On the Geometry of the Strominger System by Teng Fei B.Sc., Tsinghua University (2011) Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY MASSACHUSETTS INSTITUTE OF TECHNOLOGY JUN 1 6 2016 LIBRARIES ARCHVES June 2016 Massachusetts Institute of Technology 2016. All rights reserved. Signature redacted Author............ Sic Certified by.. Certified by. 6) Department of Mathematics A ril 9 216 gnature redacted Shing-Tung Yau William Caspar Graust 4 in Professor of Mathematics Sig nature redacted Thesis Supervisor ................... Victor Guillemin Professor of Mathematics Thesis Supervisor Signature redacted Accepted by.. William P. Minicozzi II Chairman, Department Committee on Graduate Theses p ,I
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On the Geometry of the Strominger System
by
Teng Fei
B.Sc., Tsinghua University (2011)
Submitted to the Department of Mathematics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mathematics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
MASSACHUSETTS INSTITUTEOF TECHNOLOGY
JUN 1 6 2016
LIBRARIESARCHVES
June 2016
Massachusetts Institute of Technology 2016. All rights reserved.
Signature redactedAuthor............
SicCertified by..
Certified by.
6) Department of MathematicsA ril 9 216
gnature redactedShing-Tung Yau
William Caspar Graust 4in Professor of Mathematics
Sig nature redacted Thesis Supervisor
...................Victor Guillemin
Professor of MathematicsThesis Supervisor
Signature redactedAccepted by..
William P. Minicozzi II
Chairman, Department Committee on Graduate Theses
p ,I
2
On the Geometry of the Strominger System
by
Teng Fei
Submitted to the Department of Mathematicson April 29, 2016, in partial fulfillment of the
requirements for the degree ofDoctor of Philosophy in Mathematics
Abstract
The Strominger system is a system of partial differential equations describing thegeometry of compactifications of heterotic superstrings with flux. Mathematically itcan be viewed as a generalization of Ricci-flat metrics on non-Kshler Calabi-Yau 3-folds. In this thesis, I will present some explicit solutions to the Strominger system ona class of noncompact Calabi-Yau 3-folds. These spaces include the important local
models like C' as well as both deformed and resolved conifolds. Along the way, I alsogive a new construction of non-Kihler Calabi-Yau 3-folds and prove a few results incomplex geometry.
Thesis Supervisor: Shing-Tung YauTitle: William Caspar Graustein Professor of Mathematics
Thesis Supervisor: Victor GuilleminTitle: Professor of Mathematics
3
.,--,, .......... .I'll
4
Acknowledgments
I would like to express my sincere gratitude to my advisors Prof. Shing-Tung Yau
and Prof. Victor Guillemin for their constant help and encouragement along the way.
To quote The Analects, "I looked up to them, and they seemed to become more high;
I tried to penetrate them, and they seemed to become more firm". ( e
I am greatly indebted to communications with Claude LeBrun, Li-Sheng Tseng,
Valentino Tosatti and Bao-Sen Wu. Their knowledge and insight helped shaping this
thesis.
For everyone in Yau's school, the experience of Yau's Student Seminar is unfor-
gettable. I wish to thank Yu-Wei Fan, Peng Gao, An Huang, Atsushi Kanazawa,
Siu-Cheong Lau, Yu-Shen Lin, Peter Smillie, Chung-Jun Tsai, Yi Xie, Cheng-Long
Yu, Bo-Yu Zhang, Jie Zhou and Jonathan Zhu for their inspiring talks. I benefited
tremendously from their contributions.
Thanks also go to Nate Bottman, Chen-Jie Fan, Qiang Guang, Francesco Lin,
Hai-Hao Lu, Ao Sun, Xin Sun, Guo-Zhen Wang, Hong Wang, Wen-Zhe Wei, Ben
Yang, Yi Zeng, Rui-Xun Zhang, Xin Zhou, Xu-Wen Zhu and Perverse Sheaf. You
made my years at MIT so memorable!
Special thanks are due to my family, especially my parents and grandparents, for
their everlasting support and love. Finally I would like to thank my beloved wife Yi
Consequently, the anomaly cancellation equation (3.3) reduces to
ia e 32h eg - a- a + w 7K
(4.19) (2 ( (aw, +,/w, + YWK))
4 [2(00 log B) 2 Tr(F' A F') - Tr(F A F)]
and we are free to choose functions g and h.
1 &'The simplest way to let (4.19) hold is to set g = 1 log , F = F' and choose
2 2'appropriate h such that
(4.20) (Oa log B)2 = 0.
Recall that B = s3/e 2h+9, hence (4.20) holds trivially if h is a constant say h = 0.
In this case, the metric (4.8) is conformal to the product of hyperkiihler metric on N
and the Euclidean metric on C, hence conformally Ricci-flat. It should be pointed
out that this metric is not complete, however this does not raise any problem for
the use of gluing. Intuitively we can think of w as a metric on the singular space
CP1 x N/{oo} x N. Therefore we have demonstrated that the anomaly cancellation
75
equation (3.3) can be solved by choosing both g and h to be constant and E to be
the relative cotangent bundle of p : X -+ C.
It is also possible that (4.20) holds for nonconstant h. To find such h, one usually
needs to know the explicit hyperkahler metric on N. We will give an example later
in this section.
In our last section, N is taken to be flat T4 hence F' = 0 and the Hermitian-Yang-
Mills equation (3.2) holds automatically. In general, neither N is flat nor F' trivial.
Therefore in order to solve the whole Strominger system, we need to prove
Theorem 4.3.2.
F' solves the Hermitian-Yang-Mills equation (3.2), i.e.,
F' A w2 = 0
for arbitrary g and h.
By the product structure X = C x N, we can decompose the space of 2-forms on
X as
Q2(X) = Q2 (C) E Q'(C) ® Q1(N) E q2(N).
Moreover the complex structure 3 is compatible with this decomposition, so we have
Ql,'(X) = Q1'1(C) E fiO'0 (C) 9 Q0'1(N) D Qo0'1(C) 0 Q' 0 (N) E l"(N).
We first prove the following lemma
Lemma 4.3.3.
Every entry of F' is contained in the space
Q1'0(C) 0 0'(N) f Q0'1(C) 0 Q 1'0(N) E Q1'l(N).
Proof. Recall that U = AKAT, so we have
F' = O(U-10U) = 6((AT)-1k-A-8X-kAT)+6((AT)-Ik~1 -AT)+O((AT)-1 AT)).
76
Observe that (4.12) can be rewritten as
A =i(-i)A. K1121 - 22 K1221 - K22202
K2 1 2 - K1II1 K229 2 - K1101
which does not contain any component from Q (C). Similarly Ok contains only
components in Q 1'0 (N). The lemma follows directly from these two observations. E
Now we proceed to prove Theorem 4.3.2
Proof. From (4.10), we see that w2 lives in the space
Q2 (C) ® Q 2 (N) ( Q4(N).
Therefore the only component of F' that would contribute in F' A w 2 is its Ql"'(N)-
part. The key point is that the Q1'1(N)-part of F' can be computed fiberwise.
Fix ( E C and let N be the fiber of the holomorphic twistor fibration over C.Notice that (NC, w) : -awl + 8WJ + 7WK) is hyperkhler. Moreover, E'IN is the
cotangent bundle of NC with its hyperkiihler metric. It is a well-known fact that
hyperkihler 4-manifolds are anti-self-dual, thus
F61,1(N) A wC = 0.
The theorem follows directly. l
In summary, the main theorem we have proved is the following:
Theorem A.
Let N be a hyperkshler 4-manifold and let p : Z -+ CP1 be its holomorphic twistor
fibration. By removing an arbitrary fiber of p from Z, we get a noncompact 3-fold
X which has trivial canonical bundle. It is worth mentioning that the holomorphic
structure of X depends on the choice of fiber. For such X's, we can always construct
explicit solutions to the Strominger system on them. These spaces include C3 and
the resolved conifold 0(-1, -1) as special examples.
77
Now let us take a closer look at the the solutions constructed above for N - R4
Identify R' with the space of quaternions H, then left multiplication by i, j and
k defines the standard hyperkhiler structure on R'. We can construct the space X
by removing the fiber with complex structure -I at infinity. Actually, X is biholo-
morphic to C3 . An explicit biholomorphism V : C' = C x C2 -+ X = C x R4 can be
=h"dp A (aI +/3J + yK)dp + h'(da A Idp + do A Jdp + dy A Kdp)
-4h'(aw, + 3 WJ + IWK).
One can verify that
=2(h')2 (aJdp A Kdp + 3Kdp A Idp + -yIdp A Jdp) A WcP 1 - 32h'(ph')'voR4
-8h'(ph')' ((ad3 - 3da) A *Kdp + (Od3 y - -yd/3) A *Idp + (-yda - ad'y) A *Jdp),
where * is the standard Hodge star operator on R 4. In addition we have
490h A (300 log s - &Og)
=(h"dp A (al + J + -yK)dp - 4h'(aw, + 3 WJ + YWK)) A (200 log g + 3wcpl).
By using the identity
dp A Idp + 4p -wi = Jdp A Kdp,
and assuming g is a constant, it follows that (4.22) holds if and only if h' = 0 or
h' = 2. In both cases we solve the Strominger system.2p
(a). The case h' = 0.
The metric on X = C x R4 is essentially of the form
1(cx i+ WJ +-WK +dC A
79
which is conformal to the Euclidean metric on X. However this metric has rather
complicated expression in terms of standard complex coordinates ((, wi, w 2 ) on
C3:
2iW )d( A d( + 2 (dwi A dffi + dW 2 A dfV 2
(1 + 1(12)2 2(1 +1(|2)3
+iu 2dwj A dC - iuidw2 A dC - ifl2d( A dfb1 + ifild( A dff'2
+ (Iui|2 + 1u2 12)d( A d)
where
W1 -i(iV2 W2 + i(W-1(Ul, U2) - ( 1(12 ' + 1(2)
Direct calculation shows that though its sectional curvature is not bounded
from below, however it is bounded from above by a positive constant. Same
conclusion applies to Ricci and scalar curvatures as well.
(b). The case h' .
2p
In this case eh ~ P3/2, so the metric w is only defined on C x (R4 \ {0}) e
C x (C2 \{0}). On each copy of R'\ {0}, the restricted metric is conformally flat,
with non-positive sectional curvature. The curvature properties of C x (R4 \ {0})
behave in the same way as what we have seen in the h' = 0 case.
80
Appendix A
On Chern-Ricci-Flat Balanced
Metrics
Let X be a Kdhler manifold with trivial canonical bundle. A fundamental question
is whether M admits a (complete) Ricci-flat Ksihler metric or not. For X compact,
this was answered affirmatively by Yau's famous solution to the Calabi conjecture
[116, 1181. However, when X is noncompact, the same problem is far from being
solved.
For instance, let M be any Kdhler manifold, then we know that the total space of
the canonical bundle of M, denoted by KA, is Kdhler and has itself trivial canonical
bundle. Therefore we may ask when does KM support a Kdhler Ricci-flat metric.
This problem was first studied by Calabi [20], where he showed that if M admits a
Kdhler-Einstein metric, then one can write down a Kihler Ricci-flat metric on Kk.
A relative recent progress in this direction was made by Futaki [601, where he proved
that such a metric exists if M is toric Fano.
The Calabi ansatz can be rephrased as follows. Let (M, w) be a Kdhler manifold.
By choosing a set of holomorphic coordinates {zI, .. . , z} on M, we can trivialize Km
by dzA ... A dz" locally. Let t parameterizes the fiber of KMI under this trivialization,
then {z, ... , zn, t} forms a set of holomorphic coordinates on KM and
Q = dz' A .. -A dzn A dt
81
is a globally defined nowhere vanishing holomorphic volume form.
In terms of coordinates {z1 ,...,z}, the Kdhler form w can be written as
W = ihjkdz A dzk.
It follows that h = det(hjk) is a positive function. Notice that the Khhler metric on
M naturally induces an Hermitian metric on KN, which can be expressed as
wo = w + L(dt - tO log h) A (di - fa log h).h
Let R : KAI -* R be the norm square function of fibers of KM -+ M. Clearly R th
and the metric wo has the form
wo W .i R A OR
R
In general, wo is not a Kdhler metric. It turns out that wo is Kdhler if and only if
O log R = -00 log h = 0,
i.e., (M, w) is Kihler Ricci-flat.
To get a better behaved metric, we can modify wo by some conformal factors. Let
u, v be smooth functions on M, and f, g be smooth functions of R. Then one can
cook up a new Hermitian metric
WUVfg = eu+fw + ie+g OR A ORR
It is not hard to check that Q is of constant length if and only if
(A.1) v=-nu and g=-nf+c
for some constant c. Assuming this, if we further want the metric Ju,v,fg to be Kdhler,
then u must be a constant. Without loss of generality we may assume that u = 0,
82
and we still need
e1Of A w - ie-nf+COR A O0 log R =0.
In other words,
e(n+l)f-cfw = iO log R = -iO log h = Ric(w).
We see immediately that this equation has a solution if and only if w is Kdhler-
Einstein, in which case we get the Calabi ansatz.
In this appendix, we shall consider the case that w,,,f,, is balanced and Chern-
Ricci-flat, i.e.,
d(W",vf,g) = 0 and ||0C,~ =- constant.
As we have seen, such metric appears naturally in compactification of heterotic super-
strings in the metric product model (cf. Model 2 in Section 3.1). Geometrically this
condition is interpreted as that Q is parallel under the Bismut-Strominger connection.
Now let us impose the balanced Chern-Ricci-flat
"Chern-Ricci-flat" part, i.e., =_G||W f constant,
Plugging this in and we can compute that
Wn _ n(u+f) Wn + --- fWn-1u,,f, w
condition on w,,vf,g. For the
we still need (A.1) as before.
OR AORA R
Hence
= en(u+f)Of n - iec-u-fwn-1 A OR A ac log R - ine'-fwn-1 A O9 A RR
As there are no other terms to cancel the last term in the above equation, so we need
au = 0 to make wu,,,f,g balanced. By choosing u = 0, the balancing condition is
reduced to
e"fOf A w" = inec-fwn-' A OR A 0 log R,
83
or equivalently,
e(n~lw-cf'w"l = -inw"i4 & log h = s - "
where s is the scalar curvature function of M up to a positive constant. From the
calculation we conclude that this is possible if and only if (M, w) has constant scalar
curvature.
Constant scalar curvature Kdhler metrics (cscK) has been studied extensively as a
special case of extremal Kihler metrics [211. It is believed that the existence of cscK
metrics is equivalent to certain stability condition in the sense of algebraic geometry.
Notice that in the derivation of Chern-Ricci-flat balanced metric on Km, what
we actually need is that w is balanced instead of being KAhler. In such a case, s
is known as the Chern scalar curvature, which is in general different from the scalar
curvature in Riemannian geometry. Thus we have proved the following generalization
of Calabi's result.
Theorem A.0.4.
If M admits a balanced metric with constant Chern scalar curvature, then KM admits
a Chern-Ricci-flat balanced metric.
Recall that on a complex n-fold X, a balanced metric w defines the so-called
balanced class
W"n-1 ~ _1,_- I X d-closed (n - 1, n - 1)-forms[(T Z HZj~1 X C ----(n - 1)!] 0 i-exact (n - 1, n - 1)-forms'
The balanced version of Calabi (Gauduchon) conjecture [107] in general case is still
open. In particular, for balanced manifolds with trivial canonical bundle, this conjec-
ture implies the existence of Chern-Ricci-flat balanced metrics in any given balanced
class. We refer to [105] for recent progresses on this problem.
Theorem A.0.4 implies that Chern-Ricci-flat balanced metrics should be viewed
as a balanced analogue of extremal Khhler metrics. We shall justify this claim by the
following consideration.
On a compact balanced manifold (X,w), a very useful property is that the total
84
Chern scalar curvature
s-- p n-1
x n! i (n -1)!
depends only on the complex structure of X and the balanced class. Here p =
-iO log h is the Chern-Ricci form. As an analogue of Kdhler case, it is natural to
consider the variational problem associated to the Calabi-type functional [211
S(w)= s2. W
n-1
where varies in the given balanced class. Since we can modify w by i00a,(n -i)!
where a is any (n - 2, n - 2)-form, we expect the associated Euler-Lagrange equation
is an equation of (n - 2, n - 2)-forms, or dually, a (2,2)-form equation.
Indeed, it is not hard to derive the following
Theorem A.0.5.
A balanced metric w is a critical point of S(w) if and only if it satisfies
(A.2) 2(n - 1)i&Os A p =i 0((2As + s2)W),
where A, is the complex Laplacian defined by
Af = A(i0Of) = hk 2
As an analogue of Kdhler case, we shall call such balanced metrics extremal.
From Equation (A.2), we have the following observations:
(a). If the background metric is non-Kihler, one can easily show that 0Ow 7 0.
Hence s = constant is an extremal balanced metric if and only if s = 0. In par-
ticular, Chern-Ricci-flat balanced metrics are extremal. An intuitive reason is
that there is no analogue of Kifhler-Einstein metrics with nonzero Einstein con-
stant on non-Ksihler balanced manifolds. Indeed, if there is a smooth function
f such that
p = f ,
85
one can deduce that p = 0.
(b). If there exists a (1,0)-form p such that
(A.3) 2(n - 1)s - p = (2s + s)w +p + a,
then (A.2) holds. On the other hand, if (A.2) holds, then condition (A.3) is
automatically satisfied if HA' (X) = 0, where H*4(X) is the Aeppli cohomology
group of X. In this case, by taking trace of (A.3), we get
2(n - 1)s2 = 2nAs + ns 2 + A(a + Of).
Since the last term is of divergence form, by integration over X, we get
(n - 2) s 2. W = 0.fjx n!
As we always assume that n > 2 (otherwise w is Kdhler), we conclude that
s 0.
(c). Assuming s = 0, if we further assume that 0 = c1 (X) E HBi,(X), then p = iaf
for some globally defined real function f. Therefore
0 = s = Ap = Af.
Hence by maximal principle f is a constant and p = 0.
A very important class of non-Kiihler Calabi-Yau 3-folds is of the form Xk = #k(S x
S 3) for k > 2 [50, 941, which can be constructed from projective Calabi-Yau 3-folds
by taking conifold transitions (cf. Section 2.4). Moreover, these manifolds satisfy the
00-lemma and admit balanced metrics [53]. Therefore conditions in (b) and (c) above
are satisfied. Consequently we have
Corollary A.0.6.
Extremal balanced metrics on Xk are exactly those Chern-Ricci-flat balanced metrics.
86
Hopefully this point of view will be useful in proving the balanced version of Calabi
(Gauduchon) conjecture for Xk's.
87
88
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