X − ∂ − , 3 i ∂ ( − = = 1 P 1 ∂ ) , = kl − i and p kl = P ijkl 1 , p ijkl . P ijkl P i m 3 − U ij ij = 0 ij 2 1 − ∂ ∂ p ij . X i 2 1 . p ( 3 = = = = = ∂ + 1 1 ijkl ijkl ∂ , ij i P − − ij i i 1 . = 0 ; = 0 ; = ( 1 1 S 1 8 1 4 P − U − − – 7 – 3 p ij ij P P S = = = p P , gives (for = 3 3 ij = = = m ◦ , S S ◦ . S i j S = 0 ; 1 P 1 i − − 3 ⇒ ⇒ . , P p i ) , , , ] 1 j = = 0 ; , p − 1 i i 3 − P 3 = 0 = 0 = 0 ∂ , 1 ∂ P i i i i X − 1 i ◦ 3 S S [ = 0 = 0 = = 0 ; ∂ − . ∂ i P − ∂ qr qr 3 M,T,S p P . = = ( = ⇒ ⇒ np i i i np → 1 p p S U − . ) that generalize string theory to F-theory. All of them are new except for the mnpqr mnpqr ), which generate the gauge transformations of F-gravity. (The constraints = 0 = 0 2.9 reparametrizations.) 2.4 σ mn mn p P n n ∂ ∂ Solving both sets of conditions gives S-theory: On the other hand, solving the new conditions section condition of F-gravity, which we havefield now found representation from of first principles, F-gravity. alongfor (Reduction with its the to dimensional T-theory reduction alsocurrents condition.) gives ( Their a algebra new follows covariantgenerate form from that of the (selfdual) 6 Conclusions Starting from the worldvolume theory ofconditions a ( selfdual gauge form in D=6, we have derived the describing T-theory on a 1-brane (string), in 6 (i.e., doubled) dimensions. describing M-theory, still on a 5-brane, but in 4 spacetime dimensions. Sectioning is straightforward. Solving the old sectionnew condition dimensional reduction condition 5 Sections: F JHEP02(2021)047 ) ]. 2.7 cor- 37 (6.1) (6.2) , 0 α 36 (2014) 107 06 ]. -matrix [ δ . γ m JHEP SPIRE ∂ , ” constraints in ( IN µν T [ ) m γ ( i 3 is partially supported by the . ] = 2 ν ν dl . . , µ µ . . [ arXiv:1501.02761 µν ) , ]. The tying of worldvolume and spacetime m ⇒ γ 35 ( ν 1 4 – 8 – X = m ∂ m S µν ) m γ is supported in part by National Science Foundation grant ), which permits any use, distribution and reproduction in s ]. F-theory Superspace + ( T-duality off shell in 3D Type II superspace ]. Then reduction to T-theory can be compared to the formu- 2 µ P SPIRE = IN µ ][ CC-BY 4.0 . ]. This article is distributed under the terms of the Creative Commons 34 representation of SO(5,5). Then the section condition uses a 10D Center for String & Particle Theory and National Science Foundation grants PHY- derivative: Also, use of selfdual forms would give differentof cosets than supergravities. those found Supersymmetry, in especially theshould bosonic place for sectors new the restrictions. D For the =16 D 4, = 4 6, case, and theThe bosonic 10 bosonic coordinates superstrings, covariant are derivative a then spinor resembles a fermionic supersymmetry-covariant straightforward using [ lation with Ramond-Ramond currents [ symmetries in the bosonic case suggestsSchwarz formulations that the might Green-Schwarz be and more Ramond-Neveu- directly related. the definition of truly covariant derivativescurvature on [ the worldvolume and their torsion and rections. This would require anresponsible analysis for of the the algebra dynamics.if of the part “ is A identified relatedgauge with problem fields/Lagrange the multipliers is spacetime for the vielbein, worldvolume coordinate 6D then transformations. the worldvolume remainder vielbein: might be the 6 arXiv:1403.6904 3 thanks Brenno Carlini Vallilo for discussions. W [ Of course, these results should be generalized to higher dimensions. However, simple Including the currents for SO(3,2) (or SO(3,3) for the 6D formulation) would allow We have looked at only the bosonic string. Generalization to the superstring should be The covariant 6D conformal field theory might be useful, e.g., for analyzing Many future avenues of investigation are suggested: M. Poláček and W. Siegel, W.D. Linch and W. Siegel, 4. 2. 3. 1. dl mcp [1] [2] References PHY-1316617. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. 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