NUMBER THEORY IN STRING THEORY: SKEW, MOCK, FALSE, AND QUANTUM

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NUMBER THEORY IN STRING THEORY: SKEW, MOCK, FALSE, AND QUANTUM Sarah M. Harrison based on work with Cheng, Duncan, Harvey, Kachru, Rayhaun ( 17) and Cheng, Chun, Ferrari, Gukov (to appear)

INTRODUCTION String Theory and its friends accommodate many interesting physical phenomena discussed at this conference, e.g. Black Holes Holography Gauge theories Cosmology Entanglement

<latexit sha1_base64="uk404o5c1he2qyyythimplj+xha=">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</latexit> <latexit sha1_base64="uk404o5c1he2qyyythimplj+xha=">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</latexit> <latexit sha1_base64="uk404o5c1he2qyyythimplj+xha=">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</latexit> <latexit sha1_base64="uk404o5c1he2qyyythimplj+xha=">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</latexit> INTRODUCTION Not only are there many solutions of string theory, but there are many ways to describe it, e.g STRING PERTURBATION THEORY SPACETIME ACTION/SUPERGRAVITY S = Z d d x p ge 2 (R +...) GEOMETRY/COMPACTIFICATION CONFORMAL SYMMETRY&ADS/CFT * he 11

INTRODUCTION Combinations of different viewpoints elucidate different aspects of the theory, e.g t.ir#ty PEE Field theory dualities Non-perturbative objects Black hole entropy String Dualities

INTRODUCTION I propose the following tool has been underexploited: NUMBER THEORY I will discuss 2 physical examples where I think number theory has something to teach us, but I expect there are many more: I: MSW STRINGS II: 3D N=2 QFT

INTRODUCTION These examples relate to the larger questions: What are the symmetries of string theory? Are there new ways to think about special solutions? Can we understand large classes of them? How does modularity, in its many forms, relate to the many physical phenomena in QFT and string theory?

PART I: MSW STRINGS Skew and Mock

PART I: SKEW AND MOCK 5- branes Mf P CY 3 MSW string N=(0,4) worldsheet CFT Index counting BPS states governs black hole entropy in M-theory Maldacena, Strominger, Witten 97

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sha1_base64="6a5niseixovgjhjhfo4tfvl6bpc=">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</latexit> PART I: SKEW AND MOCK Let s consider a supersymmetric index which counts worldsheet BPS states, called a modified elliptic genus: Z(, ) :=Tr R 0F 2 ( 1) F q L 0 c/24 q L 0 c/24 e 2 i Q In general it is not holomorphic, but has modular properties: a + b Z c + d, (c + d) 2 m c + d c + d e cz2 = Z(, ) c + d <latexit sha1_base64="6a5niseixovgjhjhfo4tfvl6bpc=">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</latexit> weight (-3/2,1/2) N=4 spectral flow implies the index has a theta decomposition, which depends on the geometry of P: Z(, ) = X µ2 / coefficients are modular and mock modular forms µ (, )h µ ( ) modular form Gaiotto, Strominger, Yin 06 de Boer, Cheng, Dijkgraaf, Manschot, Verlinde 06 Denef, Moore 07

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sha1_base64="3numrn/w0qg+9+zki43bzb6qjjo=">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</latexit> <latexit sha1_base64="3numrn/w0qg+9+zki43bzb6qjjo=">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</latexit> PART I: SKEW AND MOCK Example: single 5-brane on 1/2-K3 Z (1) 1 2 1 K3(R;,z)= 12 ( ) E 4( ) odd 1,1 (R;,z) where R=size of elliptic fiber H 2 ( 1 2 K3, Z) =Z ( Z) ( E 8) Minahan, Nemeschansky, Vafa, Warner 98 Special moduli: rational radii Z (1) 1 2 K3(p 2m;,z)= 1 6 ( ) ' 1 2 K3,m (,z) weight 2 skew-hol. Jacobi form R 2 =3: ' 1 2 K3, 3 2 (,z)=e 4( ) 5 ( ) 3 2, 1 2 (, z) 3 2, 1 2 (, z) M 12 moonshine function Cheng, Duncan, SH, Harvey, Kachru, Rayhaun 17

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sha1_base64="bgif6aa/xgvjsou5dio/k+owaes=">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</latexit> PART I: SKEW AND MOCK Comments: Compare with Mathieu moonshine and K3: EG(K3) = 2 1(,z) 3 ( ) (24µ(,z)+H( )) H( ) =2q 1/8 ( 1 + 45q + 231q 2 + 770q 3 +...) <latexit 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sha1_base64="3ugrlvmyrjo7qsko2dnmancejkg=">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</latexit> <latexit sha1_base64="3ugrlvmyrjo7qsko2dnmancejkg=">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</latexit> Connection between finite groups, modular forms, and complex surfaces is seemingly pervasive once we relax rules to allow signs, e.g. (V 0,n V 1,n ) f g ( ) := X q X n ( 1) i (Tr Vi,n g) n V = M n i2{0,1} Special skew-holomorphic property of BPS index appears at rational and attractor points in moduli space Can moonshine give us a new way to organize these special solutions of string theory?

PART II: 3D N=2 QFT Mock, False, and Quantum

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sha1_base64="glkgw58iuhps581rjft4xlp6u6u=">aaaccnicbvdlssnafj3uv62vqes3o43gqittqjdc0y0bpujfkiywmuzbozmhmxmhhk7d+ctuxcji1i9w5984abpq1gmxdufcy733edgjqprmt1zawv1b3yhvvra2d3b39p2drogsjkkhryzifq8jwmhiopjkrvoxjyjwgol5k+vc7z0qlmgutmuaeydao5aokuzssa5+3pchmqiqhgpe4n1l3ybytckeqqnt37onx3d1qlkzz4dlxcpifrroufrxwi9wepbqyoaesc0zlk6guksykwllkagsizxbi2irgqkaccebvtkfp0rx4tdiqkijz+rviqwfqqsbpzrzo8wil4v/exyihxdorsm4kste80xdheezwtwx6fnosgspighzqm6feiw4wlklv1ehwisvl5nuvwazneu+xm1efxguwre4awfaauegcw5ac3qabo/ggbycn+1je9hety95a0krzg7bh2ifp0dgmbm=</latexit> PART II: MOCK, FALSE, AND QUANTUM 3d-3d Correspondence: M5-branes on 3-manifold M 3 M3 brakes n # G-type Chern-Simons on M 3 Topological invariants 3- manifold 3d N = 2 theory T [M 3 ] BPS quantities We will take G = SU(2) <latexit 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sha1_base64="q29diyrkceov+z6fux99haqrmzq=">aaacehicbvc7tgjbfj31ifhclw0mghebskujjqnrqkumlpaaibpdbsbmpjjzvyubpshgx7gx0bhbszv/xufrkhissu7ouffovcelpnbo29/wwuls8spqai29vrg5tz3z2a3omfycxb7kunu8pkgkafwukkewkwc+j6hq9s9gfvuolbzhciudcjo+6waiizhdi7uyrw2ebxzpsrs0h0kv6l2qkilra81dnt24+ejxbtjkzo2cpqadj86uzmku5vbmq9eoeexdgfwyreuohwezyqoflzbmn2ineen91ow6oqhzqtet8sjdemiunu2eyrwa6vj93zewx+ub75lkn2fpz3oj8t+vhmpntjmiiioraj75qboba0m6soe2hqkocmai40qyxsnvmcu4mgztjgrn9ur5uikwhlvgxbezpfnphcmytw5injjkhjtifsktl3dysj7jk3mznqwx6936mjquwnoepfih1ucpuciccg==</latexit> <latexit sha1_base64="q29diyrkceov+z6fux99haqrmzq=">aaacehicbvc7tgjbfj31ifhclw0mghebskujjqnrqkumlpaaibpdbsbmpjjzvyubpshgx7gx0bhbszv/xufrkhissu7ouffovcelpnbo29/wwuls8spqai29vrg5tz3z2a3omfycxb7kunu8pkgkafwukkewkwc+j6hq9s9gfvuolbzhciudcjo+6waiizhdi7uyrw2ebxzpsrs0h0kv6l2qkilra81dnt24+ejxbtjkzo2cpqadj86uzmku5vbmq9eoeexdgfwyreuohwezyqoflzbmn2ineen91ow6oqhzqtet8sjdemiunu2eyrwa6vj93zewx+ub75lkn2fpz3oj8t+vhmpntjmiiioraj75qboba0m6soe2hqkocmai40qyxsnvmcu4mgztjgrn9ur5uikwhlvgxbezpfnphcmytw5injjkhjtifsktl3dysj7jk3mznqwx6936mjquwnoepfih1ucpuciccg==</latexit> Dimofte,Gaiotto,Gukov 11

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sha1_base64="tk/eoj1cfuij7ydtiaaxk9ptrzm=">aaab+nicbvbns8naen3ur1q/uj16wsycp5iuqs9c0yvhcvydmlg220m7dlojuxulxp4ulx4u8eov8ea/cdvmok0pbh7vztazl0g4u9pxvq3cyura+kzxs7s1vbo7z5f3wypojyumjxksowfrwjmapmaaqyerqkkaqzsyxu399gnixwjxq8cj+bezcbyysrsrenb5/gluspqxmmywp0k66dkvp+rmgjejm5mkytho2v9ep6zpbejttptquk6i/yxizsihscllfssejsgauoykeohys9npe3xsld4oy2lkadxtf09kjfjqhawmmyj6qba9qfif1011eo5ntcspbkhni8kuyx3jaq64zyrqzcegecqzurxtizgeapnwyytglr68tfq1qutu3zvtsv0yj6oidterokeuokn1di0aqikoektp6bw9wu/wi/vufcxbc1y+c4d+wpr8aw7ok3q=</latexit> PART II: MOCK, FALSE, AND QUANTUM We will consider the so-called half-index with boundary conditions labeled by abelian flat connections on M 3 : Ẑ a (q) =Z(D 2 S 1 ; B a ) q = e 2 i 3d N =2 theory 2d N =(0,2) boundary condition B a where a labels solutions: Z CS (M 3 )= Z DAe 2 iks(a) da + A ^ A =0 Such functions are referred to as homological blocks because, e.g., of the integrality of B 2 coefficients in this index: X Ẑ a (q) =q a a n q n, a n 2 Z n Gukov,Putrov,Vafa 16 Gukov,Pei,Putrov,Vafa 17

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sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">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</latexit> <latexit sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">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</latexit> <latexit 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sha1_base64="dkn9tfu9pofz3bfvyz+mh7glgsk=">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</latexit> <latexit sha1_base64="dkn9tfu9pofz3bfvyz+mh7glgsk=">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</latexit> <latexit sha1_base64="hz42p1dgzy5ajsf6a780qcnewmm=">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</latexit> <latexit sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">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</latexit> <latexit sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">aaacx3icbvfdaxqxfm2mh63b2o76jl4ef2ewzjlzbx0s9suxobjuw7pzw51sthsmk0mttggj+zo+cb74t8xuf9twc4hdued+nvracuuk4kes3rv/4oho7qpe3v7jg8psydnt23ag8qlrzwvok7bccsuntjjjz7xh0fssn1x18tp/ds2nfa364laazxq4vgihglhi0ez6gvrxjb8nab3yt/tnggjtatdiyrugenhdysiujt4t02co4ibqgysa+vc/ilpggevjcq6dmi7ykpalohxbq/xq8i36kxjb/vgsnu6c6xbo1i+gxsbwxvbuqr9t44rm38m8zv3dlwmsrj2whxyzd8yjjnnokc5ydaygsz6nuehd7cxv/an4vwtmenga+jtdg/bvcg+ntaumisog3nlezq3j/+wmnvu8n3mhdoe4yjedfp3e8ei12xgudgdoriiazktcfbmlggaufkkvmldepvkuob0ny2jyfn7bp/qwtwmxvuavuy5k9a4doy/obe0qqz+tnnll9pnf6u56kgy30jtz1jxd/0t6/dfpklqv</latexit> <latexit sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">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</latexit> <latexit sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">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</latexit> <latexit sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">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</latexit> <latexit sha1_base64="7h7kdjvxvjmo9pcarowoo8lb6+w=">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</latexit> PART II: MOCK, FALSE, AND QUANTUM The Ẑa are conjectured to play a role as building blocks in the following quantities: <latexit sha1_base64="k/ellj3qkoy+fgifq+gmzzfibbq=">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</latexit> <latexit sha1_base64="k/ellj3qkoy+fgifq+gmzzfibbq=">aaacshicbza/bxnbemx3nadhwh8djc0odhkvdzcgrbvbkzjicrjhw6e5vbg9eg/32j1nzfn5edqp6fizachaii49xwukgwmlp/dmd2d+zaov5yy7sjobm/fup9h6mg4/evzkaffz82nvg5m0kfzbd1qij60mdvixptpgedalppny/qhnt87iewxnes8agtc4nwqijhk0im5xncpyhc2q4vobu4coqfrzmsqhrxwwhubjahcc1qtooqxkv8pmodrwzj2kqtla8zwj1dqet8mxgcboosi/k7q9rj+tcm6lfc16yl2hrffbqliy1gryavr+mgcnj5fowelnf+koegpqznfkwygn1uthyxwic3gvnsoo4uixdcv33xtlrl1f1gxsrjfn/mbwmndlw8ctt+olmk1gmvl6o0nqlz6wkltkrwj6eqvkf1exigfouhjkn0yi+c2vb4vjvx6e9fope73992scw+kl2bgvrs7eih1xia7fqejxvxwxp8wv5dl5kfxo/ly3dpl1nrfiv+p0/gjgurd0</latexit> <latexit sha1_base64="k/ellj3qkoy+fgifq+gmzzfibbq=">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</latexit> <latexit sha1_base64="k/ellj3qkoy+fgifq+gmzzfibbq=">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</latexit> The Chern Simons partition function ( WRT invariants ): Z SU(2)k (M 3 ) / for q approaching roots of unity X a,b2{abelian} e 2 ikcs(a) S ab Ẑ b (q) q!e 2 i k And the superconformal Index: Z T [M3 ](S 2 S 1 ) X a2{abelian} Ẑ a (q)ẑa(1/q) after defining a continuation to outside the unit circle Gukov,Putrov,Vafa 16 Gukov,Pei,Putrov,Vafa 17

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sha1_base64="r0px17h7xrgxghm5/tbesmpe8kq=">aaacihicbvc7tgmxeptxdoevoksxcejqrhdpoanbqwksirfjfo05e8tcdz7spaqoyqfq8cs0fcaehxwnzqoahjesjwzm19ojuyut+f6xnzm7n7+wmfvkl6+srq0xnjavrc6mwirqsptacbavtlbckhtwuomqhwqr4d3zwk8+olfsj1futbezw20iiymannqqhaylxu0actdiqym81u1mgzhu5anrkrqsalmo+g5jeltpwf79we5xq1d0s/4qfjoey1jky1y0cp+nthzzjakjbdbwaz+lzg/cfqgwn29kflmqd3clductine2e8md+3zpkw0eaeneqnyo/p7oqwxtnw5dmgbq2elvip7n1tokjpo9maqzysjgh0wz4qt5oc3elgyfqa4jifwpundraqocxkd5v0iwefi0us6xar8uxjalj6fjonjsm+2wfrawq3bcztkfqzdbhtkze2vv3pp34r17h6pojdee2wj/4h3/aa7xoam=</latexit> 3d theory trivial=>2d elliptic genus (modular) 3d theory non-trivial =>modular properties spoiled Ẑ a Z dx 2 ix F 3d(x) (a) 2d (x) In large class of examples (certain Seifert manifolds), we find false theta functions and mock modular forms 2. How does it encode the physics? <latexit sha1_base64="ixuy1mfm11jchtnojgkbzzym6qa=">aaachhicbvc7sgnbfj2nrxhfuuubwsbyhd1yagfqjmue84akhnnzm2ti7m4yc1cnsz7exl+xsvdexklwb5w8ck08mha4517o3ophuhh03w8ns7k6tr6r3cxtbe/s7ux3d+pgjzpdjsupdnnnbqsioiycjtrjdsz0jtt84fxeb9ybnkjftzikoroyfir6gjo0ujd/virsirqngqjdbvkiuaqa4gbog+ebpwmphmccxooredyml7v5glt0p6dlxjutapmj2s1/tgpfkxai5jiz0/lcgdsp0yi4hhgunriigr+yprqsjvgippnoo8f0xcob7sltx4r0qv7esflozcj07wticgawvyn4n9dkshfrsuuuj2ivngx1eklr0ultnbaaomqrjyxryf9k+ybpxth2mbmleisnl5n6qei5re+mvchfzevikinyte6jr85jmvrildqij4/kmbysn+fjexheny/zamaz7xysp3c+fgc/iahz</latexit> <latexit sha1_base64="ixuy1mfm11jchtnojgkbzzym6qa=">aaachhicbvc7sgnbfj2nrxhfuuubwsbyhd1yagfqjmue84akhnnzm2ti7m4yc1cnsz7exl+xsvdexklwb5w8ck08mha4517o3ophuhh03w8ns7k6tr6r3cxtbe/s7ux3d+pgjzpdjsupdnnnbqsioiycjtrjdsz0jtt84fxeb9ybnkjftzikoroyfir6gjo0ujd/virsirqngqjdbvkiuaqa4gbog+ebpwmphmccxooredyml7v5glt0p6dlxjutapmj2s1/tgpfkxai5jiz0/lcgdsp0yi4hhgunriigr+yprqsjvgippnoo8f0xcob7sltx4r0qv7esflozcj07wticgawvyn4n9dkshfrsuuuj2ivngx1eklr0ultnbaaomqrjyxryf9k+ybpxth2mbmleisnl5n6qei5re+mvchfzevikinyte6jr85jmvrildqij4/kmbysn+fjexheny/zamaz7xysp3c+fgc/iahz</latexit> <latexit sha1_base64="ixuy1mfm11jchtnojgkbzzym6qa=">aaachhicbvc7sgnbfj2nrxhfuuubwsbyhd1yagfqjmue84akhnnzm2ti7m4yc1cnsz7exl+xsvdexklwb5w8ck08mha4517o3ophuhh03w8ns7k6tr6r3cxtbe/s7ux3d+pgjzpdjsupdnnnbqsioiycjtrjdsz0jtt84fxeb9ybnkjftzikoroyfir6gjo0ujd/virsirqngqjdbvkiuaqa4gbog+ebpwmphmccxooredyml7v5glt0p6dlxjutapmj2s1/tgpfkxai5jiz0/lcgdsp0yi4hhgunriigr+yprqsjvgippnoo8f0xcob7sltx4r0qv7esflozcj07wticgawvyn4n9dkshfrsuuuj2ivngx1eklr0ultnbaaomqrjyxryf9k+ybpxth2mbmleisnl5n6qei5re+mvchfzevikinyte6jr85jmvrildqij4/kmbysn+fjexheny/zamaz7xysp3c+fgc/iahz</latexit> <latexit sha1_base64="ixuy1mfm11jchtnojgkbzzym6qa=">aaachhicbvc7sgnbfj2nrxhfuuubwsbyhd1yagfqjmue84akhnnzm2ti7m4yc1cnsz7exl+xsvdexklwb5w8ck08mha4517o3ophuhh03w8ns7k6tr6r3cxtbe/s7ux3d+pgjzpdjsupdnnnbqsioiycjtrjdsz0jtt84fxeb9ybnkjftzikoroyfir6gjo0ujd/virsirqngqjdbvkiuaqa4gbog+ebpwmphmccxooredyml7v5glt0p6dlxjutapmj2s1/tgpfkxai5jiz0/lcgdsp0yi4hhgunriigr+yprqsjvgippnoo8f0xcob7sltx4r0qv7esflozcj07wticgawvyn4n9dkshfrsuuuj2ivngx1eklr0ultnbaaomqrjyxryf9k+ybpxth2mbmleisnl5n6qei5re+mvchfzevikinyte6jr85jmvrildqij4/kmbysn+fjexheny/zamaz7xysp3c+fgc/iahz</latexit> 3. How can we define Ẑa(q 1 )?

±2 1 of them is plumbing graph. There are various ways to z =1 define2 iz M3 for a(z givenqt;. q) One n2pz+a t=q to say that M3 is obtained by a Dehn surgery on the corresponding link L( ) of unknots (see Figure v.p. 5). For simplicity, also assume that is connected. Such class of 3-manifolds where stand we for principal value integration. This indeed agrees with the contains Seifert fibrations over S 2, 2which correspond to star-shaped plumbing graphs. Denote 1, cf. (3.122), function of T [M ] on D S 3 by L the number of vertices of. It is equal to the number of components of the link L( ). We will also need L L linking matrix of L( ) which will be denoted by M g: Y 8 (ui ; q)1 (ref.cs; ) b b > Z (q) = Z (q, t, u) (qt; q). 1, v, v connected, 1 a2 a 1 < (tu q; q) i (3.142) 1 Mv1,v2 = av, v1 = v2 = v, vi 2 Vertices of, L}. = {1,... i=1 t=q,ui =1 > : 0, otherwise. PART II: MOCK, FALSE, AND QUANTUM partition (3.141) How to describe a 3-manifold? 3.4 M3 = plumbed Now we move to consider anothera4very large class of 3-manifolds M3 ( ) associated to a a4 plumbing graph. There are various ways to define am for a given. One of them is 9 3 a9 a asurgery 3 to say that 3Ma is obtained by a Dehn the corresponding link L( ) of unknots a5 aon 3 6 a 5 6 Plumbing diagram: (see Figure a 5). For simplicity, we also assume that is connected. Such class of 3-manifolds a7 2 a7 2 2, which a a containsaseifert fibrations over S correspond to star-shaped plumbing graphs. Denote 8 1 a a8 by L the number of vertices of. It is 1equal to the number of components of the link L( ). We will also need L L linking matrix of L( ) which will be denoted by M : Figure 5: An example of a plumbing graph (left) and the corresponding link L( ) of framed unknots 8 in S 3 (right). The associated 3-manifold M3 ( ) can be constructed by performing a Dehn surgery onencodes topology of > < 1, v1, v2 connected, L( ). M3 and parameters Mv1,v2 = av, v1 = v2 = v, vi 2 Vertices of (3.142) = {1,..., L}. Linking matrix: in UV Lagrangian of > The first homology group 3 ( ) is given by the cokernel (over Z) of the linking matrix: : 0,of Motherwise. T[M3] H1 (M3, Z) (3.143) = Coker M = ZL /M ZL. 3d action: L <latexit sha1_base64="r/pf5n7aw5effwpgaoojhbit/nc=">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</latexit> Z a (q) p.v. <latexit sha1_base64="u2xxkrfync51vjtuslgnjtnx858=">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</latexit> Z Y zv =1 v2vertices n Z X i,j=1 dzv (zv 2 izv Mij d Ai daj 4 4 50 1/zv )2 Y (v1,v2 )2Edges Gadde,Gukov,Putrov 13 (zv1 M a (z) 1/zv1 )(zv2 1/zv2 ) Gukov,Pei,Putrov,Vafa 17

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sha1_base64="bpx5qtzx4lffwbtbhu3dr593/xq=">aaacdxicbvdlssnafj3uv62vqes3wspuldvts+1gklpxwce+oi8wmu7aozohmxohhpyag3/fjqtf3lp35984bbpq1gmxdufcy7332agjqprmt5zawl5zxuuvzzy2t7z39n29hvbdjkkd+8znlrsjwqhh6pjkrlobj8i1gwnao+uj33wgxfdfu5pjghrdnpcoqzgssrl0o8v7xlq5p4plstypcdqlssxtch6w8kuywzdxksg8sfsswtcnmbyjtegwjkhz+len7+pqjz7edanrhmyguxhikmjg4kwnfcraeiqgpk2oh1wiuth0m9g4vkrfchyuyppgvp09esfxilfrq04xyagy9ybif147le6lg1evccxx8gyrezjd+sykgqnpocgsjrvbmfn1q4ghicmsvyazfqkcf3mrniofabbgbslbvurisimdcahyaiiluau3oabqainh8axewzv2pl1o79rhrdwljtp74a+0zx9uljic</latexit> <latexit sha1_base64="bpx5qtzx4lffwbtbhu3dr593/xq=">aaacdxicbvdlssnafj3uv62vqes3wspuldvts+1gklpxwce+oi8wmu7aozohmxohhpyag3/fjqtf3lp35984bbpq1gmxdufcy7332agjqprmt5zawl5zxuuvzzy2t7z39n29hvbdjkkd+8znlrsjwqhh6pjkrlobj8i1gwnao+uj33wgxfdfu5pjghrdnpcoqzgssrl0o8v7xlq5p4plstypcdqlssxtch6w8kuywzdxksg8sfsswtcnmbyjtegwjkhz+len7+pqjz7edanrhmyguxhikmjg4kwnfcraeiqgpk2oh1wiuth0m9g4vkrfchyuyppgvp09esfxilfrq04xyagy9ybif147le6lg1evccxx8gyrezjd+sykgqnpocgsjrvbmfn1q4ghicmsvyazfqkcf3mrniofabbgbslbvurisimdcahyaiiluau3oabqainh8axewzv2pl1o79rhrdwljtp74a+0zx9uljic</latexit> PART II: MOCK, FALSE, AND QUANTUM Example: M 3 = (2, 3, 7) 3 2 7 1 (a, b, c) :=S 5 \ {(x, y, z) 2 C 3 x a + y b + z c =0} <latexit 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sha1_base64="xxqllzgyc3vgj7jhrt8wc5e9b3i=">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</latexit> <latexit sha1_base64="xxqllzgyc3vgj7jhrt8wc5e9b3i=">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</latexit> Ẑ 0 (q) =q 1/2 (1 q q 5 + q 10 q 11 + q 18 + q 30 +...) = q 83/168 42+6,14,21 1 ( ) 3 dim l SL(2,Z) irrep only 1 abelian flat connection false theta function where 42+6,14,21 1 ( ) = and m,r( ) := 0 B @ X n>0 n=r mod 2m X r2{1, 13, 29,41} X n>0 n= r mod 2m 1 42,r( ) C A q n2 /4m Holomorphic Eichler integral of wt-3/2 theta series

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sha1_base64="pgznlfwzaqok/wx51dtc/x/r6co=">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</latexit> <latexit sha1_base64="pgznlfwzaqok/wx51dtc/x/r6co=">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</latexit> PART II: MOCK, FALSE, AND QUANTUM 1X F 0 (q) := What about Ẑ 0 (q 1 )? n=0 q $ q 1 F 0 (q) =F 0 (1/q) = q n2 = q 1/2 Ẑ 0 (q) (q n+1 =1+q + q 3 + q 4 + q 5 +2q 7 +... ; q) n MOCK 1X ( 1) n q 1 2 n(n+1) (q n+1 =1 q q 5 + q 10 q 11 + q 18 +... ; q) n n=0 FALSE Ramanujan s order 7 mock theta function Ẑ(q 1 )=q 83/168 H 42+6,14,21 1 ( ) M 3 = (2, 3, 7) 3 2 7 1 mock theta function

<latexit sha1_base64="6hx7mzbhwo1kbqg1ltg2rmfm+to=">aaab8xicbvbnswmxej31s9avqkcvwsjuhlirgh6lxjxwsb/yriwbzbah2wrjsmjz+i+8efdeq//gm//gtn2dtj4yelw3w8y8iofmg9f9dpawv1bx1gsbxc2t7z3d0t5+u8tuedogkkvvdrcmnanammxw2k4uxxhaassyxk/81invmklxz0yj9wpcfyxibbsr3vfiezpf4cndsfcqu1v3crrivjyuiue9v/rqhpkkmrwgckx1x3mt42dyguy4hre7qayjjkpcpx1lby6p9rppxwn0bjuqrvlzegzn1d8tgy61hswb7yyxgeh5byl+53vse136grnjaqggs0vrypgrapi+cpmixpcrjzgozm9fziavjsagvlqhepmvl5lmwdvzq97tebl2lcdrgem4ggp4cae1uie6nicaggd4htdhoy/ou/mxa11y8pkd+apn8we5u4+i</latexit> <latexit sha1_base64="6hx7mzbhwo1kbqg1ltg2rmfm+to=">aaab8xicbvbnswmxej31s9avqkcvwsjuhlirgh6lxjxwsb/yriwbzbah2wrjsmjz+i+8efdeq//gm//gtn2dtj4yelw3w8y8iofmg9f9dpawv1bx1gsbxc2t7z3d0t5+u8tuedogkkvvdrcmnanammxw2k4uxxhaassyxk/81invmklxz0yj9wpcfyxibbsr3vfiezpf4cndsfcqu1v3crrivjyuiue9v/rqhpkkmrwgckx1x3mt42dyguy4hre7qayjjkpcpx1lby6p9rppxwn0bjuqrvlzegzn1d8tgy61hswb7yyxgeh5byl+53vse136grnjaqggs0vrypgrapi+cpmixpcrjzgozm9fziavjsagvlqhepmvl5lmwdvzq97tebl2lcdrgem4ggp4cae1uie6nicaggd4htdhoy/ou/mxa11y8pkd+apn8we5u4+i</latexit> <latexit sha1_base64="6hx7mzbhwo1kbqg1ltg2rmfm+to=">aaab8xicbvbnswmxej31s9avqkcvwsjuhlirgh6lxjxwsb/yriwbzbah2wrjsmjz+i+8efdeq//gm//gtn2dtj4yelw3w8y8iofmg9f9dpawv1bx1gsbxc2t7z3d0t5+u8tuedogkkvvdrcmnanammxw2k4uxxhaassyxk/81invmklxz0yj9wpcfyxibbsr3vfiezpf4cndsfcqu1v3crrivjyuiue9v/rqhpkkmrwgckx1x3mt42dyguy4hre7qayjjkpcpx1lby6p9rppxwn0bjuqrvlzegzn1d8tgy61hswb7yyxgeh5byl+53vse136grnjaqggs0vrypgrapi+cpmixpcrjzgozm9fziavjsagvlqhepmvl5lmwdvzq97tebl2lcdrgem4ggp4cae1uie6nicaggd4htdhoy/ou/mxa11y8pkd+apn8we5u4+i</latexit> <latexit sha1_base64="6hx7mzbhwo1kbqg1ltg2rmfm+to=">aaab8xicbvbnswmxej31s9avqkcvwsjuhlirgh6lxjxwsb/yriwbzbah2wrjsmjz+i+8efdeq//gm//gtn2dtj4yelw3w8y8iofmg9f9dpawv1bx1gsbxc2t7z3d0t5+u8tuedogkkvvdrcmnanammxw2k4uxxhaassyxk/81invmklxz0yj9wpcfyxibbsr3vfiezpf4cndsfcqu1v3crrivjyuiue9v/rqhpkkmrwgckx1x3mt42dyguy4hre7qayjjkpcpx1lby6p9rppxwn0bjuqrvlzegzn1d8tgy61hswb7yyxgeh5byl+53vse136grnjaqggs0vrypgrapi+cpmixpcrjzgozm9fziavjsagvlqhepmvl5lmwdvzq97tebl2lcdrgem4ggp4cae1uie6nicaggd4htdhoy/ou/mxa11y8pkd+apn8we5u4+i</latexit> PART II: MOCK, FALSE, AND QUANTUM Quantum Modular Forms Definition. A quantum modular form is a function Q(x) on Q such that ax + b p(x) :=Q(x) Q (cx + d) k cx + d extends to a function of R minus finitely many points and has some analyticity or continuity property. <latexit sha1_base64="9qeh7o3nhihek0b5astngsmyami=">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</latexit> HE a xxx xto 6 ' ' X @ ~ x x x Ẑ 0 (q) <latexit sha1_base64="fbqazyitoobh2aedcozn648kylg=">aaab8nicbvbns8naej34wetx1aoxxslus0le0gpri8ck9gptudbbtbt0s4m7e6gu/gwvhhtx6q/x5r9x2+agrq8ghu/nmdmvtkuw6lrfzsrq2vrgzmgrul2zu7dfojhsmittjddyihpddqnhuijeqigst1pnarxk3gqhn1o/9cs1eym6x1hkg5j2lygeo2glvzogsb66buxxrfsqu1v3brjmvjyuiue9w/rq9bkwxvwhk9qy33ntdmzuo2cst4qdzpcusihtc99srwnugvhs5ak5tuqprim2pzdm1n8tyxobm4pd2xlthjhfbyr+5/kzrlfbwkg0q67yffgusyijmf5pekjzhnjkcwva2fsjg1bngdquijyeb/hlzdi8r3pu1bu7kneu8zgkcawnuaeplqegt1chbjbi4ble4c1b58v5dz7mrstopnmef+b8/gdysjbm</latexit> <latexit sha1_base64="fbqazyitoobh2aedcozn648kylg=">aaab8nicbvbns8naej34wetx1aoxxslus0le0gpri8ck9gptudbbtbt0s4m7e6gu/gwvhhtx6q/x5r9x2+agrq8ghu/nmdmvtkuw6lrfzsrq2vrgzmgrul2zu7dfojhsmittjddyihpddqnhuijeqigst1pnarxk3gqhn1o/9cs1eym6x1hkg5j2lygeo2glvzogsb66buxxrfsqu1v3brjmvjyuiue9w/rq9bkwxvwhk9qy33ntdmzuo2cst4qdzpcusihtc99srwnugvhs5ak5tuqprim2pzdm1n8tyxobm4pd2xlthjhfbyr+5/kzrlfbwkg0q67yffgusyijmf5pekjzhnjkcwva2fsjg1bngdquijyeb/hlzdi8r3pu1bu7kneu8zgkcawnuaeplqegt1chbjbi4ble4c1b58v5dz7mrstopnmef+b8/gdysjbm</latexit> <latexit sha1_base64="fbqazyitoobh2aedcozn648kylg=">aaab8nicbvbns8naej34wetx1aoxxslus0le0gpri8ck9gptudbbtbt0s4m7e6gu/gwvhhtx6q/x5r9x2+agrq8ghu/nmdmvtkuw6lrfzsrq2vrgzmgrul2zu7dfojhsmittjddyihpddqnhuijeqigst1pnarxk3gqhn1o/9cs1eym6x1hkg5j2lygeo2glvzogsb66buxxrfsqu1v3brjmvjyuiue9w/rq9bkwxvwhk9qy33ntdmzuo2cst4qdzpcusihtc99srwnugvhs5ak5tuqprim2pzdm1n8tyxobm4pd2xlthjhfbyr+5/kzrlfbwkg0q67yffgusyijmf5pekjzhnjkcwva2fsjg1bngdquijyeb/hlzdi8r3pu1bu7kneu8zgkcawnuaeplqegt1chbjbi4ble4c1b58v5dz7mrstopnmef+b8/gdysjbm</latexit> <latexit sha1_base64="fbqazyitoobh2aedcozn648kylg=">aaab8nicbvbns8naej34wetx1aoxxslus0le0gpri8ck9gptudbbtbt0s4m7e6gu/gwvhhtx6q/x5r9x2+agrq8ghu/nmdmvtkuw6lrfzsrq2vrgzmgrul2zu7dfojhsmittjddyihpddqnhuijeqigst1pnarxk3gqhn1o/9cs1eym6x1hkg5j2lygeo2glvzogsb66buxxrfsqu1v3brjmvjyuiue9w/rq9bkwxvwhk9qy33ntdmzuo2cst4qdzpcusihtc99srwnugvhs5ak5tuqprim2pzdm1n8tyxobm4pd2xlthjhfbyr+5/kzrlfbwkg0q67yffgusyijmf5pekjzhnjkcwva2fsjg1bngdquijyeb/hlzdi8r3pu1bu7kneu8zgkcawnuaeplqegt1chbjbi4ble4c1b58v5dz7mrstopnmef+b8/gdysjbm</latexit> x captures values of CS partition function (WRT invariants) q! e 2 ir/n <latexit sha1_base64="gbke5ozatkbhvqvglwhy4jj2uiu=">aaab/hicbvbns8naen34wetxtecvi0xwvjmi6lhoxwmf+wfnljvtpf262ctdjrbc/stepcji1r/izx/jts1bwx8mpn6bywzekhcmton8wyura+sbm6wt8vbo7t6+fxdyvneqkbrozgpzdygczgs0nnmcuokeegucosh4eup3hkeqfos7nsxgr2qowmgo0ubq25uht8cy7vo6lzdmsdwtk75ddwrodhizuawpoglnvv3ldwkaria05uspnusk2s+j1ixymjs9vefc6jgmoweoibeop58dp8enrhngmjamhmyz9fdetiklsigwnrhri7xotcx/vf6qw0s/zyjjnqg6xxsmhjt/p0ngaznanc8mivqycyumiyij1savsgnbxxx5mbtrndepubfn1czveucjhafjdipcdiea6ay1uqtrlkfn9irercfrxxq3puatk1yxu0f/yh3+ajmolbm=</latexit> <latexit sha1_base64="gbke5ozatkbhvqvglwhy4jj2uiu=">aaab/hicbvbns8naen34wetxtecvi0xwvjmi6lhoxwmf+wfnljvtpf262ctdjrbc/stepcji1r/izx/jts1bwx8mpn6bywzekhcmton8wyura+sbm6wt8vbo7t6+fxdyvneqkbrozgpzdygczgs0nnmcuokeegucosh4eup3hkeqfos7nsxgr2qowmgo0ubq25uht8cy7vo6lzdmsdwtk75ddwrodhizuawpoglnvv3ldwkaria05uspnusk2s+j1ixymjs9vefc6jgmoweoibeop58dp8enrhngmjamhmyz9fdetiklsigwnrhri7xotcx/vf6qw0s/zyjjnqg6xxsmhjt/p0ngaznanc8mivqycyumiyij1savsgnbxxx5mbtrndepubfn1czveucjhafjdipcdiea6ay1uqtrlkfn9irercfrxxq3puatk1yxu0f/yh3+ajmolbm=</latexit> <latexit sha1_base64="gbke5ozatkbhvqvglwhy4jj2uiu=">aaab/hicbvbns8naen34wetxtecvi0xwvjmi6lhoxwmf+wfnljvtpf262ctdjrbc/stepcji1r/izx/jts1bwx8mpn6bywzekhcmton8wyura+sbm6wt8vbo7t6+fxdyvneqkbrozgpzdygczgs0nnmcuokeegucosh4eup3hkeqfos7nsxgr2qowmgo0ubq25uht8cy7vo6lzdmsdwtk75ddwrodhizuawpoglnvv3ldwkaria05uspnusk2s+j1ixymjs9vefc6jgmoweoibeop58dp8enrhngmjamhmyz9fdetiklsigwnrhri7xotcx/vf6qw0s/zyjjnqg6xxsmhjt/p0ngaznanc8mivqycyumiyij1savsgnbxxx5mbtrndepubfn1czveucjhafjdipcdiea6ay1uqtrlkfn9irercfrxxq3puatk1yxu0f/yh3+ajmolbm=</latexit> <latexit sha1_base64="gbke5ozatkbhvqvglwhy4jj2uiu=">aaab/hicbvbns8naen34wetxtecvi0xwvjmi6lhoxwmf+wfnljvtpf262ctdjrbc/stepcji1r/izx/jts1bwx8mpn6bywzekhcmton8wyura+sbm6wt8vbo7t6+fxdyvneqkbrozgpzdygczgs0nnmcuokeegucosh4eup3hkeqfos7nsxgr2qowmgo0ubq25uht8cy7vo6lzdmsdwtk75ddwrodhizuawpoglnvv3ldwkaria05uspnusk2s+j1ixymjs9vefc6jgmoweoibeop58dp8enrhngmjamhmyz9fdetiklsigwnrhri7xotcx/vf6qw0s/zyjjnqg6xxsmhjt/p0ngaznanc8mivqycyumiyij1savsgnbxxx5mbtrndepubfn1czveucjhafjdipcdiea6ay1uqtrlkfn9irercfrxxq3puatk1yxu0f/yh3+ajmolbm=</latexit> X +

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