HIGHER DERIVATIVE
TERMS IN M-THEORY
REDUCTIONS
based on
1408.5136 and 1412.5073 with
T.W. Grimm and M.Weissenbacher
Tom Pugh
OUTLINE
¢  Introduction
¢  M-theory
¢  The
Vacua with 3d N=2 Supersymmetry
3d Effective Theory
¢  Conclusions
OUTLINE
¢  Introduction
¢  M-theory
¢  The
Vacua with 3d N=2 Supersymmetry
3d Effective Theory
¢  Conclusions
INTRODUCTION
¢  Reductions
of M-theory to three dimensions
present many interesting features.
¢  In
general the vacua include fluxes and warping,
which require higher derivative corrections to the
11d action for global consistency.
¢  These
reductions represent the M-theory duals of
F-theory reductions to 4d and are important for
deriving the F-theory effective action.
¢  Higher
derivative corrections in M-theory can
induce corrections to the 3d effective theory and
the 4d F-theory lift.
OUTLINE
¢  Introduction
¢  M-theory
¢  The
Vacua with 3d N=2 Supersymmetry
3d Effective Theory
¢  Conclusions
THE HIGHER DERIVATIVE VACUA
¢  The
11d action and field equations are given as
an expansion in the 11d plank length
¢  So the solutions we will study exist as an
2
expansion in ↵ ⇠ (11 ) 3 ⇠ lp3
2
dŝ = e
↵2 Z
Ĝmn̄rs̄
(e
2↵2 W (2)
µ
⌫
⌘µ⌫ dx dx + 2e
↵2 W (2)
gmn̄ dy m dy n̄ )
(0)
(2)
gmn̄ = gmn̄
+ ↵2 gmn̄
= ↵G(1)
Ĝµ⌫⇢m = ✏µ⌫⇢ @m e
mn̄rs̄
3↵2 W (2)
¢  Where
3d N=2 supersymmetry at lowest order in ↵
[Becker, Becker]
implies
dJ (0) = d⌦(0) = 0 G(1) ^ J (0) = 0 G(1) 2 H 2,2 (Y4 )
O(↵2 ) CONSTRAINTS ON THE VACUA
¢  And
at second order in ↵ the solution is
constrained by the warp factor equation
d† dW (2) + G(1) ^ G(1) + X8 = 0
¢  And
the internal space part of the metric field
[Becker, Becker] [Lu, Pope, Stelle, Townsend]
equation
(2)
Rmn̄
¢  This
(0)
¯
= @m @ n̄ Z
(0)
(0)
Z = ⇤ (J
(0)
(0)
(0)
(0)
c(0)
=
iTr(R
^
R
^
R
)
3
is solved by
(0)
^ c3 )
[Grimm, TP, Weissenbacher]
(0)
(0) ¯(0)
c(0)
=
Hc
+
i@
@ F
3
3
(2)
(0)
(0)
(0)
(0)
gmn̄
= r(0)
r̄
⇤
(F
^
J
^
J
)
n̄
m
OUTLINE
¢  Introduction
¢  M-theory
¢  The
Vacua with 3d N=2 Supersymmetry
3d Effective Theory
¢  Conclusions
PERTURBATIONS FOR KÄHLER MODULI
¢  We
make an ansatz for the perturbations that
correspond to Kähler moduli where
(0)
(0)
gmn̄
= i v i !imn̄
F i = dAi
¢  The
Ĝ = F i ^ !i(0)
d ⇤(0) !i(0) = d!i(0) = 0
metric shift induces a shift of the higher
derivative parts of the ansatz
¢  And
(0)
Z = i v i Z n̄m !imn̄
+ ...
induces a shift of the warping
W (2) | ! W (2) | + @i W (2) | v i + . . .
(See Martucci’s Talk)
REDUCING THE 11D ACTION
¢  Next
we reduce the action on the background
described
S=S
¢  We
(0)
2
+↵ S
(2)
R̂4
2
+↵ S
(2)
Ĝ2 R̂3
+O(Ĝ3 ↵2 ) + O(↵3 )
2
(2)
+ ↵ S(r̂Ĝ)2 R̂2
[Liu, Minasian]
derive the effective action to second order in
3d derivatives, ↵ , v i and Ai . This is sufficient to
determine the higher derivative contributions to
3d kinetic terms and mass terms.
¢  This results in an effective action given in terms
of
(0) (0)
(0) 3
Zmn̄rs̄ = (✏8 ✏8 R
Zmm̄ = iZmm̄n n = (⇤(0) c(0)
3 )mm̄
)mn̄rs̄
Zm m = Z
KINETIC TERMS IN THE EFFECTIVE ACTION
¢  Substituting
we find
Skin =
1
211
Z
M3
¢  Where
GTij
1
⇠
V0
Z
Y4

KiT

e
into the 11d action and integrating
(GTij + VT 2 KiT KjT )Dv i ^ ⇤Dv j
R⇤1
3↵2 W (2) +↵2 Z
=
Z
Y4

e
(0)
!imn̄
!j(0) n̄m + ↵2 Zmn̄rs̄ !j(0) n̄m !i(0) s̄r ⇤(0) 1
3↵2 W (2) +↵2 Z
VT =
Z
VT2 GTij F i ^ ⇤F j
e
Y4
(0) m
!im
3↵2 W (2) +↵2 Z
Zmn̄ !i(0) n̄m ⇤(0) 1
⇤(0) 1
SCALAR MASS TERMS
¢  The
additional mass terms for the scalar
Zfluctuations cancelZ
Z
t̂8 t̂8 R̂4 ˆ
⇤1|mass =
Z
Z
R̂ˆ⇤1|mass = ↵2
¢  So
M3
(0)
v i v j ⇤3 1
rr rr Z!imn̄
!j(0) n̄m ⇤8 1
M3
Y4
Z
(0)
v i v j ⇤3 1
rr rr Z!imn̄
!j(0) n̄m ⇤8 1
Y4
that the only potential terms are
Z
2 Z
↵
Spot =
⇤3 1
(G(1) ^ ⇤0 G(1)
411 M3
Y4
G(1) ^ G(1) )
¢  These
come with the associated Chern-Simons
terms for the vectors
Z
Z
SCS = ↵
Ai ^ F j
M3
Y4
G(1) ^ !i ^ !j
OUTLINE
¢  Introduction
¢  M-theory
¢  The
Vacua with 3d N=2 Supersymmetry
3d Effective Theory
¢  Conclusions
CONCLUSIONS
¢  Effective
theories including higher derivative
corrections for 3d moduli may be derived in the
way we have described.
¢  This
process involves corrections to the vacua,
ansatz and action parameterized by the 11d
Planck length.
¢  These
computations required significant use of
computer algebra packages such as xAct.
¢  The
¢  All
masses of the 3d fluctuations are not altered.
corrections may be written in terms of
(0)
(0) 3
Zmn̄rs̄ = (✏(0)
✏
R
)mn̄rs̄ and the warp factor.
8 8
THANK YOU