Cluster algebras from 2d gauge theories

Cluster algebras from 2d gauge theories Francesco Benini Simons Center for Geometry and Physics Stony Brook University Texas A&M University Heterotic Strings and (0,2) QFT 28 April 2014 with: D. Park, P. Zhao (to appear)

Introduction 2d gauge theories with N = (2, 2) supersymmetry Seiberg-like dualities Cluster Algebra structures

Introduction 2d gauge theories with N = (2, 2) supersymmetry Seiberg-like dualities Cluster Algebra structures Cluster algebra: [Fomin, Zelevinsky 2001] to describe coordinate rings of groups and Grassmannians Other contexts: Teichmuller theory [Fock, Goncharov 03] Integrable systems (Y-systems) Wall crossing in 4d N = 2 theories Amplitudes Many interesting properties: Total positivity Laurent phenomenon Poisson structure...

Cluster algebra [Fomin, Zelevinsky 01] Commutative ring with unit and no zero divisors with distinguished set of generators called cluster variables. Set of cluster variables = (non-disjoint) union of distinguished collections of n-subsets called clusters. Exchange property (mutations): for every cluster x and x x, there is another cluster obtained by substituting x x with rule xx = M 1 + M 2 M 1,2 : monomials in n 1 variables x \ {x}, with no common divisors. Any two clusters can be obtained from each other by sequence of mutations.

Cluster algebra [Fomin, Zelevinsky 06] Seed x: Skew-symmetric matrix b ij quiver B (no 1-, 2-cycles) Coefficients y i P semifield (, ) (tropical) Cluster variables x i. i b ij : j

Cluster algebra [Fomin, Zelevinsky 06] Seed x: Skew-symmetric matrix b ij quiver B (no 1-, 2-cycles) Coefficients y i P semifield (, ) (tropical) Cluster variables x i. i b ij : j Mutation (at node k): b ij = { b ij b ij + sign(b ik ) [b ik b kj ] + if i = k or j = k otherwise k y j = { y 1 k if j = k y j y [b kj ] + k (y k 1) b kj otherwise ( yk 1 i x j = x k y k 1 x[b ik] + i + 1 ) i y k 1 x[ b ik] + i if j = k x j otherwise. Hierarchical structure. k

Cluster algebra Total positivity Cluster algebra transformations involve +, not. Canonical choice of positive submanifold of a cluster manifold.

Cluster algebra Total positivity Cluster algebra transformations involve +, not. Canonical choice of positive submanifold of a cluster manifold. Laurent phenomenon Any cluster variable x i, viewed as a rational function of the variables in a given cluster x, is a Laurent polynomial. It is conjectured that has positive coefficients.

Outline Seiberg-like dualities of 2d N = (2, 2) gauge theories S 2 partition function From dualities to cluster algebras (Speculative) applications

2d Seiberg-like dualities

2d Seiberg-like dualities [Jockers, Kumar, Lapan, Morrison, Romo 12; FB, Cremonesi 12] 2d N = (2, 2) SUSY gauge theories of vector and chiral multiplets A: U(N) with N f fundamentals, N a antifundamentals B: U ( max(n f, N a ) N ) with N a fundamentals, N f antifundamentals, N f N a gauge singlets, superpotential W dual = qmq N a N Nf N a N' N f Comments: cfr. with 4d: no gauge anomaly any N f, N a similar to Hori-Tong duality but U(N) instead of SU(N) N f = N a : flow to IR CFT (otherwise gapped) deformations: complexified FI term, twisted masses, superpotential terms t = 2πξ + iθ, m j, m f z e t, z = 1 z

Geometric interpretation [Jia, Sharpe, Wu 14] Large positive FI (assume N f N a ): geometric realization Gr(N, N f ) = Gr(N f N, N F ) Theory A: each antifundamental gives a copy of tautological bundle S Equality of bundles: S Na Gr(N, N f ) = (Q ) Na Gr(N f N, N f ) Universal quotient bundle: 0 S O N f Q 0 Theory B: mesons give (O N f ) Na, superpotential imposes short exact sequence.

Chiral ring Chiral operators modulo F-term relations Theory A: Qf Q j (no baryons) Theory B: q j q f, M fj, but W imposes q j q f = 0 Map: Qf Q j = M fj No quantum corrections in the chiral ring Add superpotential W = f( Q f Q j ) = f(m fj )

Twisted chiral ring Generators: Tr σ k k = 1,..., N or symm. polynomials in σ = diag(σ 1,..., σ N ) Q(x) = det(x σ) = x N x N 1 Tr σ +...

Twisted chiral ring Generators: Tr σ k k = 1,..., N or symm. polynomials in σ = diag(σ 1,..., σ N ) Q(x) = det(x σ) = x N x N 1 Tr σ +... Relations: effective superpotential on the Coulomb branch t = 2πξ + iθ W eff = t σ a ( )[ ( ) ] ρ(σ) mj log ρ(σ) mj 1 a j ρ R j Impose 0 = W eff / σ a and σ a σ b : (quantum equivariant coh) z e t Nf j=1 (x m f ) + i Na N f z N a f=1 (x m f ) = C(z) Q(x) T (x) T (x) has degree N = max(n f, N a ) N 1 if N f > N a C(z) = 1 + z if N f = N a i Na N f z if N f < N a

Twisted chiral ring Generators: Tr σ k k = 1,..., N or symm. polynomials in σ = diag(σ 1,..., σ N ) Q(x) = det(x σ) = x N x N 1 Tr σ +... Relations: effective superpotential on the Coulomb branch t = 2πξ + iθ W eff = t σ a ( )[ ( ) ] ρ(σ) mj log ρ(σ) mj 1 a j ρ R j Impose 0 = W eff / σ a and σ a σ b : (quantum equivariant coh) z e t Nf j=1 (x m f ) + i Na N f z N a f=1 (x m f ) = C(z) Q(x) T (x) T (x) has degree N = max(n f, N a ) N Duality: T (x) = Q (x) = det(x σ ) 1 if N f > N a C(z) = 1 + z if N f = N a i Na N f z if N f < N a

S 2 partition function [FB, Cremonesi 12; Doroud, Gomis, Le Floch, Lee 12] Any Euclidean 2d N = (2, 2) theory with R V -symmetry can be placed supersymmetrically on S 2, with no twist. Z S 2(param) = Dφ e S(param) Parameters: complexified FI term t = 2πξ + iθ, z e t Localization: Z gauge 1-loop = Z chiral 1-loop = real twisted masses m j, m f and flavor magnetic fluxes n j, ñ f R-charges R Z S 2 = 1 W roots α>0 chiral Φ ρ R j m Z N ) d N i Re W (σ+ σ e i2 m Z 1-loop ( ) α(m) 2 + α(σ) 2 4 ( Γ R[Φ] iρ(σ) if i [Φ]m 2 i ρ(m)+f i [Φ]n i ) 2 Γ ( 1 R[Φ] + iρ(σ) + if 2 i [Φ]m i ρ(m)+f i [Φ]n i ) 2

S 2 partition function We can prove: Z (A)( m j, n j, m f, ñ f ; z ) = f contact f Kt Z (B)( m f i 2, ñ f, m j i 2, ñ f ; z 1) Method: vortex partition function Z S 2 = Higgs vacua Z cl Z 1-loop Z vortex Z antivortex Mass shift: shift of R-charges, compatible with W dual r R[ Q f Q j ] = R[M fj ] R[ q j q f ] = 2 r f Kt : real function Kähler transformation FT: Lagrangian term, improvement transformation [Closset, Cremonesi 14] Geom: according to [Jockers, Kumar, Lapan, Morrison, Romo 12] Z S 2 = e K Kähler Kähler transformation (does not affect the metric)

The contact term f contact = e i Re W contact : phase Twisted superpotential, only function of parameters (twisted chirals): m j + i 2 n j, m f + i 2ñf, t Theory A: WA = t Tr σ = log z Tr σ Theory B: result depends on number of flavors N f > N a + 1: N f = N a + 1: WB = log z 1 Tr σ + log z Tr m WB = log z 1 Tr σ + log z Tr m + iz N f = N a : WB = log z 1 Tr σ + log z 1+z Tr m + log(1 + z) Tr m

Quivers

Quivers and cluster algebras Class of quiver gauge theories: no 1-cycles nor 2-cycles. Apply Seiberg-like duality to a node k Quiver: almost CA action Never generate 1-cycles. But assume that all 2-cycles are accompanied by quadratic superpotential integrate them out. Coefficients. Transformation of ranks N = max(n f, N a ) N Tropical semifield P (, ): u a u b = u a+b, u a u b = u max(a,b) u is formal variable { Ranks: r i = u Ni r j r 1 ( k i = r[bij]+ i ) i r[ bij]+ i j = k otherwise r j Beta-functions: y j i rbij i y j = u βj y j transform as CA coefficients. u interpreted as cutoff scale.

Cluster variables Transformation of FI parameters z e t : N a < N f : z a z a z z 1 z f z f z N a = N f : z a z a (1 + z) z z 1 z z f z f 1+z N a > N f : z a z a z z z 1 z f z f From cluster variables define: z 1 z j k = z j z [b kj] + Conformal case y i 1: k z j = i xbij i Transformation of FIs exactly reproduces CA General case: j = k ( yk y k 1 z k + 1 y k 1) bkj otherwise The transformation rules are the u 0 term in the expression. Can be extracted taking u limit. Cluster algebra encodes the transformations for every possible choice of ranks.

The superpotential We assumed that whenever Seiberg-like duality generates a 2-cycle, this is removed by suitable quadratic superpotential term W quad. Highly non-trivial! Q: given a quiver, there an R-symmetric superpotential W such that sequences of mutations, 2-cycles are always massive? Such W : non-degenerate graded potential Theory: non-degenerate if it admits such a potential. It is easy to produce examples of degenerate theories Complete classification of non-degenerate theories is not known E.g.: quivers dual to ideal triangulations of marked Riemann surfaces are non-degenerate.

The Q-polynomial Twisted chiral ring of the quiver: Q j (x) = det(x σ j ) i Q i(x) [bji]+ + i Na(j) N f (j) z j Q i(x) [ bji]+ = C j (z j ) Q j (x) T j (x) j i Under duality, identify Q k (x) = T k(x).

The Q-polynomial Twisted chiral ring of the quiver: Q j (x) = det(x σ j ) i Q i(x) [bji]+ + i Na(j) N f (j) z j Q i(x) [ bji]+ = C j (z j ) Q j (x) T j (x) j i Under duality, identify Q k (x) = T k(x). Define dressed Q-polynomials : Q j (x) x j det(x σ j ) Q-polynomials transform as cluster variables! Q i Q i(x) [b ki] + + i Q i(x) [ b ki] + j(x) = Q k (x) Q j (x) j = k otherwise

Some applications

The quantum Kähler moduli space Conformal quivers flow to IR CFTs: NLSM on (non-compact) Calabi-Yau s Complexified FI parameters z i control Kähler moduli Metric on Kähler moduli space computed by [Jockers, Kumar, Lapan, Morrison, Romo 12] Z S 2(t i, t i ) = e K Kähler(t i, t i) Quantum Kähler moduli space has cluster algebra structure Also has Poisson structure. Compact example: Gulliksen-Negård CY 3 X = {φ P 7 rank(a a 4 4φ a ) 2} (h 1,1, h 2,1 ) = (2, 34) Can be realized by a U(1) U(2) quiver [Jockers, Kumar, Lapan, Morrison, Romo 12] 8 1 2 4 There is a cubic superpotential that breaks the flavor symmetry.

Quantum integrable systems (spin chains) Nekrasov-Shatashvili: N = (2, 2) gauge theory quantum integrable system Q: Integrable systems for our quivers? Seiberg-like dualities?

Quantum integrable systems (spin chains) Nekrasov-Shatashvili: N = (2, 2) gauge theory quantum integrable system Q: Integrable systems for our quivers? Seiberg-like dualities? N = (2, 2) SQCD: { U(N), Nf hypers (fund + antifund), one adjoint W = QΦQ Param: cplx FI t = 2πξ + iθ, twisted masses m Φ, m Q, m Q = m Φ m Q SU(2) periodic twisted inhomogeneous XXX 1 2 spin chain, N f nodes, sector with S z = N f 2 + N (N-particle states), inhomogeneous def s ν a H = J N f a=1 S a S a+1 SNf +1 = e i 2 ϑσ3 S 1 e i 2 ϑσ3 Dictionary: z = e t = e iϑ, m Φ = iu, m Q = ν a u + i 2 u, m Q = ν a u + i 2 u Twisted chiral ring relations algebraic Bethe ansatz equations

Quantum integrable systems (spin chains) [in progress] Strategy: Construct N = (2, 2) quivers Look for a duality Limit of parameters N = (2, 2) cycle-free quiver Interpret duality in integrable system context N = (2, 2) : diagrams with no arrows N N f Particle-hole duality: U(N) U(N f N) T Gr(N, N f ) = T Gr(N f N, N f ) Limit: m Φ, m Q with m Q fixed. It is crucial to expand in the correct vacuum. Duality involves non-trival map of vacua. N N f Limit does not reproduce Seiberg-like duality directly, but its compatible with it. Integrable system: highly quantum and inhomogeneous limit (u, ν a ).

Conclusions

Open directions Cluster algebra in systems of Picard-Fuchs equations? GKZ-systems, A-systems,... well-understood only for Abelian theories B-side of the story and Hori-Vafa mirror symmetry Relation of 2d N = (2, 2) quivers to other physical systems? Teichmüller theory and class S theories (what is Z S 2? what is K Kähler?) Wall crossing of BPS states in 4d N = 2, DT invariants and quiver quantum mechanics

Thank you!