Action Principle for Self-dual Forms
Ashoke Sen
Harish-Chandra Research Institute, Allahabad, India

Allahabad, April 2021

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Plan:
1. A brief introduction to the issues
2. A brief review of closed superstring field theory
3. Action for self-dual forms
4. Hamiltonian formulation
5. Some speculations
References:
A.S., arXiv:1511.08220, arXiv:1903.12196
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In 4n+2 space-time dimensions we can consider theories of2n-form fields C with self-dual field strength F:
F = dC,∗F = F*: Hodge dual
There is no conventional manifestly Lorentz invariant action
describing this theory.
Normally we impose the self-duality constraint by hand.

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There are several proposals to get around this problem.
1. Give up manifest Lorentz invariance
Henneaux, Teitelboim; Schwarz; Belov, Moore; · · ·2. Introduce infinite number of auxiliary fields
McClain, Yu, Wu; Wotzasek; Martin, Restuccia; Devecchi, Henneaux
Faddeev, Shatashvili; Bengtsson, Kleppe; Berkovits

3. Introduce a non-polynomial action containing 1/(∂µ φ∂ µ φ)
multiplying a term in the action
Pasti, Sorokin, Tonin, Bandos, Lechner, Nurmagambetov, Dall’Agata, · · ·4. Other approaches

Siegel; Floreanini, Jackiw; Castellani, Pesando; Witten

In this talk we shall follow a different approach motivated bysuperstring field theory.
4Why superstring field theory?
For a long time, construction of an action for superstring field
theory had suffered from an obstruction in the RR sector.
If we could construct a field theory for type IIB string theory,
then by taking its low energy limit we could get a field theory fortype IIB supergravity– but this should not be possible since type IIB supergravity hasa chiral four form field which does not have an action.
However eventually an action for superstring field theory wasformulated– suggests that we should be able to construct an action for
type IIB suergravity!
5A brief review of the structure of classicaltype II superstring field theory:
Htotal = ⊕m,n Hm,n : A graded vector space,m,n ∈ Z/2

hA|Bi: an inner product between A ∈ Hm,n with B ∈ H−2−m,−2−nString field has two components, |ψi and |φi|ψi ∈ H ≡ H−1,−1 ⊕ H−1,−1/2 ⊕ H−1/2,−1 ⊕ H−1/2,−1/2e ≡ H−1,−1 ⊕ H−1,−3/2 ⊕ H−3/2,−1 ⊕ H−3/2,−3/2|φi ∈ H6

|ψi ∈ H ≡ H−1,−1 ⊕ H−1,−1/2 ⊕ H−1/2,−1 ⊕ H−1/2,−1/2e ≡ H−1,−1 ⊕ H−1,−3/2 ⊕ H−3/2,−1 ⊕ H−3/2,−3/2|φi ∈ H
ActionS = hφ|Q|ψi −X 1
1hφ|Q G|φi +{ψ n }
2n!
ne 7→ H,e
Q: maps H 7→ H, He 7→ H
G: maps H(identity on H−1,−1 ){A1 · · · An }: maps Hn 7→ R.
QG = GQ

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S = hφ|Q|ψi −X 11hφ|Q G|φi +{ψ n }
2n!
nDerive equations of motion from S:
Q(|ψi − G|φi) = 0
X 1
Q|φi +[ψ N ] = 0N!
Ne is defined via:
For A1 , · · · , AN ∈ H, [A1 · · · AN ] ∈ H
hA|[A1 · · · AN ]i = {A A1 · · · AN }Combine the two equations to get the interacting field equation:
Q|ψi +X 1NN!

G [ψ N ] = 08

Q|ψi +X 1N!

N

Q|φi +G [ψ N ] = 0

X 1
NN!

[ψ N ] = 0

Once we solve the first equation for |ψi, the solution for |φi tothe second equation is fixed up to addition of solutions to freefield equations:Q|φi = 0
Furthermore which solution we choose does not affect theequations of motion for |ψi.
Therefore even though we have doubled the number of degreesof freedom, · · ·· · · one set of degrees of freedom describe free fields thatcompletely decouple from the interacting sector, and areunobservable.
9Q(G|φi − |ψi) = 0
X 1[ψ N ] = 0
Q|φi +N!
N|ψi ∈ H ≡ H−1,−1 ⊕ H−1,−1/2 ⊕ H−1/2,−1 ⊕ H−1/2,−1/2e ≡ H−1,−1 ⊕ H−1,−3/2 ⊕ H−3/2,−1 ⊕ H−3/2,−3/2|φi ∈ HOn H−1,−1 , G is identity
Therefore, in this sector we can solve the first equation by
setting φ = ψ– doubling of the degrees of freedom not needed
NSNS sector fields, including the metric, reside in H−1,−1 andtherefore doubling of the metric degrees of freedom is notneeded.

10Gauge transformationseThe gauge transformation parameters are also elements of H, H.

|δψi = Q|θi +X|θi ∈ H,e ∈He|θiG |[θψ n ]i,e +|δφi = Q|θinX|[θψ n ]ine free field gauge transformation|θi– does not affect the interacting field |ψi
Non-trivial gauge transformations like general coordinatetransformation are in θ.
Peculiar property: Transformation law of φ involves ψ, not φ– strange for general coordinate transformation

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This construction holds also for type IIB string theory.
It contains a self-dual 5-form field strength in D=10.
Furthermore compactification of type IIB on K3 has self-dual
3-form field strengths in D=6.
IIB compactification on K3×K3 has chiral scalar fields in D=2.
Therefore the low energy effective action of superstring field
theory must contain Lorentz invariant action for these chiralfields.
This suggests that we should be able to construct an action forchiral fields in 4n+2 dimensions by doubling the degrees offreedom.
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In principle it should be possible to begin with the string field
theory action and take the low energy limit to get the action for
type IIB supergravity.
But we shall use the insights from string field theory and directly
write down an action for type IIB supergravity and other theories
with chiral field.
We shall take the minimalist approach where we only double thesector containing the chiral field, treating all other fields as if
they are in H−1,−1 where we can avoid doubling.
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Proposed field content:– a 2n-form potential P (like |φi)– a (2n+1)-form self-dual field strength Q (like |ψi)
Proposed action:
1
S=2Z

ZdP ∧ ∗dP −ZdP ∧ Q +LI (Q, χ) ,χ: all other fields including the metric
Q satisfies ∗Q = Q,*: Hodge dual with metric ηµν

LI : A general interacting Lagrangian density(assumed to not contain derivatives of Q when we discuss
Hamiltonian formalism)
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1
S=2

Z

ZdP ∧ ∗dP −ZdP ∧ Q +LI (Q, χ) ,Equations of motion:d(∗dP − Q) = 0,dP − ∗dP + R(Q, χ) = 0,δδχZ
LI = 0

R is an anti-selfdual (2n+1)-form defined through:ZZ
1R ∧ δQ
δ LI = −2Adding the exterior derivative of the second equation to the firstequation we getd(R − Q) = 0 .

15d(∗dP − Q) = 0,dP − ∗dP + R(Q, χ) = 0,d(Q − R) = 0

Once we find Q by solving the third equation, the anti-self-dualpart of dP is fixed by the middle equation.
Writing the first equation asd(dP + ∗dP − Q) = 0
we see that the self-dual part of dP is also fixed except for thefreedom of adding a self-dual form A satisfying
dA = 0This is a free field degree of freedom.
Furthermore the choice of A does not affect the interacting fieldequations d(Q − R) = 0.
Net content: An unobservable free field and an observableinteracting field Q.

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S=12

ZQ satisfies ∗Q = Q,ZdP ∧ ∗dP −ZdP ∧ Q +LI (Q, χ) ,*: Hodge dual with metric ηµν

Eventually we need a field that is self-dual with respect to thedynamical metric contained in χ, not with respect to ηµν .
Also the field strength involves Chern-Simons terms involving
fields χ.
Nevertheless, string field theory suggests that it should bepossible to encode this into the framework described here.
Peculiar feature: Under general coordinate transformation, bothδP and δQ must involve Q, not P.
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A brief summary of the construction
In the usual formulation of a self-dual form coupled to themetric, we consider an action of the form:
S = S1 + S2 (χ)
S1 ≡ −12Z

Z(dC + Y) ∧ ∗g (dC + Y) +dC ∧ Y ,Constraint(dC + Y) = ∗g (dC + Y)
Y: a (2n+1)-form constructed out of the other fields χS2 is a function of the other fields χ, including the metric∗g : Hodge dual with respect to the metric gµν .
Question: How do we replace the self-duality constraint with gµν
for dC+Y by self-duality constraint with respect to ηµν for Q?
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It turns out that it is possible, via a non-standard coupling of themetric to Q.
ZZ
1S1 ≡ −(dC + Y) ∧ ∗g (dC + Y) + dC ∧ Y ,2
can be replaced byZZ
1
S01 =dP ∧ ∗dP − dP ∧ Q2
Z1 T
1 T 1
1 T
1 T4n+2+ dx
Q MQ + Q
M Y − (ζ − ε) Y − Y ζ Y + Y M Y .
162224
4n+2M: a 4n+22n+1 × 2n+1 matrix determined in terms of the metric.
ζ, ε: constant matrices
Identification of variables: Q = dC + ∗dC∗dY acts as the source for the (2n+1)-form field strength.
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This construction has also been generallized to the case when
the self-duality constraint is non-linear in the field strength,
generalizing the DBI type action to chiral fields.
Vanichchapongjaroen201S=2ZZdP ∧ ∗dP −ZdP ∧ Q +LI (Q, χ) ,Note that the decoupling does not happen at the level of theaction but only at the level of the equations of motion– may seem a little unsatisfactory.
We shall now show that the same decoupling can be seen at thelevel of the Hamiltonian.
In the following we shall treat P and Q in the canonicalformulation but the other fields χ in the Lagrangian formulation.
Result: Routhian (although we shall denote it by H)
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Conjugate momenta:
ΠiP1 ···i2n = −(∂0 Pi1 ···i2n + ∂[i1 Pi2 ···i2n ]0 ) −X

i1 ···i2n j1 ···j2n+1 Qj1 ···j2n+1 ,j1 ,··· ,j2n+1
j1 <j2 <···<j2n+10i ···i2n−1

ΠP 1= 0,i ···i2n+1

ΠQ1= 0.

Note: The last two are constraint equations
H=XZ

d4n+1 x ΠiP1 ···i2n ∂0 Pi1 ···i2n − L

i1 ,··· ,i2n
i1 <i2 ···<i2nup to addition of constraints.
1. Compute the Poisson brackets of the constraints with H togenerate more constraints
2. Analyze Poisson brackets of the constraints with each otherto sort out first and second class constraints.
22Define new variables:1  i1 ···i2nΠi±1 ···i2n ≡ΠP± i1 ···i2n j1 ···j2n+1 ∂[j1 Pj2 ···j2n+1 ]2Result: Second class constraints:
i ···i2n+1

ΠQ1=0i1 ···i2n
Qj1 ···j2n+1 + 2 i1 ···i2n j1 ···j2n+1 Π−+∂LI=0∂Qj1 ···j2n+1i ···i2n+1– can be used to eliminate Qj1 ···j2n+1 and ΠQ1.

First class constraints generating gauge transformations:
0i ···i2n−1

ΠP 1= 0,∂i1 ΠPi1 ···i2n = 0

The second constraint is equivalent to:i1 ···i2n⇒ ∂i1 Π±=0

Π± are gauge invariant.
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The Routhian is given by H = H+ + H− whereZ
i1 ···i2n i1 ···i2nH+ = − d4n+1 x Π+Π+,Z
H− =d4n+1 x Πi−1 ···i2n Πi−1 ···i2n +Z

d4n+1 x g(Π− , χ)
 1g(Π− , χ) = 
− 2X
j1 ···j2n+1
j1 <j2 <···<j2n+1

∂LI∂Qj1 ···j2n+1

2
− LI (Q, χ)
evaluated at the solution to the constraint:
i1 ···i2nQj1 ···j2n+1 + 2 i1 ···i2n j1 ···j2n+1 Π−+∂LI=0∂Qj1 ···j2n+124

Dirac brackets:
1 i1 ···i2n j1 k1 ···k2n ∂δ(~x−~y) ,2∂xj11∂
k1 ···k2n(t, ~y)}DB = − i1 ···i2n j1 k1 ···k2n j δ(~x−~y) ,{Πi−1 ···i2n (t, ~x), Π−2∂x 1i ···ij ···j
{Π+1 2n+1 (t, ~x), Π−1 2n+1 (t, ~y)}DB = 0 .
k1 ···k2n(t, ~y)}DB ={Πi+1 ···i2n (t, ~x), Π+Recall that H = H+ + H− whereZ
i1 ···i2n i1 ···i2n,
Π+H+ = − d4n+1 x Π+Z
H− =d4n+1 x Πi−1 ···i2n Πi−1 ···i2n +Z

d4n+1 x g(Π− , χ)

– sum of a free Hamiltonian H+ and an interacting Routhian H−– confirms the result based on the analysis of equations ofmotion.
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Using this formalism one can address some apparent puzzles
involving compactification of theories of self-dual forms.
e.g. if we start with a self-dual 3-form in D=6, uponcompactification on T2 we should get U(1) gauge theory in D=4
In standard compactification the dimensionally reduced action ismultiplied by the area R1 R2 of T2– should act as the inverse coupling.
But the expected coupling from S-duality is R1 /R2 !

Witten26

In the new formulation, due to the non-standard coupling of themetric, the dimensionally reduced action no longer has R1 R2 as
an overall multiplicative factor.
A detailed analysis shows that the dimensionally reduced action
and the associated Routhian are indeed S-duality invariant.

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Speculations

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This formalism allows us to construct the action of a single
M5-brane
What about multiple M5-branes?
– expected to have tensionless strings carrying charge under
the three form field strength
The possibility of including these degrees of freedom exist in
this formulation since ∗dY acts as the source for the self-dual
field.
The rest of the talk will be some speculations on how we might
construct appropriate Y representing tensionless string source.
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Go back to type IIB on K3.
The self-dual 3-forms arise from components of self-dual 5-form
along a harmonic two form on K3.
D3-branes wrapped on the dual 2-cycles are charged under theresulting 3-form field strength in 6D.
The area modulus X of the 2-cycle is part of the same
supermultiplet as the 3-form field.
When X approaches zero, the D3-brane wrapped on the 2-cycle
has zero tension– a tensionless string– related to the interacting part of the theory of multiple
M5-branes.

30Question: Is there a field theory that can describe these strings,
but does not introduce other fields?
Strings can be generated as solitons, e.g. instanton solutions in
6D Yang-Mills theory.
· · · , Kim, Kim, Park
However this also introduces Yang-Mills fields in the theory.
One example where this does not happen is tachyon
condensation in open string theory.
Tachyons can produce solitons without generating perturbative
excitations around the vacuum.

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In type IIB on K3, we can consider a pair of D7-branes and a pairof anti-D7 branes wrapped on the 2-cycle of size X.
Tachyon condensation on this system can produce a D3-brane
wrapped on the 2-cycle – the string that we want to generate.
Open string field theory on the D7-D7 brane system, coupled tothe abelian tensor multiplet, will contain the desired CFT
The relevant part of the open string field theory action:X g−1s Sopen
gs : string coupling (eventually taken to 0)
Sopen : Open string field theory action*dY: related to ‘Ellwood invariant’ measuring D3-brane charge
wrapped on 2-cycle of size X32

Question: Is there a truncation to field theory with finite numberof fields?
Can we write down a tachyon action of the form
Stachyon = X S(T, · · · )with T a 2 × 2 complex matrix valued scalar, describing thephysics of tachyon condensation?
We could then identify the source dY appearing earlier to the4-form constructed from T that measures the D3-brane chargedY ∝ Tr(dT ∧ dT† ∧ dT ∧ dT† )33

At present the only known perturbatively UV finite theory of thiskind is p-adic open bosonic string field theory with actionZ 
1 −/2
1p+1− Tp
T+T,p : a prime number,
2p+1with a real field T.

Freund, Olson; Freund, Witten; Brekke, Freund, Olson, Witten– captures the physics of tachyon condensation in open bosonicstring theory quite well
Ghoshal, A.S.
A generalization of this to superstring theory could give us aquantum field theory on multiple M5-branes

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