The exact quantum corrected moduli space for the universal hypermultiplet

Transcription

1 The exact quantum corrected moduli space for the universal hypermultiplet Bengt E.W. Nilsson Chalmers University of Technology, Göteborg Talk at "Miami 2009" Fort Lauderdale, December 15-20, 2009 Talk based on: Instanton Corrections to the Universal Hypermultiplet and Automorphic Forms on SU(2,1) Ling Bao, Axel Kleinschmidt, Bengt E.W. Nilsson, Daniel Persson and Boris Pioline arxiv: [hep-th]

2 Introduction: Motivation Motivation: In field theory one normally computes quantum corrections order by order in perturbation theory and then tries to add instanton corrections one by one each with its own perturbation series. Is it possible to collect all these quantum corrections into one function obtained from some basic principles and express the effective action in closed form? In some cases this can be done: We will generalize an approach to this problem based on the discrete symmetries appearing in string theory! [Green, Gutperle] This gives non-perturbative information about the theory May provide a way to define quantum field theories Mathematics: Seems to link physical properties to fundamental areas in pure mathematics (number theory and perhaps the Langlands program)

3 Introduction: The approach Our approach is to: Consider 4d supergravity theories with 2 supersymmetries obtained from the type IIA superstring compactified on Calabi-Yau 3-folds (CY3). Restrict to rigid CY3 for simplicity and derive the relevant function, i.e. the Eisenstein series, for the universal hypermultiplet. Fourier expand this function and compare the result to the structure of the various types of perturbative and non-perturbative quantum corrections mentioned above.

4 Type II supergravity compacified on Calabi-Yau threefolds: common sector Compactification on a CY3 gives N = 2 supergravity in 4d Mink! Low-energy type IIA and IIB supergravity in 10d: the bosonic sector NS-NS sector common to both cases: g MN, B MN, φ The Hodge numbers h 0,0 = 1, h 1,1 1, h 2,1 0, h 3,0 = 1 imply the following massless fields in four dimensions: g µν, g i j (2h 2,1 scalars), g i j (h 1,1 scalars) B µν, B i j (h 1,1 scalars) φ Note: we will dualize B µν to ψ. no isometries means h 1,0 = 0 and thus no vectors in this sector also h 2,0 = 0 means no massless fields from B i j Summary: the NS-NS sector provides one metric, one tensor and one scalar plus 2h 2,1 + 2h 1,1 scalars, i.e. the geometric moduli of CY3 (with the Kahler ones complexified)

5 Type II supergravity compacified on Calabi-Yau threefolds: the IIA RR sector Additional physical moduli from embedding CY3 in supergravity: R-R sector type IIA: C M, C MNP give in 4d C µ (one graviphoton from h 0,0 = 1) C µ i j (h 1,1 vectors) C i j k (2h 2,1 scalars) C i j k (2 scalars from h 3,0 = 1) >denoted χ and χ Summary of N = 2 multiplets in type IIA: one graviton multiplet one tensor mult. = the univ. hypermult. (UHM): φ, ψ, χ and χ h 1,1 vectormultiplets h 2,1 hypermultiplets => Moduli space: M A = M VM A (2h 1,1) M HM A (4h 2,1 + 4) is a product up to discrete groups which follows from susy and holonomy arguments (see e.g. [Aspinwall])

6 Type II supergravity compacified on Calabi-Yau threefolds: the IIB RR sector Similar statements true for type IIB: R-R sector type IIB: C, C MN, C + MNPQ give in 4d C (one scalar) C µν (one tensor), C i j (h 1,1 scalars) C µν i j (h 1,1 scalars), C µ i j k (2h 2,1 vectors), C µ i j k (h 3,0 = 1 vector, not 2 since to selfdual), C i j k l (h 2,2 scalars, but not counted due to selfduality) Summary type IIB: one graviton multiplet, one tensor multiplet (UHM) h 2,1 vector multiplets, h 1,1 hypermultiplets => Moduli space: M B = M VM B (2h 2,1) M HM B (4h 1,1 + 4)

7 Comments Comments: The moduli spaces are in general complicated: N = 2 susy not enough to make them coset spaces, but we know that M VM is a 2n-dimensional special Kahler (SK) space which is Kahler M HM is a 4n-dimensional quaternionic-kahler (QK) space which is NOT Kahler in general torus compactifications give more susy and moduli spaces which are coset spaces (see e.g. [Aspinwall]) Simplifications arise if we consider rigid CY3 s: [Cecotti, Ferrara, Girardello] They have h 2,1 = 0 There are many examples of such CY3 s Type IIA case gives then M UHM A = SU(2, 1)/(SU(2) U(1)) This is the case we will study here: M UHM A is homogeneous and Kahler, a unique exception!

8 Type IIA supergravity on rigid CY3 The N = 2 supergravity effective action at tree level no scalar potential on any CY3 (from vector fields only in RR sector and thus no minimal couplings) M UHM A = SU(2, 1)/(SU(2) U(1)) enters in the scalar field σ-model having a target space metric with SU(2, 1) isometry but not in the matrix coupling the kinetic terms of the vector fields The effective action in perturbation theory [Antoniadis et al],[strominger] is believed to have only a one-loop term [Vandoren et al] Non-perturbative corrections are due to D- and NS5-brane instantons: compare IIB in 10d (see [Green, Gutperle]) the relevant double coset is SL(2; Z)\SL(2, R)/U(1) D(-1) instanton effects are encoded by the non-holomorphic Eisenstein series E s (τ, τ) = Σ (Imτ) s (m,n) m+τn, with s = 3 2s 2 the instanton action is S p=mn D( 1) = 2π mn e φ 2πimnC 0

9 Instanton corrected Type IIA supergravity on rigid CY3 At the non-perturbative level the type IIA effective action in 4d coming from a rigid CY3 depends on the double coset SU(2, 1; Z[i])\SU(2, 1)/(SU(2) U(1)) where the gaussian integers Z[i] = m + in, with m, n integers. That SU(2, 1; Z[i]) is the correct discrete group is based on: the non-abelian Heisenberg group of SU(2, 1) is known to be relevant [Becker, Becker] and assuming that SL(2, Z) is a subgroup acting on χ + ie φ and an associated Eisenstein function; a special function living on this double coset (the fundamental domain) The quantum corrected σ-model metric is no longer a homogeneous space: How do we determine its exact metric when it is not Kahler but quaternionic? > try contact potential on the twistor space Z MUHM which is Kahler and project back to M UHM Given this double coset the Eisenstein function can be obtained as a Poincare series, from an adelic construction and spherical vectors, or >

10 Construction of the Eisenstein function Given the double coset there is a standard way to get the Eisenstein function [Obers, Pioline] applied here to the UHM: E s (K) := Σ ω η ω=0(ω K ω) s ω is a non-zero 3-dim vector of gaussian integers Z[i] K is the metric on the coset SU(2, 1)/(SU(2) U(1)) obtained from the Iwasawa decomposition of SU(2, 1) the constraint on the summation needed to make the Eisenstein series an eigenfunction of the Laplace operator on the coset space the coset is similar to the upper half plane SL(2, R)/U(1) with a metric involving Im(τ) > 0: here we have R 3 x R + denoted CH 2 with a metric involving F(z 1, z 2 ) = Im(z 1 ) 1 2 z 2 2 > 0 z 1, z 2 are related to the UHM fields φ, ψ, χ, χ it has also zero third Casimir => in the principle discrete series of SU(2, 1) (see [Bars, Teng])

11 Fourier expansion of the Eisenstein function: tree level Dividing the lattice summation (over six integers) into sectors => E s = E const s + E abelian s + E non abelian s (φ) = the tree and 1-loop terms E abelian s (φ, χ, χ): encodes D2-brane instantons E non abelian s (φ, χ, χ, ψ): encodes NS5-brane instantons and bound states with D2-brane instantons E const s This sum of terms is obtained by first doing the ω 3 = 0 part of the sum over six integers not all zero, which means summing only over ω 1 = 0 since the constraint sets also ω 2 = 0 => the tree level term equals 4ζ Q[i] (s)e 2sφ, where ζ is the Dedekind zeta function 1 4 Σ 1 (m,n) (m 2 +n 2 ) s the relevant value for s is 3 2 so the corrections enter via eφ E s

12 Fourier expansion of the Eisenstein function: the perturbative term at one loop The structure of the perturbative terms and Langlands functional relation: Turning to the terms with ω 3 non-zero we need to solve the constraint: use Bezout s identity q 1 p 2 q 2 p 1 = d (for which solutions q i exist only if d = gcd(p 1, p 2 )) m 1, m 2 then replaced by m, d, after a Poisson resummation in m > m the non-abelian term comes from the sum with m non-zero the m = 0 then needs another resummation involving n 1, n 2 which produces two new integers l 1, l 2 to sum over which gives two kinds of terms: First l 1, l 2 both zero provides the second constant term i.e. the one loop term: 4ζ Q[i] (s) Z(2 s) Z(s) e 2(2 s)φ which leads to a generalized Langlands functional relation with s = 3 2 we get e( φ) and thus no φ dependence in e φ E s and second: l 1, l 2 not both zero >

13 Fourier expansion of the Eisenstein function: the non-perturbative terms l 1, l 2 not both zero: the abelian non-perturbative terms : e 2φ ( ) (l 1,l 2 ) Z 2 C(A) l 1,l 2 (s) K 2s 2 2πe l φ l2 2 e 2πi(l 1χ+l 2 χ) encodes the effects of D2-brane instantons with charges (l 1, l 2 ) expanding the Bessel function at weak coupling gives the D-brane instanton action computing the coefficient C gives instanton measures involving double sums over gaussian divisors which generalize previously found D2 single sum instanton measures the non-abelian terms involve Hermite and Whittaker functions ([Ishikawa]) and are much more complicated due to the non-abelian structure of the Heisenberg group (ψ gives a twisted bundle over χ, χ which can be seen from the σ-model metric) the NS5 brane instanton measures are not yet understood expanding these functions now gives NS5 brane instanton actions

14 Summary Final comments Langlands functional relation: supported by the constant and abelian terms Z(s)P s = Z(2 s)p 2 s where Z(s) := ζ Q(i) (s)β (2s 1) where the first factor is the completed Dedekind zeta function and the second is the completed Dirichlet beta function (like the Riemann zeta function but with alternating sum only over odd integers) and E s (K) = 4ζ Q(i) (s) P s (Z) S k,q = 2π( k e 2φ + 2 k ( χ n ) 2 iqχ + 2ik(ψ + χ χ)) the abelian measure: µ s (l 1, l 2 ) = ω 3 Λ ω 3 2 2s z Λ ω 3 where the first sum is over primitive gaussian divisors and the second over all divisors z 4 4s