Constraining gravitational interactions in the M theory effective … · 2014-01-17 · Constraining gravitational interactions in the M theory effective action Anirban Basu HRI,

Introduction and motivationA class of local M theory interactions

The spacetime structure of the interactionsConstraints from supersymmetry and S–duality in IIB

The D12R4 interactionConclusions

Constraining gravitational interactions in the Mtheory effective action

Anirban BasuHRI, Allahabad

January 17, 2014

Anirban Basu




Outline of the talk

Introduction and motivationA class of local M theory interactionsThe spacetime structure of the interactions in M/stringtheoryConstraints from supersymmetry and S–duality in IIBThe D12R4 interactionConclusions

Anirban Basu




Outline of the talk


Anirban Basu




Outline of the talk


Anirban Basu




Outline of the talk


Anirban Basu




Outline of the talk


Anirban Basu




Outline of the talk


Anirban Basu




It is important to understand the effective action of stringtheory/M theory in various backgrounds.The effective action encodes important information aboutthe various duality symmetries of the theory, which allowsus to calculate various perturbative as well asnon–perturbative effects. Every term in the effective actionencodes non–trivial information about the S matrices of thetheory.This effective action leads to duality covariant equations ofmotion.

Anirban Basu





Anirban Basu





Anirban Basu




Calculating the effective action is difficult in general, butcertain terms in the effective action in maximallysupersymmetric theories can be calculated exactly, theseare BPS protected.The non–BPS terms are much more difficult to determine. Ishall discuss a particular case in detail where string theorygives some information about such a term in the M theoryeffective action.

Anirban Basu




Calculating the effective action is difficult in general, butcertain terms in the effective action in maximallysupersymmetric theories can be calculated exactly, theseare BPS protected.The non–BPS terms are much more difficult to determine. Ishall discuss a particular case in detail where string theorygives some information about such a term in the M theoryeffective action.

Anirban Basu




Our aim is to look at the local, purely gravitationalinteractions in the M theory effective action in 11dimensional flat spacetime, perhaps the simplest case tostudy.This theory has 32 supersymmetries, which shouldconstrain its effective action, like other theories with such alarge amount of supersymmetry.

Anirban Basu




Our aim is to look at the local, purely gravitationalinteractions in the M theory effective action in 11dimensional flat spacetime, perhaps the simplest case tostudy.This theory has 32 supersymmetries, which shouldconstrain its effective action, like other theories with such alarge amount of supersymmetry.

Anirban Basu




Let us consider local, purely gravitational interactions in theM theory effective action of the form

S = l2k−311

∫d11x

√−GD2kR4.

Compactify on a circle of (dimensionless) radius R11 suchthat

l11 = eφA/3ls, R311 = e2φA .

The length element is given by

ds2 = GMNdxMdxN = gµνdxµdxν + R211(dx11 − Cµdxµ)2.

Anirban Basu





S = l2k−311

∫d11x

√−GD2kR4.


l11 = eφA/3ls, R311 = e2φA .



Anirban Basu





S = l2k−311

∫d11x

√−GD2kR4.


l11 = eφA/3ls, R311 = e2φA .



Anirban Basu




This leads to a purely gravitational interaction in the typeIIA effective action of the form

S = 2πl2k−2s

∫d10x

√−ge2kφA/3D2kR4.

Thus the coefficient of the M theory interaction is known ifthe coefficient of the e2kφA/3 term is known in thecorresponding interaction in the type IIA theory at strongcoupling.

Anirban Basu




This leads to a purely gravitational interaction in the typeIIA effective action of the form

S = 2πl2k−2s

∫d10x

√−ge2kφA/3D2kR4.

Thus the coefficient of the M theory interaction is known ifthe coefficient of the e2kφA/3 term is known in thecorresponding interaction in the type IIA theory at strongcoupling.

Anirban Basu




For k ≤ 3, the IIA interactions are BPS and receive only afinite number of perturbative contributions. Thus the Mtheory interactions are easily read off from the coefficientsof the perturbative amplitudes.Thus the R4 and D6R4 interactions are non–vanishing inthe M theory effective action, while the D4R4 interactionvanishes.

Anirban Basu




For k ≤ 3, the IIA interactions are BPS and receive only afinite number of perturbative contributions. Thus the Mtheory interactions are easily read off from the coefficientsof the perturbative amplitudes.Thus the R4 and D6R4 interactions are non–vanishing inthe M theory effective action, while the D4R4 interactionvanishes.

Anirban Basu




In M theory, the coefficient of the R4 (D6R4) term is fixedby the coefficient of the genus 1 (2) type IIA R4 (D6R4)amplitude.The interactions are

l−311 ζ(2)

∫d11x

√−GR4

andl311ζ(2)2

∫d11x

√−GD6R4

dropping overall numerical factors.The coefficients have a very precise transcendental nature.

Anirban Basu





l−311 ζ(2)

∫d11x

√−GR4

andl311ζ(2)2

∫d11x

√−GD6R4


Anirban Basu





l−311 ζ(2)

∫d11x

√−GR4

andl311ζ(2)2

∫d11x

√−GD6R4


Anirban Basu




The interactions D2kR4 for k ≥ 4 are non–BPS. Hence thetype IIA interactions are expected to receive perturbativecontributions from all orders in the genus expansion.Thus the M theory interactions are difficult to determinebecause the type IIA coefficients have to extracted atstrong coupling.Can we still make some statements about these non-BPSinteractions, for small values of k?

Anirban Basu





Anirban Basu





Anirban Basu




For k = 3n, the type IIA interaction is of the form

S = 2πl6n−2s

∫d10x

√−ge2nφAD6nR4.

At weak coupling, this has the structure of the genus(n + 1) string amplitude.

Anirban Basu




For k = 3n, the type IIA interaction is of the form

S = 2πl6n−2s

∫d10x

√−ge2nφAD6nR4.

At weak coupling, this has the structure of the genus(n + 1) string amplitude.

Anirban Basu




Though the complete coefficient of the M theory interactionis difficult to determine, it is plausible that for low values ofn, a part of the coefficient will have features qualitativelydescribed by the genus (n + 1) amplitude, namely thetranscendentality.We shall do the analysis for the D12R4 interaction. Theanswer we shall get generalizes the transcendentalstructure for n = 0 and n = 1. Also there is agreement witha particular supergravity calculation that is valid at strongcoupling (to be reviewed later).

Anirban Basu




Though the complete coefficient of the M theory interactionis difficult to determine, it is plausible that for low values ofn, a part of the coefficient will have features qualitativelydescribed by the genus (n + 1) amplitude, namely thetranscendentality.We shall do the analysis for the D12R4 interaction. Theanswer we shall get generalizes the transcendentalstructure for n = 0 and n = 1. Also there is agreement witha particular supergravity calculation that is valid at strongcoupling (to be reviewed later).

Anirban Basu




We shall proceed with this assumption for the D12R4

interaction.Thus we want to analyze the genus 3 amplitude for thetype IIA D12R4 interaction. This is the same in the type IIBtheory as well.

Anirban Basu




We shall proceed with this assumption for the D12R4

interaction.Thus we want to analyze the genus 3 amplitude for thetype IIA D12R4 interaction. This is the same in the type IIBtheory as well.

Anirban Basu




The spacetime structure of the R4 interaction in flat spacehas two kinds of contributions:(i) t8t8R4

(ii) ±ε10ε10R4

These follow directly from the string amplitude calculations.The perturbative contributions to (i) are the same in IIA andIIB, hence this is the part we calculate.

Anirban Basu




The spacetime structure of the R4 interaction in flat spacehas two kinds of contributions:(i) t8t8R4

(ii) ±ε10ε10R4

These follow directly from the string amplitude calculations.The perturbative contributions to (i) are the same in IIA andIIB, hence this is the part we calculate.

Anirban Basu




Thus we analyze the genus 3 type IIB D12R4 amplitude.We shall perform the analysis using the constraintsimposed by supersymmetry and S–duality of the type IIBtheory.

Anirban Basu




Thus we analyze the genus 3 type IIB D12R4 amplitude.We shall perform the analysis using the constraintsimposed by supersymmetry and S–duality of the type IIBtheory.

Anirban Basu




The basic argumentTheR4 couplingThe D6R4 coupling

In the Einstein frame, the moduli dependent coefficientsfk (τ, τ) of the D2kR4 interactions in the action

l2k−2s

∫d10x

√−gfk (τ, τ)D2kR4

are SL(2,Z) invariant modular forms.Now fk (τ, τ) can be constrained using supersymmetry andS–duality.

Anirban Basu





In the Einstein frame, the moduli dependent coefficientsfk (τ, τ) of the D2kR4 interactions in the action

l2k−2s

∫d10x

√−gfk (τ, τ)D2kR4

are SL(2,Z) invariant modular forms.Now fk (τ, τ) can be constrained using supersymmetry andS–duality.

Anirban Basu





The analysis is done using the Noether procedure.The action is expanded as

S = S(0) +∞∑

n=3

l2ns S(n).

The supersymmetry transformation is also expanded as

δ = δ(0) +∞∑

n=3

l2ns δ(n).

Anirban Basu






S = S(0) +∞∑

n=3

l2ns S(n).


δ = δ(0) +∞∑

n=3

l2ns δ(n).

Anirban Basu






S = S(0) +∞∑

n=3

l2ns S(n).


δ = δ(0) +∞∑

n=3

l2ns δ(n).

Anirban Basu





ImplementingδS = 0

order by order in the ls expansion gives the desired result,on using

δ(0)S(n) + δ(n)S(0) +∑

p+q=n

δ(p)S(p) = 0.

To actually implement this procedure in a useful way, at afixed order in the momentum expansion, one looks at themaximally fermionic terms of the form G2kλ16 andG2kψ∗λ15 which should be in the same supermultiplet asthe D2kR4 term.

Anirban Basu





ImplementingδS = 0

order by order in the ls expansion gives the desired result,on using

δ(0)S(n) + δ(n)S(0) +∑

p+q=n

δ(p)S(p) = 0.

To actually implement this procedure in a useful way, at afixed order in the momentum expansion, one looks at themaximally fermionic terms of the form G2kλ16 andG2kψ∗λ15 which should be in the same supermultiplet asthe D2kR4 term.

Anirban Basu





These interactions mix with no other terms in S(k+3) underδ(0), and one has to find terms in δ(k+3), as well as terms inδ(m) and S(n) with m + n = k + 3 such that the totalsupervariation vanishes.The couplings of these terms in the action as well as thesupervariations are SL(2,Z) modular forms of fixedweights, which are further constrained using the closure ofthe superalgebra.

Anirban Basu





These interactions mix with no other terms in S(k+3) underδ(0), and one has to find terms in δ(k+3), as well as terms inδ(m) and S(n) with m + n = k + 3 such that the totalsupervariation vanishes.The couplings of these terms in the action as well as thesupervariations are SL(2,Z) modular forms of fixedweights, which are further constrained using the closure ofthe superalgebra.

Anirban Basu





These lead to first order differential equations satisfied bythe G2kλ16 and G2kψ∗λ15 couplings, which also holds forother interactions in the same supermultiplet.These equations are of the form

Df ∼ f ′ +∑

i

gihi ,

andDf ′ ∼ f +

∑i

ki li .

Anirban Basu





These lead to first order differential equations satisfied bythe G2kλ16 and G2kψ∗λ15 couplings, which also holds forother interactions in the same supermultiplet.These equations are of the form

Df ∼ f ′ +∑

i

gihi ,

andDf ′ ∼ f +

∑i

ki li .

Anirban Basu





Here f and f ′ are coefficients of terms in the samesupermultiplet which differ in SL(2,Z) weight by 1 unit, andthe other coefficients involve terms at lower orders in the lsexpansion.Iterating these two equations, we find that the D2kR4

coupling should be expressed as sums of SL(2,Z)invariant modular forms, each of which satisfies thePoisson equation

4τ22∂2

∂τ∂τf ∼ f +

∑i

risi +∑

i

minipi

on the fundamental domain of moduli space.Thus, for a fixed value of k the coupling can be solvedrecursively once the couplings at lower values of k areknown.

Anirban Basu







4τ22∂2

∂τ∂τf ∼ f +

∑i

risi +∑

i

minipi


Anirban Basu







4τ22∂2

∂τ∂τf ∼ f +

∑i

risi +∑

i

minipi


Anirban Basu





Fromδ(0)S(3) + δ(3)S(0) = 0,

we get that the R4 coupling satisfies the Laplace equation

4τ22∂2

∂τ∂τf (0) =

34

f (0).

Thus

f (0) = E3/2(τ, τ) = 2ζ(3)τ3/22 + 4ζ(2)τ

−1/22 + . . . ,

leading to the

l−311 ζ(2)

∫d11x

√−GR4

term in the M theory effective action.

Anirban Basu





Fromδ(0)S(3) + δ(3)S(0) = 0,

we get that the R4 coupling satisfies the Laplace equation

4τ22∂2

∂τ∂τf (0) =

34

f (0).

Thus

f (0) = E3/2(τ, τ) = 2ζ(3)τ3/22 + 4ζ(2)τ

−1/22 + . . . ,

leading to the

l−311 ζ(2)

∫d11x

√−GR4


Anirban Basu





Fromδ(0)S(6) + δ(6)S(0) + δ(3)S(3) = 0,

we get that the D6R4 coupling satisfies the Poissonequation

4τ22∂2

∂τ∂τf (6) = 12f (6) − 6E2

3/2.

Thus

f (6) =23ζ(3)2τ3

2 +43ζ(2)ζ(3)τ2+

85ζ(2)2τ−1

2 +32

945ζ(2)3τ−3

2 +. . . ,

leading to the

l311ζ(2)2∫

d11x√−GD6R4


Anirban Basu





Fromδ(0)S(6) + δ(6)S(0) + δ(3)S(3) = 0,

we get that the D6R4 coupling satisfies the Poissonequation

4τ22∂2

∂τ∂τf (6) = 12f (6) − 6E2

3/2.

Thus

f (6) =23ζ(3)2τ3

2 +43ζ(2)ζ(3)τ2+

85ζ(2)2τ−1

2 +32

945ζ(2)3τ−3

2 +. . . ,

leading to the

l311ζ(2)2∫

d11x√−GD6R4


Anirban Basu




IR divergences in loop amplitudesThe analysis for generic λiThe analysis for specific λi

Consider the constraints coming from

δ(0)S(9)+δ(9)S(0)+δ(3)S(6)+δ(6)S(3)+δ(4)S(5)+δ(5)S(4) = 0.

For every SL(2,Z) invariant modular form in the D12R4

coupling, δ(3)S(6) + δ(6)S(3) contributes source terms of theform

µE3/2f (6) + νE33/2

in the Poisson equation.What about the source terms from δ(4)S(5) + δ(5)S(4)?Recall S(4) and δ(4) vanish on–shell. Hence visible only insome off–shell formalism.

Anirban Basu






δ(0)S(9)+δ(9)S(0)+δ(3)S(6)+δ(6)S(3)+δ(4)S(5)+δ(5)S(4) = 0.



µE3/2f (6) + νE33/2


Anirban Basu






δ(0)S(9)+δ(9)S(0)+δ(3)S(6)+δ(6)S(3)+δ(4)S(5)+δ(5)S(4) = 0.



µE3/2f (6) + νE33/2


Anirban Basu






δ(0)S(9)+δ(9)S(0)+δ(3)S(6)+δ(6)S(3)+δ(4)S(5)+δ(5)S(4) = 0.



µE3/2f (6) + νE33/2


Anirban Basu





However they provide source terms needed for unitarity.The genus 1 four graviton amplitude has a non–localcontribution of the schematic form

ζ(2)sln(−l2s s)R4

which in the Einstein frame gives a local interaction of theform

ζ(2)lnτ2(s + t + u)R4

which vanishes on–shell.Thus off–shell, the D2R4 interaction has an SL(2,Z)invariant coupling

Y (τ, τ) = ζ(2)lnτ2 + . . . .

Anirban Basu








ζ(2)lnτ2(s + t + u)R4


Y (τ, τ) = ζ(2)lnτ2 + . . . .

Anirban Basu








ζ(2)lnτ2(s + t + u)R4


Y (τ, τ) = ζ(2)lnτ2 + . . . .

Anirban Basu





Thus δ(4)S(5) + δ(5)S(4) contributes source terms

YE5/2

to the Poisson equation, since E5/2 is the D4R4 coupling.

Hence the D12R4 coupling f (12) is given by (the structure isthe same for either spacetime structure)

f (12) =∑

i

f (12)i ,

where each f (12)i satisfies the Poisson equation

4τ22∂2

∂τ∂τf (12)i = λi f

(12)i − µiE3/2f (6) − νiE3

3/2 − ηiYE5/2.

Anirban Basu





Thus δ(4)S(5) + δ(5)S(4) contributes source terms

YE5/2

to the Poisson equation, since E5/2 is the D4R4 coupling.

Hence the D12R4 coupling f (12) is given by (the structure isthe same for either spacetime structure)

f (12) =∑

i

f (12)i ,

where each f (12)i satisfies the Poisson equation

4τ22∂2

∂τ∂τf (12)i = λi f

(12)i − µiE3/2f (6) − νiE3

3/2 − ηiYE5/2.

Anirban Basu





To be consistent with perturbative string amplitudes, wehave that

λi = si(si − 1)

where si is half–integral.We can thus solve the equation, whose perturbative part isgiven by

Anirban Basu





To be consistent with perturbative string amplitudes, wehave that

λi = si(si − 1)

where si is half–integral.We can thus solve the equation, whose perturbative part isgiven by

Anirban Basu





f (12)i ∼ c1iτ

si2 + c2iτ

1−si2 + αiζ(3)3τ

9/22 + βiζ(2)ζ(3)2τ

5/22

+γiζ(2)2ζ(3)τ1/22 + ζ(2)3(εi + σiζ(3))τ

−3/22 + ωiζ(2)4τ

−7/22

+ηiζ(2)(

2ζ(5)τ5/22 +

83ζ(4)τ

−3/22

)lnτ2

+ηiζ(2)ζ(5)τ5/22 + ηiζ(2)3τ

−3/22 + . . . .

It agrees with known calculations in string perturbationtheory.

Anirban Basu





f (12)i ∼ c1iτ

si2 + c2iτ

1−si2 + αiζ(3)3τ

9/22 + βiζ(2)ζ(3)2τ

5/22

+γiζ(2)2ζ(3)τ1/22 + ζ(2)3(εi + σiζ(3))τ

−3/22 + ωiζ(2)4τ

−7/22

+ηiζ(2)(

2ζ(5)τ5/22 +

83ζ(4)τ

−3/22

)lnτ2

+ηiζ(2)ζ(5)τ5/22 + ηiζ(2)3τ

−3/22 + . . . .

It agrees with known calculations in string perturbationtheory.

Anirban Basu





In the string frame, leads to analytic terms at genus 3 ofthe form

ζ(2)3(Ω1 + Ω2ζ(3))e4φAD12R4,

and at genus 4 of the form

ζ(2)4e6φAD12R4.

Calculations of the 4 graviton amplitude in regularizedmaximal supergravity at 1 and 2 loops have been done inthe limit of large eφA , and yield(

ζ(2)3ζ(3)e4φA + ζ(2)4e6φA + ζ(2)6e10φA)

D12R4.

Anirban Basu





In the string frame, leads to analytic terms at genus 3 ofthe form

ζ(2)3(Ω1 + Ω2ζ(3))e4φAD12R4,

and at genus 4 of the form

ζ(2)4e6φAD12R4.

Calculations of the 4 graviton amplitude in regularizedmaximal supergravity at 1 and 2 loops have been done inthe limit of large eφA , and yield(

ζ(2)3ζ(3)e4φA + ζ(2)4e6φA + ζ(2)6e10φA)

D12R4.

Anirban Basu





Natural to assume that the transcendental structuresurvives at large coupling, and the M theory coupling is ofthe form

ζ(2)3(Ω1 + Ω2ζ(3))l911

∫d11x

√−GD12R4.

We also have to do the analysis for λi = 63/4,15/4,−1/4.

Anirban Basu





Natural to assume that the transcendental structuresurvives at large coupling, and the M theory coupling is ofthe form

ζ(2)3(Ω1 + Ω2ζ(3))l911

∫d11x

√−GD12R4.

We also have to do the analysis for λi = 63/4,15/4,−1/4.

Anirban Basu





In short, for λi = 63/4,−1/4 no change in the genus 3answer.For λi = 15/4, we solve

4τ22∂2

∂τ∂τh =

154

h − σ1E3/2f (6) − σ2E33/2 − σ3YE5/2.

Apart from the terms for generic λi , we also get (includingthe c1i and c2i parts)

Anirban Basu






4τ22∂2

∂τ∂τh =

154

h − σ1E3/2f (6) − σ2E33/2 − σ3YE5/2.


Anirban Basu






4τ22∂2

∂τ∂τh =

154

h − σ1E3/2f (6) − σ2E33/2 − σ3YE5/2.


Anirban Basu





h = c1τ5/22 + c3τ

−3/22 − 4

(σ1

3+ 3σ2

)ζ(2)ζ(3)2τ

5/22 lnτ2

+8(1

5

1 +

2189

ζ(3)σ1 + 2σ2

)ζ(2)3τ

−3/22 lnτ2

−σ3

4ζ(2)ζ(5)τ

5/22 (lnτ2)2 +

σ3

3ζ(2)ζ(4)τ

−3/22 lnτ2 + . . . .

We can fix the coefficients using string theory data.

Genus 1 non–analytic piece ∼ ζ(2)ζ(5)τ5/22 lnτ2, hence

σ3 = 0, σ1 = −9σ2.

Anirban Basu





h = c1τ5/22 + c3τ

−3/22 − 4

(σ1

3+ 3σ2

)ζ(2)ζ(3)2τ

5/22 lnτ2

+8(1

5

1 +

2189

ζ(3)σ1 + 2σ2

)ζ(2)3τ

−3/22 lnτ2

−σ3

4ζ(2)ζ(5)τ

5/22 (lnτ2)2 +

σ3

3ζ(2)ζ(4)τ

−3/22 lnτ2 + . . . .



σ3 = 0, σ1 = −9σ2.

Anirban Basu





h = c1τ5/22 + c3τ

−3/22 − 4

(σ1

3+ 3σ2

)ζ(2)ζ(3)2τ

5/22 lnτ2

+8(1

5

1 +

2189

ζ(3)σ1 + 2σ2

)ζ(2)3τ

−3/22 lnτ2

−σ3

4ζ(2)ζ(5)τ

5/22 (lnτ2)2 +

σ3

3ζ(2)ζ(4)τ

−3/22 lnτ2 + . . . .



σ3 = 0, σ1 = −9σ2.

Anirban Basu





Unitarity implies E5/2Y non–analytic coupling, hence a

ζ(2)3τ−3/22 lnτ2 contribution, thus σ1 = 0.

The final equation is

h ∼ ζ(2)E5/2 ∼ ζ(2)ζ(5)τ5/22 + ζ(2)3τ

−3/22 + . . . .

Hence the genus 3 contribution remains the same.

Anirban Basu








h ∼ ζ(2)E5/2 ∼ ζ(2)ζ(5)τ5/22 + ζ(2)3τ

−3/22 + . . . .


Anirban Basu








h ∼ ζ(2)E5/2 ∼ ζ(2)ζ(5)τ5/22 + ζ(2)3τ

−3/22 + . . . .


Anirban Basu





There could be other contributions to the D12R4 term in Mtheory.Our calculation and matching with the supergravityanalysis suggests that the interaction has at least the terms

ζ(2)3(Ω1 + Ω2ζ(3))l911

∫d11x

√−GD12R4.

Anirban Basu





There could be other contributions to the D12R4 term in Mtheory.Our calculation and matching with the supergravityanalysis suggests that the interaction has at least the terms

ζ(2)3(Ω1 + Ω2ζ(3))l911

∫d11x

√−GD12R4.

Anirban Basu




Natural to try to generalize to higher values of n.D18R4 interaction is the next one. The couplings for thesecoefficients satisfy Poisson equations with non–BPSsource terms like the couplings for the D8R4,D10R4 andD12R4 interactions.Analogous analysis using known perturbative amplitudesusing supersymmetry yields a genus 4 amplitude withcoefficient ∼ ζ(2)4(Ω1 + Ω2ζ(3) + Ω3ζ(5)).It is plausible the structure of transcendentality continuesat strong coupling showing that the D18R4 M theoryinteraction has at least the terms

l1511ζ(2)4(Ω1 + Ω2ζ(3) + Ω3ζ(5))

∫d11x

√−GD18R4.

Anirban Basu





l1511ζ(2)4(Ω1 + Ω2ζ(3) + Ω3ζ(5))

∫d11x

√−GD18R4.

Anirban Basu





l1511ζ(2)4(Ω1 + Ω2ζ(3) + Ω3ζ(5))

∫d11x

√−GD18R4.

Anirban Basu





l1511ζ(2)4(Ω1 + Ω2ζ(3) + Ω3ζ(5))

∫d11x

√−GD18R4.

Anirban Basu




Important to understand better the role of supersymmetry.Non–BPS interactions in theories with maximalsupersymmetry might be tightly constrained.

Anirban Basu




Important to understand better the role of supersymmetry.Non–BPS interactions in theories with maximalsupersymmetry might be tightly constrained.

Anirban Basu

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Constraining gravitational interactions in the M theory effective … · 2014-01-17 · Constraining gravitational interactions in the M theory effective action Anirban Basu HRI,