CC BY
22
UDC 530.1; 539.1
U(N) spinning particles and higher spin fields on Kâhler backgrounds
R. Bonezzi
Dipartimento di Fisica, Università di Bologna and INFN, sezione di Bologna, via Irnerio 46, 1-40126 Bologna, Italy
E-mail: [email protected]
In this short contribution we will review the quantization of U(N) spinning particles with complex target spaces, producing equations for higher spin fields on complex backgrounds. We will focus first on flat complex space, and subsequently present our model on an arbitrary Kahler manifold. In the final section, we will specialize to (p, q)-forms on arbitrary Kahler spaces and present their one-loop effective actions as well as issues related to Hodge duality.
Keywords: sigma models, higher spins, gauge symmetry.
1 Introduction
Spinning particle models [1—4] have been a useful framework to study fields of different spins in a first-quantized approach. Wordline techniques can indeed give manageable representations of one-loop quantities in quantum field theories, such as effective actions, amplitudes and anomalies (see for a review [5,6] and references therein), and allow to describe ordinary higher spin fields in first quantization. We will present here a class of spinning particles enjoying a U(N)-extended supersymmetry on the worldline [7,8], that naturally live on complex manifolds and give rise to complex higher spin equations. Although a direct spacetime interpretation of these models is prevented by the complex nature of the target space, they are useful playgrounds to study issues related to the quantization of higher spins. They can as well provide insights in the study of Kahler geometries, supersymmetric field theories and QFT’s in curved backgrounds.
More precisely, spinning particles are quantum mechanical models, enjoying local supersymmetries on the
N
persymmetries describe spin N/2 fields in four dimensional spacetime. The constraints on the Hilbert space of the particle theory, arising from the gauging, translate into a set of differential equations for the spacetime field, viewed as the wave function of the quantum mechanical model. Interactions with scalars, gauge fields and gravity can be achieved, when allowed, by coupling the particle theory to suitable backgrounds. Quantizing the particle, one can recover in a rather simple way various useful objects of the related QFT, such as effective actions, n-point functions, anomalies and so on. The path integral quantization for these non-linear sigma models, arising in gravitational backgrounds, requires regularization [6,9,10]. This is essentially related to ill-defined products of distributions in perturbative computations, and we will use Time Slicing regulariza-
tion (TS) in all the computations that will be showed. We organize the paper as follows: in the next sec-U(N)
Dirac quantization in flat complex space. The end of the section will be devoted to the coupling to a curved Kahler background. In the last section we will focus on the U(2) model, studied in [11], describing (p,q)-forms on an arbitrary Kahler space. The local proper time expansion of the one-loop effective action will be briefly sketched, along with the first Seeley-DeWitt coefficients. We will comment at the end on exact relations between Hodge dual forms that have been extracted from the particle model.
U(N)
The graded phase space of the model is spanned bycomplex coordinates of Cd and momenta: xp(t), xp(t) and pp(t), pp(t), along with the fermionic superpartners Vf(t) mid Vp(t), with i = 1, 2,..., N. They obey canonical (anti)-commutation relations:
[xp,pv] = iSp . [xp.p,} = iSp . {Vp. VV} = SppS3i .
(1)
Quadratic operators constructed from the basic variables provide generators for the U(N)-extended world-line SUSY:
h = PpPp , Qi = v,p Pp , Qi = Vpi Pp ,
J = 2[*?.i;j = vp *l - 2 j (2)
where we raise and lower indices by means of the flat complex metric SpV and to inverse. H, Qi wid Qi are the hamiltonian and supercharges, respectively generating worldline translations and supersymmetries, while Jj generates U(N) R-symmetry rotations. Given the (anti)-commutation relations (1), the above generators obey an extended supersymmetry algebra with
R-symmetry group U(N):
[j Qk] = Sf Qj, [j Qk] = —Sf Qj,
[J,Jk] = s? Jj - sf Ji, {Qj,Qj} = Sj H (3)
the other (anti)-commutators being zero. In order to construct a worldline action that is invariant under local symmetries generated by (2), we couple the generators to one-dimensional gauge fields: an einbein e(t) for local worldline translations, complex graviti-nos Xi(t) and Xj(t) f°r local supersymmetries and a one-dimensional U (N) gauge fie Id aj (t). Together with the symplectic kinetic terms, we obtain the phase space U(N)
the spacetime fields the supercharges Qj and Qj act as Dolbeault operators1 and their hermitian conjugates, generalized to multi-forms such that Qk — — id(k) meaning that it antisymmetrizes the derivative only among the indices of the k-th block. The remaining constraints have then the form of generalized Bianchi and Maxwell equations, i.e.
Qj |F) =0 - d[MFMl...Mm],..,Vl...Vm =0 ,
Qj |F) =0 - dPFMM2...Mm,...,Vi...Vm =0 .
(5)
S :
dt
PpXp + Pp xp + ¿V’u — eH — ixjQ
It is natural to interpret the F fields as higher spin curvatures that obey Maxwell-like equations. It is indeed possible to solve the first of (5) by introducing a gauge potential ^ as |F) = QiQ2...Qn |^) or, in tensor language:
—*XiQi — aj ( Jj — sSj
(4)
Fp
P1 [m],
dp1 ...dpN ^p1 [m-1],
(6)
where we added a Chern-Simons coupling s that is quantized as s = m — | for integer m. The equations of motion for the one-dimensional gauge fields constrain the classical generators to vanish. At the quantum level, the operators (2) impose constraints on the Hilbert space, by requiring that they annihilate physical states:
|$) € Hphys ^ Ta |$) =0 ,
where TA = (H, Qj, Qj, Jj — sSj). This is allowed since (3) is a first class superalgebra, and amounts to Dirac quantization. The above constraints will be the aforementioned higher spin equations obeyed by spacetime fields, that sit as x-dependent coefficients inside |$).
2.1 Dirac quantization in Bat space
In a Schroedinger picture, we shall realize the fermionic oscillator algebra in (1) treating the ¿P operators as Grassmann-odd variables, and the ¿p as odd derivatives thereof: ¿p — • ^he states in
the Hilbert space will have a finite Taylor expansion in powers of ¿’s, so that the coefficients of the expansion are multi-form spacetime fields with only holomorphic indices. The Jj — sSj constraints are purely algebraic on the multi-forms, and impose irreducibility conditions. After imposing them, the only states that survive in
N
of m antisymmetric indices, with s = m —
where each set of indices is antisymmetrized:
[m—i] := ...Mm].
Let us notice that for N > 1 the Maxwell equation (5) on the potential is higher derivative. We also mention that for N = 1 we have ordinary (p, 0)-forms ^(p,0), carefully studied in [12], while for m = 2 the gauge potentials are completely symmetric tensors ^Pl...PN. Since (Qj)2 = 0 for each i, it is easy to see that the curvature F is indeed invariant under a gauge transformation of the form S |^>) = Qj |Aj), that is
S^pMp],...,pN [p] = dp1 Ap1
p1 Ap1[p-i],...,pN [p]
+ ... + dpN Ap1)p],...,pN [p-i]
(7)
We already noticed that the Maxwell-like equations are higher derivative in terms of the gauge field: QjQ1...Qn |^) = 0. It is actually possible to introduce a second order wave operator, analogous to the Fronsdal-Labastida one, and reduce the field equations to second order: (—H + Qj<3j) |^) = QijQj |pjj i bymeans of an auxiliary compensator |pjj). In tensor language it reads
N
dvd ^p1 [p] ,...,pN [p] ^ , dpzd ^p1 [p] ,..,vpz[p-1] ,...,pN [p]
j=1
^ ./ dpi dpj P pj[p],..,pi[p-1],..,pj [p-1],..,pN [p] . j=j
The field equations are invariant under the gauge transformations (7), provided that the compensator field transforms as the divergence of the gauge parameter. It turns out that it is possible to gauge fix the compensators to zero, at the price of having transverse gauge parameters2: d • A = 0.
On a complex (p,q)-form AM1...Mpi>1...i>q dzv1 A ...dz,p A dzV1... A dzl'q the Dolbeault operator acts as an holomorphic exterior
|F) — fp1 [m],...,pN [m] (x, x) ,
where we denoted ^[k] := [^1...^k]. Symmetry between block exchanges ensures they belong to a U (d) rectan-
mN
derivative: d := dz^d.
2For related issues on real spacetime, see for instance [13,14]
1
o
2.2 Coupling to curved space
We analyze here the changes needed to couple the spinning particle to an arbitrary background metric. Let us consider as target space a D = 2d dimensional Kahler manifold, equipped with a metric gp.(x,x) in holomorphic coordinates. Having in mind the minimal coupling, it is sufficient to replace suitably covari-antized constraints in the action (4). To this aim, we define U(d) “Lorentz” generators Mp = ^ [¿f, 9] so that we can construct covariant momenta and supercharges3 :
np = Pp + i rp„ ma , np = pp , (9)
Qj = g2 npg-2 , Qj = ¿p gpi/ g1 n.g-1 (10)
The superalgebra (3) is deformed by the target space geometry, namely one has
are realized now as Dolbeault operators and their her-mitian conjugates: d, d mid d^, d^, while the superalgebra (3) closes on the hamiltonian that acts as the Hodge laplacian
21
A = —{d, dt} = — + - RpPAff MppM^ .
Bianchi equations can be locally integrated by introducing a potential: F(p+1,q+1) = ddA(p,q). In this particular model, even if we are dealing with differential forms, Maxwell equations (d^F = d^F = 0) are higher derivative with respect to the potential A. As we did in the general case, it is possible to have second order field equations by introducing a compensator:
(A + + <9df) .
(12)
(H)
The equations (12) are gauge invariant under the combined transformations of the gauge field and compensator
where the minimally covariantized hamiltonian reads H0 = g1 gpi/npnpg- 2. One can see from (11) that the algebra is no longer first class, and hence the model is inconsistent, on general Kahler backgrounds. Important exceptions are the cases N = 1, 2, presented in the next section, that can be quantized on any curved back-
N > 2
one can still quantize the model on particular backgrounds. For instance, it is possible to quantize the
N
“spin”, on Kahler spaces with constant holomorphic curvature [8], i.e.
RppA<r A (gpPgA<r + gp.gA.) .
3 N=2, (p,q)-forms on Kahler spaces
We shall focus in this section to the model with
N=2
gpP. We decide to realize the fermionic operators in a slightly different way: here we will treat ¿p and ¿2p as odd coordinates, and ¿p, ¿2p as derivatives thereof. In this way, states in the Hilbert space are (p, q)-forms. In [11], several theories of differential forms were in-
U(2)
we restrict ourselves to present one of those models. To obtain the model we are interested in, it is sufficient to gauge only the U (1) x U (1) subgroup of the R-symmetry U(2) generated by J1 md J|. In such a case one is free to have two different Chern-Simons s1 s2 s
J11 J22
constraints, and physical states will be forms F(m,n), for given and arbitrary m, n. The four supercharges
¿A
(p>q)
¿p(p- 1,q-1) =
dA(p-1,q) + dA(p,q-1) , ^tA(p-1,9) - dt A(p,q-1)
(13)
and one can see again that gauge fixing the compensator to zero would restrict the gauge parameters to be transverse.
Let us turn now to the path integral quantization of
U(2)
on a circle with external gravity, one finds an expansion for the QFT one-loop effective action of the (p, q)-form in a gravitational background. We will present the heat kernel expansion in terms of local Seeley-DeWitt coefficients. In order to quantize the spinning particle, we have to take into account the gauge fixing of local worldline symmetries and Faddeev-Popov determinants. Because of the topology of the worldline circle, one is left with three modular integrals: the usual one over ¡3, being the proper length of the circle, and two angular integration over 0 and taking into account
U(1)
tors. The resulting partition function is given by
[g]
f f f dn f «
ddxoddxo
(2n,0)d
g(xo, xo) (e Sin
(14)
where x0 is an arbitrary fixed spacetime point, S;nt is the interaction part of the spinning particle action and ^(^, 0) is the modular measure given by
M(^, 0) = e-i(p+1-d/2)0ei(q+1-d/2)e
2 cos —
d-2
0
2 cos —
x
d-2
(15)
3The g factors ensure that (Qi) = Qi
4In the modular integration one encounters poles along the integration path. Detailed explanation of the prescription to deal with such poles can be found in [11]
X
X
After performing the worldline perturbative computation, and evaluating the modular integrals4 we organize the effective action expansion, up to order r2j as follows:
[g] = -r
dr
r
ddxoddxc
(2nr)d
g(xo,xo) V1 {1 + V2rR
+r2
v3 (RuPA<t) + v4 (RpP) + v5 R2 + v6V2^ .
(16)
The Seeley-DeWitt coefficients vi are given by
V1 =
d 2 d 2
V2 = 1 - 12k1 , 6
p j \ q
V3 = - k2 + k3 , V4 = - — - k1 + 4k2 - 2k3 ,
180 360
V5 = -1 + k1 - 3k2 + k3 , V6 = -1 - k1 ,
72
60
(17)
where the numerical factors ki read
k1
k2
k3
p(d - 2 - q) + q(d - 2 - p) 24(d - 2)2 p(d - 2 - p) + q(d - 2 - q) 24(d - 2)(d - 3) p(d - 2 - p)q(d - 2 - q) 2(d - 2)2(d - 3)2 .
(18)
ki
p(1 - q) + q(1 - p) 24
k2
p + q 24
k3 = pq (19)
kj
used to make manifest the symmetry under the exchanges p ^ ^d p ^ d — 2 — q. The first exchange is related to the symmetry under complex conjugation, that states the equivalence Zp,q [g] = Zq,p[g] and is exact. The second exchange relates Hodge dual forms and is more subtle. Despite the manifest symmetry in the coefficients (18), it is not an exact symmetry at the quantum level, and one can already see that in d = 3 the kj (19) are not invariant under p ^ 1 — q.
d
pears in higher order Seeley-DeWitt coefficients. The spinning particle model, however, allows us to find an exact non-perturbative result for the mismatch, that is
aWe denoted Zp,q = fc
purely topological. The derivation can be found in [11], and one has5
Zd-2-q,d-2-p(3) — Zp,q (3) = ( —)q+d2^1,, (3)
+(—)p+dzp,P-1(3) + (—)p+q Z°_pM_1(3)
q
+(d — 1 — p)(—)p+q Y, (—)m(q +1 — m) ind(nm,°, d)
m=0
p
+(d — 1 — q)( — )p+^(—)n(p +1 — n) ind(Q",°, d)
+(-)
n=o
d \ ( d \ d2 p+1-2 q+1-2 -T
x(M)
(20)
In the above formula ind(^m,°, <9) is the Dolbeault in-
( m, 0)
sion can be found in [11,15]. x(M) is the Euler characteristics of the manifold, and the top forms Zp,d-1, Zd-1,q, Zd-1,d-1 are related to the Ray-Singer analytic torsion [16] via
Z0d-1 =^(-)”+1(n +1)lnTd-p+n(M) .
(21)
A few comments are now in order. First of all, let us stress that the overall coefficient v1 gives the number of
pq
d - 2
Other coefficients can and indeed are non-vanishing, and represent the topological contribution of a nonpropagating form. As a second remark we should notice that the result (18) holds only for d > 3. In d =2 only scalars propagate, and all the ki vanish. In d =3 instead, they have the form
Having presented the expansion for the effective actions r2
d = 2 d = 3 sidered in [11] and agree with (20), giving a nontrivial check.
4 Conclusion
In this short contribution, we presented the U(N) spinning particles, introduced in [7] and subsequently-studied in [8], where they were shown to describe a class of gauge invariant higher spin equations, close in form to Fronsdal-Labastida equations for real space-times and to Maxwell-like equations recently introduced in [13]. We briefly described the possibility of defining the model on a Kahler manifold, and finally fo-U(2)
(p, q)
as well as the heat kernel expansion of their effective actions and issues related to Hodge duality. Even if a genuine spacetime interpretation is somehow prevented by the complex target space, these models can provide interesting insights in the general problem of higher spin field theories, sharing crucial properties such as the appearance of constrained gauge invariance and compensator fields [17]. When quantized on Kahler backgrounds, they show quite a rich structure, such as the topological issues related to Hodge duality, or the U(1)
could be a useful instrument in Kahler geometry.
CXJ
o
Acknowledgement Francia for valuable discussions.
I would like to thank Fiorenzo Bastianelli and Carlo Iazeolla for working with me on this topic, and Dario
References
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Received 01.10.2012
P. Вонецци
U(N) СПИНОВЫЕ ЧАСТИЦЫ И ПОЛЯ ВЫСШИХ СПИНОВ НА КЭЛЕРОВОМ ФОНЕ
Даётся краткий обзор квантования U(N) спиновых частиц на комплексном фоне. В начале рассматривается плоское комплексное пространство и затем рассматривается модель на произвольном Кэлеровом многобразии. В последнем разделе рассматриваются (p, д)-формы на произвольных Кэлеровых пространствах и даётся их однопетлевое эффективное действие а также вопросы связанные с дуальностью Ходжа.
Ключевые слова: спиновые частица, поля высших спинов, Кэлсровы многообразия, дуальность.
Вонецци, Р.
'Университет Болоньи.
via Irnerio 46, 1-40126 Bologna, Италия.
E-mail: [email protected]
