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42
UDC 530.1; 539.1
INFINITE-DIMENSIONAL SYMMETRIES OF THE TWISTOR STRING MODELS
D. V. Uvarov
NSC Kharkov institute of Physics and Technology, 1 Academicheskaya str., 61108 Kharkov, Ukraine.
E-mail: d_ uvarov@hotmail.com
It is shown that similarly to massless superparticle models, space-time symmetry of the classical action of the Berkovits twistor string is infinite-dimensional. Its superalgebra contains finite-dimensional subalgebra that includes the generators of psl(4|4, R) superalgebra. In quantum theory this infinite-dimensional symmetry breaks down to SL(4|4, R) one.
Keywords: supertwistor, twistor string, inSnite-dimensional symmetry, anomaly.
1 Introduction
Twistor string theory [1, 2] inspired remarkable progress in understanding spinor and twistor structures underlying scattering amplitudes in gauge theories and gravity. To gain further insights into the properties of twistor strings it is helpful to identify their symmetries both classical and quantum. It was shown in [1,3] that except for an obvious PSL(414, R) global symmetry twistor strings are also invariant under its Yangian extension that is closely related to infinite-dimensional symmetry of integrable N = 4 super Yang-Mills theory.
In this note we exhibit infinite-dimensional global symmetry of the world-sheet action of the Berkovits twistor string and its generalization with ungauged GL(1, R) symmetry. In the quantum theory infinite-dimensional symmetries break down to SL(4|4,R) one, whose consistency was proved in [3]. Let us mention that infinite-dimensional nature of massless superparticles' symmetries was revealed already in [4].
2 Twistor string models and their symmetries
For Lorentzian signature world sheet the simplest twistor string action can be presented as
S = Jdrda(LL + LR) : LL = -2(Yad-Za + nid-f)+ LL_mat, Lr = -2(Y«d+Za + ñd+e)+ LR-mat,
where d± = 2 (dT ± da )
a± = t
Y_
= Ya
n+i = r\i, n-i = n
± a, Y+a = Y and LL
twistor-string model [2] global scale symmetry for both left- and right-movers
SZa = AZ 0 SZa =AZ 0
SYa = -AY«
SY« = -AY„
se=Aei se=Ae
Sri = -Ar¿; Sri = -Ani
(2)
is gauged by adding to the action (1) appropriate constraints T = YaZa + niC - 0 and T = YaZa + n^® - 0 with the Lagrange multipliers
Sgl(i,r) = dtda(xt + xt ).
(3)
(1)
L(R)-mat
are Lagrangians for left- and right-moving non-twistor matter variables, whose contribution to the world-sheet conformai anomaly equals c = c = 26 to cancel that of (b, c)—ghosts. Such variables may contain a current algebra for some Lie group (see, e.g., [5]). In Berkovits
This necessitates add two units to the central charges of the matter variables to compensate that of (b, c)-ghosts and ghosts for the gauged GL(1, R) symmetry.
Definition of the open string sector, that to date is the only one well-understood, is based on the conditions Za = ZA, YB = imposed on the world-sheet boundary on the supertwistors ZA = (Za,£®), ZA = (Za,c) and their duals YB = (Yp), Yb = (Y/3, n j). So taking into account reality condition of the Lagrangian one is led to consider left(right)-moving supertwistors ZA (ZA) ^^d ^^^^upertwistors YB (YB) as independent variables with real components. Such supertwistors are adapted for the description of fields on D = 4 N = 4 superspace for the space-time of
signature (+ +---)1. Conformal group of Minkowski
space-time of such a signature is SO(3,3) ~ SL(4, R) N=4
PSL(4|4, R) with the bosonic sub group SL(4, R) x SL(4, R)
ZA
SL(4,R)l x SL(4,R)l, whereas bosonic and odd
components of YA belong to the antifundamental
Z A
and YA transform according to the (anti)fundamental representation of SL(4,R)r x SL(4, R)r.
1 Detailed discussion of the reality conditions of the twistor string Lagrangian for both Lorentzian and Euclidean world sheets, and different real structures in the complex supertwistor space associated with D = 4 space-times of various signatures can be found, e.g., in [6].
a
Focusing on the sector of left-movers of the model generating gl(4\4, R) superalgebra
(1) and applying the Dirac approach yields equal-time D.B. relations
{Z a(a),Yp (a')} d.b. {?(a),j (a')} d.b.
S$S(a - a'), SijS(a - a')
{Tab (a),TcD (a' )}d.b.
= (SBTad - (-T
(H)
that in terms of the PSL(4l4,] written as
^SDtcb)(a)S(a - a'),
(4) where eba = (-)a+b.
Irreducible components of gl(4l4, R) current supertwistors can be densities (10) are
tab (a) = [Tj, Tij; Qj, Qf; T, U} :
{ZA(a), Yb (a')}d.b. = SAS(a - a'). Similar relations hold for the right-movers.
2.1 Classical symmetries of twistor strings
(5) TJ=YaZp - 1SP(YZ), Tij=nig - 4sj(nO;
(12)
Qaj = Ya$, Qip = niZp; t = YaZa + mi\ u = YaZa - m?.
On D.B. they generate infinitesimal SL(4, R) x
Global symmetry of the left-moving part of the SL(4, R) rotations of the supertwistor components
SZa(a) = Aa$ Z $ (a),
(6)
action (1) is generated on D.B. by the function G = J da £ G{L)(a),
L>0
G(L)(a) = Yb (a)ABAL...A1 ZAl (a) ■ ■ ■ Zal (a).
L
for the supertwistors read SZa(a) = £aiC(a), S^i(a) = -Ya(a)eai;
SZA(a) = Aab(l)ZB(L)(a), SYa(a) = -Vi(a)eia, S?(a) = eiaZa(a), SYA(a) = -LYc(a)KCabl-i...bI ZB(L-1)(a),
SYa(a) = -Yp(a)Apa, Aaa = 0; (13)
Sé(a) = Aijij(a), Sm(a) = (a)Aji, Aii = 0
(14)
(7)
where eai mid eia are independent odd parameters with 16 real components each. T(a) generates GL(1, R) where convenient notation to be used below is ZA(L) = transformations (2) and U(a) - 'twisted' GLt(1, R)
ZA ■•■Zal (Za(0) = 1 Za(1) = Za
Z
Ai ■ ■ ■
ZAl iZA(0)
1 2
A(1)
ZA
and Za(l) = )2. Associated
Noether current densities up to irrelevant numerical factor are given by the monomials
transformations SZa(a) = AtZa(a), SYa(a) = -AtYa(a), SC(a) = -At?(a), Sni(a) = Atm(a).
(15)
T(L)bA(L)(a) = YbZA(L)(a), L > 0
(8) 2.2 Quantum symmetries of twistor strings
that enter generating functions G(L). On D.B. they generate the twistor string algebra (TSA)3
{T(l)BA(L)(a), T(m)dc(m)(a')}D.B. = (SA(1)T (l+m-I)ba(l-I)c(m )
-SC(1)T(l+m-1)da(l)c(m-1))(a)S(a - a').
(9)
It was shown in [3] that SL(4l4, R) symmetry-is preserved at the quantum level, whereas the generator U of the 'twisted' GLt(1,R) symmetry-has anomalous OPE with the world-sheet stress-energy tensor implying that corresponding symmetry-is broken in twistor string theory. Thus possible type of infinite-dimensional symmetry that could survive in the quantum theory is restricted to that The finite-dimensional subalgebra of TSA is based 011 sl(M4, R) as finite-dimensional subalgebra. spanned, apart from the order 0 generator YA(a) that Since sl(M4, R) superalgebra belongs to the family-is responsible for constant shift of the supertwistor of sl(M^, R) superalgebras, whose properties differ components, by quadratic monomial from those of sl(MR) superalgebras with M = N,
one is forced to take components of supertwistors as TAB (a) = YAZb (a) (10) building blocks of their generators.
Composite objects like Z A(L) and ZA(L) are graded symmetric in their indices. In general it is assumed graded symmetry in supertwistor indices denoted by the same letters. Similarly one defines the products of supertwistor bosonic and fermionic components as Za(m) = Za 1 ■■■ Z, Za(m) = Zai ■■■ Zam and Si[n] = T1 ••• , £i[n] = ••• (n < N = 4) that are
(anti)symmetric. Antisymmetry in a set of n indices is indicated by placing n in square brackets. Both symmetrization and antisymmetrization are performed with unit weight.
3To be more precise one has to introduce TSA as an infinite-dimensional Lie superalgebra and then consider its loop version pertinent to twistor-string global symmetry. Let us also note that the subscript L in the notation of symmetry groups and algebras will be omitted as the discussion is concentrated on the sector of left-movers only. On the boundary left- and right-moving variables are identified and thus also no subscripts are needed.
2.2.1 Superalgebraic perspective on quantum higherspin symmetries
In the bosonic limit TSA reduces to TSAb -an infinite-dimensional Lie algebra, whose generators are obtained from (8) by setting to zero fermionic components of the supertwistors. Order 0 and 1 generators are given by the dual bosonic twistor Ya and g1(4, R) generators YaZp. The latter divide into s1(4, R) Tj aid g1(1, R) To = YaZa ones. Higherorder generators YaZ P(L) divide into
T„P(i) = YaZ-
L + 3
(YZ )^(1)Z p(
L-1)
{QiP (a),T/(2V)}D.B.
¿PQi'(2) - 1 ¿Y'1 Qi'2)p) (aM(a - a')
where
Qi'(2)
®Z '(2)
{Qaj (a),T/(2V)}D.B.
= - (¿a'1 C?y'2)j - 1 ¿Y'1 Qa'2)j) (^¿(a - a'),
where another order 2 supersymmetry generator
Q
'i
'¿i
Ty° e
{Qaj (a),Qfc'(2V)}D.B. = ¿jTa'(2)(a^(a - a') + ¿a'1
Tj'2)j
+¿jj (40T - 40^Z'2) (a^(a - a')
and similarly
{Qip (a),(?Y"(a')}D.B.
¿PTi'1 - 4¿YT/1) (a)¿(a - a')
Typ' + (40T - 40U)
(22)
x (¿PZ' - 4¿YZp)] (a)¿(a - a'),
where
T'Pj = Tj Zp.
(23)
(16)
and T0ZP(L 1). Expression (16) is an obvious generalization of Tap from (12) to the case L > 1.
Proceeding to TSA superalgebra, from (9) one infers that the D.B. relations of order L mid M generators close on order L + M - 1 generators. So that order 1 generators, i.e. g1(4|4, R) ones (12), play a special role: D.B. relations of the generators of an L
L
irreducible higher-order generators.
Thus the form of irreducible order 2 generators can be found by D.B.-commuting corresponding bosonic
generator (16) with order 1 supersymmetry generators
QiP Qaj
r.h.s. into irreducible SL(4, R) x SL(4, R) tensors, then
QiP Qaj
such a way we obtain
Continuing further one recovers the set of irreducible order 2 generators
T P(2) T aj t j[2] = YP'[2]-
a i i i a — 1 ?
Qia(2), QaPj, Qj[2] = ^j[2] - 3 (^¿!j1 ej2]
and
TZa
UZa, t£®, Uf.
(24)
(25)
(17)
(18)
The operators associated with the generators (25), as will be shown below, are not the primary fields in the world-sheet CFT and hence corresponding symmetries are broken at the quantum level. Since these generators appear on the r.h.s. of (21), (22) this
Qia(2)
Qapj and, in view of (17), (19) breaking of the bosonic symmetry generated by TY'(2). So that classical odrer 2 symmetries break in the quantum theory.
For order L > 2 calculation of D.B. relations of the corresponding bosonic generator (16) and order 1 supersymmetry generators gives
{QiP (a),TY'(L)(a')}D.B.
= (¿PQi'(L) - L+3¿Y(1)Qi'(L-1)P)(aM(a - a'),
(26)
QiP
D.B. relations of TY'(2) and Qaj yields
and
{Qaj (a),TY'(L)(a')}D.B.
(19)
(20)
= - (¿a(1)(?Y'(i-1)j
-L+3¿'(1)Qa'(L-1)j) (a)¿(a - a'),
(27)
L
by the expressions
Q
'(L)
Z'(L), Q '(L-1)j = T '(L-1)£j.
(28)
D.B.-commutes with Qaj. Applying Qaj to Qk'(2) gives
Their D.B. relations with order 1 supersymmetry
generators read
{Qaj(a), Qfc'(L)(a')}_D.B.
(21) = ¿jTa'(L) (a^(a - a') + ¿i
'(1)
a
+j sL+3)T- «(Li) U)Z'(L-1)
Tj'(L-1)j (29)
(a)¿(a - a')
1
and 2.2.2 Higher-spin symmetries from the world-sheet
~ , „ CFT perspective
[Qi$(a), Q~/S (a')}d.b.
, _ <5(i) ~ ^ This subsection is devoted to the twistor part of the
T^(L-1) l++2 Sj rTips(L-2')l\(a)S(a — a') left-moving world-sheet CFT justifying the arguments
+si
,, ( r \ (30) above discussion relied on. To apply the 2d CFT
ty + ^ 8(l+3) T — 8(l+3) U) technique to the model (1) it is helpful to carry out
x [sPZs(l- 1 )__1— S5( 1)zP(a)S(a-a') Wick rotation to Euclidian signature world-sheet
2 1 t —> ta2, a a1 =>■
where (36)
-u 1.9 — — / 1 ■ 9 \ ^ ^
a+ ^ z = a1 +ta2 ,a ^-z = -(a1 - ta2).
Tks(L-1)j = Tkj Zs(L-1). (31)
Continuing further calculation of D.B. relations of
gl(4l4, R) supersymmetry generators and order L d+ ^ dz = 1 (d1 - id2) = c1, generators allows to find complete set of irreducible L
The following changes of the world-sheet derivatives
(37)
d- ^ -d-z = -1 (d1 + td2) = -Z, 2d volume element
(32)
'TiPiP)j\q\ = Tij[q]ZP(p), q = 1, 3, p + q = L V 7 dTda ^ ida1da2 = id2z, (38)
TTap(P)j[q] = Tap(p) q = 0, 2, 4,p + q = L;
p
and fermionic generators anf[ supertwistor components
QJ(P)M= T*P(pj\q = 1, 3,P + q = Ya ^Ya(z), Ya ^-yA(z), (39)
QMriM = Qj^ZP(p), q = 0, 2, 4, p + q = L.
result in the Euclidean action
Se =i d2z(YAdZA + YAdZA). (40)
Relevant (traceless) products of bosonic components of supertwistors are defined in (16) and the definition of (traceless) products of fermionic components is given
in (12), (24) and by the expressions Non-trivial OPE for the supertwistors of the left
moving sector is
j = nj - 1 (nOsj j j], Qi = ni, Qj4 = nj. ZA(z)YB (w) - . (41)
(34) ^ ^w ^ SA
There are also generators of the form The supertwistor part of the left-moving stress-energy
, , ., , / n -r i tensor equals4 TZa(p)^i[q\, UZa(p)C[q\ 0 < q < 4,p+q = L—1.(35)
T ■ U U i Ltw(z) = —Yad ZA (42) It is these generators that correspond to non-tensor
operators in the world-sheet CFT. They are present on so that Yb and Za are primary fields of conformal
the r.h.s. of (29) and (30) implying breaking of order weight 1 and 0 respectively. L
In Berkovits twistor string theory GL(1, R) neCessary condition for the considered global
symmetry is gauged so that generators carrying the symmetries to survive in the quantum theory is that
factor of T are set to zero. However, glt(1, R) their generators become primary fields, i.e. their OPE's
symmetry, being anomalous, cannot be gauged thus with the stress-energy tensor are anomaly free. As we
the generators carrying the factor of U cannot be put find the generators containing the factor of T or U fail
L
possible to find a set of generators with closed D.B. Direct caicuiatiori using the OPE definition (41)
relations that would correspond to the primary fields. showg that ^4, r) generators Tj, Tj, Qai, Qi$ and
As a result the quantum symmetry of the twistor string t are primary fields of unit weight, while U is not [31
reduces to the symmetry SL(4|4, R) x SL(4l4, R) for lJ
the sector of closed strings and its diagonal subgroup —8 _2
for the sector of open strings. Ltw(z)U(w) ~ {z — w)3 + O((z — w) ). (43)
4 It is assumed that composite operators depending on a single argument are normal-ordered but normal ordering signs : : will be omitted.
Higher-order generators (32), (33) also become primary fields of unit weight. While OPE's of the generators (35) with the stress-energy tensor are anomalous
Ltw (z)TZ a(p)ei[q](w)
~ - Za(p)ei[q] (w) + O((z - w)-2),
-
Ltw (z)UZ a(p)ei[q](w)
~ -S-w)3Za(p)ei[q] (w) + O((z - w)-2).
In the case p = q = 0 one recovers the OPE's of g1(1, R) and g1t(1, R) generators with the stress-energy tensor. p = 0 q = 0
associated symmetries are broken. Since generators
L
by order 1 supersymmetries (cf. Eqs. (26), (27), (29), (30)) it appears that higher-spin symmetry is broken for arbitrary value of L except for L = 1, for which quantum-mechanically consistent global symmetry is isomorphic to SL(4|4, R).
3 Conclusion
The main result reported in this note is the identification of the inifinite-dimensional classical symmetry of the Berkovits twistor string model and
its extension with ungauged GL(1, R) symmetry. Associated Noether current densities have been constructed in terms of PSL(4|4, R) supertwistors. For the generalized twistor string model the D.B. relations of the Noether current densities have been shown to form the TSA inifinite-dimensional Lie superalgebra with the finite-dimensional subalgebra spanned by the g1(4|4, R) generators and the generator of constant shifts of the supertwistor components. The full classical symmetry of the twistor string action is generated by the direct sum of two copies of TSA superalgebra for the left- and right -movers. Clsssicsl symmetry of the Berkovits model is described by the subalgebra of TSA obtained by going on the constraint shell YaZa + niei — 0. Its finite-dimensional subalgebra is spanned by ps1(4|4, R), 'twisted' (1,R) generators and that of shifts of the supertwistor components.
More detailed account of the material covered in the present note can be found in Ref. [7].
Acknowledgement
It is a pleasure to thank the Organizing Committee of QFTG-2014 conference and for warm hospitality in Tomsk and support.
References
[1] Witten E., 2004 Comm. Math. Phys. 252 189, [hep-th/0312171],
[2] Berkovits N.. 2004 Phys. Rev. Lett. 93 011601 [hep-th/0402045],
[3] Corn J., Creutzig T. and Dolan L., 2010 J. High Energy Phys. 1010 076 [arXiv: 1008.0302 [hep-th]].
[4] Townsend P. K., 1991 Class. Quantum Grav. 8 1231.
[5] Berkovits N. and Witten E„ 2004 J. High Energy Phys. 0408 009 [hep-th/0406051],
[6] Abou-Zeid M., Hull С. M. and Mason L. J., 2008 Comm. Math. Phys. 282 519 [hep-th/0606272],
[7] Uvarov D. V., 2014 Nucl. Phys. 889 207 [arXiv: 1405.7829 [hep-th]].
Received, 11.11.2014
Д. В. Уваров
БЕСКОНЕЧНОМЕРНЫЕ СИММЕТРИИ МОДЕЛЕЙ ТВИСТОРНЫХ СТРУН
Показано, что, подобно моделям безмассовой суперчастицы, пространственно-временная симметрия классического действия твисторпой струны Берковпца является бесконечномерной. Ее супералгебра содержит конечномерную подалгебру, которая включает генераторы psl(4|4, R) супералгебры. В квантовой теории данная бесконечномерная симметрия нарушается до SL(4|4, R) симметрии.
Ключевые слова: супертвистор, твисторная струна, бесконечномерная симметрия, аномалия.
Уваров Д.В., кандидат физико-математических наук, старший научный сотрудник. ННЦ Харьковский физико-технический институт HAH Украины.
Ул. Академическая, 1, 61108 Харьков, Украина. E-mail: d_uvarov@hotmail.com; uvarov@kipt.kharkov.ua
