Pure spinors in classical and quantum supergravity Martin Cederwall Department of Physics Chalmers Univ. of Technology SE- four.oldstyleone.oldstyletwo.oldstyle nine.oldstylesix.oldstyle Gothenburg Sweden

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Pure spinors in classical and quantum supergravity

Martin Cederwall

Department of Physics, Chalmers Univ. of Technology, SE-412 96 Gothenburg, Sweden

and

NORDITA, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

Abstract: This is an overview of the method of pure spinor superﬁelds,

written for “Handbook of Quantum Gravity”, eds. C. Bambi, L. Modesto

and I. Shapiro. The main focus is on the use of the formalism in maxi-

mal supergravity on a ﬂat background. The basics of pure spinor super-

ﬁelds, and their relation to standard superspace, is reviewed. The pure

spinor superstring model of Berkovits is brieﬂy discussed. Consequences

for divergence properties of loop diagrams in maximal supergravity are

restated. Some ﬁnal remarks are made concerning desirable development

of the theoretical framework.

email: martin.cederwall@chalmers.se

arXiv:2210.06141v1 [hep-th] 12 Oct 2022

Contents

1Introduction 3

2Pure spinor superﬁeld theory 3

2.1From superspace to pure spinor superspace . . . . . . . . . . . . . . 5

2.2Non-minimal variables, integration, BV actions . . . . . . . . . . . . 8

2.3Othermodels ............................... 10

2.4Pure spinor partition functions and superalgebras . . . . . . . . . . . 11

3D“11 supergravity 12

3.1Geometry vs. 4-form........................... 13

3.2BVaction................................. 15

3.3Twisting.................................. 16

4Superstrings 17

5Quantum theory 18

5.1Gaugeﬁxing ............................... 18

5.2Perturbativeresults ........................... 20

6Remarks 21

References 23

1Introduction

Pure spinor superﬁeld theory [1] provides a solution to the long-standing problem of

covariant quantisation of (Brink–Schwarz) superparticles [2,3] or (Green–Schwarz)

superstrings [4] with manifest supersymmetry, or roughly equivalently, to the prob-

lem of ﬁnding oﬀ-shell superspace formulations of maximally supersymmetric ﬁeld

theories, including supergravity.

Concretely, the diﬃculties with space-time supersymmetric particles and strings

manifest themselves as a mixture of ﬁrst and second class constraints in the same

spinor. This is the famous κ-symmetry [5–7], which is necessary for the superparti-

cle/superstring action to describe the dynamics of a 1

2-BPS object.

In the present overview, we will not start with these superparticle or -string

actions. Rather, the introduction of pure spinor variables will be motivated by the

structure of the (on-shell) multiplets of maximal super-Yang–Mills theory (SYM)

and supergravity (SG) in their traditional treatment on superspace. The relation of

the pure spinor formulation to the Green–Schwarz superstring is explained in ref. [8].

The basics of the formalism is laid out in Section 2. In Section 3it is applied

to supergravity, with maximal supergravity as main focus. A brief account of the

pure spinor superstring theory of Berkovits is given in Section 4. Quantum theory is

sketched in Section 5, and some convergence results for loop diagrams are restated.

Finally, some remarks are made in Section 6concerning possible reﬁnement and

development of the formalism.

The technical level of the presentation is kept at a minimum. Instead, we aim at

collecting results from the sources in the reference list and present them as concisely

and coherently as possible, while emphasising concepts rather than techniques.

2Pure spinor superﬁeld theory

Before going into a more precise derivation of pure spinor superﬁeld formulations

of speciﬁc supersymmetric models, we would like to sketch what lies at the heart of

the formalism. The supersymmetry algebra (which of course is a subalgebra of the

super-Poincaré algebra) takes the generic form tQα, Qβu “ 2γa

αβBa. Here, αis some

(possibly multiple) spinor index, and Qα“B

Bθα` pγaθqαBa. Covariant fermionic

derivatives Dα“B

Bθα´ pγaθqαBasatisfy tQα, Dβu “ 0. They anticommute among

themselves as tDα, Dβu “ ´2γa

αβBa— ﬂat superspace in endowed with torsion

Tαβa“2γa

αβ.

Suppose we introduce a bosonic spinor λαsubject to the constraint

pλγaλq “ 0.(2.1)

Then we may form a fermionic operator

Q“λαDα,(2.2)

which, thanks to the constraint on λis nilpotent: Q2“0.

It seems meaningful to consider the cohomology of Q, acting on functions of x,θ

and λ. This cohomology is guaranteed to be supersymmetric, since Qanticommutes

with the supersymmetry generators. It thus describes some supermultiplet. As it

turns out, any linear supermultiplet in any dimension may be obtained this way. In

the case of on-shell multiplets the virtue of the formalism is even greater, since it

seems to oﬀer a natural way to an oﬀ-shell formulation by relaxation of the linear

“equation of motion” QΨ“0. The correspondence will be made more precise below,

ﬁrst for D“10 super-Yang–Mills theory and later for D“11 supergravity.

A word of caution: We will refer to a spinor λsubject to the constraint (2.1)

as a “pure spinor”. This is a slight misuse of the mathematical terminology. A pure

spinor, in the sense of Cartan [9], is a chiral spinor in even dimension D“2n,

constrained to lie in the minimal Spinp2nqorbit of the spinor module. This implies

that, if the Dynkin label of the spinor module in question is p0. . . 01q, monomials of

degree of homogeneity pin λbelong to the single module p0. . . 0pq. The concept of a

pure spinor is not deﬁned in other cases, neither for odd dimensions or for extended

supersymmetry. In certain cases, our constrained spinors coincide with Cartan pure

spinors. This happens notably in D“10. There, the spinor bilinears are a vector

p10000qand a self-dual 5-form p00002q, and the constraint in the vector immediately

puts λin the minimal orbit. In other situations, for example D“11 which we will

encounter later, where the symmetric spinor bilinears are a vector, a 2-form and

a5-form, the vector constraints puts λin (a completion of) an intermediate orbit,

which is not the minimal one.

2.1From superspace to pure spinor superspace

Although we ultimately aim at addressing supergravity, the introduction of pure

spinor superspace is much simpler in the setting of super-Yang–Mills theory [10],

ﬁrst treated in superspace in ref. [11].

Flat p10|16q-dimensional superspace, appropriate for D“10 super-Yang–Mills,

has coordinates ZM“ pxm, θµq. There is no metric on superspace, but a super-

vielbein EMA. The Lorentz frame indices A“ pa, αqconsist of a Lorentz vector and

a chiral spinor. The non-vanishing superspace torsion is

Tαβa“2γa

αβ ,(2.3)

where the components are converted to Lorentz frame using the inverse vielbein:

TBC A“ pE´1qCNpE´1qBMTMN A.

Let us now recollect some known facts about D“10 super-Yang–Mills. A

connection on superspace, taking values in the adjoint of some gauge group, is

written A“dZMAM“dZMEMAAA. There is a priori two superﬁelds, Aapx, θq

(bosonic, dimension 1) and Aαpx, θq(fermionic, dimension 1

2)1. The ﬁeld strength is

F“dA `A^A, and due to the presence of torsion we have

Fαβ “2DpαAβq`2ApαAβq`2γa

αβAa.(2.4)

The symmetric product of two spinors can be decomposed into a vector Fa“

16 γαβ

aFαβ and a self-dual 5-form Fabcde “1

2¨5!¨16 γαβ

abcdeFαβ . Obviously, from eq. (2.4),

setting Fa“0expresses Aain terms of Aα, leaving only the latter as an independent

superﬁeld. This goes under the name of “conventional constraint”. Note that it is

natural, since Fαβ has dimension 1, and there are no physical and gauge-covariant

ﬁelds of this dimension in the supermultiplet we want to derive (the spinor χαhas

1As is standard, dimension is in terms of powers of inverse length.

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PurespinorsinclassicalandquantumsupergravityMartinCederwallDepartmentofPhysics,ChalmersUniv.ofTechnology,SE-634;8Gothenburg,SwedenandNORDITA,HannesAlfvénsväg34,SE-328;3Stockholm,SwedenAbstract:Thisisanoverviewofthemethodofpurespinorsuperelds,writtenforHandbookofQuantumGravity,eds.C.Bambi,L.Modest...

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Pure spinors in classical and quantum supergravity Martin Cederwall Department of Physics Chalmers Univ. of Technology SE- four.oldstyleone.oldstyletwo.oldstyle nine.oldstylesix.oldstyle Gothenburg Sweden

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