Gauged Linear Sigma Model and Hemisphpere Partition Function [M. Romo, E. Scheidegger, JK: arxiv:1602.01382 [hep-th], in progress] [K. Hori, JK: arxiv:1612.06214 [hep-th]] [R. Eager, K. Hori, M. Romo, JK: in progress] [D. Erkinger, JK: in progress] Johanna Knapp TU Wien CERN, January 31, 2017
Outline CYs and GLSMs Definitions Examples B-branes and Z D 2 Matrix Factorizations Hemisphere Partition Function Applications of Z D 2 Analytic Continuation of Periods D-brane Transport Landau-Ginzburg Central Charge 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.
GLSM Data G... a compact Lie group (gauge group) V... space of chiral fields φ i V ρ V : G GL(V )... faithful complex representation CY condition: G SL(V ) R : U(1) GL(V )... R-symmetry R i... R-charges R and ρ V commute charge integrality: R(e iπ ) = ρ V (J) for J G T G... maximal torus Lie algebras: g = Lie(G), t = Lie(T ) Qi a tc... gauge charges of chiral fields
GLSM Data (ctd.) t gc... FI-theta parameter t a = ζ a iθ a t a Kähler moduli of the CY σ t C g C... scalar component of the vector multiplet W Sym(V )... superpotential G-invariant R-charge 2 non-zero for compact CYs Classical Potential U = 1 8e 2 [σ, σ] 2 + 1 ( σφ 2 + σφ 2) + e2 2 2 D2 + F 2
Higgs branch: σ = 0 D-terms Classical Equations of Motion µ(φ) = ζ µ : V g... moment map G is broken to a subgroup F-terms dw = 0 Phases: parameter space gets divided into chambers Classical Vacua X ζ = {dw 1 (0)} µ 1 (ζ)/g Ideal I ζ : φ V where the quotient is ill-defined
Singularities Coulomb branch: φ = 0, ζ = 0 σ fields can take any value classically. One-loop corrections generate en effective potential. W eff = t(σ) i Q i (σ)(log(q i (σ)) 1) + πi α>0 α(σ) Q i : weights of the matter representation ρ V α > 0: positive roots Singularities are lifted except for points, lines, etc. discriminant One can smoothly interpolate between the phases. The CY undergoes a topological transition. Mixed Branches if rkt > 1.
Example 1: Quintic G = U(1) Field content: φ = (p, x 1,..., x 5 ) C( 5) C(1) 5 Potential: W = pg 5 (x 1,..., x 5 ) D-term: 5 p 2 + 5 i=1 x i 2 = ζ F-terms: G 5 (x 1,..., x 5 ) = 0, p G 5 x i = 0 ζ 0: p = 0, Quintic G 5 = 0 in P 4 ζ 0: Landau-Ginzburg orbifold with potential G 5 Landau-Ginzburg/CY correspondence Moduli space [Witten 93][Herbst-Hori-Page 08][Kontsevich,Orlov] LG CY G Z 5 ζ e (ζ iθ) = 1 3125 ζ G 1
Example 2: Rødland model G = U(2) Field content: (p 1,..., p 7, x 1,... x 7 ) (det 1 S) 7 S 7 with S C 2 fundamental representation Potential: D-terms: W = 7 2 i,j,k=1 a,b=1 A ij k pk x a i ε ab x b k = 7 p i 2 + i=1 7 A ij (p)[x i x j ] i,j=1 7 x i 2 = ζ j=1 xx 1 2 x 2 1 2 = 0
Example 2: ctd. ζ 0: Complete intersection of codimension 7 in G(2, 7): 7 i,j=1 Aij k [x ix j ] = 0 ζ 0: Pfaffian CY on P 6 : 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) ζ ζ Grassmannian CY G 1 The two CYs are not birational, i.e. no relation via a flop transition.
Example 3: Non-abelian two parameter model A two-parameter non-abelian model with two geometric phases. [Hori-JK arxiv:1612.06214 [hep-th]] G = (U(1) 2 O(2))/H free Z 2 -quotient of a codimension 5 complete intersection in a toric variety determinantal variety Phases hybrid hybrid geometric X Ỹ geometric unknown hybrid
Example 3: (ctd.) The CYs in the geometric phases are not birational. There is a phase which has both strongly and weakly coupled components. The discriminant locus has a mixed Coulomb-confining branch that is mapped to a mixed Coulomb-Higgs branch under the 2D version of Seiberg duality. [Hori 11] There is an extra phase boundary due to non-abelian D-term equations. Mirror symmetry calculations work here because one phase is almost toric.
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 P 11222 [Hori-JK 13] Models with G = U(1) SU(2) and U(2) Hybrid models Pfaffian CYs in weighted P 4 [Kanazawa 10][Hori-JK 13] PAX/PAXY [Jockers et al. 12] SSSM [Gerhardus-Jockers 15]
B-branes in the GLSM B-branes in the GLSM are G-invariant matrix factorizations of the GLSM potential with R-charge 1 [Herbst-Hori-Page 08][Honda-Okuda,Hori-Romo 13] Data: Z 2 -graded Chan-Paton space: space M = M 0 M 1 Matrix Factorization: Q End 1 (M) with Q 2 = W id M G-action: ρ : G GL(M) with ρ(g) 1 Q(gφ)ρ(g) = Q(φ) R-action: r : u(1) R gl(m) with λ r Q(λ R φ)λ r = λq(φ)
Examples No classification for matrix factorizations! However, there are some canonical examples. Example 1: Quintic Example 2: Quintic Q 1 = pη + G 5 (x) η {η, η} = 1 Q 2 = 5 x i η i + p 5 i=1 Example 3: Rødland G 5 x i η i {η i, η j } = δ ij Q 3 = 7 i=1 p i η i + W p i η i Studied mostly in the context of B-twisted Landau-Ginzburg models.
Hemisphere Partition Function SUSY localization in the GLSM yields the hemisphere partition function. [Honda-Okuda,Hori-Romo 13] Z D 2(B) = C d rk G σ α(σ) sinh(πα(σ)) ( Γ iq i (σ) + R ) i e it(σ) f B(σ) γ 2 α>0 i α > 0 positive roots σ t C twisted chirals R i... R-charges, Q i... gauge charges t = ζ iθ... complexified FI (Kähler) parameter(s) γ... integration contour (s.t. integral is convergent) Brane factor f B (σ) = tr M ( e iπr e 2πρ(σ)) M... Chan-Paton space The brane input is obtained by restricting the matrices ρ(g) and λ r to the maximal torus.
What does Z D 2 compute? Z D 2 computes the fully quantum corrected D-brane central charge. In large volume of U(1) GLSMs (CY hypersurface X in toric ambient space), it reduces to the Gamma class: [Hori-Romo 13] Z LV D 2 = C e nt n=0 ˆΓ(x) = Γ ( 1 x ) 2πi... Gamma class: ˆΓˆΓ =  X ω... Kähler class ch(b LV )... Chern character of the LV brane B H 2 (X, Z)... B-field X ˆΓ X (n)e B+ 1 2π ω ch(b LV )
Mirror Symmetry Instanton corrections can be expressed in terms of the periods of the mirror CY. Example: Structure sheaf O X Z(O X ) = mirror = H 3 3! ( ) it 3 ( ) it c2 H + 2π 2π 24 + i ζ(3) (2π) 3 χ(x ) + O(e t ) H 3 3! ϖ 3 + c 2H 24 ϖ 2 + i ζ(3)χ(x ) (2π) 3 ϖ 0 This is exactly what one gets when one evaluates Z D 2 ζ 0-phase for Q 1 and Q 3. However, no mirror symmetry required! in the
Analytic continuation of periods Z D 2 is a Mellin-Barnes integral. Quintic: Z D 2(B) = C dσ Γ(iσ) 5 Γ(1 5iσ)e itσ f B (σ) Z D 2 satisfies a Picard-Fuchs equation: LZ D 2 = 0 Quintic: θ = z d dz, z = e t. L = θ 4 5z(5θ + 1)(5θ + 2)(5θ + 3)(5θ + 4) Depending on how we close the integration contour we get an expansion in terms of periods near the large volume/lg point. Z D 2 can be used for analytic continuation of periods.
The quintic and the conifold point What about the conifold point? [Romo-Scheidegger-JK arxiv:1602.01382 [hep-th]] Rewrite the integrand of Z D 2 such that the integral becomes convergent near the conifold point when the contour is closed in a certain way. For one-parameter CY hypersurfaces in (weighted) P N we can use the fact that Z D 2 with suitable brane factor is the integral representation of certain hypergeometric and MeijerG functions Use identities like ( ) ( ) a, b Γ(c)Γ(c a b) a, b 2F 1 ; z = c Γ(c a)γ(c b) 2F1 1 + a + b c ; 1 z ( ) Γ(c)Γ(a + b c) c a, c b + 2F 1 Γ(a)Γ(b) 1 a b + c ; 1 z
The quintic and the conifold point (ctd.) To tackle the quintic one needs recursion formulas to reduce the expressions to such basic identities. Recursions relating n+1 F n to n F n 1 [Bühring 92] Analytic continuation of fundamental period/d0 brane Recursions for MeijerG functions [Nørlund 55][Scheidegger arxiv:1602.01384 [math.ca]] Analytic continuation of the remaining periods/b-branes Analytic continuation of a basis of B-branes/periods to the conifold point Convergent power series. closed form? Direct check that a particular D6-brane becomes massless.
D-brane transport One can use the GLSM to transport D-branes from one phase to another. Solved for abelian GLSMs. [Herbst-Hori-Page 08] Transport is non-trivial due to singularities in the moduli space. Quintic Rødland ζ θ 0 2π 2π 0 2π
Paths and grade restriction Not every GLSM brane can be transported along a given path in a well-defined way. Brane may become unstable/break SUSY. Which branes are allowed is determined by the grade restriction rule. [Herbst-Hori-Page 08] Given a path (i.e. choice of θ-angle) only certain gauge charges of the brane are allowed. Given a brane in a phase, one can always find a GLSM brane that is in the charge window. Understood in the abelian case. New approach: The hemisphere partition function knows about the grade-restriction rule. [Eager-Hori-Romo-JK in progress]
Z D 2 and grade restriction We can obtain the grade restriction rule from the asymptotic behavior of the hemisphere partition function. Z q D 2 = d l G σe Aq(σ), σ = σ 1 + iσ 2 γ A q(σ) = ζ(σ 2 ) (θ 2πq)(σ 1 ) + + i { Q i (σ 2 ) (log Q i (σ) 1) + Q i (σ 1 ) ( π 2 + arctan Q )} i (σ 2 ) Q i (σ 1 ) Condition: Z D2 has to be convergent for a given contour γ. Depending on the path, this restricts the allowed charges q of the brane. A q is the effective boundary potential on the Coulomb branch. Difficult to analyze because we have to find a parametrization for γ.
Results on grade restriction For the quintic, the known grade restriction rule is reproduced. [Hori-Romo 13] A q(σ) = (ζ 5 log 5) }{{} ζ eff σ 2 + (5π sgn(σ 1 )(θ + 2πq)) σ 1 ζ eff 0: σ 2 0 is admissible. ζ eff = 0: 5 2 < θ 2π + q < 5 2 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.
Z D 2 in non-geometric phases In geometric phases Z D 2 computes the quantum corrected central charge of the D-brane. What about Landau-Ginzburg orbifold phases? [Lerche-Vafa-Warner 89][Intriligator-Vafa 90][Fan-Jarvis-Ruan-Witten][Chiodo-Iritani-Ruan] In one-parameter models with a LG-phase the hemisphere partition function computes ( ) ZD LG = I 2 FJRW, ˆΓ FJRW ch(q) [Romo-Scheidegger-JK] I FJRW... I -Function (periods) ˆΓ FJRW... Gamma-class ch(q)... Chern character [Walcher 04] (, )... pairing
Landau-Ginzburg Central Charge (ctd.) Works for a basis of branes/periods on the quintic. Does this give a definition of a quantum-corrected central charge, like in the geometric phases? Does it define a stability condition? [Douglas-Fiol-Römelsberger 00][Aspinwall-Douglas 01][Douglas 02][Bridgeland] At the Landau-Ginzburg point this should reduce to R-stability [Walcher 04] Compare stable vs. unstable configurations of branes and their bound states.
Summary GLSMs are useful physics tools to study CYs and their moduli spaces. Non-abelian GLSMs lead to exotic CYs that are not complete intersections in toric ambient spaces. Supersymmetric localization gave us new tools to calculate quantum corrections in CY compactifications. Without mirror symmetry Applies to exotic CYs: determinantal varieties, etc. Applies also to exotic phases: strongly coupled, hybrid, etc. The hemisphere partition function computes the quantum corrected central charge of a D-brane Analytic continuation of periods D-brane transport and monodromies Landau-Ginzburg central charge
Open Problems Systematic construction of exotic CYs via non-abelian GLSMs Central charge in non-geometric phases. Hemisphere partition function as ( global ) stability condition? Better understanding of hybrid phases D-branes and D-brane transport on exotic CYs Mirror symmetry for exotic CYs Analytic continuation of periods of CYs with more than one conifold point