3d&5d holomorphic blocks and q -CFT correlators Sara Pasquetti - PowerPoint PPT Presentation

3d&5d holomorphic blocks and q -CFT correlators Sara Pasquetti University of Surrey GGI, Florence based on: arXiv:1303.2626 with F. Nieri and F. Passerini, work in progress with F. Nieri, F. Passerini, A. Torrielli

In recent years many exact results for gauge theories on compact manifolds have been obtained by the method of SUSY localisation initiated by Pestun. The idea is that by adding a Q -exact term to the action it is possible to reduce the path integral to a finite dimensional integral: � � D ψ e − S [Ψ] = D Ψ 0 e − S [Ψ 0 ] Z 1-loop [Ψ 0 ] Localisation: Z M = ◮ Ψ 0 : field configurations satisfying localising (saddle point) equations ◮ with a clever localisation scheme, Ψ 0 is a finite dimensional set ◮ Z 1-loop [Ψ 0 ] is due to the quadratic fluctuation around Ψ 0 ⇒ useful to study holography ⇒ connect to exactly solvable models such as 2d CFTs and TQFTs

2 generalized฀quiver 2 2 2 2 2 2 2 building฀block 2 2 2 2 2 2 linear฀quiver 2 2 2 2 The AGT correspondence [Alday-Gaiotto-Tachikawa] , [Wyllard] maps S 4 partition functions of 4d N = 2 theories T g , n obtained wrapping M5 branes on Σ g , n (class S -theories [Gaiotto] ) to Liouville correlators: ☛ ✟ � � � � 2 α ( ζ ) | 2 = � � n � � d α C · · · C |F α i i V α i � Liouville Z S 4 [ T g , n ] = [ d a ] Z cl Z 1 loop � Z inst = � C g , n ✡ ✠ generalised N = 2 S-duality ⇔ CFT modular invariance ◮ Associativity of the operator algebra requires crossing symmetry ◮ Partition functions are invariant under generalised N = 2 S-dualities (different pant-decompositions of Σ g , n )

b 2 L 2 1 Simple surface operators ⇔ degenerate primaries � � L − 2 + V − b / 2 = 0 − 1 [Alday-Gaiotto-Gukov-Tachikawa-Verlinde] ◮ Several results: degenerate conformal blocks ↔ vortex counting [Dimofte-Gukov-Hollands] , [Kozcaz-Pasquetti-Wyllard] , [Bonelli-Tanzini-Zhao] . ◮ Recent proposal [Droud-Gomis-LeFloch-Lee] (see also [Benini-Cremonesi] ) ✞ ☎ � V α 4 V α 3 (1) V − b / 2 ( z ) V α 1 � = Z SQED S 2 ✝ ✆ flop symmetry ⇔ crossing symmetry

Liouville theory can be completely solved by the conformal bootstrap approach which only uses Virasoro symmetry & crossing symmetry. Now considering that: ◮ there is an action of the W-algebra on the equivariant cohomology of the moduli space of instantons, [Maulik-Okounkov] , [Schiffmann-Vasserot] ◮ N = 2 S-duality, flop symmetry are gauge theory avatars of crossing symmetry, we could say that S 2 and S 4 gauge theory partition functions and CFT correlators are constrained by the same bootstrap equations! Today I will argue that a similar story holds in 3d and 5d: ◮ 3d partition functions ⇔ degenerate q -CFT correlators ◮ 5d partition functions ⇔ non-degenerate q -CFT correlators.

Plan of the talk ◮ Block-factorisation of 3d & 5d partition functions ◮ q -CFT correlators via the bootstrap approach ◮ 3d and 5d partition functions as q -CFT correlators ◮ Conclusions and open issues

N = 2 theory on S 3 b b 2 | z 1 | 2 + 1 b 2 | z 2 | 2 = 1 S 3 b : Coulomb branch localization scheme [Hama-Hosomichi-Lee] . SQED: U (1) gauge group, N f chirals m i , N f anti-chirals ˜ m k , with FI ξ . � � N f � s b ( x + m j + iQ / 2) Z SQED dx e 2 π ix ξ = dx G cl · G 1-loop = S 3 s b ( x + ˜ m k − iQ / 2) j , k The 1-loop contribution of a chiral multiplet is: mb + nb − 1 + Q � 2 − ix s b ( x ) = 2 + ix , Q = b + 1 / b . mb + nb − 1 + Q m , n ∈ Z ≥ 0

Higgs-branch-like factorized form: [S.P.] � � � � N f � 2 � � � � G ( i ) cl G ( i ) � Z ( i ) Z SQED = � � � S 3 1-loop V S i ◮ G ( i ) cl , G ( i ) 1 loop evaluated on the i -th SUSY vacuum of the effective (2 , 2) theory: � N f s b ( m j − m i + iQ / 2) G ( i ) G ( i ) cl = e − 2 π i ξ m i , 1-loop = m k − m i − iQ / 2) , s b ( ˜ j , k ◮ Vortices on R 2 × S 1 satisfy basic hypergeometric equations: N f � � ( y k x − 1 ; q ) n z n = N f Φ ( i ) Z ( i ) i V = N f − 1 ( � x , � y ; z ) . ( qx j x − 1 ; q ) n i n j , k � � � � 2 � � � � ◮ S -pairing: � f ( x ; q ) S = f ( x ; q ) f (˜ x ; ˜ q ) � � � q = e 2 π i ω 2 x i = e 2 π m i /ω 1 , y i = e 2 π ˜ m i /ω 1 , z = e 2 πξ/ω 1 , ω 1 , q = e 2 π i ω 1 x i = e 2 π m i /ω 2 , y i = e 2 π ˜ m i /ω 2 , z = e 2 πξ/ω 2 , ω 2 , ˜ ˜ ˜ ˜ ω 1 = b , ω 2 = 1 / b

Classical (mixed Chern-Simons) terms can be factorized: � � � � 2 � � � � G ( i ) � G ( i ) cl = � � � cl S using that: � � � � e − (log x )2 2 π 2 � � � � 2 log q + log q 1 1 24 − 6 log q = 2 x ) ∞ ( − q 2 x − 1 ) ∞ θ ( x ; q ) := ( − q � � θ ( x ; q ) � � S , to obtain ☛ ✟ � � � � = � N f 2 � � � � G ( i ) � G ( i ) cl Z ( i ) Z SQED � � � S 3 i 1-loop V S ✡ ✠ Finally we can factorize the 1-loop part too using that � � � � 2 � � � � i π 2 ( iQ / 2+ z ) 2 s b ( iQ / 2 + z ) = � ( qe 2 π z /ω 1 ; q ) ∞ S , e � � � and obtain the block factorized form: ☛ ✟ � � � � = � N f 2 � � � � ( i ) := G ( i ) cl G ( i ) 1-loop Z ( i ) Z SQED � B 3 d B 3 d � � � S , S 3 i ( i ) V ✡ ✠ Blocks are expressed in terms of periodic variables e 2 π z /ω 1 , e 2 π z /ω 2 , (invariant under shift z → z + k ω i ).

◮ In the semiclassical limit, q = e βǫ , ǫ → 0, finite β , we find: � 1 � B ( i ) ǫ → 0 � ∼ W | s ( i ) ( x ) exp ǫ where � W | s ( i ) ( x ) is the twisted superpotential evaluated on the i -th SUSY vacuum. ◮ Blocks form a basis of solutions to a system of difference equations, in this case basic hypergeometric operator. ◮ The factorization is not unique, blocks are defined up to q -constants c ( x ; q ) satisfying: � � � � 2 � � � � c ( qx ; q ) = c ( x ; q ) , � � c ( x ; q ) � � S = 1 Notice that multiplication by c ( x ; q ) does not change the semiclassical limit (asymptotics of solutions).

N = 2 theory on S 2 × q S 1 Computes the (generalised) super-conformal-index [Imamura-Yokoyama] , [Kapustin-Willet] , [Dimofte-Gukov-Gaiotto] . SQED with fugacities: U (1) N f , ( φ i , r i ) , i = 1 , · · · N f , (+) flavor U (1) N f , ( ξ i , l i ) , i = 1 , · · · N f , ( − ) flavor ( ω, n ) , topological U (1) , ( t , s ) , gauged U (1) . � N f N f � � � dt χ ( t − 1 ξ − 1 2 π it t n ω s Z S 2 × S 1 = χ ( t φ j , s + r j ) k , − s − l k ) . s ∈ Z j =1 k =1 The 1-loop contribution of a chiral multiplet is: � ∞ (1 − q l +1 ζ − 1 q − m / 2 ) χ ( ζ, m ) = ( q 1 / 2 ζ − 1 ) − m / 2 (1 − q l ζ q − m / 2 ) k =0

Higgs-branch-like factorized form [Beem-Dimofte-S.P.] , [Dimofte-Gaiotto-Gukov] � � � � � N f 2 � � � � G ( i ) G ( i ) Z SQED � Z i S 2 × S 1 = � � � cl 1-loop V id i =1 ◮ G ( i ) cl G ( i ) 1-loop are evaluated on the i -th SUSY vacuum. � � � � 2 � � � � ◮ The id -pairing is defined by � � f ( x ; q ) � � id := f ( x ; q ) f (˜ x ; ˜ q ) with: q r i / 2 , q l i / 2 , x i = φ i q r i / 2 , x i = φ − 1 y i = ξ i q l i / 2 , y i = ξ − 1 ˜ ˜ i i z = ω q n / 2 , z = ω − 1 q n / 2 , q = q − 1 ˜ ˜ As before we can factorize the classical and 1-loop term and obtain: ☛ ✟ � � � � � � � � S 2 × S 1 = � N f 2 id = � N f 2 � � � � � � � � i =1 G ( i ) � G ( i ) Z SQED cl Z i � B 3 d � � � � � � 1-loop V i ( i ) id ✡ ✠

to summarize: � � � � N f N f � � 2 � � � � G S 3 , ( i ) � G ( i ) cl Z ( i ) ||B 3 d ( i ) || 2 Z S 3 = S = � � � 1-loop S V i i � � � � N f N f � � 2 � � � � G S 2 × S 1 , ( i ) � G ( i ) cl Z ( i ) ||B 3 d ( i ) || 2 Z S 2 × S 1 = id = � � � id 1-loop V i i Same blocks with different pairing gives Z S 3 , Z S 2 × S 1 “like” S 3 , S 2 × S 1 are obtained by gluing solid tori with S , id ∈ SL (2 , Z ). S 1 S 1 S 1 × q id × × ˜ q q � S 2 q D 2 D 2 q ˜ S 1 D 2 q ˜ S 3 × q S � × ˜ q b q D 2 S 1 Holomorphic blocks B 3 d are Melvin cigar D × q S 1 partition functions. [Beem-Dimofte-S.P.]

Observe the following flop symmetry of SQED partition functions: � N f � s b ( x + m j + iQ / 2) Z SQED dx e 2 π ix ξ = S 3 s b ( x + ˜ m k − iQ / 2) j , k is invariant under : m i ↔ − ˜ m k and ξ ↔ − ξ exchanges phase I and phase II � � � N f � N f dt 2 π it t n ω s χ ( t − 1 ξ − 1 Z S 2 × S 1 = χ ( t φ j , s + r j ) k , − s − l k ) s ∈ Z j =1 k =1 is invariant under : ω ↔ ω − 1 , n ↔ − n , φ j ↔ ξ − 1 , r j ↔ − l j j exchanges phase I and phase II