D-branes in non-abelian gauged linear sigma
models
with R. Eager, K. Hori and M. Romo
Johanna Knapp
TU Wien
LMU, January 15, 2015
Outline
CYs and GLSMs
D-branes in GLSMs
Matrix Factorizations
Hemisphere Partition Function
D-brane transport
Conclusions
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
CYs and GLSMs
• A Calabi-Yau space (CY) can be realized as the low energy
configuration of an N = (2, 2) supersymmetric gauge theory
in two dimensions - the gauged linear sigma model (GLSM).
[Witten 93]
• The choice of gauge groups and matter content determines
the CY.
• Hypersurfaces and complete intersections in toric ambient
spaces are realized by GLSMs with gauge groups G = U(1)k .
• Non-abelian gauge groups lead to exotic CYs (e.g. Pfaffian
CYs).
• Phases:
• The FI-parameters and the θ-angles can be identified with the
the Kähler moduli of the CY.
• By tuning these couplings we can probe the Kähler moduli
space.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
GLSM Data
• Gauge group G :
• Should contain at least one U(1) ↔ at least one Kähler
parameter
• Chiral fields:
• p i , i = 1, . . . , M: charged under the U(1)s: Qp i
• xia , i = 1, . . . , N: in fundamental representations of G and
U(1)s: Qxia
• CY condition: U(1)-charges sum to 0.
• Twisted chiral fields:
• σi , i = 1, . . . , k: take values in the maximal torus of G .
• Superpotential W
• G -invariant polynomial in chirals with R-charge 2.
• For W 6= 0, the CY is compact.
Conclusions
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Equations of Motion I
• Classical e.o.ms: minimize the GLSM potential
N
V
=
1
e2
1X † †
† 2
2
Tr[σ,
σ
]
+
|D|
+
xj {σ , σ}xj + |F |2
2e 2
2
2
j=1
+
M
X
Qp2i |σ|2 |p i |2 +
i=1
N
X
Qx2j |σ|2 ||xj ||2
i=j
• σ = 0: Higgs branch → G broken (to a subgroup)
• σ=
6 0, all/some chirals 0: Coulomb/mixed branch G broken to
(subgroup of) maximal torus
• F-terms
∂W
=0
∂p i
∂W
= 0.
∂xia
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Equations of motion II
• D-terms:
•
l
U(1) :
M
X
i=1
Qpl i |p i |2
+
N
X
Qxl ja ||xj ||2 = rl
j=1
• Further conditions for non-abelian factors, e.g. O(k) :
xx † − (xx † )T = 0
• rl ∈ R are the FI parameters. Together with the θl -angle they
can be identified with the complexified Kähler modulus of the
CY: tl = rl − iθl
• Phases: Different solutions for different values of rl .
• At low energies (IR), the CY can be described by different
theories.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Phase Diagram
hybrid model
non−linear sigma model
II
G II = Z2
hybrid model
I
GI = 1
III
G III
IV
G IV = SU(2)
non−linear sigma model
V
G V = Z4
r2
LG−model
r1
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Singularities
• At the phase boundaries the theory becomes singular.
• σ fields can take any value classically.
• One-loop corrections generate en effective potential.
feff = −
W
X
χ(σ)(log(χ(σ)) − 1) + . . .
χ
• χ: characters of the representation the fields transform in with
respect to the maximal torus
• “. . .”: further terms linear in σ, the FI parameters and θ-angle
contributions
[Hori,JK 13]
• Singularities are lifted except for points, lines, etc. → conifold
singularities
• One can smoothly interpolate between the phases.
• The CY undergoes a topological transition.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Example 1: Quintic - G = U(1)
• Field content: (p, x1 , . . . , x5 ) with Q = (−5, 1, . . . , 1)
• Potential: W = pG5 (x1 , . . . , x5 )
• D-term: −5|p|2 +
P5
2
i=1 |xi |
=r
5
• F-terms: G5 (x1 , . . . , x5 ) = 0, p ∂G
∂xi = 0
• r 0: p = 0, Quintic G5 = 0 in P4
• r 0: Landau-Ginzburg orbifold with potential G5
• Landau-Ginzburg/CY correspondence
[Witten 93][Herbst-Hori-Page 08][Kontsevich,Orlov]
• Moduli space
LG
G = Z5 r → −∞
1
e −(r +iθ) = − 3125
r →∞
CY
G =1
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Example 2: Rødland model - G = U(2)
• Field content: M=N=7, g ∈ U(2)
pi
→ det−1 gp i
xi
→ gxi
• Potential:
W =
7
2
X
X
Aijk p k xia εab xkb
=
7
X
Aij (p)[xi xj ]
i,j=1
i,j,k=1 a,b=1
• D-terms:
−
7
X
i=1
i 2
|p | +
7
X
|xi |2 = r
j=1
1
xx † − ||x||2 12 = 0
2
Conclusions
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Example 2: ctd.
• r 0: Complete intersection of codimension 7 in G (2, 7):
ij
i,j=1 Ak [xi xj ]
P7
=0
• r 0: Pfaffian CY on P6 : rkA(p) = 4; G broken to SU(2)
→ strongly coupled!
• Grassmannian/Pfaffian correspondence
[Hori-Tong 06][Rødland,Kuznetsov,Addington-Donovan-Segal,Halpern-Leistner,Ballard-Favero-Katzarkov]
• Moduli space
Pfaffian CY
G = SU(2)
r → −∞
r →∞
Grassmannian CY
G =1
• The two CYs are not birational, i.e. no relation via a flop
transition.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Further non-abelian examples
• One-parameter models with (U(1) × O(2))/H
• Reye congruence ↔ determinantal quintic [Hosono-Takagi 11-14][Hori 11]
• Hybrid model ↔ determinantal variety in P11222
[Hori-JK 13]
• Models with G = U(1) × SU(2) and U(2)
• Hybrid models ↔ Pfaffian CYs in weighted P4
[Kanazawa 10][Hori-JK 13]
• Two-parameter models
[Hori-JK unpublished]
• Systematic construction with GLSM in principle possible, but
• Phases hard to analyze.
• Smooth geometric phases hard to find.
• Some clever combinatorial technology required.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Matrix Factorizations
• B-type D-branes are gauge-invariant, R-invariant matrix
factorizations of the GLSM potential.
[Herbst-Hori-Page ’08][Honda-Okuda,Hori-Romo ’13]
• Take a square matrix Q with polynomial entries such that
Q2 = W 1
• Gauge invariance:
ρ(g )−1 Q(g φ)ρ(g ) = Q(φ)
• R-invariance:
λr∗ Q(λR φ)λ−r∗ = λQ(φ)
• GLSM branes map to (trivial or non-trivial) branes in the
individual phases.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Elementary matrix factorizations I
• To construct matrix factorizations for the Rødland example we
start with a toy model with gauge group U(2).
• Superpotential
W = p[x1 x2 ] = p(x11 x22 − x12 x21 ).
a b
• Gauge transformations with g =
∈ U(2):
c d
p
→ (ad − bc)−1 p
xi1
→ axi1 + bxi2
xi2
→ cxi1 + dxi2 .
• R-charge: can choose [p] = 2, [xia ] = 0.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Elementary matrix factorizations II
• Simplest one:
Q=
0
x11 x22 − x12 x21
p
0
ρ=
1
0
0
det g
• Using a basis of 4 × 4 Clifford matrices {ηi , η̄j } = δij :
Q = x11 η1 + px22 η̄1 + x12 η2 − px21 η̄2

0
 0
Q=
 px22
−px21
0
0
x12
−x11
x11
−px21
0
0

x12
2 
−px2 
0 
0
1
 0
ρ=
 0
0

0
1
det g
0
0
0
0
d
det g
− detb g
0
0


− detc g 
a
det g
• For the models of interest we can obtain matrix factorizations
by taking tensor products of these two elementary
constructions.

CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
D-brane charges
• SUSY localization in the GLSM yields the hemisphere
partition function.
[Honda-Okuda,Hori-Romo ’13]
Y
Y Ri
d lG σ
ZD 2 (B) = C
α(σ) sinh(πα(σ))
Γ iQi (σ) +
e it(σ) fB (σ)
2
γ
α>0
i
Z
•
•
•
•
•
•
α > 0 positive roots
σ =∈ tC with tC = Lie(TG ) ⊗ C (maximal torus)
lG = rk(G )
Ri . . . R-charges, Qi . . . U(1)-charges
t = r − iθ . . . complexified FI (Kähler) parameter(s)
γ . . . integration contour (s.t. integral is convergent)
• Brane factor
fB (σ) = trM e iπr∗ e 2πρ(σ)
• M . . . Chan-Paton space
• The brane input is obtained by restricting the matrices ρ(g ) and λr∗ to
the maximal torus.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
What does ZD 2 compute?
• ZD 2 computes the fully quantum corrected D-brane central
charges.
• In large volume of U(1) GLSMs (CY hypersurface X in toric
ambient space), it reduces to the Gamma class:
ZDLV2
=C
∞
X
e
n=0
•
•
•
•
−nt
Z
[Hori-Romo ’13]
1
Γ̂X (n)e B+ 2π ω ch(BLV )
X
x
Γ̂(x) = Γ 1 − 2πi
. . . Gamma class: Γ̂Γ̂∗ = ÂX
ω . . . Kähler class
ch(BLV ) . . . Chern character of the LV brane
B ∈ H 2 (X , Z) . . . B-field
• Instanton corrections can be expressed in terms of the periods
of the mirror CY.
• No mirror symmetry required!
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Structure Sheaf
• The structure sheaf encodes the topological data of the CY.
Z (OX )
=
mirror
=
H 3 it 3
it c2 H
ζ(3)
+
+i
χ(X ) + O(e −t )
3! 2π
2π 24
(2π)3
H3
c2 H
ζ(3)χ(X )
$3 +
$2 + i
$0
3!
24
(2π)3
• Different topological data for different (geometric) phases in
the GLSM.
• This is a way to extract topological data of exotic CYs.
• Can we identify the brane factor/matrix factorization
associated to the structure sheaf?
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Structure Sheaf II
• For the Rødland model matrix factorizations discussed above
lead to a brane factors of type
fB (a, b) = (1 − e 2π(σ1 +σ2 ) )a (1 − e 2πσ1 )b (1 − e 2πσ2 )b
a, b ∈ Z≥0
• In r 0-phase the structure sheaf is given by the matrix
factorization with brane factor f (7, 0)
Q(7, 0) =
7
X
i=1
p i ηi +
∂W
η̄i
∂p i
• In the strongly-coupled r 0-phase no simple matrix
factorization has been found. However,
• We can construct matrix factorizations with D6-charges from
Lascoux complexes.
• In the weakly-coupled dual GLSM the structure sheaf is
described by a canonical matrix factorization.
[Hori 11]
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
D-brane transport
• The CY undergoes a topological transition when one goes
from one phase to another.
• Also the D-brane charges change when crossing the phase
boundaries.
• Given a D-brane in one phase, how can be transport it to
another phase?
• Solved for abelian GLSMs.
[Herbst-Hori-Page 08]
• Depending on the choice of poles (or the integration contour
γ), the hemisphere partition function computes the brane
charges in different phases.
• Main problem: How to get the brane past the conifold
singularity?
• New problem: Branes in strongly coupled phases?
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
How to transport from phase A to B?
1. Start with a brane in phase A and lift it to a GLSM brane.
2. Select a charge window corresponding to a specific path from
A to B that avoids the conifold singularity.
3. If the brane does not fit the window, grade restrict by binding
with a GLSM brane that is trivial in A.
4. Push down to a brane in phase B.
• The hemisphere partition function knows about the grade
restriction rule.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
GLSM branes vs. branes in phases
• There are infinitely many GLSM branes that map to the same
brane in a given phase.
• There are branes in the GLSM that are “empty” in a given
phase.
• To make the brane transport between phases well-defined one
has to restrict to a subset of GLSM branes.
Rødland
Quintic
D(GLSM)
D(GLSM)
∪
πLG
∪
πCY
πPf
D(LG )
D(CY )
πGr
S
Tw
D(Pf )
∪
Tw
D(Gr )
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Why is there a grade restriction rule?
• To get from one phase to another one has to choose paths
that avoid the conifold singularity.
• Different paths correspond to different choices of the window
categories.
• To be an object in a given window category that GLSM charges
(determined by ρ(g )) have to be in a certain charge window.
• Paths differ by conifold monodromies of the brane.
Rødland
Quintic
r
θ
0
2π
−2π
0
2π
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
How to find the grade restriction rule?
• We can derive the grade restriction rule from the asymptotic
behavior of the hemisphere partition function.
Z
ZD 2
=
Aq (σ)
=
d lG σe −Aq (σ) ,
σ = σ1 + iσ2
γ
−
X
|α(σ1 )| +
α>0
i
+|Qi (σ1 )|
X
Qi (σ2 )(log Qi (σ) − 1)
π
Qi (σ2 )
+ arctan
+ r (σ2 ) − (θ + 2πq)(σ1 )
2
|Qi (σ1 )|
• Condition: ZD2 has to be convergent for a given path γ.
• Depending on the path, this restricts the allowed charges q of
the brane.
• Difficult to analyze because we have to find a parametrization
for γ.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Results on grade restriction
• For the quintic, the known grade restriction rule is reproduced.
Aq (σ) = (r − 5 log 5) σ2 + (5π − sgn(σ1 )(θ + 2πq))|σ1 |
|
{z
}
r eff
• r eff ≷ 0: σ2 ≷ 0 is admissible.
θ
• r eff = 0: − 52 < 2π
+ q < 25
• Rødland CY:
• We find four different windows corresponding to the four
inequivalent paths.
• We can verify the correct monodromy behavior for the paths
around the three conifold points.
• In the strongly coupled Pfaffian phase, there is a grade
restriction rule deep inside the phase.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
How to grade restrict?
• Grade restriction by binding (via tachyon condensation) empty
branes.
• Binding with an empty brane changes the GLSM brane but
not the brane in the IR.
• For every phase a set of empty branes has to be identified.
• For the quintic empty branes are certain Koszul complexes
completed to matrix factorizations.
• For the Rødland example the empty branes in the
Grassmannian phase are twisted Lascoux complexes.
[Donovan-Segal 12]
• GLSM matrix factorizations associated to the Lascoux
complexes can be constructed.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
More on Rødland
• Given a Grassmannian G (S, V ) a Lascoux complex is given by
a complex of terms of the form
Symh S ∗ (i) ⊗ ∧j V ∗ ⊗ C[S ⊗ V ] i ∈ Z,
j, h ∈ Z≥0
• Example of an empty brane with (h, i, j):
0 → (3, −3, 7) → (2, −3, 6) → (1, −3, 5) → (0, −3, 4) → (0, −2, 2) → (1, −1, 1) → (2, 0, 0) → 0
• By adding backwards arrows, this can be completed to a
matrix factorization of the GLSM potential.
• There are six empty branes in the Grassmannian phase.
• By binding certain Lascoux complexes, we can construct
matrix factorizations for D6-branes in the Pfaffian phase.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Summary
• Non-abelian GLSMs lead to CYs which are not complete
intersections in toric ambient spaces.
• Phases of non-abelian GLSM are not necessarily birational but
derived equivalent.
• D-branes in non-abelian GLSMs are G -invariant matrix
factorizations of the GLSM potential.
• D-brane charges plus quantum corrections are computed by
the hemisphere partition function.
• Convergence of the hemisphere partition function determines
the grade restriction rule.
• Empty branes (Grassmannian), required for D-brane transport,
are twisted Lascoux complexes.
CYs and GLSMs
D-branes in GLSMs
D-brane transport
Conclusions
Open Questions
• Pfaffian branes and their lift to the GLSM.
• Empty branes in the Pfaffian phase.
• D-brane transport for examples other than Rødland.
• Explicit parametrizations for the integration path γ and
numerical convergence checks.
• D-branes and hemisphere partition function in non-geometric
phases.