Free fermion and wall-crossing Jie Yang School of Mathematical Sciences, Capital Normal University yangjie@cnu.edu.cn March 4, 202
Motivation For string theorists It is called BPS states counting which means counting the D brane bound states. 2/35
Motivation For string theorists It is called BPS states counting which means counting the D brane bound states. For mathematician Invariants of moduli spaces of virtual dimension zero associate with Calabi-Yau 3-fold X. holomorphic curves in X with fixed genus and degree, e.g. Gromov-Witten invariants 2. coherent sheaves with a fixed Chern character, e.g. Donaldson-Thomas (DT) invariants 2/35
The world of wall Algebra wall-crossing gauge theory Seiberg- Witten geometry supergravity Donaldson- Thomas black hole 3/35
Kontsevich-Soibelman wall-crossing formula There is a symplectomorphism transformation for functions over algebraic torus over lattice Γ U γ : X γ X γ (+σ(γ)x γ ) γ,γ where U γ can be expressed as the operator ( ) σ(nγ) U γ = exp {X n 2 nγ, } n () (2) Wall-crossing formula γ U Ω(γ,u+) γ = γ U Ω(γ,u ) γ (3) 4/35
Motivic wall-crossing formula On the quantum algebraic torus the associative algebra is generated by ê γ s.t. ( ) [ê γ,ê γ2 ] = q 2 γ,γ 2 q 2 γ,γ 2 ê γ +γ 2 (4) The wall-crossing formula is γ A mot γ (u + ) = γ A mot γ (u ) (5) where A mot γ is a quantum analog of the classical symplectomorphism U Ω(γ) γ. 5/35
Dualities of BPS counting Free fermion for C 3 Refined wall-crossing for O( ) P O( ) Open questions 6/35
D5 NS5 toric CY quiver dimer conjecture 2d Young crystal top vertex free fermion 3d Young 7/35
D5-NS5 brane and toric Calabi-Yau geometry T-duality (mirror symmetry) between D5-NS5 brane configuration and toric diagram 0 2 3 4 5 6 7 8 9 D5 * * * * - - NS5 * * * * - - NS5 * * * * - - 8/35
Quiver and dimer model The quiver, universal quiver, and dimer diagram for conifold are 2 2 2 2 2 2 2 2 2 2 2 2 9/35
Crystal q 2 0q q0 2 q 2 q 2 0q q0 2 q0q 2 q0 2 q 2 0q q 0 q q0q 2 q 0 q q 2 0q q 0 q q 2 0q q 0 q q0 q 0 q q0 q 0 q q0 q 0 q q0 q0 q0 q0 q q q q q0q q q0q q q0q q q0q q 0 q 2 q0q q 0 q 2 q0q q 0 q 2 q0q q 0 q 2 q 2 0q 2 q 0 q 2 q0q 2 2 q 0 q 2 q 2 0q 2 q 0 q 2 q 2 0q 2 (e) a screenshot of paper by Young and Bryan arxiv:0802.3948 (f) dimer assigned with weight Conjecture: The partition function of statiscal model of crystal melting is related to the BPS partition function. 0/35
Dualities of BPS counting Free fermion for C 3 Refined wall-crossing for O( ) P O( ) Open questions /35
Build up the quiver diagram for C 3 The universal quiver for C 3 The dimer model for C 3 2/35
Free fermion and 2D Young diagram correspondence 2D Young diagram λ a a2 a3 b2 b3 The correspondence b where λ λ = d(λ) i= ψ (ai )ψ (b i ) 0 (6) a i = λ i i + 2, b i = λ t i i + 2 (7) 3/35
3D Young diagram Y 3 Plane partitions Definition: 2D Young diagram with weakly decreasing number filling in rows and columns 5 3 4 3 4 2 MacMahon function is the generating function of Y 3 n= ( q n ) = q π n π all Y 3 (8) 4/35
2D and 3D Young diagram The diagonal diagram y x 2 2 3 t = 0 Interlacing condition of the diagonal slices { λ(t) λ(t +) t > 0 λ(t) λ(t ) t < 0 where (9) λ µ λ µ λ 2 µ 2 (0) 5/35
Vertex operators Γ ± (x) = exp { n= } x n n α ±n () α ±n are the creation and annihilation operator for the bosonization of free fermion. Γ can be treated as the creation operator while Γ + the annihilation one of nearest neighbor slices, i.e. Commutation relation Γ + () µ = λ µ λ µ λ, (2) Γ + (x)γ (y) = xy Γ (y)γ + (x) (3) 6/35
Crystal melting for C 3 The statistics of a melting cubic crystal is identified with the partition function of D0-D6 bound states of C 3 MacMahon partition function n= ( q n ) n = (4) q L 0 Γ + () q L 0 Γ + () q L 0 Γ }{{} ()q L0 Γ ()q L 0 }{{} where we perform commutation relation of vertex operators. λ( 2) λ( ) λ(0) λ() λ(2) 7/35
Dualities of BPS counting Free fermion for C 3 Refined wall-crossing for O( ) P O( ) Open questions 8/35
Marginal stability wall SUSY condition for mass and central charge of a state when =, it is called a BPS state. Marginal stability wall M Z(γ) (5) γ = γ +γ 2, Z(γ) = Z(γ )+Z(γ 2 ) (6) 2-particle BPS states M 2 particle = Z +Z 2 satisfies M 2 particle = Z +Z 2 Z + Z 2 = M +M 2 (7) Therefore 2-particle BPS states are separated from -particle BPS states M = Z,M 2 = Z 2 unless Z +Z 2 = Z + Z 2, i.e., at the wall. 9/35
Charge lattice D0 D2 D4 D6 H 0 H 2 H 4 H 6 q 0 Q P p 0 n m 0 D0-D2-D6 bound states Central charge of BPS states Z(γ) = γ e t (8) where the complexified Kähler moduli is t = zp +Λe γ = ndv mβ+ iϕ P Marginal stability walls in the moduli space of BPS states ArgZ(γ) = ArgZ(γ ) (9) W m n X ϕ 20/35
Define an index by Index of BPS states Ω(γ) = Tr H γ BPS,u ( )2J, (20) where γ Γ the charge lattice and J so(3) is the generator of rotations around any axis. Ω(γ) Euler characteristic of the stable sheaf moduli space Donaldson-Thomas Invariants Refined index is defined by [Dimofte & Gukov arxiv:0904.420] Ω ref (γ;y) = Tr H γ BPS,u ( y)2j (2) Ω ref (γ;y ) = Ω(γ) (22) 2/35
Some description of refined refined quantum motivic y q 2 L 2 Physically we distinguish left and right spin, (corresponding to turing on Ω background in Nekrasov s formalizm) Mathematically Ω ref (γ;y) Poincare polynomial of the stable sheaf moduli space Refined/Motivic Donaldson-Thomas Invariants 22/35
Wall-crossing for the resolved conifold Wall W 0 W W 2 W Wn 0 W W2 W W0 W C C2 C2 DT DT MacMahon Core 2-colored stone diagram for chamber 0 and ε ε 2 The relation between 2D partition and free fermion is preserved. But for different slices the vertex operators have different arguments. Figure: The red (dotted) lines denote the left or right moving of the slices, and the blue (solid) lines denote the up or down moving 23/35
Crystal melting for resolved conifold Toric diagram Γ± Q Γ ± Vertex operators { } x n Γ ± (x) = exp n α ±n n { } Γ ( ) n x n ±(x) = exp α ±n n n Γ + () µ = λ, λ t µ t λ t µ t 24/35
How do we count refined in the crystal melting? arxiv: 00.0348 collaborated with Haitao Liu Arrow diagrams for chamber n are [ ] Γ+ q i ( Q) 2 [ ] Γ + q i 2 +n q 2 2 ( Q) 2 [ ] Γ q j+n 2 ( Q) 2 [ ] Γ q 2 q j 2 2 ( Q) 2 Figure: Arrow diagrams for chamber n of the conifold Stone diagrams with arrows are n=0 n= 25/35
Vertex Operators If we define A ± (x) by A + (x) := Q 2 2 0 Q ( Q 2 0 Γ + (x) Q 2 0 ) Q ( Q 2 02 Γ +(x) Q 2 02 ) Q 2 Q 2 0, A (x) := Q 2 2 02 Q ( Q 2 02 Γ (x) Q 2 02 ) Q ( Q 2 0 Γ (x) Q 2 0 ) Q 2 Q 2 02 Then we can show that Z cystal NCDT := 0 A +() A + ()A () A () 0 = Z ref BPS(q,q 2,Q) NCDT (up to refined MacMahon). 26/35
For chamber n we construct Z crystal = 0 i= Γ + [ q i ( Q) 2 ] Γ [q 2 ( Q) 2 ] Γ [q2( Q) 2 2 Γ + Γ + Γ [q2( Q) n 2 j= ] [ ] Γ + q i 2 +n q 2 2 ( Q) 2 [ [ q n 2 q 2 q n 3 2 q 2 ] 2 ( Q) 2 ] 2 ( Q) 2 ] [ ] Γ + q 2 q 2 2 ( Q) 2 [ ] [ Γ q j+n 2 ( Q) 2 Γ q 2 q j 2 2 ( Q) 2 = M δ= (q,q 2 )M δ= (q,q 2 ) i+j>n+ i,j ( q i 2 q j 2 2 Q ) ( q i 2 i,j= ] 0 q j 2 2 Q) 27/35
Generalized conifold Γ + Γ + Γ + Γ + Γ + Γ + We have some progress on constracting the free fermion formalism of crystal melting for NCDT chamber and DT chamber and computing the partition function of D0-D2-D6 bound states. 28/35
The refined MacMahon function M(q,q 2 ) is defined by [Behrend, Bryan, Szendrői arxiv:0909.5088] M(q,q 2 ) = 6 ( d=0 Md 3 2 (q,q 2 )) ( ) d b d, where b d is the Betti number of the Calabi-Yau threefold X of degree d and M δ (q,q 2 ) is the refined MacMahon function defined by M δ (q,q 2 ) = ( q i 2 +δ 2 i,j= q j 2 δ 2 2 ). 29/35
Dualities of BPS counting Free fermion for C 3 Refined wall-crossing for O( ) P O( ) Open questions 30/35
Refined MacMahon We have seen that by using crystal melting model we can recover the refined BPS partition function except for the refined MacMahon function (defined by Behrend et. al.). Is it possible to adjust the counting weight to recover the exact refined BPS partition function? Crystal and motivic DT What is the intrinsic reason that counting crystal can reproduce the refined BPS partition function? Invariants and symmetries How is Heisenberg algebra categorification going to enter this story? Namely, how can we find operators which satisfy KS motivic wall-crossing formula? 3/35
Next: Topological string/gauge theory duality Geometric engineering arxiv: hep-th/9609239 Katz, Klemm, and Vafa Topological string theory on a Calabi-Yau with ADE type of singularities gauge theory of gauge group of the same ADE type Topological strings and Nekrasov s formulas arxiv: hep-th/030235 Eguchi and Kanno 32/35
Free fermion and gauge theory arxiv: hep-th/0306238 Nekrasov and Okounkov For U() gauge field we can build the Maya diagram for a state µ (charge is 0) 9 7 5 3 2 2 2 2 2 2 3 2 5 2 7 2 2 3 2 33/35
For SU(N) group we need N-component fermion field where ξ r = N+ (r ). N 2 ψ (r) k = ψ N(k+ξr), ψ (s) l = ψ N(l ξ s) (23) {ψ (r) k,ψ(s) l } = δ r,s δ k+l,0 (24) The sets of charged Maya diagrams of µ and λ (r) satisfy {x i (µ)+n;i } = {N(x ir (λ (r) )+p r +ξ r );i } (25) arxiv: hep-th/042327 Maeda, Nakatsu, Takasaki and Tamakoshi 34/35
Thank you! 35/35